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

Version for a finite group
Suppose $$G$$ is a fact about::finite nilpotent group and $$H$$ is a normal subgroup of $$G$$. Then, there exists a fact about::chief series for $$G$$ that has $$H$$ as one of its members, i.e., there is a series:

$$\{ e \} = K_0 \le K_1 \le K_2 \le \dots \le K_r = G$$

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