"Romance" seems like an obvious genre choice for Lady Chatterley's Love. Connie and Oliver are almost nauseatingly in love, although sometimes that love is very poetically expressed: "But the little forked flame between me and you: there you are! That's what I abide by, and will abide by" (19.167). This "little forked flame" of love—but also of sex—is at the heart (so to speak) of the novel.
But D.H. Lawrence doesn't write romance like Danielle Steele writes romance. He writes romance like this: "And as she melted small and wonderful in his arms, she became infinitely desirable to him, all his blood-vessels seemed to scald with intense yet tender desire, for her, for her softness, for the penetrating beauty of her in his arms, passing into his blood" (12.145)." 50 Shades of Grey this isn't.
Lawrence's literary prose, detailed focus on character, and psychological exploration tells us that he's aiming for something more than sexy writing. This is Literary Fiction with a capital L (and F). The plot really doesn't matter—in fact, it matters so little that there isn't even a resolution. What matters is (1) the characters, (2) the writing and (3) the philosophical exploration of …
More a style than a genre, modernism is what gives us the repetitive, chant-like prose (see "Writing Style" for more on this); the lack of realism, at least with the upper-class characters (you never see them eat or dress or do anything normal except talk at each other); and the frustrating ending.
This work isn't modernist like some of the classic texts—Mrs Dalloway or Ulysses, to name some obvious contenders—but it's definitely in the same vein. Lawrence isn't out to give us a satisfactory little plot and round it off with a moral. The rejection of tidy plot and neat life lesson situate Lady Chatterley's Lover firmly in the modernist camp.