  As a natural continuation of study $LCM$-groups, we explore other properties
of $LCM$-groups and $LC$-series. We obtain some characterizations of finite
groups which are not LCM-groups but all proper sections are $LCM$-groups. Also,
for a $p$-group $G$, we prove that $G$ is a $LC$-nilpotent group and we obtain
a bound for its $LC$-nilpotency. Finally, as an application, we prove that a
finite supersolvable group, groups of order $pq, pq^2$ and $pqr$ are
$LC$-nilpotent groups, where $p,q$ and $r$ are prime numbers.
