Let's denote the price of each sweater as \( x \). Darren bought three sweaters, so the total cost for the sweaters is \( 3x \).
He then pays five times the amount he paid for the sweaters to settle his credit card bill. Therefore, the payment toward the credit card bill is \( 5(3x) = 15x \).
Initially, Darren had $579 in his checking account. After buying the sweaters and paying the credit card bill, he ended up with $165 in his account. We can set up the following equation:
\[ 579 - 3x - 15x = 165 \]
This simplifies to:
\[ 579 - 18x = 165 \]
Now, we solve for \( x \):
\[ 579 - 165 = 18x \] \[ 414 = 18x \] \[ x = \frac{414}{18} = 23 \]
Now that we have the price of each sweater, we can find the total amount paid for the sweaters:
\[ 3x = 3 \times 23 = 69 \]
Next, we can calculate the amount paid toward the credit card bill:
\[ 15x = 15 \times 23 = 345 \]
Therefore, Darren's credit card dues amount to:
\[ \boxed{345} \]