The Americans. The British having the somewhat easier beaches (Juno, Sword etc.) While the Americans were left the task of Omaha and Utah. It was the American Rangers that disabled the guns at Pointe Du Hoc. Also the 101st and 82nd airborne that provided the support at Utah by sealing the Causeways controlled by the Nazis. Not to mention them taking German towns behind the beaches (Carentan, Foy etc.) Also I beleive it was American Bombs that softened up the beaches.

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"When we come to a minefield, our infantry attacks as if it were not there. The losses we get from personnel mines we consider equal to those we would have gotten from machine guns and artillery if the Germans had chosen to defend that particular area." - General Zukhov to Eisenhower