What is the generally accepted range of the flute? I know it starts at middle C (or B below that), which is C4 if I understand it correctly.....

One of my books has a fingering chart up to 2 Gs above middle C - which would be G6...? But another of my books has fingering up to the C above that, so C7, i.e. C four ledger lines above the staff. What would be an acceptable range for most players to aim for? I am approaching grade 2 standard (I think) and have played fairly reliably up to C6 and its sharp. If I try any of the notes above that they don't come out at all! I assume it's simple case of working on my embouchure to attain those higher notes? At the moment they sound a bit like when you play second octave D followed by second octave E and forget to put your first finger down again.
I hope that all made sense, and would welcome advice.