Latnok is a very secret (and very rich) group of scientists who share the dream of maximizing human brain power with the hope of an advanced civilization. The word is Hungarian for "prophet" or "visionary". When we see the big table, there is always an empty seat, possibly representing Adam Baylin. The circle with a cross is similar to the Zzyzx symbol that Adam had designed.
Before Season 1
Latnok was the financial support for Zzyzx, the research facility that created 781227. Adam Baylin was a member of Latnok and supported what he thought they were doing. Sarah Emerson, Adam's girlfriend in college, was a member too. She became disillusioned by their strictly scientific approach and attempts to control her life, and left. Kyle learns that the reason he, as a gestation experiment, was ordered "terminated" is that he refused to process strategic military information and calculations that were William Kern's intentions for 781227 as a living super-computer. It was that intention, supported by what we later learn is Latnok, that produced the expulsion of Adam from Zzyzx when he protested.
Season 2
The name is first mentioned to Kyle in the first episode of Season 2 by Adam, who says they are "visionaries, great humanitarians". Kyle rejects Latnok from the beginning. Adam too eventually realizes that the society has lost its original moral purpose. When he finally leaves, he says one reason is to get away from Latnok.
Latnok is next mentioned in connection with Madacorp. Brian Taylor and Julian Ballantine each wear a Latnok ring when meeting at Madacorp. Julian, whose father had been a member, is removed from Madacorp by orders from Latnok after his unauthorized attack on Kyle's mind. That makes Latnok more powerful than Madacorp, a "billion-dollar multi-national" company.
The box that Kyle risks his life to retrieve from the Zzyzx compound is covered with etched symbols that he does not investigate, but which are similar to the data that Jessi reveals at Madacorp. It can be opened only with a Latnok ring, as he learns after losing the one Adam had left to him. It contains a puzzling photograph that sends Kyle and Jessi on a quest in Leap of Faith.
Latnok sends a social worker to investigate a complaint about Kyle at the Trager home. It seems that Latnok also directs the subsequent threat to Nicole's psychotherapist license so that she cannot counsel Jessi, who then turns to Brian.
We see the board in session after they hear from the social worker.
Kyle tells Adam about Jessi, and Adam tells Latnok about her. Brian is removed from Latnok for having created Jessi without authorization, and he then tries to use Jessi's super abilities to get back in.
When Kyle and Jessi visit JP's Diner, Kyle discovers the Latnok symbol with "A + S Forever June '82" scratched into the table. The old man who had known Adam and Sarah says that Adam did that and never came back.
At UW, Kyle and Jessi find a painting that Sarah made on the ceiling of a lecture hall, containing the answers to an exam done in two colors to show the Latnok symbol.
Somehow, Latnok knows about the diner and is able to monitor the events there. They are so impressed with what Kyle does for Jessi, that they arrange to further test him and then meet with him.
Season 3
Latnok becomes the central plot in season 3, starting with the abduction of Amanda. Kyle and Jessi both have to try to their limits to handle what turns out to be only a vicious test of Kyle. He is brought before Cassidy and the assembled board to join with them, but is more resolved than ever to have nothing to do with them. The rest of the season is Cassidy's continued attempt to involve Kyle and even Jessi.
There is a club at UW of students chosen to serve the purposes of that season's villain, Cassidy. Mark is seen speaking with Cassidy in the Latnok meeting room after Amanda is abducted; and Nate is aware of the Latnok connection, having the symbol as a tattoo on his forearm.
The final episode has Kyle and Jessi as a team attempting to wreck the main project by Latnok, hoping to "bring down" Latnok as they had done to Madacorp.