the number of odd sylow numbers in a non-abelian simple group

Let $G$ be a non-abelian simple group.Is there exist a positive integer number $k$ such that there are at most $k$ odd sylow numbers in $G$?

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I suppose you mean: is there a fixed number $k$ which works for all non-Abelian finite simple groups- for any given such group $G,$ there obviously is such a $k.$ You might also explain to the reader that you are really asking for bound of the number of odd primes $q$ such that a Sylow $2$-subgroup $S$ of$G$ normalizes a Sylow $q$-subgroup of $G.$ – Geoff Robinson Dec 26 at 17:40