  Iwasawa's theorem indicates that a finite group $G$ is supersolvable if and
only if all maximal chains of the identity in $G$ have the same length. As
generalizations of Iwasawa's theorem, we provide some characterizations of the
structure of a finite group $G$ in which all maximal chains of every minimal
subgroup have the same length. Moreover, let $\delta(G)$ be the number of
subgroups of $G$ all of whose maximal chains in $G$ do not have the same
length, we prove that $G$ is a non-solvable group with $\delta(G)\leq 16$ if
and only if $G\cong A_5$.
