So it's pretty common knowledge that Eisenhower and Nixon had a very strained relationship... as in they both hated each other. And Nixon took the VP spot because he wanted to President, while Eisenhower dreaded a Nixon presidency and did everything he could to drop Nixon from the ticket in 56' going as far as promising him any cabinet position "outside of Defense and State".
So, say in 56' Eisenhower is successful in dropping Nixon off the ticket, who does he choose as his Vice President? Does he still need to solidify the Conservative voters, or is his re-election so secure he could pick a ham sandwich?