Dragon Math in 'House of the Dragon': these are the ones each team has, for now

"With his death, the war of ravens, envoys, and marital pacts ended, which led to the war of fire and blood proper." This is the phrase from George R.R. Martin's book that refers to what comes after the death of Lucerys Velaryon, with which the plot of the first season of House of the Dragon  concludes.

With the murder of Jacaerys Targaryen at the hands of Blood and Cheese, and commissioned by Daemon, there is no turning back. The civil war has officially begun. It will be dark, turbulent, brutal, and bloody. And the dragons will be protagonists.

Just as Viserys told his daughter Rhaenyra in the first episode of the series (and as Aemond discovered): "The idea that we control the dragons is an illusion", they are weapons of war. Daemon knows this well, as in the final of the previous installment he took stock of the living dragons in Westeros and which side each one belongs to. Because this war will be fought by land, sea, and air. And in its wake, it will leave fire and blood.

Rhaenys riding Meleys.

Dragons on the Black side 

The Blacks have three adult dragons in their possession: Caraxes, Syrax, and Meleys, ridden by Daemon, Rhaenyra, and Rhaenys, respectively.

After the death of Lucerys and Arrax, they have Vermax, Jacaerys Velaryon's dragon; Tyraxes, young Joffrey Velaryon's, who is still locked up in Dragonpit; and Moondancer, of Bela.

The majestic and imposing Vhagar

Dragons on the Green side

The Greens, on the other hand, have three adult dragons: Vhagar (Aemond), Sunfyre (Aegon II), and Dreamfyre (Helaena). They would also have a fourth dragon slightly smaller that we haven't seen yet.

This last one, Tessarion, is a female dragon known as the Blue Queen, and in Fire and Blood is with Daeron, the fourth son of Alicent and Viserys, who hasn't appeared in the series or been mentioned yet, but is theoretically growing in Oldtown, as tradition dictates for the youngest children of the court. This has been confirmed by  George R.R. Martin.

Mist (Seasmoke)

Unclaimed Dragons

In addition to the six dragons on Rhaenyra's side and the four on Aegon's (and the pending dragon eggs), there are riderless dragons at Dragonstone ready to be claimed, so now anyone with a bit of Targaryen blood, even if a bastard, known as "Dragonseeds," will be highly sought after. 

On one hand, there are three wild dragons that have never been ridden, named Sheepstealer, Grey Ghost, and Cannibal (as they are named in the Spanish edition of Fire and Blood).

There are also riderless Seasmoke, the dragon left by Laenor Velaryon; Silverwing, Queen Alysanne's dragon; and Vermithor, six powerful weapons of war waiting to be claimed. 

The Mighty Vermithor

Vermithor, a new dragon in 'House of the Dragon'.

In the last episode of the previous season, Daemon goes to visit this legendary dragon because he knows it is the only one that could stand up to Vhagar, as it is about a hundred years old and is the second oldest and largest dragon still alive.

Vermithor is an old and powerful male dragon with bronze and cinnamon-colored scales, and eyes described as "pools of molten bronze." Legend says his flame can melt steel and stone.

He has been secluded at Dragonstone for 30 years since his rider, King Jaehaerys I Targaryen, predecessor of Viserys, died. Until now, when Daemon establishes contact by singing to him in High Valyrian, the language used to communicate with dragons, a song he could have learned from his grandfather, precisely, Jaehaerys I. 

Vermithor will be claimed during the Dance of Dragons and his rider will be a Targaryen blooded bastard.