The program files for LrC only take up about 2+ Gigabytes of space on your drive.
The catalog file can grow to around 1 + GBs and the Previews folder can get quite big. But the files taking up most of the space on your drive are the actual Image files.
You can move both all your images files and the LrC Catalog file, along with the support files & folders for the catralog file (they are all in the same base folder as the catalog file), over to the external drive but you can not and Should NOT attempt to move the program files over to the external drive.
Your best bet is to get an external drive that is a SSD (Solid State Drive). They are the fastest drives.