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

From Groupprops

## Statement

Suppose is a finite nilpotent group and is a normal subgroup of . Suppose is a natural number dividing the order of . Then, contains a subgroup of order such that is normal in .

## Related facts

### 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.