I sometimes wish ancient languages didn't have pronouns and just repeated names. Here's my interpretation of verse 26's pronouns (which may be utterly irrelevant):
26 And may the grace of God the Father, whose throne is high in the heavens, and our Lord Jesus Christ, who sitteth on the right hand of his God the Father's power, until all things shall become subject unto him Jesus Christ, be, and abide with you forever. Amen.
The immortality of man is brought to pass as a gift through the resurrection of Jesus Christ, so clearly there is no conflict here. One can be subject to Christ and immortal at the same time. No problem.
Eternal life is exaltation, living in the presence of God, living as God lives, inheriting all that God has. While it may seem contradictory that the way to accomplish all that is to become subject to Christ, I think it's fairly obvious why that's the only way. Being "subject" to Christ is being lifted by Him, not oppressed by Him. It is being led, encouraged, taught, forgiven, and perfected by Him. It is being made one with Him. God's thoughts and ways are different from man's (quite possibly always the reverse of man's) and thus, God's subjection, rather than oppression or limitation, is exaltation.