I am working on showing that there are exactly 4 non-isomorphic groups of order 30. I'm stuck on showing that this group of order 30,

must have a normal subgroup,

of order 15.

I have

Then by using Sylow-theorems I have the possibilities for Sylow p-subgroups of G as followed:

Then I use counting argument to show that it can not be the case that

and

. So, either

or

. Then it follows that

has either a normal Sylow 3-subgroup or a normal Sylow 5-subgroup. But I can't argue that it must have both of those.

I notice that by Cauchy, G must also have a subgroup of order 2, but didn't see how this helps.

I really appreciate any help.