Let G be a finite group. In general case, given two normal subgroups N and K of G, we need not to have N < K or K< N. The easiest example is the Klein 4-group V4 and its subgroups of order 2. So ...
It is well known that if $K$ is a finite index subgroup of a group $H$, then there is a finite index subgroup $N$ of $K$ which is normal in $H$. Indeed, one can observe that there are only finitely ...