The building blocks of English

Affixes

-agogue

In the US often -agog.

Leading or promoting.

Greek agōgos, leading.

A demagogue (Greek demos, the people) is a political leader who seeks support by appealing to popular desires and prejudices; a pedagogue is a teacher, especially a strict or pedantic one (originally from Greek paidagōgos, a slave who accompanied a child to school).

The ending appears in medical terms for a substance that induces expulsion or secretion: cholagogue (Greek cholē, bile), something that promotes the flow of bile; emmenagogue (Greek emmēna, menses), one that stimulates or increases menstrual flow; secretagogue, a substance which promotes secretion.

However, synagogue comes from the Greek word for a meeting, from sun‑, together, plus the related agein, bring.