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How many subgroups of order $n$ does $D_n$ have?

My work:

Since subgroups of $D_n$ are either cyclic or dihedral, the subgroups of order $n$ of $D_n$ are $\left< r \right>$ (cyclic) and $D_{n/2}$ (dihedral). However, this works only for $n$ is even and $n\geq 2$. I don't know how to do when $n$ is odd.

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  • $\begingroup$ Note that any subgroup of order $n$ is normal. I think it will be easier to see from this angle (and you are correct in the even case) $\endgroup$ – TokenToucan Mar 6 '16 at 17:27
  • $\begingroup$ @TokenToucan I haven't learnt anything about normal groups. This question is listed as "question to ponder" in the cyclic group chapter. I wonder in the case when n is odd, if there is only one such subgroup $<r>$. $\endgroup$ – Kenneth.K Mar 6 '16 at 17:32
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All subgroups and all normal subgroups are classified in K. Conrad's text on Dihedral Groups II, Theorem $3.1$ and $3.8$.

Section $3$ gives all subgroups of $D_n$, for $n$ odd and $n$ even. In particular, Theorem $3.1$ gives a complete listing of all subgroups, namely all $⟨r^d⟩$ with $d|n$, and all $⟨r^d,r^is⟩$, where $d|n$ and $0≤i≤d−1$.

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