does the symmetric group s_n always contain a subgroup of index n-1?

for s_2 we have s_2 itsself as a subgroup with index 1,

for s_3 we have the alternating group a_3 as a subgroup with index 2,

for s_4 we have the dihedral group d_4 as a subgroup with index 3,

for s_5 i cant find any subgroup with index 4, does there exist one? and what about n greater then 5?

also for fixed n what is the biggest index k smaller then n such that there exists a subgroup of s_n with index k?

submitted by /u/kai21351
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Nevin Manimala

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