I came across this statement on page 24 of these notes:
[Every finite group] G has $\leq |G|^{\log |G|}$ subgroups (because every subgroup has at most $\log |G|$ generators).
I figured out why the part in parentheses is true, but I’ve been having trouble seeing why the main statement follows from that fact. I’ve searched for a while but have only seen the bound mentioned as “trivial” in some more advanced papers, so I’m guessing it’s an elementary fact in finite group theory that I don’t have the background in to see. If anyone could clarify the bound a bit that would be great.