Normality-large normal subgroup

Definition
A normality-large normal subgroup is a subgroup that satisfies the following equivalent conditions:


 * 1) Its intersection with every nontrivial normal subgroup is a nontrivial normal subgroup
 * 2) It is normal and normality-large: its intersection with every nontrivial normal subgroup is nontrivial

For a finite group (or more generally a group in which every nontrivial normal subgroup contains a minimal normal subgroup), there is another equivalent condition: it is a normal subgroup containing the socle.

Metaproperties
A finite intersection of normality-large normal subgroups is normality-large normal.