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.
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 $0≤i≤d−1$.