# Cyclic isomorph-containing subgroup

This article describes a property that arises as the conjunction of a subgroup property: isomorph-containing subgroup with a group property (itself viewed as a subgroup property): cyclic group
View a complete list of such conjunctions

## Definition

A subgroup $H$ of a group $G$ is termed a cyclic isomorph-containing subgroup if $H$ is a cyclic group and $H$ is an isomorph-containing subgroup of $G$, i.e., every subgroup of $G$ isomorphic to $H$ is contained in $H$.

## Relation with other properties

### Stronger properties

property quick description proof of implication proof of strictness (reverse implication failure) intermediate notions
Cyclic homomorph-containing subgroup

### Weaker properties

property quick description proof of implication proof of strictness (reverse implication failure) intermediate notions
1-automorphism-invariant subgroup |
* Cyclic 1-automorphism-invariant subgroup |
Quasiautomorphism-invariant subgroup |
* Cyclic quasiautomorphism-invariant subgroup |
Cyclic characteristic subgroup |
Cyclic normal subgroup |
Characteristic subgroup |
Normal subgroup |