Solution:

The fifth-largest divisor is $2,014,000,000$ divided by the fifth smallest divisor.

The prime factorization of $2,014,000,000$ is: $$2^7\cdot 5^6 \cdot 19\cdot 53.$$ This makes the first $5$ smallest divisors $1,2,4,5,8$ Therefore, the fifth smallest divisor is $8,$ and the fifth largest divisor must be: $$\dfrac{2014000000}8 =251750000.$$

Thus, the correct answer is C .