Normal-hereditarily characteristic subgroup

Definition
A subgroup of a group is termed a normal-hereditarily characteristic subgroup if every normal subgroup of the subgroup is a [[characteristic subgroup] of the whole group.

Stronger properties

 * Characteristic subgroup which is a group in which every normal subgroup is characteristic
 * Weaker than::Hereditarily characteristic subgroup

Weaker properties

 * Stronger than::Characteristic subgroup
 * Stronger than::Transitively normal subgroup
 * Stronger than::Hereditarily normal-to-characteristic subgroup