Given a class equation of a group H of 20=1+4+5+5+5, does H have a subgroup of order 5? Of order 4?
The Attempt at a Solution
Order 5 I can't get, but for order 4 I think I am correct in saying that H does have a subgroup of order 4 because each summand in the class equation is either |Z(H)| or is the index of the normalizer subgroup. So H has 3 subgroups of order 4.
Is this enough of an answer for the subgroup of order 4 part?
And any help on the order of 5 part?