Is there an easy way to find the derived group of $S_5$?

Question

In order to find the derived group of $S_5$ I've tried using Lagrange’s Theorem to find the order of the possible subgroups but $O(S_5)=2^3\cdot 3 \cdot 5$ so there are too many possible subgroups to look for.

Is there a smarter way to calculate the derived group?

Thanks in advance.

Answer 1

The derived subgroup of $S_5$ is normal, and there's only one proper normal subgroup of $S_n$ for $n\geq 5$.