Maximal subgroup of Sylow subgroup

Definition
Let $$G$$ be a finite group and $$p$$ be a prime number. A subgroup $$H$$ of $$G$$ is termed a maximal subgroup of $$p$$-Sylow subgroup' if it satisfies the following equivalent conditions:


 * $$H$$ is a maximal subgroup of some $$p$$-Sylow subgroup of $$G$$.
 * $$H$$ is a subgroup such that the largest power of $$p$$ dividing the index of $$H$$ in $$G$$ is $$p$$ itself.
 * $$H$$ is a subgroup of order $$p^{m-1}$$ where $$p^m$$ is the largest power of $$p$$ dividing the order of $$G$$.

Stronger properties

 * Weaker than::Maximal Sylow intersection: An intersection of two (usually distinct) Sylow subgroups that is a maximal subgroup in both.
 * Weaker than::Characteristic maximal subgroup of Sylow subgroup

Weaker properties

 * Stronger than::Normal subgroup of Sylow subgroup: $$H$$ is normal in any of the Sylow subgroups containing it.