Let $m$ 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 $m$. To this end, we
introduce a family of non-semisimple radically graded quasi-Hopf algebras
$A(H,s)$, constructed as subalgebras of Hopf algebras twisted by a quasi-Hopf
twist, which are not twist equivalent to Hopf algebras.