What would the diagram look like for a horizontal cleanout?

Although illegal in mosts states that I know of it can be done in the following manner (see image).

AAVS (see image below) are mechanical vents that again, may be illegal in your area, so you may want to run a dedicated 1.5" vent from the new setup over to an existing vent to at least attempt to make this setup as legal as possible.

You may also find difficulty in removing the cleanout plug, or if you are successful in removing the cleanout plug you may not easily match the threads up to a PVC adapter, so measure things out and double check that you can transition from the cleanout to PVC before you purchase everything.

Finally, a better and LEGAL job would be to cut a new fitting into the horizontal drain line. Glad to help with that if you want to do that, OK?

Mark