Full Name: Clarence Marylebone
Died: 1746
The Duke of Sandringham was first mentioned by Frank Randall in the S01-E01 Sassenach episode when Frank and Rev. Wakefield were talking about Jonathan Randall having a secret patron who protected him from punishment by his superiors.
He was mentioned again in the S01-E08 Both Sides Now episode by Claire when she was being questioned by Jack Randall at Fort William, causing Jack to choke on his wine and stain his cravat. She was using Frank's supposition about the duke being Jack's patron and it paid off. But Jack countered with mentioning the dutchess, but of course, Sandringham never married.
He shows up in the latter half of the first season in the S01-E10 By the Pricking of my Thumb episode, where his full name is revealed: Clarence Marylebone (also mentioned in Voyager). The duke likes young boys, and was after Jamie when he was a lad. In the book Jamie told a funny story about how he drank a whole bottle of laxative to avoid the duke's attention, then swiftly left Castle Leoch. When they met again in 1743, the duke remembers the episode and asked Jamie if he was still having problems with his bowels.
There's more in Dragonfly in Amber about the Duke of Sandringham, including his holding Claire captive, his niece Mary Hawkins, and his demise (by Murtagh) in S02-E11.