Normal subgroups of dihedral groups

In relation to my previous question, I am curious about what exactly are the normal subgroups of a dihedral group $D_n$ of order $2n$.

It is easy to see that cyclic subgroups of $D_n$ is normal. But I suspect that case analysis is needed to decide whether dihedral subgroups of $D_n$ is normal.

A little bit of Internet search suggests the use of semidirect product $(\mathbb Z/n\mathbb Z) \rtimes (\mathbb Z/2\mathbb Z) \cong D_n$, but I do not know the condition for subgroups of a semidirect product to be normal.

I would be grateful if you could suggest a way to enumerate the normal subgroups of $D_n$ that does not resort to too much of case analysis.

Solution 1:

Here is a nice answer: the dihedral group is generated by a rotation $R$ and a reflection $F$ subject to the relations $R^n=F^2=1$ and $(RF)^2=1$. For $n$ odd the normal subgroups are given by $D_n$ and $\langle R^d \rangle$ for all divisors $d\mid n$. If $n$ is even, there are two more normal subgroups, i.e., $\langle R^2,F \rangle$ and $\langle R^2,RF \rangle$.