The Dominican Friars of the Province of St. Albert the Great currently administer the Shrine of St. Jude Thaddeus. Since the Shrine was founded in 1929, however, its history joins together the Province of St. Albert the Great (Midwest) and the Province of St. Joseph (Eastern U.S.). At the time of its founding, there were only two provinces of Dominican friars in the United States: Province of the Most Holy Name of Jesus and the Province of St. Joseph. The Province of the Most Holy Name of Jesus extended from the West Coast to the Rockies and the Province of St. Joseph extended from the Rockies to the East Coast. Being located in Chicago, therefore, the Shrine of St. Jude was located in the Province of St. Joseph from its foundation in 1929 until the division of the Province of St. Joseph in 1939 to form the Midwest Province of St. Albert the Great.
“Dominicans” is the informal name given to members of the Order of Preachers and is based upon the name of the founder of the Order of Preachers, St. Dominic de Guzman O.P. In fact, the initials O.P. which appear after the name of every professed Dominican be it a friar (priest or lay brother and formerly called 1st Order), nun (cloistered woman in a monastery and formerly called 2nd Order), sister (active sister who does ministry in the community and formerly called 3rd Order), or lay member (ordinary non-ordained Catholics and also formerly called 3rd Order) stand for Ordinis Praedicatorum which is simply Order of Preachers in Latin.
Find out more at the websites of the Dominican Province of St. Albert the Great, USA and the Order of Preachers.