The Privy Council
The Privy Council constituted the governing executive and chief advisors of the monarch; Privy Councillors were all chosen personally by the Queen (or King), to whom they swore an oath of personal loyalty.
Under Elizabeth there were about 18 members*, drawn from the nobility and gentry, but most business was handled by a minority of leading officials.
The most important and active members of the Council were usually the Lord Treasurer, Lord Chancellor, Lord Privy Seal and, most influential of all, the Secretary.
Footnotes
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A growing bureaucracy
In a classic demonstration of Parkinson's Law, this number was increased to 25 and 30 under James and Charles I.