# Decomposing a subgroup

Let $G$ be the symmetric group of degree six. Identify a subgroup of $G$ of order 16 and decompose it as much as you can. This came up on and old exam. How would you do this without having access to listing of all the subgroups of $Sym(6)$

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You should be able to find a copy of the dihedral group (the group of symmetries of a square, an $8$-element group) inside the symmetric group on $4$ letters. With the two letters left over, you can form a transposition, and bring it up to a group of order $16$.