Upper bound on groups of order 60

Question

I am aware of the fact that there are 13 non-isomorphic groups of order 60 but the proof of this is really long and something that I cannot present in a few minutes. Hence, I need to give a short proof that there are less than 65 non-isomorphic groups of order 60. The reason I want this bound is because in the numerical example mentioned in Brauer's lecture on Representation of Finite Groups compiled in Lectures on Modern Mathematics Vol. I, it has been said that there are 65 systems satisfying certain properties but not every system has a group corresponding to it. So is there any quick way to show that there have to be less than 65 non-isomorphic groups?