Exactly one nontrivial proper subgroup Question: Determine all the finite groups that have exactly one nontrivial proper subgroup.
MY attempt is that the order of group G has to be a positive nonprime integer n which has only one divisor since any divisor a of n will form a proper subgroup of order a. Since 4 is the only nonprime number that has only 1 divisor which is 2, All groups of order 4 has only 1 nontrivial proper subgroups (Z4 and D4)

Hints/questions to consider: Have you done Sylow's theorem? How many Sylow subgroups can your group have? Show that your group must be cyclic- (hint for this: if not, all its element are in the unique proper subgroup). How many subgroups does a cyclic group of order $p^{n}$ have, when $p$ is prime and $n$ is a positive integer?
