I like carebear's answer. I once wrote to Crisco Canada to ask, and they reminded me that their recipes are 'trade secrets' and they wouldn't disclose anything.
but most best guesses are a combination of palm, soy and/or cottonseed oils.
I find Crisco makes my soaps too soft, and palm is harder.
In our SoapMaker program, you can find a table of 70 or more oils, with the four 'qualities' (Hardness, Fluffy Lather, Creamy Lather and Moisturizing or Conditioning factor) .....and if you click on any of the four headers, the list will re-organize from high to low, or low to high for any quality you choose.
e.g if you want to know which oils give harder soap, click on the Hardness tab, and the list will show you top to bottom the hardest to softest. If you click twice, it reverses: softest to hardest....and likewise for lathers, moisturizing..
It's so handy, I use it all the time for determining a new soap recipe.