Order-dominating Hall subgroup

Definition with symbols
A subgroup $$H$$ of a finite group $$G$$ is termed an order-dominating Hall subgroup if it satisfies the following equivalent conditions:


 * It is both an order-dominating subgroup and a Hall subgroup: in other words, it is a Hall subgroup such that any subgroup $$K$$ of $$G$$ whose order divides the order of $$H$$ is contained in some conjugate of $$H$$.
 * It is a $$\pi$$-subgroup and is $$\pi$$-dominating for some set of primes $$\pi$$: In other words, $$H$$ is a $$\pi$$-subgroup of $$G$$ and every $$\pi$$-subgroup of $$G$$ is contained in some conjugate of $$H$$.