I am asked the find the possible orders of subgroups of a group of order 60.
By Lagrange's theorem: $\left | H \right | | \left | G \right |$
Any positive integer n that is a divisor of $\left | G \right |=60$ is a possible order of a subgroup of a group of order 60.
The possible orders are 1,2,3,4,5,6,10,12,15,20,30,60.
Question: Find a group of order 60 that has subgroup of all possible orders.
Is there a quick way to determine this?
Best Answer
A finite supersolvable group has a subgroup for every possible order. $11$ of the $13$ groups of order $60$ are supersolvable, so you can use any of them.
The remaining groups, $A_5$ and $C_5\times A_4$, do not have the required property.
Note, that there are finite groups having a subgroup for every possible order which are not supersolvable, so the converse of the initial claim is false.