When someone who is not a member of a main group joins - or attempts to - a sub-group (directly, with first joining the main group), there are various issues. See also https://groups.io/g/GroupManagersForum/topic/inviting_new_member_to_a/81357919
Bottom line over this issue is that when somebody not a member of a main group wishes to join a sub-group, two seperate processes have to be gone through for that to happen: (1) join main group and (2) join sub-group - both requiring authorisation (explicit or implicit), and visibility (to member and moderator(s)/owner(s)) notifications by e-mail and activity log entries), as specified by settings.
When a non-member applies directly to join a sub-group, by sending an e-mail to Subgroup+subscribe@... , the Main group owner gets an approval request, and if approved, the Main group is joined - with a 'Welcome to Main@...' e-mail. to new member, notification to Main group owner and Main group activity log entry. From the subscribers point of view, the process they have undertaken has failed: they have not joined the Subgroup they asked to, rather the Main group they didn't: (without any positive indication that they haven't joined the subgroup, or what they need to do to join the subgroup).
If the non-member is invited to join the subgroup (by a subgroup moderator who is not a main group moderator), and accepts that invitation, then they do join the subgroup, with a 'Welcome to Subgroup' e-mail, notification to subgroup moderator, and entry in subgroup activity log. And also they become a member of the Main group, without either any approval by a Main group moderator, or with any notification e-mails (to new member or Main group moderator), or entry in main group activity log. The first they will know is they receive a post from a group (Main) that neither they, nor anyone else, have had any notification of having joined.
Similarly, when a member leaves a main group, they also should be going through a process of leaving any sub-groups they are members of (including appropriate notifications). What actually happens is that they just disappear from the sub-group, without any notification e-mail or activity log entry.