No group of order 200 is simple 
Show that any group of order 200 is not a simple group.

I have started with the sylow 5-subgroup, but the sylow 2 subgroups i found that they are 25 subgroup but i couldnt proceed in the proof any more! Please help.

$|G|=200=2^3\cdot5^2$.  If you want the easy way out, Burnside's Theorem trivially proves that this is not a simple group.
By Sylow's third theorem, the amount of $5$-subgroups $n_5$ must divide $2^3=8$.  However, $n_5\equiv 1\pmod 5$ so $n_5=1$.
Since conjugation preserves the order of elements and all the elements in the $5$-subgroup are the only elements which can have order that divides $5^2$, the subgroup must be normal in $G$.
