How many sylow 5 subgroups in S7?

Question

Number of sylow 5 subgroups in $S_{7}$ is? $|S_{7}| =5040= 2^4 \times 3^2 \times 5\times 7$. number of sylow 5 subgroups is by sylow 3rd theorem n5=1+5k=possibilities are$\left\{1,6,16,21,36,56,126,336,1008\right\}$.we know that sylow subgroup exist of order 5 is cyclic . O(Sylow 5 subgroup)=5.Suppose we have 6 sylow subgroup then each sylow subgroup order 5 elements is 4 totally we have 24 elements of order 5 is it possible?