Let $G \leqslant S O(3)$ be a finite group. Suppose $G$ does not preserve any plane in $\mathbb{R}^{3}$. Show that for any point $p$ in the unit sphere $S^{2} \subset \mathbb{R}^{3}$, the stabiliser $\operatorname{Stab}_{G}(p)$ contains at most 5 elements.