Let $\G(M)$ be the set of $S$-subsets of 
a centered $S$-set $M$ with a zero, where 
$S$ is a semigroup. In general, minimal 
$\z$-subsets of an $S$-set $M$ fall into 
three types, where $\z$ is a conjugate map 
on $\G(M).$ Now, for the $\z$-core $K_\z$ 
of a maximal $S$-subset $K$ of an $S$-set 
$M,$ the $\b{\z}$-socle of $M/K_\z$ consists 
of the only minimal $\b{\z}$-subset of $M/K_\z$ , 
where $\b{\z}$ is a conjugate map on $\G(M/K_\z)$ 
naturally induced by $\z.$ Here we use this fact 
to introduce the three $\z$-types of maximal 
$S$-subsets of $M$ and we give a characterization 
of a maximal $S$-subset of $M$ of $\z$-type 
$i(i = 1, 2, 3).$ Now, it is known that a finite 
group $G$ is solvable if and only if every maximal 
subgroup of $G$ is $c$-normal in $G.$ On the other 
hand, a concept of a $c_\z$-subset of  an $S$-set 
is analogous to that of a $c$-normal subgroup of 
a group and here we show that for any maximal $S$-subset 
$K$ of an $S$-set $M,~K$ is a $c_\z$-subset of $M$ 
if and only if $K$ is either of $\z$-type 1 or of 
$\z$-type 2. Continuously, we give some properties 
about an $S$-set whose maximal $S$-subset is always 
a $c_\z$-subset.