The intersection of subgroup and normal subgroup: the greatest common divisor?

Question

Is the order of intersection of subgroup $H$ and normal subgroup $N$ of group $G$ the greatest common divisor of $\lvert H\rvert$ and $\lvert N\rvert$?

Answer 1

You can also take $H$ and $N$ to be two subgroups of order $4$ in $Q_{8}$. Then both $H$ and $N$ are normal and their intersection is the center which has two elements and so the result fails in this case.

Alternatively, take $N = V_{4}$ - which is the subgroup made by cycles of the type (2, 2) and so it is normal in $S_{4}$. Then let $H$ be any other subgroup of order 4. Then the intersection has two elements once again and you get a contradiction.