For a group $G$ the length $l(G)$ is defined as the supremum of the lengths of all composition series of $G$. If $N$ is a normal subgroup of $G$, then $l(G)=l(N)+l(G/N)$. This, I believe, is a consequence of the Jordan-Hölder theorem.
For a subgroup $H$ of $G$ that is not normal in $G$, maybe even self-normalizing, is it still true that $l(H)\leq l(G)$?