# Normal-hereditarily characteristic subgroup

From Groupprops

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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

## Definition

A subgroup of a group is termed a **normal-hereditarily characteristic subgroup** if every normal subgroup of the subgroup is a [[characteristic subgroup] of the whole group.

## Relation with other properties

### Stronger properties

- Characteristic subgroup which is a group in which every normal subgroup is characteristic
- Hereditarily characteristic subgroup