One of the missing numbers nust be another 52 in order to make
the mode be 52.

So the numbers so far are 52,52,72,78
  
Those have sum 254.  The sum of all 6 must be 360 in order for the mean
of the 6 numbers to be 60. So the remaining two have to have sum 360-254 
or 106.

One of them cannot be larger than 78, because that would
make the remaining one have to be smaller than 106-78 or 28, making the 
range larger than 78-28 or 50.  BUt the range is 28,  So 78 is the largest
value, To make the range 28, the smallest number must be 78-28 or 50.  
And that makes the trmaining number 106-50 or 56.

So the numbers are

50, 52, 52, 56 72, 78.  The missing ones are 52, 50, and 56

Edwin