If you're looking at the main parts of the big 4 that's pretty much right. It's important to remember that LNER also included what is now 'one' though. FGW is the only TOC that is closes to one of the big 4, although parts of Arriva Trains Wales and Cross Country would have been GWR.
A more comprehensive list would be something like:
LMS - Virgin West Coast, Silverlink (the entire lot, including the NNL and GOBLIN, although the latter would have run from Kentish Town on the MML), FCC (Thameslink north of Moorgate),the majority of Central Trains, Merseyrail, parts of Arriva Trains Wales (North Wales Coast etc), parts of Northern, parts of First Scotrail, parts of Virgin Cross Country.
LNER - GNER, FCC(GN), one, c2c, parts of Central (Nottingham to Skegness), parts of Northern, Chiltern (former GCR), parts of Scotrail, parts of Virgin Cross Country
SR - South Eastern, Southern, FCC (Thameslink south of Blackfriers), SWT.
GWR - First Great Western, parts of Arriva Trains Wales (south and central Wales), parts of Chiltern (lines to Birmingham), parts of Virgin Cross Country.