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Madame Aubain isn't a terrible person, but she can be a little bit toxic. She's the widow of a man and never seems to get past that fact; she's haunted by death throughout her life. It's even in her house:
[T]he memory of 'Monsieur' hung over everything! (2.18)
This hanging on to Monsieur's memory tells us that being married to him gave Madame Aubain her status. Being a widow is important for her standing in the community.
And she's part of a powerful community—the bourgeoisie, aka the middle class. Part of keeping her standing in this community is her ability to manage her household, which really just means choosing a good maid:
For half a century, the bourgeois ladies of Pont-l'Évêque envied Madame Aubain her maid Félicité. (1.1)
That envy is important to Madame Aubain; it lets her know that she has something everyone else wants.
While Félicité is important to her standing, she's not important to her as a person. This tells us something that's not so great about Madame: she's pretty much a big old snob. Check out this scene, where she's worried because she hasn't heard from her daughter in four days:
To console her with her own example, Félicité said, 'I haven't had any for six months!'
'Why…my nephew!' Félicité replied softly.
'Oh, your nephew!' Shrugging her shoulders, Madame Aubaine resumed her pacing, as if to say, 'I'd forgotten all about him! And why should I care anyway? A ship's boy, a rogue, so what? Whereas my daughter… Think of that!' (3.35-38)
This separation between Félicité's nephew and Madame Aubain's daughter, in the eyes of Madame Aubain herself, is a class separation. The lady of the house cannot imagine that her maid would have feelings like hers, or that Félicité's family members matter as much as Madame Aubain's do. She's incapable of sympathy for anyone outside of her own social class.