Let $m$ be greater than or equal to 3 and $n$ be a multiple of $m$.
An $m$-vertex cycle graph is denoted $C_m$.
We will call a spanning subgraph whose components are $C_m$ of
the complete graph $K_{n}$ a $C_m$-spanning subgraph of $K_{n}$.
The Dihedral group $D_{n}$ acts on the complete graph $K_{n}$ naturally.
This action of $D_{n}$ induces the action on the set of the $C_m$-spanning
subgraphs of the complete graph $K_{n}$ .
In [4], we calculated the number of the equivalence classes of
the $K_m$-spanning subgraphs of the complete graph $K_n$ by using
Burnside's Lemma.
In this paper we calculate the number of the non-equivalent
$C_m$-spanning subgraphs of $K_{n}$ for all $m$ and $n$.
In the special case we have the number of the non-equivalent
Hamiltonian cycles of $K_m$ for all $m$.