The Trinity, Part 5: Personality Traits of the Persons of the Trinity...

...further establishing that the Persons of the Trinity are persons and not inanimate things: ...

Question: Is there a hierarchy in the Trinity? Is the Father on top, and the Son and the Holy Spirit subject to him?
No hierarchy; it is a perfect synergysm, like three equal strands of a rope.
1 John 5:7 - These Three are One.
John 10:30 - I and the Father are one
Isaiah 45:5 - I am the Lord. There is no other. Apart from me there is no God.

But that's not the whole story. Maybe there is a hierarchy. (complex, but don't let it get in your way).
At the last supper, Jesus was trying to help the disciples understand what was going to happen.
John 14:5 - [Thomas]: "We don't know where you're going... how can we know the way?"
John 14:8 - [Phillip]: "Show us the Father". [Jesus]: "Anyone who has seen me has seen the Father".

Generation: defined as the eternal relationship between the Father and the Son. (It does not denote that God Made or created the Son.) Not a matter of chronology. It is a matter of logic: the process set up for decision-making. Father-Son-Holy Spirit in that order does not imply chronological order, nor level of importance.

Also The Holy Spirit does not send the Son. The Father sends the Son.
John 1:14 -- "Sent by the Father". So the Father has a slightly different role from the other two.

Procession: The Holy Spirit can be sent by either the Father or the Son.
Acts 2:32 - Received from the Father the promised Holy Spirit.
John 3:34 - God sends the Spirit without limit.
John 15.26 - Jesus: "...[The Holy Spirit] whom I will send to you, from the Father".
John 21 - "As the Father sent me, so send I you." ... And he breathed on them, and they received the Holy Spirit.

Conclusion: The Father Sends the son.