  Every finite group $G$ has a normal series each of whose factors is either a
solvable group or a direct product of nonabelian simple groups. The minimum
number of nonsolvable factors attained on all possible such series is called
the nonsolvable length of the group and denoted by $\lambda(G)$. For every
integer $n$, we define a particular class of groups of nonsolvable length $n$,
called \emph{$n$-rarefied}, and we show that every finite group of nonsolvable
length $n$ contains an $n$-rarefied subgroup. As applications of this result,
we improve the known upper bounds on $\lambda(G)$ and determine the maximum
possible nonsolvable length for permutation groups and linear groups of fixed
degree resp. dimension.
