Finite nilpotent implies every normal subgroup is part of a chief series

Statement

Version for a finite group

Suppose $G$ is a Finite nilpotent group (?) and $H$ is a normal subgroup of $G$. Then, there exists a Chief series (?) for $G$ that has $H$ as one of its members, i.e., there is a series:

$\{e\}=K_{0}\leq K_{1}\leq K_{2}\leq \dots \leq K_{r}=G$

such that each $K_{i}$ is a normal subgroup of $G$ and there exists some value of $i$ for which $K_{i}=H$.