Page "Lagrange's theorem (group theory)" Paragraph 11

There are partial converses to Lagrange's theorem.

For general groups, Cauchy's theorem guarantees the existence of an element, and hence of a cyclic subgroup, of order any prime dividing the group order ; Sylow's theorem extends this to the existence of a subgroup of order equal to the maximal power of any prime dividing the group order.

For solvable groups, Hall's theorems assert the existence of a subgroup of order equal to any unitary divisor of the group order ( that is, a divisor coprime to its cofactor ).