Interesting. I was a decent runner in high school, way back. I'm a cyclist now, but I found that cycling groups tend to either be focused on athletic performance or activism and I don't particularly care for either at this point. I'll have to try some running groups as there are a lot of local ones.
15 years ago I started cycling, by going to amateur XC races. I did that for almost 10 years and it was fun, I got some friends doing it and it worked great for a while. A combination of cycling injury and many people leaving the country ended that endeavor, but it is an example of some simple and practical approach. I was not even looking to socialize, but improve my sedentary life, I found others with similar interests.
I have a childhood friend that is cycling, but he lives in the Netherlands. There people who are cycling more than daily drivers are interested in athletic performance or activism and it kind of sucks, but he got friends skiing and scuba diving, as long as there are common interests you will find some decent people to socialize with.
That's interesting, but in my experience cycling groups are the most social individual sports groups (even more social than many team sports even). Even the performance focused groups tend to stop at the coffee shop for some banter after the ride, and some less performance oriented groups seem to be more focused on the coffee than even the ride itself.
Are you talking about road cycling or mountain biking? My experience is definitely with the former. I think it helps that in group rides you automatically end up riding next to someone new and chatting along. Easily breaks the ice.