# Complemented isomorph-containing subgroup

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This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: complemented normal subgroup and isomorph-containing subgroup

View other subgroup property conjunctions | view all subgroup properties

## Definition

A subgroup of a group is termed a **complemented isomorph-containing subgroup** if it satisfies the following equivalent conditions:

- It is both a permutably complemented subgroup and an isomorph-containing subgroup.
- It is both a lattice-complemented subgroup and an isomorph-containing subgroup.
- It is both a complemented normal subgroup and an isomorph-containing subgroup.

## Relation with other properties

### Stronger properties

- Fully invariant direct factor:
`For full proof, refer: Equivalence of definitions of fully invariant direct factor` - Normal Hall subgroup
- Normal Sylow subgroup
- Complemented homomorph-containing subgroup