Not counting the mom-and-pop operations; the big chains.
I grew up with Safeway, IGA, and Jack 'n Jill, all of which seemed to have vanished from both the prairies and the Sandhills of Nebraska a long time ago.
Around here, on the eastern slope of the Sandhills, the major grocery distributor is Affiliated Foods, but the owners of the individual stores name their stores anything they wish to name them.
The "big names" in Nebraska used to be Safeway and Hinky-Dinky. Hinky-Dinky no longer exists, and Safeway has but one solitary store in the state, in Scottsbluff, clear over near the border with Wyoming.
In Omaha and Lincoln, the "big names" are Hy-Vee, Baker's, Russ's, and Food4Less; there may still be some Albertson's around, but I dunno.
I suppose Hy-Vee is generally the "biggest" chain in Nebraska, given the number of locations.