It's difficult to define "groups" as a couple of the teams overlap a lot, or some actors leave the show and the role is re-casted. For Example, The Turbo Rangers are most of the Zeo Rangers, whom are most of the Mighty Morphing Power Rangers in the first place.
Team 1 through 5 are slight cast, costume and setting changes, considered the Zordon Era. So it can be considered 7 seasons of a single show. (Alien Rangers being a filler mini-season). Everything forward of Lost Galaxy were pretty stand alone seasons/teams, until Samurai and Super Samurai, which are two seasons of the same team. The same for Megaforce & Super Megaforce, and Dino Charge & Dino Super Charge.
Given that, there are 20 distinct major groups (From Wikia):
- Mighty Morphin Power Rangers
- Aquitar Rangers (Might Morphin Alien Rangers)
- Zeo Rangers
- Turbo Rangers
- Space Rangers
- Galaxy Rangers
- Lightspeed Rangers
- Time Force Rangers
- Wild Force Rangers
- Ninja Rangers (3 intertwined teams: Wind Rangers, Thunder Rangers, & Green Samurai Ranger)
- Dino Rangers
- SPD Rangers
- Mystic Rangers
- Overdrive Rangers
- Jungle Fury Rangers
- Ranger Operators
- Samurai Rangers
- Mega Rangers
- Dino Charge Rangers
- [TBD] Ninja Steel Rangers, probably will have a unique name to not be confused with the Ninja Rangers
Spread among 23 individually titled series:
- Mighty Morphin Power Rangers (3 Seasons)
- Mighty Morphin Alien Rangers
- Power Rangers Zeo
- Power Rangers Turbo
- Power Rangers in Space
- Power Rangers Lost Galaxy
- Power Rangers Lightspeed Rescue
- Power Rangers Time Force
- Power Rangers Wild Force
- Power Rangers Ninja Storm
- Power Rangers Dino Thunder
- Power Rangers S.P.D.
- Power Rangers Mystic Force
- Power Rangers Operation Overdrive
- Power Rangers Jungle Fury
- Power Rangers RPM
- Power Rangers Samurai (Season 1)
- Power Rangers Super Samurai (Season 2)
- Power Rangers Megaforce (Season 1)
- Power Rangers Super Megaforce (Season 2)
- Power Rangers Dino Charge (Season 1)
- Power Rangers Dino Super Charge (Season 2)
- Power Rangers Ninja Steel [TBD]
*This does not include any secondary groups, evil clone groups, history/past groups (Like the Legendary Rangers (actors from past shows), or alternative power sets (like the Ninja Rangers for the MMPR, or Shogun Rangers for the Samurai Rangers).
This is also limited to the Americanized show, not the original Japanese shows that the scenes are cut from. Which is much longer, and disjointed, as it's a genre of multiple unconnected shows, not just a franchise.