Objection 1. It would seem that we may not say that Christ was subject to the Father. For everything subject to the Father is a creature, since, as is said in De Eccles. Dogm. iv, "in the Trinity there is no dependence or subjection." But we cannot say simply that Christ is a creature, as was stated above (Question 16, Article 8). Therefore we cannot say simply that Christ is subject to God the Father.
Objection 2. Further, a thing is said to be subject to God when it is subservient to His dominion. But we cannot attribute subservience to the human nature of Christ; for Damascene says (De Fide Orth. iii, 21): "We must bear in mind that we may not call it" (i.e. Christ's human nature) "a servant; for the words 'subservience' and 'domination' are not names of the nature, but of relations, as the words 'paternity' and 'filiation.'" Hence Christ in His human nature is not subject to God the Father.
Objection 3. Further, it is written (1 Corinthians 15:28): "And when all things shall be subdued unto Him, then the Son also Himself shall be subject unto Him that put all things under Him." But, as is written (Hebrews 2:8): "We see not as yet all things subject to Him." Hence He is not yet subject to the Father, Who has subjected all things to Him.
On the contrary, Our Lord says (John 14:28), "The Father is greater than I"; and Augustine says (De Trin. i, 7): "It is not without reason that the Scripture mentions both, that the Son is equal to the Father and the Father greater than the Son, for the first is said on account of the form of God, and the second on account of the form of a servant, without any confusion." Now the less is subject to the greater. Therefore in the form of a servant Christ is subject to the Father.
I answer that, Whoever has a nature is competent to have what is proper to that nature. Now human nature from its beginning has a threefold subjection to God. The first regards the degree of goodness, inasmuch as the Divine Nature is the very essence of goodness as is clear from Dionysius (Div. Nom. i) while a created nature has a participation of the Divine goodness, being subject, so to say, to the rays of this goodness. Secondly, human nature is subject to God, as regards God's power, inasmuch as human nature, even as every creature, is subject to the operation of the Divine ordinance. Thirdly, human nature is especially subject to God through its proper act, inasmuch as by its own will it obeys His command. This triple subjection to God Christ professes of Himself. The first (Matthew 19:17): "Why askest thou Me concerning good? One is good, God." And on this Jerome remarks: "He who had called Him a good master, and had not confessed Him to be God or the Son of God, learns that no man, however holy, is good in comparison with God." And hereby He gave us to understand that He Himself, in His human nature, did not attain to the height of Divine goodness. And because "in such things as are great, but not in bulk, to be great is the same as to be good," as Augustine says (De Trin. vi, 8), for this reason the Father is said to be greater than Christ in His human nature. The second subjection is attributed to Christ, inasmuch as all that befell Christ is believed to have happened by Divine appointment; hence Dionysius says (Coel. Hier. iv) that Christ "is subject to the ordinance of God the Father." And this is the subjection of subservience, whereby "every creature serves God" (Judith 16:17), being subject to His ordinance, according to Wisdom 16:24: "The creature serving Thee the Creator." And in this way the Son of God (Philippians 2:7) is said to have taken "the form of a servant." The third subjection He attributes to Himself, saying (John 8:29): "I do always the things that please Him." And this is the subjection to the Father, of obedience unto death. Hence it is written (Philippians 2:8) that he became "obedient" to the Father "unto death."
Reply to Objection 1. As we are not to understand that Christ is a creature simply, but only in His human nature, whether this qualification be added or not, as stated above (Question 16, Article 8), so also we are to understand that Christ is subject to the Father not simply but in His human nature, even if this qualification be not added; and yet it is better to add this qualification in order to avoid the error of Arius, who held the Son to be less than the Father.
Reply to Objection 2. The relation of subservience and dominion is based upon action and passion, inasmuch as it belongs to a servant to be moved by the will of his master. Now to act is not attributed to the nature as agent, but to the person, since "acts belong to supposita and to singulars," according to the Philosopher (Metaph. i, 1). Nevertheless action is attributed to the nature as to that whereby the person or hypostasis acts. Hence, although the nature is not properly said to rule or serve, yet every hypostasis or person may be properly said to be ruling or serving in this or that nature. And in this way nothing prevents Christ being subject or servant to the Father in human nature.
Reply to Objection 3. As Augustine says (De Trin. i, 8): "Christ will give the kingdom to God and the Father, when He has brought the faithful, over whom He now reigns by faith, to the vision," i.e. to see the essence common to the Father and the Son: and then He will be totally subject to the Father not only in Himself, but also in His members by the full participation of the Godhead. And then all things will be fully subject to Him by the final accomplishment of His will concerning them; although even now all things are subject to Him as regards His power, according to Matthew 28:18: "All power is given to Me in heaven and in earth."
Source: Summa Theologica (New Advent)