Let $D_{506}$ denote the dihedral group with $1012 = 4 \cdot 23 \cdot 11$ elements. The Sylow theorems tell us that the number of 2-Sylow-subgroups (that is, subgroups of order 4) of $D_{506}$ divides 253.
But there are (at least) 506 elements of order 2 in $D_{506}$, namely all the reflections. These all induce subgroups of order 2, which have to be contained in a subgroup of order 4 due to another one of the Sylow theorems.
Since every group of order 4 contains exactly one element of order 2, we would need at least 506 2-Sylow-subgroups, contradiction.
Can you help me figure out what I did wrong? Thanks!
The part where you say that a group of order 4 contains exactly one element of order 2 is wrong. They are isomorphic either to $\mathbb{Z}_4$ with 1 element of order 2 or to $\mathbb{Z}_2 \times \mathbb{Z}_2 $ with 3 elements of order 2.