Some more context would be helpful. What does the author ordnen?

Genau, kannst Du bitte die entsprechenden Sätze und Textstellen hier einstellen ?

As mentioned by the others above, a context would be helpful.
In my interpretation the words share certain meanings, so it really depends on the context they're used in. But in general they signify the following:

"unterordnen" has a hierarchic touch to it. I would use when trying to establish ranks. This may include people (e.g. "Der Soldat muss sich dem General unterordnen") or priorities (e.g. "Geld ist von untergeordneter Wichtigkeit. Gesundheit ist wesentlicher.")

"einordnen" may have several meanings: Concerning people "Bitte ordnen Sie sich in eine Gruppe ein" ('Please allocate yourself into one of the groups"). It shares this meaning with "zuordnen".
It also refers to categorising items into groups. E.g. "Einen Ferrari würde ich als Sportauto einordnen")

"nachordnen" shares the hierachical aspect with "unterordnen" (e.g. "Der Soldat ist dem General nachgeordnet"). In terms of priorities it has a sense of time to it, especially when one thing comes after another ('afterwards' = 'danach' -> nachordnen). Fairly formal, though.

Hope this helps.

Actually, I think these are synonyms

According to my Dietl/Lorenz Dictionary:

Nachgeordnet:
Nachgeordnete Behörde: subordinate authority, authority at a lower level;
Nachgeordnete Stelle: subsidiary (or subordinate body)

Untergeordnet:
Von untergeordnetem Interesse: of minor interest
Untergeordnete Stellung: subordinate position

Unterordnung:
Subordination

Unterordnen: subordinate