# Subisomorph-containing subgroup

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
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]
If the ambient group is a finite group, this property is equivalent to the property: variety-containing subgroup
View other properties finitarily equivalent to variety-containing subgroup | View other variations of variety-containing subgroup |

## Definition

A subgroup $H$ of a group $G$ is termed subisomorph-containing if whenever $K$ is a subgroup of $H$ and $L$ is a subgroup of $G$ such that $K$ and $L$ are isomorphic, then $L$ is also a subgroup of $H$.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
variety-containing subgroup contains every subgroup of the whole group in the variety generated by it Subhomomorph-containing subgroup|FULL LIST, MORE INFO
subhomomorph-containing subgroup contains every subgroup of the whole group isomorphic to a subquotient of it |FULL LIST, MORE INFO
normal Sylow subgroup normal subgroup of prime power order whose order and index are relatively prime Variety-containing subgroup|FULL LIST, MORE INFO
normal Hall subgroup subgroup whose order and index are relatively prime Variety-containing subgroup|FULL LIST, MORE INFO

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
isomorph-containing subgroup contains every isomorphic subgroup Right-transitively isomorph-containing subgroup|FULL LIST, MORE INFO
injective endomorphism-invariant subgroup invariant under all injective endomorphisms Isomorph-containing subgroup|FULL LIST, MORE INFO
intermediately injective endomorphism-invariant subgroup injective endomorphism-invariant in every intermediate subgroup Isomorph-containing subgroup|FULL LIST, MORE INFO
characteristic subgroup invariant under every automorphism Intermediately characteristic subgroup, Isomorph-containing subgroup, Right-transitively isomorph-containing subgroup, Transfer-closed characteristic subgroup|FULL LIST, MORE INFO
intermediately characteristic subgroup characteristic in every intermediate subgroup Isomorph-containing subgroup, Transfer-closed characteristic subgroup|FULL LIST, MORE INFO
transfer-closed characteristic subgroup intersection with any subgroup is characteristic in that subgroup |FULL LIST, MORE INFO