Suppose that $N$ is a normal subgroup of a finite group $G$, and $H$ is a subgroup of $G$. If $|G/N| = p$ for some prime $p$, then show that $H$ is contained in $N$ or that $NH = G$.
I imagine this is related to the fact that $|NH| = |N||H|/|N \cap H|$, but this is not really helping me. I considered the fact that since $N$ is normal, we get that $NH \leq G$, and I then used Largrange, but I'm stuck, and some help would be nice.