There are Five rivers according to Greek Mythology:
Acheron (woe), Cocytus (wailing), Phlegethon (fire), Lethe (forgetfulness), Styx (across with which spirits of the dead were ferried also the river of dreams lost because of death)
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the river Lethe and the river Styx. if there are more then I don't know them
Hades is a term that can be applied to the Underworld (the abode of the dead) or as a name of the God of the Underworld. There are five rivers in the Underworld, but the most well known is the river Styx. The other four are Acheron, Cocytus, Lethe, and Phlegethon.
Hades' Underworld is called The Underworld.
The Underworld is in the Underworld. It is a different realm of existence, not part of this earth.
The Underworld