For general $n$, what is the largest subgroup of the symmetric group $S_n$ that has odd order?
I have a feeling that it might be the Sylow 3-subgroup...ADDED: but it isn't, as Mark Bennet points out in a comment. If I calculate right, it isn't true for large $n$ either, as $\mathrm{Syl_3}(S_{n-5}) \times C_5$ beats $\mathrm{Syl_3}(S_n)$ when $n=9k+5$ for integer $k$. But is it the biggest?