If we're talking real lye-based soap as opposed to soap-syndet cleansers, my notes show the typical coconut oil and/or palm kernel oil content is around 20% for modern commercial bath soaps. I also gather the CO/PKO content can range from 10% to 40%, so it's not like 20% is the only percentage used.
edit: The other main ingredient I usually see in commercial soap is tallow (sodium tallowate). An 80:20 blend of tallow and coconut is pretty common in the old soap makers' manuals I've collected. I can't say I've ever seen lard (sodium lardate) in the ingredients list for any commercial soap, modern or older.