What is the primary form of social organization for a Christian society?
Certainly not 'democracy' nor a republic - but neither a monarchy, nor a theocracy.
Not rule-of-law. Not arbitrary rule by a dictator.
Indeed not anything 'organized' or imposed or abstract.
Instead a Christian society should be organized by the prime Christian value of love; which means that love of God and Jesus Christ is what makes possible and sustains Christian love of fellow Men; as it generates creation itself...
And the most powerful and lasting form of 'group' Christian love is that which emerges from the fundamental reality and primacy of the family - which ideally means the natural, 'extended' family.
The family is spontaneous, and the ideal family coheres by love. This predates Christianity.
Then, by free agency, Christianity became possible through the work of Jesus Christ.
A Man chooses to become Christian.
When the members of a family become Christian, we get the only form of love-rooted Christian social group.
Christianity is therefore added-to the natural social form that is the family; and because love of God is the first principle; the family cohere by this love of God; and are given purpose and direction by their commitment to follow Jesus Christ to resurrected life eternal in Heaven.
Families may be extended and joined by Marriage (when marriage is a genuinely spiritual commitment).
Men and women may move between families by marriage; and 'adoption' into another family - when adoption is a love (not legal) relationship (this is rare - but sometimes happens).
Or by committed friendship (for example: the love that bound Jesus and his disciples) may become a type of family (albeit genuine loving friendships as strong as family love are rare, and seldom stable - even on a mortal timescale).
Family is the true and relationship of love; as we each may experience in Heaven (Heaven is 'organized' in families).
And in this earthly mortal life; the above concept of Christian family is is the partial and intermittent model of the true and mystical church.
Or, to put matters the other way around; unless a human group is organized on the basis of love; then that group is not primarily a Christian group (even when it calls itself a Church, or a Christian Nation or Empire).
We are Christians as a spiritual group only in the context of loving familial relations (as defined above).
Such groups are sufficient for Christian life, and have spiritual priority above all other groupings (Empire, Nation, Church) that lack their origin in love.