关键词： 5次对称群, 子群, Lagrange定理, 循环置换

Abstract: Because the inverse of Lagrange theorem of finite group does not hold,it is difficult to determine all the subgroups of  Sn,to determine whether  S5 has the same order subgroups for any positive factor of the absolute value of  ｜Sn｜.Using Lagrange's theorem and the concept of n-letters symmetric group,some of subgroups of 5-letters symmetric group  S5 are proved.

Key words: 5-letters symmetric group, subgroup, Lagrange's theorem, cycle permutation