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.