Which is more important to Sophocles, family or authority? Look at Antigone, Ismene, Creon, and Haimon and at what happens to them and their...

While both family and authority are important to Sophocles, the importance of honoring family seems a bit more important to him when the fates of Antigone and her family are considered. Creon's realization that he caused the tragedy in the story by defying the gods implies that honoring family is the will of the gods—because the gods were displeased when he chose to honor the laws of the land as the King of Thebes.

Antigone chooses to bury her brother, whom her Uncle Creon, the new King of Thebes, ordered to be left unburied and unsanctified as punishment for warring against his brother. She doesn't deny her crime and accepts her death sentence when it is handed down. In the end, she kills herself before the death sentence is rescinded. Antigone values family over authority, saying:

I urge no more; nay, wert thou willing still,
I would not welcome such a fellowship.
Go thine own way; myself will bury him.
How sweet to die in such employ, to rest,--
Sister and brother linked in love's embrace--
A sinless sinner, banned awhile on earth,
But by the dead commended; and with them
I shall abide for ever. As for thee,
Scorn, if thou wilt, the eternal laws of Heaven.

To Antigone, honoring her family is also honoring the gods.

Ismene refuses to help Antigone bury Polyneices. She says it's not possible to do so, because his corpse is being guarded. When Antigone is caught, Ismene falsely confesses —but Antigone defends her. Nevertheless, Ismene is imprisoned by Creon and later released before Antigone's death. Ismene chooses to obey the law and preaches the same to Antigone, saying:

Shall we not perish wretchedest of all,
If in defiance of the law we cross
A monarch's will?—weak women, think of that,
Not framed by nature to contend with men.
Remember this too that the stronger rules;
We must obey his orders, these or worse.

Ismene survives the play, but many of the people she loves are dead. 

Haemon, Creon's son and Antigone's fiance, appears to support his father but subtly tries to convince him to spare Antigone, claiming the city is on her side. He tries to appeal to Creon, saying that Creon isn't considering true justice and is immature in thought. Haemon insists: "A State for one man is no State at all. As Monarch of the desert thou wouldst shine," telling his father that his thoughts are flawed. They exchange harsh words, and Haemon swears he'll never speak to Creon again as he leaves. After Antigone is found dead, Haemon attempts to kill his father and then kills himself.

Creon orders Antigone to be put to death for burying Polyneices in defiance of his order. Though he spares Ismene, he has Antigone sealed in a cave. When a blind prophet warns him the gods are displeased with his treatment of Polyneices and Antigone, he sends men to free her—only to find that she's dead. He loses his son and wife to suicide and blames himself for what has happened, saying:

Lead me away. I have been rash and foolish.
I have killed my son and my wife.
I look for comfort; my comfort lies here dead.
Whatever my hands have touched has come to nothing.
Fate has brought all my pride to a thought of dust . . .

In the end, Sophocles respects both family and authority but considers honoring family to be the will of the gods, which must be respected.