You can make \({7\choose 2}{7\choose 2} = 441\) quadrilaterals.
There are 5 squares.
There are \({7 \choose 2}\) rectangles.
There are \({7 \choose 2} + ({7 \choose 2}-1) + ({7 \choose 2} - 3) + ({7 \choose 2} - 6) + ({7 \choose 2} - 10) + ({7 \choose 2} - 15) = 91\) parallelograms.
All of the quadrilaterals are trapezoids. To count the non-parallelogram trapezoids, compute \({7\choose 2}{7\choose 2} - \left[ {7 \choose 2} + ({7 \choose 2}-1) + ({7 \choose 2} - 3) + ({7 \choose 2} - 6) + ({7 \choose 2} - 10) + ({7 \choose 2} - 15) \right]\text{.}\)