Finding subgroups of order m in $S_n$

Question

For example On finding the subgroups of order 60 ,in $S_6$.

I can calculate order of elements but how to find the subgroup of order 60.

Is it the one of the form like a group of any 60 elements of $S_6$.

Or is it like a group having 10 elements of order 6 or 10 6-cylces or is it anything else.

Answer 1

Since in $S_6$ you have chains of length $6$, the most most basic thing that you can do is to keep one of the members constant and permute the rest. In this manner we can get basically the following six configurations

$$ (2,3,4,5,6) (1,3,4,5,6) (1,2,4,5,6) (1,2,3,5,6) (1,2,3,4,5) (1,2,3,4,6)$$

Now you can easily find the $A_5$ corresponding to them to get six subgroups of order 60 for $S_6$