A finite group $G$ with conjugacy class nX is said to be $nX$-complementary generated if, for any arbitrary $x\in G-\{1\}$, there is an element $y\in nX$ such that $G = \langle x, y\rangle$. In this paper we prove that the Conway's group $Co_1$ is $nX$-complementary generated for all $n\in \Pi_e(Co_1)$. Here $\Pi_e(Co_1)$ denotes the set of all element orders of the group $Co_1$.