# Subgroup having finitely many automorphic subgroups

From Groupprops

## Contents

## Definition

A **subgroup having finitely many automorphic subgroups** is a subgroup of a group such that there are only finitely many subgroups of the group that are automorphic to it, i.e., the orbit of the subgroup under the action of the automorphism group of the whole group is finite.

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]

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Characteristic subgroup | no other automorphic subgroups | |FULL LIST, MORE INFO | ||

Subgroup of finite index in finitely generated group | subgroup of finite index in a finitely generated group | finitely generated implies every subgroup of finite index has finitely many automorphic subgroups | |FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Nearly normal subgroup | finitely many conjugate subgroups | |FULL LIST, MORE INFO |