C4 is actually 1 4 b7 in BIAB which is just an inversion of the sus chord

The other quartals currently missing are 1 4 Maj7 and 1 b5 Maj7.

In a four part harmony situation C 1,4,6 7 in BIAB is achieved by F5b/C
It blends quite well with Maj7b5 and 7sus chords

Regards

Alan