The name of the order which she joins

They deliberately didn't name the order in the film, and I'm not sure it was every named in the book, either. The woman who was the real-life inspiration for Sister Luke, Marie-Louise Habets, joined the Congregation of the Sisters of Charity of Jesus and Mary in Belgium in 1927 and left the order in 1944. (Her religious name was Sister Marie-Xaverine.)

The order was founded in 1803 in the Belgian city of Ghent by Father Peter Joseph Triest. There are other orders with "Sisters of Charity" in their name, and this particular order is not affiliated with those. Most notably, St. Vincent de Paul founded the Daughters of Charity in 1633 and some congregations related to that order are known as Sisters of Charity.

The habits in this film closely resemble the ones worn in the era depicted by the Belgian order of the Sisters of Charity of Jesus and Mary.