Finding the total number of proper subfields of $F$? I was thinking about the following problem:


  Let $F$ be a field with $5^{12}$ elements.Then how can I find the total number of proper subfields of $F$?


Can someone point me in the right direction? Thanks in advance for your time.
I take it you mean, proper subfield. 

Can you show that any subfield of $F$ contains the field of $5$ elements? 

Can you show that any subfield must contain $5^r$ elements, for some $r$? 

Can you show that the degree of such a subfield (over the field of $5$ elements) must be $r$? and must be a divisor of the degree of the field of $5^{12}$ elements? 

Can you show that a finite field has at most one subfield of any given number of elements? 

If you can do all those, you have your answer.
The number of divisors of 12 are 1,2,3,4,6,12
Therefore 6=subfields exists.
Proper subfield means except 12 then number of proper subfield of 5^12 is 5.