When I brine my 20 lb + birds, I use a 5 gallon bucket and usually only need to make 2 gallons of brine. That means it displaced 3 gallons or 12 qts..
I don't deep fry my birds but based on the math that would mean at least an 8 gal pot. Plus a safety factor. So I would think you would need at minium a 36 qt pot.