The ordering of the Christian angelic hierarchy in choirs, according to St. Thomas Aquinas in Summa Theologica (1225–1274) [as noted by @Jesse] is, in descending order:
- Seraphim, Cherubim, and Thrones;
- Dominations, Virtues, and Powers;
- Principalities, and Angels.
(For symmetry perhaps 3. is Principalities, Archangels, and Angels. Distinguishing Arch-angel from Archangel.)
The place of the Arch-angels is in the heavenly court. Arch-angel meaning being above all angels [of the 8 (or 9) orders], of any Choir [3 choirs].
Ans. The being, who later became known as, the primeval serpent, the devil or satan, did not belong to any of these 3 choirs.
The Day [morning] Star, Lucifer, son of Dawn [cf. Is 14:12], was member of the heavenly court; with him the four living creatures, and the seven spirits of God. These were the morning stars [Jb 38:7], first created by God, it seems, in that order.
Then came the rest of the 8 (or 9) orders of the heavenly hosts revealed in scripture: Seraphim, Cherubim, and Thrones; Dominations, Virtues, and Powers; Principalities, [Archangels] and Angels.
The seven spirits of God are the Arch-angels [cf. Tb 12:15: "I am Raph′ael, one of the seven holy angels who present the prayers of the saints and enter into the presence of the glory of the Holy One."]. Only two others are named in scripture: Arch-angel Michael [Jude 1:9] and Arch-angel Gabriel [Lk 1:19].
How this was deduced:
From the book of revelation, the four living creatures are present, and the seven spirits of God. This adds up to 11 and not 12 which represents totality, wholeness, and the completion of God's purpose. The missing one must have been one of the heavenly court.
The Day Star, Lucifer, son of Dawn. While the rest of the stars flee as it dawns, only he remains.
“I saw Satan fall like lightning from heaven;"/“How you are fallen from heaven, O Day Star, son of Dawn! [...]"
Paradise Lost. This is sad ...