Air carrier First Officer reported an inflight diversion due to fumes in the cabin. Flight landed at diversion airport.

Narrative

We were flying from ZZZ1 to ZZZ2 and when the event started we were cruising at FL350 off the coast. Everything in the flight before that had proceeded normally; but the event started with a call from FA3; I believe; informing of us of a strong and terrible odor between rows 12 and 23. It had caused on the FAs (flight attendants) to vomit and feel light headed to the point of passing out. We later found out that 2 of the other FAs noticed the smell and were getting headaches. At that point the Captain and I knew we were going to need to divert; thinking this a potential fume event; especially with the FA getting so ill. I took control of the aircraft and radios while the Captain called Dispatch through his iPad to discuss with them. Dispatch agreed with out plan of diverting to ZZZ; so I informed ATC of our request as well as the reason for the diversion. The Captain then elected to request priority handling; we decided with the information we had; it would be prudent to get priority handling to get the aircraft on the ground and receive any needed medical assistance. I was flying at this point and got the aircraft setup for the approach in ZZZ while the Captain spoke with the FAs and informed the pax of what was going on. The Captain took control of the aircraft again while I kept the radios and helped with communication back and forth with dispatch through ACARS. Once we were caught up getting everything set up and briefed for the diversion; we started looking at the QRH for a fume event. At this point we hadn't smelled anything and we both felt fine; we followed the QRH and since the odor was determined to be in the cabin elected to turn off pack 2; I think we were around 11000 to 13000 feet on the arrival at this point. Pack 2 stayed off the rest of the flight; partly was for the time required in the procedure and then we were getting into the approach phase and our focus was more on getting the aircraft on the ground safely. We had received our gate; so we requested XXC to minimize taxi time. It was until short final the we both started getting a faint odor; it had apparently been spreading throughout the cabin at that point as well; as reported by the FA in the rear of the aircraft. We had not received any complaints from passengers about feeling ill; and it was intermittent on whether they smelled it or not. Once we landed and taxied off the runway; I turned pack 1 off as well out of precaution since we had started to smell a dirty socks type odor. We taxied to the gate without issue; where we were met with the requested medics.Cause: One thing the Captain and I discussed after the event; was it probably would have been a good idea to proactively don the masks even though we didn't start to smell anything until the very end of the flight.