At the head are the imperial couple, Emperor Yongan (Zhao Chang) and Empress Cixi, who preside over the Imperial Cabinet, comprised of the ministers of each ministry or ministry-level formation (and some organizations that are technically sub-ministry level, such as the Mobilization Directorate), the Chancellor and Grand Secretary (unofficially, one is to be Qian and the other Wu) and their deputies, the First Councillor, the governor of Kaifeng (or the governors of Hangzhou and Chang'an, if the Cabinet is meeting in either city), the viceroys for each region (though they are not always present), and the chiefs of staff of Imperial Supreme Command. Each major ministry has two co-ministers, one Qian (known as the "Western Minister" or "Western President") and one Wu (known as the "Eastern Minister/President"). The two Empresses Dowager also technically have seats on the Cabinet, but in practice only the Dowager Empress Xianghua, the Emperor's mother, regularly attends, due to animosity between the Empress and her mother.
The Chancellor oversees the Chancellery, which disseminates imperial directives down to the bureaucracy, oversees the running of the palace and the capital, and formulates or proposes policy. The Chancellery also, critically, screens all documents submitted to the throne and the Cabinet, and runs Cabinet meetings. The chiefs of staff of the Supreme Command also report to the Chancellor on military matters, meaning the Chancellor is one of the principal advisers to the throne and most senior government officials.
The First Councillor is elected by the members of the Grand Council, a body formed after the consolidation of the Wu Assembly of Notables and the various ad hoc Qian kurultais over the years- essentially, a large body of aristocrats, oligarchs, elder statesmen, and other luminaries and influential persons. This body holds no formal authority, but provides advice and counsel to the Cabinet and the monarchs.
The Grand Secretary oversees the Grand Secretariat, essentially serving as head of the civil service (including, perhaps most importantly for the office's power, the imperial secret police and spies, as well as diplomats, ambassadors, and envoys). They are heavily involved in the day-to-day operation of all ministries and departments both in the capital and around the country, and work closely with the Chancellor and the Chancellery as well as the viceroys, governors, and ministers. All ministers report to the Grand Secretary, who acts as a filter between them and the Chancellor much as the Chancellor acts as a filter between the Emperor and Empress and all others.
The ministries and ministry-level or Cabinet-level formations are as follows: the War Ministry (also containing Cabinet-level Mobilization Directorate, Shipyards Directorate, Armaments Directorate, and Planning Directorate), the Foreign Ministry, the Trade Ministry (with the Inspection Directorate as a Cabinet-level department), the Public Works Ministry (with the Reservoirs Directorate as a Cabinet-level subdivision), the Commerce Ministry (with the Industry, Labor, Finance, and Customs Directorates as Cabinet-level units), the Post Ministry (with the Roads, Pigeonry, Telegraph, and Packet Directorates as Cabinet-level units), the Culture Ministry (along with the Weights and Measures, Museums, and Library Directorates), the Education Ministry (with the Examination Directorate and the Discipline Directorate), the Railways Ministry, and the Security Directorate (an independent office which reports to the Grand Secretary). Because there is are both "western" and "eastern" ministers for each ministry (as well as two vice-ministers per Cabinet-level directorate), the Cabinet, when fully convened, is of somewhat untenable size (with 58 ministers or vice-ministers alone). This has led the monarchs to form a smaller Standing Committee [CUE MEMES], consisting of the Chancellor, the Grand Secretary, the War, Foreign, Works, and Commerce ministers, chiefs of staff of the Army and Navy, and the Vice-Minister for Security.
The country is divided into a number of provinces, which are overseen by governors. Several provinces and governors are then overseen by the viceroys, who report directly to the capital and the Cabinet. There are seven viceroys- the East Sea Viceroy, based in Tamaki (RL Auckland, overseeing the Pacific colonies), the Yangtze River Viceroy, based in Hangzhou (overseeing the old State of Wu), the Hongze Lake Viceroy (based in Huai'an, overseeing the old State of Ming), the Western Regions Viceroy, based in Kashgar (overseeing Xinjiang, Tibet, and Central Asia), the Northern Regions Viceroy, based in Karakorum (overseeing Mongolia and Tuva), the Yellow River Viceroy, based in Tianjin (overseeing the old states of Yan and Qi), and the Central Viceroy (based in Chang'an, overseeing the old Qian Chinese provinces). This viceroys report to the Cabinet on issues in their regions and keep tabs on problems in various provinces (as well as monitoring governors or local rulers and making sure they stay in line), and in some cases (especially the far west, but even moreso in the eastern sea thanks to a lack [thus far] of transoceanic telegraph cables) have significant autonomy or leeway to enforce imperial rule and formulate policy on their own.
Formally, the former State of Ming is an autonomous region, known as the Eastern Feudatory, ruled by the same line of Ming emperors (now demoted to the rank of "king" and pledging fealty to and recognizing the authority of the imperial throne). Its ruler officially has a place on the Cabinet of equivalent importance to the Chancellor, but in practice, since the Cabinet never meets in Ming cities, this is rarely relevant unless he makes the trip to Kaifeng, Hangzhou, or Chang'an. Additionally, a great deal of administrative authority over regional governance has passed from the King of Ming to the Hongze Lake Viceroy, who wields tremendous power at the Ming court and is usually more influential than the king (operating much as a British resident might have in an Indian princely state in the late 1800s IRL- as the de facto ruler). Ming technically retains its own ministries, but custom now dictates that the Ming king appoints the Hongze Lake Viceroy as Grand Secretary.