Prove that the order of Z(G) is either <e> or G

Let be a group of order , with and (not necessarily distinct) primes. Prove that the center or .

I started off by doing a proof by contradiction.

Suppose that ,

Then divides either or , which implies that or

We know that is a normal subgroup.

Consider the quotient group .

Using Larange's Theorem, we know that (*).

I broke it into two cases: 1: , 2: .

If under case 1, (*) evaluates to . Otherwise, in case 2, (*) evaluates to .

This is as far as I've gotten and I'm pretty stumped as how else to follow up after.