  The $H~\mathrm{mod}~K$ theorem gives all possible periodic solutions in a
$\Gamma-$equivariant dynamical system, based on the group-theoretical aspects.
In addition, it classifies the spatio temporal symmetries that are possible. By
the contrary, the equivariant Hopf theorem guarantees the existence of families
of small-amplitude periodic solutions bifurcating from the origin for each
$\mathbf{C}-$axial subgroup of $\Gamma\times\mathbb{S}^1.$ In this paper we
identify which periodic solution types, whose existence is guaranteed by the
$H~\mathrm{mod}~K$ theorem, are obtainable by Hopf bifurcation, when the group
$\Gamma$ is finite cyclic.
