If your question is about pure possibility then the answer is 5. If you sacrifice 96.8% of your daily standing you can be with 5 different syndicates.
But a more efficient approach would be to start with 3 (no sacrifices made) and then expand to 4 which sacrifices only 11.1% of your daily standing.
I'd recommend to play 2 days red veil and one day new loka then repeat to be max standing with red veil, new loka, steel meridian and perrin sequence
Or you can play 2 days steel meridian and 1 day cephalon suda then repeat to be max standing with steel meridian, cephalon suda, red veil and arbiters of hexis