Find subgroup(s) of order 2 of $D_{2p}$

Question

Suppose that $p$ is a prime greater than 2. Dihedral group $$ D_{2p}=\left\{ 1,a,a^2,\cdots ,a^{p-1},b,ab,a^2b,\cdots ,a^{p-1}b|ord\left( a \right) =p,ord\left( b \right) =2,ab=ba^{p-1} \right\} . $$ Now I can figure out that $\{1,b\}$ is a subgroup of order 2, and it is not normal, as $aba^{-1}=ba^{p-1}a^{-1}=ba^{p-2}\ne 1\ or\ b$. But if I consider Sylow Theorem, there exist more than one Sylow-2 subgroups since we have got a non-normal subgroup of order 2. Then what is the second Sylow-2 subgroup? Equivalently, I am looking for another element of order 2 in $D_{2p}$ other than $b$. Any help would be appreciated! Thanks!

Answer 1

All the elements with $b$ in them are of order $2$, corresponding to a reflection of a regular polygon in some line of symmetry; this is true of any dihedral group.