We consider the class
C
\mathcal {C}
which consists of the groups
M
M
with
M
/
M
′
M/M’
finitely generated which satisfy the maximal condition on direct factors. It is well known that any
C
\mathcal {C}
-group has a decomposition in finite direct product of indecomposable groups, and that two such decompositions are not necessarily equivalent up to isomorphism, even for a finitely generated nilpotent group. Here, we show that any
C
\mathcal {C}
-group has only finitely many nonequivalent decompositions. In order to prove this result, we introduce, for
C
\mathcal {C}
-groups, a slightly different notion of decomposition, that we call
J
J
-decomposition; we show that this decomposition is necessarily unique. We also obtain, as consequences of the properties of
J
J
-decompositions, several generalizations of results of R. Hirshon. For instance, we have
Z
×
G
≅
Z
×
H
\mathbb {Z} \times G \cong \mathbb {Z} \times H
for any groups
G
G
,
H
H
which satisfy
M
×
G
≅
M
×
H
M \times G \cong M \times H
for a
C
\mathcal {C}
-group
M
M
.