(Joint with C. Gorodski, C. Lange, A. Lytchak) We show the existence of C>0 such that, for all n>1 and every closed non-transitive subgroup G of O(n+1), the diameter of the quotient of the unit sphere S^n by G is larger than C. The novelty is that C is independent of the dimension n. The classification of finite simple groups is used in the proof.