Homomorphism from $S_n$ to $D_k$

Question

Let $k$ and $n$ be natural numbers so that $k>2$, $n>4$ and $k$ is not divisible by $3$. How many homomorphisms from the symmetric group $S_n$ to the dihedral group $D_k$ are there?

Answer 1

There are only three normal subgroups of $S_n$. The resulting quotients have orders $1$, $2$, and $n! $. The third option is not possible because $k$ is not divisible by $3$. So the answer is one more than the number of elements of $D_k$ of order $2$.