Finite nilpotent implies every normal subgroup contains normal subgroups of all orders dividing its order

Statement
Suppose $$G$$ is a finite nilpotent group and $$H$$ is a normal subgroup of $$G$$. Suppose $$m$$ is a natural number dividing the order of $$H$$. Then, $$H$$ contains a subgroup $$K$$ of order $$m$$ such that $$K$$ is normal in $$G$$.

Similar facts

 * Finite nilpotent implies every normal subgroup is part of a chief series
 * Finite nilpotent Lie ring implies every ideal contains ideals of all orders dividing its order
 * Finite nilpotent Lie ring implies every ideal is part of a chief series

Stronger facts

 * Congruence condition on number of subgroups of given prime power order is a somewhat roundabout way of proving this fact. It says that the number of subgroups of a given prime power order is congruent to 1 modulo the prime.