The Frattini subgroup is defined for arbitrary group in at least two ways:
It is the intersection of all maximal subgroups.
It is the set of all non-generators of the group (i.e. it contains those $x\in G$ such that if a $S\cup \{x\}$ generates $G$ then $S$ also generates $G$).
If we move in the world of $p$-groups, the Frattini subgroup has two more characterizations:
It is the smallest normal subgroup such that quotient by it is elementary abelian $p$-group.
It is the product $[G,G]G^p$, where $G^p$ is the subgroup of $G$ generated by $p$-th powers of all elements.
Question: What are the other subgroups of a group (or $p$-group) which has several characterizations?
(Of course, "normal subgroups" have some characterizations).