Let mm a positive integer, not divisible by 2,3,5,7. We generalize the classification of basic quasi-Hopf algebras over cyclic groups of prime order given in \cite{EG3} to the case of cyclic groups of order mm. To this end, we introduce a family of non-semisimple radically graded quasi-Hopf algebras A(H,s)A(H,s), constructed as subalgebras of Hopf algebras twisted by a quasi-Hopf twist, which are not twist equivalent to Hopf algebras. Any basic quasi-Hopf algebra over a cyclic group of order mm is either semisimple, or is twist equivalent to a Hopf algebra or a quasi-Hopf algebra of type A(H,s)A(H,s).Comment: 32page

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