Sylow's Theorems and Simple Groups

[noora]

I am wondering if some one can help it this:
Suppose G is a group with 316 [itex]\leq[/itex]|G|[itex]\leq[/itex] 325. Given that G is simple, find the possible value(s) for |G|. Be sure to explain your reasoning for each number. You'll need Sylow's Theorems of course.

This is what I have done:
the prime factorization of |G|:

316 2 2 79
317 317
318 2 3 53
319 11 29
320 2 2 2 2 2 2 5
321 3 107
322 2 7 23
323 17 19
324 2 2 3 3 3 3

If p is the largest prime factor, and |G| = mp^k where p doesn't divide m, the the p-Sylow subgroup is normal (The number of p=Sylow subgroups, n_p, = 1).

All the results except for 320 and 324 are straightforward:

316 n79 = 1
317 G is Z/317 and simple
318 n53 = 1
319 n29 = 1

321 n107=1
322 n23 = 1
323 n19=1

Thanks in advance

 

[Asim Riaz]

For 320, m = 2^6 and p = 5. Since n_5 = 1, the 5-Sylow subgroup is normal. For 324, m = 3^3 and p = 2. Since n_2 = 1, the 2-Sylow subgroup is normal. Therefore, the possible values of |G| are 316, 317, 318, 319, 320, 321, 322, 323, and 324.