Is G a CLT group if its order is product of distinct prime?

Question

Let G be group whose order is square-free. Is G always CLT group?

Trying to apply Hall's theorem but not conclusive.

Answer 1

Every finite group of square-free order is supersolvable, and hence a CLT-group. For proofs and references see this question, and this one. One should explain that a group is called $CLT$-group, or Lagrangian group, if for each positive divisor of the group order, there exists at least one subgroup of that order.

Remark: For a longer and more detailed account on CLT-groups here on MSE see this question.