It is a monster. Let me try to describe it in a rough but understandable way (knowing that I am not an experimentalists and do not belong to any huge collaboration of more than 3000 people).

You have different groups focusing on different aspects (data analysis, data quality monitoring, data taking, etc.). Within each group, you have subgroups and within each subgroups, you have other groups. It is a kind of giant tree. At each leaf, if I can speak that way, you have responsible taking care of what is going one below them. And those are renewed regularly.

You also have teams taking care of transversal activities with respect to all leafs. For instance, each analysis group will produce a physics analysis note. Internal referees that have no connection with the analysis group are nominated (and a good number of them) to review the work. There is something like an editorial board, and a publication committee. Etc.

At the end, a lot of administration but it works well.