it is not a matter of "what impedance does it support" as impedance (resistance) of the headphone circuit is not the only important factor. how easy they are to drive (efficiency.. often rated in volume level at a set input wattage value) is also very important. some headphones also have varying amounts of power needed depending on the tone/frequency of audio. for example, cans which are capable of heavy bass often require more power to generate this than treble/midrange. headphones of lower impedance can actually be much harder to drive than high impedance models if we are comparing high to low efficiency drivers which is why going on one statistic alone is impossible. motherboards which state things like "can power up to 600ohm headphones" do not help the confusion as well.
while i havent heard much about the 1770, from what i've heard this is how it compares to the older 770...
-somewhat better build quality, removable cable
-larger soundstage, especially for a closed headphone
-less bass, very bloated and noticible sounding. the 770 had more quantity of bass and was less bloated
-flatter response curve. the 770 was very pronounced with its v-shaped accentuated treble/bass signature.
it seems to me that the new 1770 is a bit more efficient than the 770 so i would say "good" quality onboard or most soundcards should drive them as the dt770 was the same way. on the dt770 getting bass quantity was the main question but since the 1770 lacks much of that bass it may be less of an issue.
since onboard varies so much, the best i can say is that it may drive them fine although using onboard really is not the best solution. going with a decent soundcard or dac/amp solution would give you the best quality audio. in such cases it is often best to test your onboard first with just the headphones and make a decision once you try them out.