Depends. There's 3 different groups, each with own claimant to the throne: Legitimists (House of Bourbon as the eldest branch of Capetians; but a different branch of Bourbons from the one currently ruling Spain), Orléanists (House of Orléans, a cadet branch of Bourbons, descended through Louis XIV's brother; Louis Philippe I of this line was king 1830-1848), and Bonapartists (descendants of Napoléon – here's even an internal dispute inside the family, between father and son, because the deceased grandfather thought his grandson is preferrable to his son as the head). There's a lot of genealogy autism involved, and a lot of old nobility going full larp, not only regarding monarchism in France but also regarding traditionalism in Catholic Church, opposing Vatican 2 and such.