Center Controller reported an aircraft flew below minimum instrument altitude during a period of high traffic volume.

Narrative

I cant really give a detailed description of what happened. If you pull up the falcon (Radar Display Relay) on the session you will see why. We did have multiple aircraft on vectors to get them into ZZZ. It is the busiest day of the year there. I as did point out to the R-side a few minutes before Aircraft X was going to get into the 121 MIA (Minimum IFR Altitude) and told him to turn him. He gave them a heading. I didn't verify that the heading was going to be enough. I got pulled away with everything else going on. I saw it when the R-side saw the msaw alert. It is a very crazy situation every year with this fly in. One thing that was brought to my attention after the fact was that Aircraft X was a ZZZ1 arrival and somehow got sequenced in with the ZZZ arrivals. I don't know if they came from Sector X on a heading and that's why we missed it or if we just overlooked it. But he was never cleared to the ZZZ airport. The pilot never said anything either. I believe that there are numerus things that we could do to make this a safer operation. First we were briefed one thing about how many will be allowed into the airspace at once and how arrivals and departures were going to be handled. That was not followed and way to many aircraft were allowed into sector XY then what should have been. During the time we had Aircraft X below the MIA was when they shut the airport off because the airport couldn't take anymore aircraft. It throw off the entire plan of the R-side. We then had to start issuing holding and the time that took with aircraft needing to divert for fuel and just in general it was a mess. The tower was claiming we needed to allow aircraft to depart but they weren't requesting departures. Please review the day and help come up with ways to make this a better operation