Proving that if a finite group $G$ has a composition series of length 2, then any composition series has length 2

Question

I'm having trouble proving this. If $G$ has a composition series of length 2, then that means there is a maximal normal subgroup $H$ of $G$ such that $H$ is simple, i.e. we would have a composition series $$\{e\}\lhd H\lhd G.$$ Suppose there is another maximal subgroup $H'$ of $G.$ I'm guessing I'd want to show $H'$ is simple as well, but I have no idea how to do it. I tried proving it by way of contradiction. So assume $H'_1$ is a nontrivial maximal subgroup of $H'.$ Then something should go wrong, but I don't know what and how. Any hints would be much appreciated.

Answer 1

Suppose there is a proper, nontrivial normal subgroup $K$, which is distinct from $H$.

What can you say about the normal subgroup $H K$? What can you say about $H \cap K$?

Hint

$H K/ H$ is a non-trivial normal subgroup of the simple group $G/H$.