### On the number of isomorphism classes of derived subgroups

We show that a finite nonabelian characteristically simple group $G$ satisfies $n=\left|\pi \right(G\left)\right|+2$ if and only if $G\cong {A}_{5}$, where $n$ is the number of isomorphism classes of derived subgroups of $G$ and $\pi \left(G\right)$ is the set of prime divisors of the group $G$. Also, we give a negative answer to a question raised in M. Zarrin (2014).