The Number of Isomorphism Classes of Finite Groups with Given Element Orders

Let $G$ be a finite group and $\pi_e(G)$ the set of element orders of $G$. Denote by $h(\pi_e(G))$ the number of isomorphism classes of finite groups $H$ satisfying $\pi_e(H)=\pi_e(G)$. We prove that if $G$ has at least three prime graph components, then $h(\pi_e(G))\in\{1, \infty\}$.