  We associate a graph $\mathcal{C}_G$ to a non locally cyclic group $G$
(called the non-cyclic graph of $G$) as follows: take $G\backslash Cyc(G)$ as
vertex set, where $Cyc(G)=\{x\in G | < x,y> \text{is cyclic for all} y\in G\}$
is called the cyclicizer of $G$, and join two vertices if they do not generate
a cyclic subgroup.
  For a simple graph $\Gamma$, $w(\Gamma)$ denotes the clique number of
$\Gamma$, which is the maximum size (if it exists) of a complete subgraph of
$\Gamma$. In this paper we characterize groups whose non-cyclic graphs have
clique numbers at most 4. We prove that a non-cyclic group $G$ is solvable
whenever $w(\mathcal{C}_G)<31$ and the equality for a non-solvable group $G$
holds if and only if $G/Cyc(G)\cong A_5$ or $S_5$.
