Ezekiel 16:49
Behold, this was the guilt of your sister Sodom: she and her daughters had pride, excess of food, and prosperous ease, but did not aid the poor and needy.
Jude 1:7
Sodom and Gomorrah and the surrounding cities, which likewise indulged in sexual immorality and pursued unnatural desire, serve as an example by undergoing a punishment of eternal fire.
The underlying sin is serving self rather than serving God. People like that rape, bugger, flaunt their luxuries and turn their backs on the poor.
The Latin word "luxuria" actually means both gluttony and lust. Luxury, gluttony and uncontrolled lust usually go hand in hand.
One of the points of the story, too, was that homosexuality was so accepted and widespread that the Sodomites saw nothing wrong with asking for the angels.