On Imperial Amarr megacorps:
Amarr Constructions: See Imperial Constructions
Carthum Conglomerate: 35 % Sarum, 20 % Zoar (via Ducia), 45 % unknown
Ducia: 80 % Zoar, rest unknown (essentially controlled by Zoar). Risk taker, known to mine in null.
Further Foodstuffs: Privately owned (by Ardishapur and Sarum)
HZO Refinery: 51 % unknown, 35 % Nefantar Mining Association, 14 % Sarum (so essentially Ardishapur and Sarum have a large stake, though Zoar has almost as much influence as Sarum - they control Ducia, which has a large stake of NMA)
Imperial Armaments: 15 % owned by the Empire (via the Chancellor), rest unknown. Largest weapon manufacturer in the cluster. (So yes, larger than KK.)
Imperial Constructions: 15 % owned by the Empire (via the Chancellor), 55 % majority stake self-owned, rest unknown (30 %)
Imperial Shipment: 91 % Unknown, 9 % Imperial Armaments (assumed to be controlled by the Emperor)
Inherent Implants: 18 % controlled by the Emperor (via the Chamberlain), rest unknown. The #1 corp in implants in the cluster (possibly add "outside the Nation", though of course their capsuleer implant line isn't the most competitive one if we forget "fluff implants")
Joint Harvesting: Privately held, owners unknown - possibly the agricultural & mining companies that form the Joint Harvesting. Largest Amarr industrial corporation, though Zoar-Ducia is almost as large.
Noble Appliances: A gigantic mess. 55 % unknown, 25 % directly to the Empire (via Chancellor), rest 40 % various via Royal Amarr Institute (incl. some more Empire, Lai Dai, Sarum and Zoar)
Nurtura: Privately held. A major exporter.
Viziam: 51 % privately held (by self), 30 % Ardishapur, 19 % Imperial Shipment (Emperor & various)
Zoar & Sons: 60 % controlling stake owned by Zoar family. 35 % stake by Lai Dai, rest 5 % unknown. Historically used to be larger. Controls Ducia Foundry, has stakes in Khanid Innovation and Nefantar Mining Association. Together with Zoar competes with Joint Harvesting in size (but is less focused on resource gathering).
For other effects, would need to check where the stations are located - though Zoar, at least, has influence all over the Empire.
For comparative size, Joint Harvesting and Zoar-Ducia do both dwarf such corporations as Lai Dai and Wiyrkomi (even when the corporate police and subsidiaries are calculated in). For size comparison, I used the amount of stations, though that estimate may be very flawed. (Based on the amount of stations controlled directly and by subsidiaries, Hyasyoda is pretty damn huge, for example.)