Order-dominating subgroup

Symbol-free definition
A finite subgroup of a group is termed order-dominating if every other subgroup whose order divides its order, is conjugate to a subgroup contained in it.

Definition with symbols
Let $$G$$ be a group and $$H$$ be a finite subgroup. Then, $$H$$ is termed order-dominating in $$G$$ if, for any subgroup $$K \le G$$ such that the order of $$K$$ divides the order of $$G$$, there exists $$g \in G$$ such that $$gKg^{-1} \le H$$.

Stronger properties

 * Weaker than::Sylow subgroup (in a finite group)
 * Hall subgroup in a finite solvable group

Weaker properties

 * Stronger than::Order-conjugate subgroup