Complemented homomorph-containing subgroup

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


 * 1) It is both a complemented normal subgroup and a homomorph-containing subgroup of the whole group.
 * 2) It is both a complemented normal subgroup and a conjunction involving::normal subgroup having no nontrivial homomorphism to its quotient group.
 * 3) It is both a conjunction involving::permutably complemented subgroup  and a homomorph-containing subgroup of the whole group.
 * 4) It is both a conjunction involving::lattice-complemented subgroup  and a homomorph-containing subgroup of the whole group.

Stronger properties

 * Weaker than::Normal Sylow subgroup
 * Weaker than::Normal Hall subgroup
 * Weaker than::Fully invariant direct factor

Weaker properties

 * Stronger than::Complemented fully invariant subgroup
 * Stronger than::Complemented isomorph-containing subgroup
 * Stronger than::Complemented characteristic subgroup
 * Stronger than::Complemented normal subgroup
 * Stronger than::Homomorph-containing subgroup
 * Stronger than::Fully invariant subgroup
 * Stronger than::Isomorph-containing subgroup
 * Stronger than::Characteristic subgroup
 * Stronger than::Normal subgroup