Permutably complemented Hall subgroup

Definition
A subgroup $$H$$ of a finite group $$G$$ is termed a permutably complemented Hall subgroup if it satisfies the following two conditions:


 * $$H$$ is a Hall subgroup of $$G$$: the order and index of $$H$$ are relatively prime.
 * $$H$$ is a permutably complemented subgroup of $$G$$: there exists a subgroup $$K$$ of $$G$$ such that $$H \cap K$$ is trivial and $$HK = G$$ (in other words, $$H$$ and $$K$$ are permutable complements). Note that $$K$$ also must be a Hall subgroup of $$G$$: if $$H$$ is $$\pi$$-Hall, $$K$$ is $$\pi'$$-Hall.

Stronger properties

 * Weaker than::Hall retract
 * Weaker than::Normal Hall subgroup:
 * Weaker than::Hall direct factor

Weaker properties

 * Stronger than::Lattice-complemented Hall subgroup