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It sounds like your minyan is misapplying the opinion of the Rambam.

There are 4 opinions regarding reciting "Half Hallel". ("Half-Hallel" is said on Rosh Chodesh and Pesach starting Chol Hamoed.) The disagreement is about Half Hallel; Whole Hallel all agree to recite with a bracha even with no minyan.

1. Rambam: No Half Hallel is said without a minyan. With a minyan,Half Hallel is said without a bracha.

2. Rashi (which is followed by Mechaber and Sephardim): Half Hallel is said even with no minyan. A bracha is not recited for Half Hallel

3. The Rif (Rabenu Yitzchak Alfasi): a bracha is recited only with a minyan. With no minyan Half Hallel is recited albeit without a bracha

4. The Tur (followed by Rema and Ashkenazim, in general): to say Half Hallel with a bracha even with no minyan.

(The Rema concludes that to satisfy the opinion of Rif, one should be careful to recite Half Hallel with a minyan.)

                  Minyan    | No Minyan
         Rambam | no bracha | Dont say 
         Rashi  | no bracha | no bracha
         Rif    | bracha    | no bracha
         Tur    | bracha    | bracha

"Whole Hallel" is said with a bracha under all circumstances. On Chanukah, when whole hallel is said, it is recited with a bracha even without a minyan contrary to the synagogue referenced in the question. According to Rif and Rambam, when saying half hallel no bracha is said even with a minyan; contrary to synagogue referenced in the question.

Source: Shulchan Aruch 422,2 with Be'er Hagola and Mishne Berurah, Tur ibid and Rambam (Hilchos Chanukah 3:7)

It sounds like your minyan is misapplying the opinion of the Rambam.

There are 4 opinions regarding reciting "Half Hallel". (Half Hallel is said on Rosh Chodesh (except Tishrei and Tevet) and the last 6 days of Pesach.) The disagreement is about Half Hallel; all agree to recite Whole Hallel with a bracha even with no minyan.

1. Rambam: No Half Hallel is said without a minyan. With a minyan, Half Hallel is said without a bracha.

2. Rashi (which is followed by Mechaber and Sephardim): Half Hallel is said even with no minyan. A bracha is not recited for Half Hallel

3. Rif (Rabenu Yitzchak Alfasi): a bracha is recited only with a minyan. With no minyan Half Hallel is recited albeit without a bracha

4. The Tur (followed by Rema and most Ashkenazim): say Half Hallel with a bracha even with no minyan.

(Rema concludes that to satisfy the opinion of Rif, one should be careful to recite Half Hallel with a minyan.)

                  Minyan    | No Minyan
         Rambam | no bracha | don't say 
         Rashi  | no bracha | no bracha
         Rif    | bracha    | no bracha
         Tur    | bracha    | bracha

"Whole Hallel" is said with a bracha under all circumstances. On Chanukah, when Whole Hallel is said, it is recited with a bracha even without a minyan contrary to the synagogue referenced in the question. According to Rif and Rambam, when saying Half Hallel no bracha is said even with a minyan; contrary to synagogue referenced in the question.

Source: Shulchan Aruch 422,2 with Be'er Hagola and Mishne Berurah, Tur ibid., and Rambam (Hilchos Chanukah 3:7)

It sounds like your minyan follows a misapplication of the opinion of the Rif.

There are 3 opinions cited in Shulchan Aruch regarding reciting a bracha over "Half Hallel". The disagreement is only about the bracha -- All agree to recite Half Hallel. ("Half-Hallel" is said on Rosh Chodesh and Pesach starting Chol Hamoed.) Also, the only disagreement is abut Half Hallel; Whole Hallel all agree to recite with a bracha even with no minyan.

1. The Rif (Rabenu Yitzchak Alfasi): a bracha is recited only with a minyan. With no minyan Half Hallel is recited albeit without a bracha

2. Rashi and Rambam (which is followed by Mechaber and Sephardim): not to say bracha even with a minyan for Half Hallel

3. The Tur (followed by Rema and Ashkenazim, in general): to say Half Hallel with a bracha even with no minyan.

The Rema concludes that to satisfy the opinion of Rif, one should be careful to recite Half Hallel with a minyan.

"Whole Hallel" is said with a bracha under all circumstances. Half Hallel is always said even with no minyan according to all opinions. The disagreement is only about the bracha over Half Hallel.

Note: There is another opinion cited in Tur (and not codfied in Shulachan Aruch) that opines that Half Hallel is not said at all without a minyan. Since in the questioner's synagogue they do not recite Whole Hallel on Chanukah without a minyan, this opinion is not completely consistent with what they do.

Source: Shulchan Aruch 422,2 with Be'er Hagola and Mishne Berurah

It sounds like your minyan is misapplying the opinion of the   Rambam.

There are 4 opinions regarding reciting "Half Hallel". ("Half-Hallel" is said on Rosh Chodesh and Pesach starting Chol Hamoed.) The disagreement is about Half Hallel; Whole Hallel all agree to recite with a bracha even with no minyan.

1. Rambam: No Half Hallel is said without a minyan. With a minyan,Half Hallel is said without a bracha.

2. Rashi (which is followed by Mechaber and Sephardim): Half Hallel is said even with no minyan. A bracha is not recited for Half Hallel

3. The Rif (Rabenu Yitzchak Alfasi): a bracha is recited only with a minyan. With no minyan Half Hallel is recited albeit without a bracha

4. The Tur (followed by Rema and Ashkenazim, in general): to say Half Hallel with a bracha even with no minyan.

(The Rema concludes that to satisfy the opinion of Rif, one should be careful to recite Half Hallel with a minyan.)

                  Minyan    | No Minyan
         Rambam | no bracha | Dont say 
         Rashi  | no bracha | no bracha
         Rif    | bracha    | no bracha
         Tur    | bracha    | bracha

"Whole Hallel" is said with a bracha under all circumstances. On Chanukah, when whole hallel is said, it is recited with a bracha even without a minyan contrary to the synagogue referenced in the question. According to Rif and Rambam, when saying half hallel no bracha is said even with a minyan; contrary to synagogue referenced in the question.

Source: Shulchan Aruch 422,2 with Be'er Hagola and Mishne Berurah, Tur ibid and Rambam (Hilchos Chanukah 3:7)

It sounds like your minyan follows a misapplication of the opinion of the Rif.

There are 3 opinions cited in Shulchan Aruch regarding reciting a bracha over "Half Hallel". The disagreement is only about the bracha -- All agree to recite Half Hallel. ("Half-Hallel" is said on Rosh Chodesh and Pesach starting Chol Hamoed.) Also, the only disagreement is abut Half Hallel; Whole Hallel all agree to recite with a bracha even with no minyan.

1. The Rif (Rabenu Yitzchak Alfasi): a bracha is recited only with a minyan. With no minyan Half Hallel is recited albeit without a bracha

2. Rashi and Rambam (which is followed by Mechaber and Sephardim): not to say bracha even with a minyan for Half Hallel

3. The Tur (followed by Rema and Ashkenazim, in general): to say Half Hallel with a bracha even with no minyan.

The Rema concludes that to satisfy the opinion of Rif, one should be careful to recite Half Hallel with a minyan.

"Whole Hallel" is said with a bracha under all circumstances. Half Hallel is always said even with no minyan according to all opinions. The disagreement is only about the bracha over Half Hallel.

Source: Shulchan Aruch 422,2 with Be'er Hagola and Mishne Berurah

It sounds like your minyan follows a misapplication of the opinion of the Rif.

There are 3 opinions cited in Shulchan Aruch regarding reciting a bracha over "Half Hallel". The disagreement is only about the bracha -- All agree to recite Half Hallel. ("Half-Hallel" is said on Rosh Chodesh and Pesach starting Chol Hamoed.) Also, the only disagreement is abut Half Hallel; Whole Hallel all agree to recite with a bracha even with no minyan.

1. The Rif (Rabenu Yitzchak Alfasi): a bracha is recited only with a minyan. With no minyan Half Hallel is recited albeit without a bracha

2. Rashi and Rambam (which is followed by Mechaber and Sephardim): not to say bracha even with a minyan for Half Hallel

3. The Tur (followed by Rema and Ashkenazim, in general): to say Half Hallel with a bracha even with no minyan.

The Rema concludes that to satisfy the opinion of Rif, one should be careful to recite Half Hallel with a minyan.

"Whole Hallel" is said with a bracha under all circumstances. Half Hallel is always said even with no minyan according to all opinions. The disagreement is only about the bracha over Half Hallel.

Note: There is another opinion cited in Tur (and not codfied in Shulachan Aruch) that opines that Half Hallel is not said at all without a minyan. Since in the questioner's synagogue they do not recite Whole Hallel on Chanukah without a minyan, this opinion is not completely consistent with what they do.

Source: Shulchan Aruch 422,2 with Be'er Hagola and Mishne Berurah