# Class two characteristic subgroup

From Groupprops

This article describes a property that arises as the conjunction of a subgroup property: characteristic subgroup with a group property (itself viewed as a subgroup property): group of nilpotency class two

View a complete list of such conjunctions

## Contents

## Definition

A subgroup of a group is termed a **class two characteristic subgroup** if it is a group of nilpotency class two and is a characteristic subgroup of the whole group.

Note that *group of nilpotency class two* means group whose nilpotency class at *most* two, so this includes abelian characteristic subgroups.

## Examples

VIEW: subgroups satisfying this property | subgroups dissatisfying property characteristic subgroup | subgroups dissatisfying property group of nilpotency class twoVIEW: Related subgroup property satisfactions | Related subgroup property dissatisfactions

## Relation with other properties

### Stronger properties

- Abelian characteristic subgroup
- Cyclic characteristic subgroup
- Critical subgroup
- Aut-abelian characteristic subgroup