Naval HQs and admin command

I’m in need of some admin command for various jump ships and harvesting operations on a planet 4 jumos from Sol

do I keep building naval hqs on sol to expand the range to cover this planet or do I just build a naval hq on the distant planet itself

the range system of multiple hqs has confused me and I’m not sure I understand them properly

You can do either.
The range is very simple. If the system in question is four jumps from Sol you need 8 NHQs in Sol to reach it.
1 NHQ - 1 jump
2 NHQ - 2 jumps
4 NHQ - 3 jumps
8 NHQ - 4 jumps
So you could build 8 NHQs on Earth and then make an Admin Command chain, all located on Earth, incorporating your highest rank all the way down to your second-lowest rank (commercial vessels are normally at the lowest rank), and then put your harvesters in that, and you’ll get the maximum bonuses.
You could also put a single NHQ on that planet and then build an admin command chain located at that planet down to the second-lowest rank. If the starting admin command located on Earth doesn’t have the range to reach that system (i.e. has fewer than 8 NHQs) then you just won’t get bonuses from the overall commander back on Earth. If you assign a highest-rank member to that admin command, I’m pretty sure your overall admin commander back on Earth will stop applying bonuses everywhere because he won’t be high enough rank for his position.
Finally, you could put 2 NHQs on Earth and 2 NHQs on a new colony 2 jumps between Sol and your harvester system, and then make an admin command chain there, so that it covers the harvester system and is covered by the Earth admin command.

I think the system is intended to make branching commands located on separate locations as you branch out into the galaxy, and that’s how I do it. However, I would guess that the “optimal” way to play is to just stack NHQs on Earth because exploring JPs gives a chance of linking back to a system near Sol and bringing that part of the galaxy back within admin command range. Somebody has probably done the math but it wouldn’t surprise me if something like 95% of systems are within ~15 jumps of Earth regardless of how many systems you explore. This would also minimize micromanagement of fleets between commands.

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So I can bring the distant planet under the command radius of earth to receive the bonuses of earths hierarchy

if it isn’t in range of earths command and I build a hq on the distant planet it will form its own command structure and utilise the benefits of all the officers in its own chain?

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That’s perfect thank you