Prove that if is a normal subgroup of of prime index then for all subgroups of either

(i) is a subgroup of , or

(ii) and intersect

I tried to do this in 2 cases. The first case is to let be a subest of , then since is a subgroup of , it is a subgroup of . For case 2, I let not contained in , then try to show and intersect , but don't know how... please help, thank you.