Group Theory

A group $G$ is solvable if it has a chain of subgroups $$1 = G_0 < G_1 < \dots < G_k = G$$ such that for $j=1,2,\dots,k$, the group $G_{j-1}$ is normal in $G_j$, and $G_j/G_{j-1}$ is an abelian group.

For the purposes of this problem, call such a chain good. A solvable group may have more than one good chain (i.e. good chains may not be unique).