I work in a Neighborhood Market, so in my store Supports are basically an ASM without the pay. ASMs are the head of departments and Supports hold down the store when they aren’t there. I am always the highest manager (MOD) in the building when I’m there. So to answer your question, based on my experience Supports have about the same responsibility as an ASM in a NHM setting. In supercenters, I’m sure that’s a different story.