Is there a neat way to show that a group of order 160 is not simple without directly quoting Poincare's theorem?
I am thinking of maybe using the Sylow theorems to say that in order that the group is not simple we must have $n_2=5,\,\,\,n_5=2^4$ (due to the Sylow constraints). So then there are $2^4\times 4$ order 5 elements. But then I can't say that there are $5\times 2^5$ elements with not-equal-to-5 orders since these subgroups may intersect because they don't have prime orders. So what can we do next, or is this not even the right direction to start?