No she doesn't.
She CAN NOT choose a PM
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Yes she can, she chose one back in the 60s I believe. Theoretically she has more power over England than the President does over the USA. However, she is constitutional so we'll never see her exercise this power. And obviously she wouldn't choose a PM without prior consultation with parliament first because they are the ones who can grant him the means to govern.
If, for example, she owned a retro and Sulake sued her nothing would happen as she is susceptible to prosecution. A crime is done against her, therefore she is the only one above the law. Obviously if she went around stabbing people she would be removed from power and locked up at the least though. What I'm saying is her lawful power, no one knows what would possibly happen because the notion of the Queen committing a crime is not very common.
Make up your own mind, its your life.
It'd be rather pointless if they did arrest her considering she can pardon convicts. As already stated by the Virgin Mary, the monarch has the power to:
Choose the Prime Minister.
Dismiss ministers and governments.
Dissolve Parliament.
Refuse to agree to legislation passed by Parliament.
Dismiss the governments of other countries of which she is monarch.
Pardon convicted criminals.
Declare a state of emergency.
Issue proclamations.
Command the army and raise a personal militia.