We introduce, in the abstract framework of finite isometry groups on a Hilbert space, a generalization of antiperiodicity called N-cyclicity. The non-existence of N-cyclic solutions of a certain type for the autonomous ODE x '' g(x) = 0 implies the existence of N different subharmonic solutions for some forced equations of the type x '' + g(x) + cx' = epsilon f(t) where c and epsilon are sonic positive constants and f is, for instance, a sinusoidal function.