Each club has a member group of its own. Joining the club puts you in it, leaving takes you out, and the club's forum, album, calendar and blog are restricted to it. That is the entire mechanism — clubs reuse the permission system rather than having one of their own.
🚨 Those groups are deliberately kept out of the Groups screen and out of every group picker. A site with two hundred clubs would otherwise put two hundred tick boxes on every restriction form.
They are also refused if submitted directly, which matters more than the tidiness: without that, a crafted request could restrict a site-wide page to the members of somebody's private club.
Delete the club and its group goes with it.