Suppose $N\trianglelefteq G$ and $H\leqslant G$. If $\vert G/N \vert$ is prime prove $H\subseteq N$ or $ NH=G$
I believe I want to make use of this fact that if $H,K\leqslant G$ that $HK=H \iff K\subseteq H$.
So here I would proceed by cases either $H\subseteq N$ or it is not. Case 1 there is nothing to prove.
For $H\not\subseteq N$ then $NH\not = H$ but I'm not sure how to proceed from here. I'm thinking something about $N$ being normal should give me a reason that $NH=G$.