Subnormalizer

Definition

Suppose $H$ is a subgroup of a group $G$. A subnormalizer of $H$ in $G$ is a subgroup $K$ of $G$ containing $H$ such that $H$ is a subnormal subgroup of $K$, and further, if $H\le L\le G$ is such that $H$ is subnormal in $L$, then $L\le K$.

Since subnormality is not upper join-closed, not every subgroup need have a subnormalizer.