Left-transitively complemented normal subgroup

Symbol-free definition
A subgroup of a group is termed a left-transitively complemented normal subgroup if whenever the whole group is a complemented normal subgroup of a bigger group, the subgroup is also a complemented normal subgroup of that group.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed a left-transitively complemented normal subgroup if, for any group $$K$$ containing $$G$$ such that $$G$$ is a complemented normal subgroup of $$K$$, then $$H$$ is also a complemented normal subgroup of $$K$$.

Stronger properties

 * Weaker than::Sylow direct factor
 * Weaker than::Hall direct factor
 * Weaker than::Characteristically complemented characteristic subgroup
 * Weaker than::Complete characteristic direct factor
 * Weaker than::Complemented normal subgroup of complete characteristic direct factor

Weaker properties

 * Stronger than::Characteristic subgroup:
 * Stronger than::Complemented characteristic subgroup:
 * Stronger than::Complemented normal subgroup

Related properties

 * Right-transitively complemented normal subgroup