how to use advanced filter to filter "column header"

I have a large table 22004 (rows) x 561 (columns).
553 of the 561 columns represent unique subject IDs.
Each row represents a data point of those 553 subjects.
(so yes, we have 22004 unique data points for each subject)

I need to extract data from this table by selecting a subset of those subjects.
Generally I'd use advanced Filter to select them. But in this case,

(a) I fail to transpose the table - maybe because Excel cannot have 22004 columns ??
(b) The 553 subjects do not have a "common header" to match to the header in the criteria range.

Any approach to subsection a large table and extract a subset of the data from the table ?