Question

Suppose that G is a group of order 60. What are the possible orders of a subgroup
H ≤ G?

Answer 1

The answer to this question is provided by Lagrange's theorem : it states that if $G$ is a finite group, $H \leq G$ a subgroup, then $|H| | |G|$ , i.e. order of $H$ divides order of $G$ . Therefore the possible orders are : $1, 2, 3, 4, 5, 6, 10,12,15,20,30,60$ .