$H\cap K$ is normal maximal subgroup of $H$.

Question

if $G$ is a group and $H$ and $K$ are different normal maximal subgroup of $G$ .Prove that:

$H\cap K$ is normal maximal subgroup of $H$.

If we suppose that $L$ is subgroup of $G$ which $H\cap K \le L \le H$ now how can I prove that $L = H\cap K$ or $L =H$?

Answer 1

Hint: Use the isomorphism $HK/K \cong H/H\cap K$.