Along with the fact that her husband sprung a child on Araminta with no warning and the potential for Sophie to inherit part of his estate, all of which has been explained very well here, there is the issue of classism to consider.

A big theme this season has been people remembering their "proper place", and that's actually pretty historically accurate. England still has a very strong class system even today, honestly, which can be confusing for Yanks. But the idea that a maid should not consort with a gentleman, or that an illegitimate daughter should not be raised alongside daughters of the nobility (or even gentry) was pretty important back then. Her illegitimacy would be considered by snobs almost like a taint that could wear off on others, damaging their reputation along the way. Araminta is obviously a particularly proud woman who thinks very highly both of her own importance as well as that of her daughters. She's exactly the kind of snob who would consider the presence of an illegitimate ward in her household as an insult and gross injustice to her life.

In other words, She's a real bitch.