Paranormal subgroup

This article is about a definition in group theory that is standard among the group theory community (or sub-community that dabbles in such things) but is not very basic or common for people outside.
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Definition

Definition with symbols

A subgroup $H$ of a group $G$ is termed paranormal if for any $g\in G$ , $H$ is a contranormal subgroup of $\langle H,H^{g}\rangle$ ; in other words, the normal closure of $H$ in $\langle H,H^{g}\rangle$ is $\langle H,H^{g}\rangle$ .

Here $H^{g}=g^{-1}Hg$ is a conjugate of $H$ , and the angled braces are for the subgroup generated.

Relation with other properties

Stronger properties

Weaker properties

Polynormal subgroup: It has been conjectured that for finite groups, the two notions coincide; however this has neither been proved nor disproved.

Metaproperties

Intermediate subgroup condition

YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
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Testing

References

On the arrangement of intermediate subgroups by M. S. Ba and Z. I. Borevich
On the arrangement of subgroups by Z. I. Borevich, Zap. Nauchn. Semin. tOMI, 94, 5-12 (1979)
On the lattice of subgroups by Z. I. Borevich and O. N. Macedonska, Zap. Nauchn. Semin. LOMI, 103, 13-19, 1980
Testing of subgroups of a finite group for some embedding properties like pronormality by V. I. Mysovskikh