TITLE: Solvability of finite groups of order coprime to 15 -- proof without using CFSG?
QUESTION [8 upvotes]: Is the solvability of finite groups of order coprime to 15
essentially easier to prove than the entire Classification of Finite Simple Groups?

REPLY [4 votes]: As Derek Holt has pointed out, the answer to the question is yes. --
Thompson proved that the only finite simple groups of order coprime to 3 are the
Suzuki groups, and Glauberman later extended this to a classification of
simple groups that do not have ${\rm S}_3$ as a subgroup.
Both of these results are pre-classification, though they might not have been published.