How does the United Club membership from Chase Club card work?

Do I get a formal United Club subscription under wallet? Or the linked Chase Club card in my MileagePlus account suffice for entry to United Club?

The reason I am asking, is I don't see an official United Club subscription under my wallet section (however, if i attempt to subscribe to a new United Club membership, it tells me I already have that subscription and wouldn't let me subscribe a new one), even though my Chase United Club card is listed in my MileagePlus account... so I wanna make sure I can access United Club without a formal subscription under wallet section, and a linked Chase United Club card in my MileagePlus account will be sufficient for entry (i.e. Club reception agent can see I have a Chase United Club card in my account and grant me entry without a subscription)...

So i wanna make sure I understand how the United Club access works from Chase Club card....