Switzerland in 7 Days

I would not do this solo, so I cannot imagine it with a group of 13. I would choose no more than three stops in a week, and that is only if you need a one-night stay in Zurich before your flight out.
If the flight is noon or later, you can stay in Lucerne and depart from there in the morning for an easy journey to the airport. That could help a bit, but if it is a morning flight, you'll need to stay in Zurich.

Welcome jan, IMO too much time on the train. Not to mention trying to please such a large group and manage their movements. If it were me (again), I'd:

Fly to Zurich, immediately taking the train 1.25 hours from the airport to Luzern, three nights there. Then train to the Lauterbrunnen valley, and three nights there. IF you must see Lausanne, make that your last night before flying home. If not, either Bern or back to Zurich to fly out, or add a night to either Luzern or Lauterbrunnen.

I think Luzern has the best combination of outdoor beauty and larger city activities (such as shopping) allowing your group to safely divide up. Bern is a nice city, but it's not in the mountains so strictly indoor things to do, and thats not what I go to Switzerland for.

Zurich is better to fly in and out of, because there are actually direct flights to North America from there. With Geneva, you have to make a connection.

For your group here are my suggestions (I am the "leader" for 10, but we all have similar interests, so much easier to plan!)
Day 1: KEF to ZRH, take train to Luzern
Day 2: Luzern
Day 3: Train to Interlaken. This location allows for a variety of options such as day trip to Bern, boat cruises on the lakes, easy walks in the Berner-Oberland (or just riding mountain transport), the Ballenburg Folk Museum, etc. The Berner-Oberland Pass will allow each group member to do her own thing. https://www.berneseoberlandpass.ch/all-in-1-ticket-bernese-oberland/
Day 4: Interlaken
Day 5: Interlaken
Day 6: Interlaken
Day 7: Train to Zurich
Day 8: Flight home