Making a Blanket with Cloth Property not fall through floor

I want to make a blanket on a bed, and currently it is falling straight through the floor. I have made it a rigid body with a cloth collider, and have highlighted all of the vertices in the mesh, but this makes the blanket act as a piece of wood. How do I make sure the blanket does not fall through the ground when it's lying on the bed, but acts like a cloth when I pick it up?