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I think that counting the number of subgroups of various groups is usually very difficult. I was wondering about the number of subgroups of $(C_2)^n$. For example, there are 5 subgroups of $C_2 \times C_2$ and there are 16 subgroups of $C_2 \times C_2 \times C_2$. I found these counts in the corresponding Math World articles.

Is it very difficult to count the subgroups of $(C_2)^n$ ? Is this sequence in Sloane's OEIS ? I would be satisfied in just knowing a few more terms or to see some references about how these might be counted.

Michael Hardy
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Geoffrey Critzer
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    Have you heard of $q$-binomial coefficients? If so, the answer is $\sum_{r=0}^n,\binom{n}{r}_q$ evaluated at $q=2$. – Batominovski Mar 28 '16 at 17:05
  • Thanks Batominovski, I am NOT familiar with q-binomial coefficients. Is there any way to explain them or can the answer be given in another form? I should have mentioned in my question that a Mathematica code might also help me. – Geoffrey Critzer Mar 28 '16 at 17:09
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    Well, you can identify a subgroup $C_2^n$ as a vector subspace of $\mathbb{F}2^n$. Hence, the number of subgroups of $C_2^n$ is the number of subspaces of $\mathbb{F}_2^n$. There are precisely $\binom{n}{r}{q=2}$ subspaces of $\mathbb{F}_2^n$ of dimension $r$. – Batominovski Mar 28 '16 at 17:12
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    See also http://math.stackexchange.com/questions/142589/how-to-count-number-of-bases-and-subspaces-of-a-given-dimension-in-a-vector-spac. – Batominovski Mar 28 '16 at 17:13
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    And sure enough, here is the OEIS entry: https://oeis.org/A006116 – Ivan Neretin Mar 28 '16 at 17:49

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