Detecting the prime divisors of the character degrees and the class sizes by a subgroup generated with few elements

We prove that every finite group $G$ contains a three-generated subgroup $H$ with the following property‎: ‎a prime $p$ divides the degree of an irreducible character of $G$ if and only if it divides the degree of an irreducible character of $H.$ There is no analogous result for the prime divisors of the sizes of the conjugacy classes‎.

Keywords:

‎Character degrees , class sizes