D-property for a prime set

Definition
Let $$G$$ be a finite group, and $$\pi$$ be a set of primes. We say that $$G$$ satisfies $$D_\pi$$ if $$D$$ has a $$\pi$$-Hall subgroup $$H$$ (viz, a Hall subgroup whose order contains only primes from $$\pi$$ and whose index contains no primes from $$\pi$$) and every $$\pi$$-subgroup (subgroup whose order contains only primes from $$\pi$$) is contained in a conjugate of $$H$$.

The D here stands for Domination and relates to the problem of whether we can extend Sylow's theorems to Hall subgroups.

Weaker properties

 * C-property for a prime set
 * E-property for a prime set