In his book Abelian groups, L. Fuchs raised the question as to whether, in general, in the factorization of a finite (cyclic) abelian group one factor may always be replaced by some subgroup. The answer turned out to be negative in general, but positive in certain cases. In this paper the complete answer for cyclic groups is given. In all previously unsolved cases, the answer turns out to be positive. It is shown that a cyclic group has the property that in every factorization, one factor may be replaced by a subgroup if and only if the group has order equal to the product of a prime and a square-free integer. Certain results are also given in non-cyclic cases.