Both Annas and Caiaphas were Sadduccees - Jesus was not (likely) a part of "their inner circle". (Don't believe shows or movies based on nothing other than -
the easiest way to dramatize a story).
The Sadduccees were despised by most of the common people and the Pharisees alike. They were associated with "the elite/rich class" within society - flaunted gold, silver and other possessions as a way for them to gain favor with God. The Pharisees, on the other hand, were associated with the middle/lower classes and lived simple , working lives, devoid of riches.
Could this be why most of the Gospels refer to the Jewish religious leaders as Pharisees but in the Gospel of John, Jesus' accusers are referred to simply as "Jews"?
John 10:
Once again, the Jews picked up rocks to stone him, but Jesus said to them, “I have performed in your presence many good works from my Father. For which of these works are you going to stone me?” The Jews answered, “We are not going to stone you for any good work you have done, but for blasphemy. Even though you are a man, you are claiming to be God.;Jesus replied,
“Is it not written in your Law,
‘I said: You are gods’?
If those to whom
the word of God was addressed
are called ‘gods’
—and Scripture cannot be set aside—
how can you say, ‘You blaspheme,’
to the one whom the Father has consecrated
and sent into the world
for saying, ‘I am the Son of God’?
“If I am not performing
the works of my Father,
then do not believe me.
However, if I am doing them,
then even if you do not believe me,
at least believe my works,
so that you may realize and understand
that the Father is in me
and I am in the Father.”
They again tried to seize him, but he escaped from their clutches.
It is clear that Jesus is making his case in front of "unbelieving accusers" - the same type of accusers who hold the Law in highest regard (adding dozens of rules of their own) but ignore most of the (miraculous) works of the Prophets. (The Sadduccees).
The fact that he was able to "escape their clutches" screams that he likely had help from other Jewish leaders (The Pharisees - who would have helped him out of spite of the Sadduccees alone).
All this said - this is just my opinion - an "educated guess" based on 40+ years of devout Bible study.