Simple groups with cyclic odd Sylow subgroups

Question

I know that every odd Sylow subgroups of $PSL(2,p)$ is cyclic. Is there any other simple group with cyclic odd Sylow subgroups.

Thank you in advance

Answer 1

Another family of examples is $\operatorname{PSL}(2,2^n)$, where $n > 1$.

Also, the Janko group J1 (wikipedia) has order $175560 = 2 ^ 3 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 19$, so its Sylow subgroups of odd order are cyclic of prime order.

I found these examples from the following paper, which discusses simple groups with the desired property that have abelian Sylow $2$-subgroups.

Gagen, Terence M.. "On groups with abelian Sylow 2-groups.." Mathematische Zeitschrift 90 (1965): 268-272. link to article

Are there more examples? With the classification it should be possible to find all finite simple groups that have cyclic Sylow $p$-subgroups for all odd primes $p$. I don't know if anyone has done this.

EDIT: Another family of examples is given by the Suzuki groups $\operatorname{Sz}(2^{2n+1})$, where $n \geq 1$ (wikipedia). It is possible to show that every subgroup of $\operatorname{Sz}(2^{2n+1})$ that has odd order is cyclic. This is shown by Suzuki in the following paper (pg. 138, after Theorem 9):

M. Suzuki, On a Class of Doubly Transitive Groups, Annals of Mathematics Second Series, Vol. 75, No. 1 (Jan., 1962), pp. 105-145 JSTOR

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LATER EDIT (2022): See this answer of Geoff Robinson for the complete solution: link.