Describe the structure of the Sylow $2$-subgroups of the symmetric group of degree $22$ 
Describe the structure of the Sylow $2$-subgroups of the symmetric group of degree $22$.

The only thing I've managed to deduce about the structure of $P\in \operatorname{Syl}_p(G)$ is that $|P| = 2^{12}$. 
Help please :)
edit: I obviously can't count. Silly, I (for some reason) only counted $8$ as one $2$ and $16$ as one $2$. But as far as I can tell now, $2^{17}$ is the highest power of $2$ dividing $22$.

First let's consider the case of Sylow 2-subgroups for symmetric groups of degree $2^k$.  I claim that the Sylow 2-subgroups of $S_{2^k}$ are isomorphic to the automorphism group $G_k$ of $T_k$, the complete, rooted binary tree of depth $k$ (having $2^k$ leaves).
Since an automorphism of $T_k$ is determined uniquely by automorphisms of the two subtrees of the root, together with a decision of whether to exchange the subtrees of the root, we have $|G_k| = 2 |G_{k-1}|^2$.  Since $|G_0|=1$, by induction, $|G_k|$ is always a power of 2.  Furthermore, $G_k$ maps injectively into $S_{2^k}$ (by sending an automorphism to the permutation it induces on the leaves of $T_k$), so $S_{2^k}$ has a 2-subgroup with the order of $G_k$.  Finally, it is easy to check that the order of the Sylow 2-subgroup of $S_{2^k}$ satisfies the same recurrence relation that is satisfied for $|G_k|$.  This proves the claim.
For the general case of $S_n$, write $n$ as a sum of distinct powers of 2, and take the direct product of the Sylow 2-subgroups for each.  For example, the Sylow 2-subgroups of $S_{14}$ are isomorphic to the direct product of the Sylow 2-subgroups of $S_8$, $S_4$, and $S_2$.
