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Since nontrivial unipotent elements have order $p$, you will have plenty of subgroups, doc. I don't think they are understood.

For instance, the answer to your second question is no as soon as $K$ contains at least $p^3$ elements. Just take two different $C_p^2$ in the additive group of $K$ and consider two upper triangular groups with off-diagonal elements in your $C_p^2$-s...

Having said that, it may be an interesting questions what finite simple groups sit there. Are they all $PSL(2,q)$?

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Since unipotent elements have order $p$, you will have plenty of subgroups, doc. I don't think they are understood.

For instance, the answer to your second question is no as soon as $K$ contains at least $p^3$ elements. Just take two different $C_p^2$ in the additive group of $K$ and consider two upper triangular groups with off-diagonal elements in your $C_p^2$-s...

Having said that, it may be an interesting questions what finite simple groups sit there. Are they all $PGL(2,q)$?PSL(2,q)$? 1 Since unipotent elements have order$p$, you will have plenty of subgroups, doc. I don't think they are understood. For instance, the answer to your second question is no as soon as$K$contains at least$p^3$elements. Just take two different$C_p^2$in the additive group of$K$and consider two upper triangular groups with off-diagonal elements in your$C_p^2$-s... Having said that, it may be an interesting questions what finite simple groups sit there. Are they all$PGL(2,q)\$?