Republican National Committee Chairman Reince Priebus fought Sunday to downplay the notion that the GOP is divided, particularly over immigration, after House Majority Leader Eric Cantor (R-Va.) lost his primary last week.

The same day that economics professor Dave Brat beat Cantor in a shocking upset, Sen. Lindsey Graham (R-S.C.) won his primary, creating a debate over whether support for immigration reform measures could doom GOP members. While Cantor was part of a GOP leadership that has not allowed votes on reform bills, he also has been supportive of measures like legal status for undocumented young people, prompting attackers to say he supports "amnesty." Graham was an even easier target on that front -- he helped draft and pass the Senate's comprehensive immigration bill -- but he won, anyway.

CBS "Face the Nation" host Bob Schieffer started out by asking Priebus about those two primaries and to what degree they show divisions within the GOP.

"I don't think it's divided at all," Priebus replied. "You have districts that are 85 percent Republican, and more than one Republican wants to be congressman, and in some cases more than one person wants to be a senator."

Priebus said he agreed with other Republicans that Cantor may have been hurt in his district by his leadership position, which required him to travel the country and think on a more national level.

"It takes you out of your district, and pretty soon that good work you're doing nationally becomes liability locally," he said. "I think it's a local issue."

Priebus gave somewhat conflicting answers on whether his party is divided on immigration. When Schieffer noted that some Republicans are for immigration reform and some are opposed, Priebus said, "The Democrats don't agree on everything either." But he went on to say the GOP may be more united on the issue than they seem.