Draw an equilateral triangle. The internal angle at a corner is 1/6 of a circle, 1.0471975511965 radians.
That can be divided by 60 into degrees, which are 0.0174532925199 radians
Those can be divided by 60 into minutes, which are 0.0002908882086 radians
Those can be divided by 60 into seconds, which are 0.00000484813681109 radians
Those can be divided by 60 into thirds, which are 8.08022801849E−8 radians
I think going any further is likely to be meaningless in a modern context, but fourths and fifths were used in the past.