Groups with 5 nontrivial conjugacy classes of non self-normalizing subgroups are solvable

For any $n$ nonnegative integer a family of groups, denoted by ${\mathcal{D}}_n$, was introduce by Bianchi et al., as the collection of all finite groups with exactly $n$ conjugacy classes of nontrivial, non self-normalizing subgroups. It was conjectured that ${\mathcal{D}}_5$ consists of solvable groups with derived length at most $3$. In this note we verify their conjecture.