Eligius Corporation is a business that collapsed over a hundred years ago. The page is for a historical organization, compare to Second Dawn or 12 Nations.
Eligius IV is a space ship which could be compared to each of the space stations that combined to make the Ark or to a bunker. A hundred years ago, these shelters were owned by the historical organizations.
In the series' present day, the groups of people aren't beholden to the historical companies/governments that no longer exist. While the former prisoners of Eligius IV were affiliated (likely not by their own choice) to Eligius Corporation, they're not a sub-organization of a collapsed company.
It's way too early to discuss Eligius III group. We don't even have confirmation that they're prisoners. Nor, if they do appear, do we know they're in "coalition" with the former Eligius IV prisoners. While both Eligius III & IV groups are connected to Eligius Corporation that doesn't define them being a singular group. Both Mount Weather and Alpha Station were controlled by US government, and they're not a singular group.
In other words, pages can be related without one being a sub-group of another. I think Eligius Corporation page should focus on the historical business, while a separate page focuses on the present day group of people. Just like Second Dawn focuses on the historical organization while Wonkru focuses on the present day group.