When you look at the chariot seat's mesh in wireframe mode, you'll notice that one half of the mesh is a brighter white. That means that there are overlapping faces. This also messes up the subdivision on your object.
So, to fix this you'll need to use the select tool on the the vertices (not drag select) because basically what is happening is that there are "double points". There are two points, both in the same spot, and they are all connected to faces. So select once, and delete a point, if it deletes and the structure of your mesh is intact, move to the next. If not, undo and find another point to delete until you get a clean mesh again (remember there may be some points you might miss because after you delete some points, they'll be left over but not connected to faces.)
That's how I solved the problem.