On Ditsman's lemma

Let $H$ be a subgroup of a group $G$ generated by a finite $G$-invariant subset $X=\bigcup_{i=1}^kC_i$ that consists of elements of finite order, where $C_i$ is the class of conjugate elements of $G$ with representative $a_i$. We prove that
$$ |H|\leq\prod_{i=1}^ko(a_i)^{|C_i|}, $$
where $o(a_i)$ is the order of the element $a_i\in C_i$. Best estimates are obtained for some important special cases.