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See http://en.wikipedia.org/wiki/N-group_%28finite_group_theory%29 where it talks about minimal simple groups. You'd need to check in particular that the example of 2x2 projective special linear groups can't have order divisible by 60. But I don't believe that.

Edit: I was however using the wrong formula for the order, which can indeed avoid divisibility by 5.
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See http://en.wikipedia.org/wiki/N-group_%28finite_group_theory%29 where it talks about minimal simple groups. You'd need to check in particular that the example of 2x2 projective special linear groups can't have order divisible by 60. But I don't believe that.