Given a finite group $G$, we denote by $\pi_s(G)$ the set of orders of its subgroups. Which finite groups $G$ can be characterized by the set $\pi_s(G)$, i.e. $\pi_s(H)=\pi_s(G)$ implies $H\cong G$? Elementary examples of such groups are cyclic groups of prime order, $A_4$, ... and so on. Is this property true for finite simple groups? (similarly with the recognition of finite simple groups by their spectrum). Any suggestions or references are welcome.

Given a finite group $G$, we denote by $\pi_s(G)$ the set of orders of its subgroups. Which finite groups $G$ can be characterized by the set $\pi_s(G)$, i.e. $\pi_s(H)=\pi_s(G)$ implies $H\cong G$? Elementary examples of such groups are cyclic groups of prime order, $A_4$, ... and so on. Is this property true for finite simple groups? (similarly with the recognition of finite simple groups by their spectrum). Any suggestions or references are welcome.

Additional question: Let $G$ and $H$ be two finite groups. Assume that $\pi_s(G)=\pi_s(H)$ and $card\{K\leq G \mid |K|=d\}=card\{K\leq H \mid |K|=d\}, \forall\, d\in\pi_s(G)$. Is it true that $G\cong H$?