It is known that for each (composite) n n every group of order n n contains a proper subgroup of order greater than n 1 / 3 {n^{1/3}} . We prove that given 0 > δ > 1 0 > \delta > 1 , for almost all n ⩽ x n \leqslant x , as x → ∞ x \to \infty , every group G G of order n n contains a characteristic cyclic subgroup of square-free order > n 1 − 1 / ( log ⁡ n ) 1 − δ > {n^{1 - 1/{{(\log n)}^{1 - \delta }}}} , and provide an upper bound to the number of exceptional n n . This leads immediately to a like density result for a lower bound to the number of conjugacy classes in G G .