Subgroup having a subnormalizer

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed a subgroup having a subnormalizer if $$H$$ has a defining ingredient::subnormalizer in $$G$$: there exists a subgroup $$K$$ of $$G$$ containing $$H$$ such that $$H$$ is a defining ingredient::subnormal subgroup of $$K$$, and if $$H \le L \le G$$ such that $$H$$ is subnormal in $$L$$, then $$L \le K$$.

Not every subgroup has a subnormalizer, because subnormality is not upper join-closed.

Stronger properties

 * Weaker than::Intermediately subnormal-to-normal subgroup
 * Weaker than::Subnormal subgroup
 * Weaker than::Normal subgroup
 * Weaker than::Self-normalizing subgroup
 * Weaker than::Pronormal subgroup
 * Weaker than::Weakly pronormal subgroup
 * Weaker than::Paranormal subgroup
 * Weaker than::Polynormal subgroup
 * Weaker than::Weakly normal subgroup