Center Controller reported a heavy workload with congested airspace led to an aircraft descending below the minimum IFR altitude.

Narrative

Sector X had an aircraft that was flashing at me landing ZZZ at 100 heading for a 107 MIA. I was busy with a large volume of aircraft and multiple ZZZ1 arrivals. The aircraft checked on and I did not run the vector lines to make sure that the aircraft was going to go in the MIA block. Once I noticed he was close to it I gave him a left turn to avoid the terrain with the word immediately at the end. He did not read it back and turned left instead. Another aircraft that I thought was him said that they would like to cancel IFR and I removed strips on the aircraft. I then realized what I had done and tried to tag the aircraft back up but was not able to. I gave him a left turn again and he then said he would like to cancel IFR. Note that I did not give him a low altitude warning when I dropped the tag.The Supervisor was not engaged in the area at all. The sector needed to be split but he continued to have controllers hit the list. I was asked for a break and all I had time to answer was with a simple 'no.' This Supervisor is often not engaged in the area that he is supervising. It is not fair to hold a controller that is extremely busy to the standard that they should ask for help. Sometimes a controller does not have the time to ask; or they're preoccupied with making the sector run smoothly. The supervisors need to be more engaged.