<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>Na</mml:mi><mml:mn>5</mml:mn></mml:msub><mml:msub><mml:mi>Co</mml:mi><mml:mrow><mml:mn>15.5</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mi>Te</mml:mi><mml:mn>6</mml:mn></mml:msub><mml:msub><mml:mi mathvariant="normal">O</mml:mi><mml:mn>36</mml:mn></mml:msub></mml:mrow></mml:math> : An <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>S</mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</…

$S$ = 1/2 Ising kagome antiferromagnets (KAFs) are rare in reality. In this work, we report the magnetic ground state and field-induced transitions of an antiferromagnet with a stacked geometry, Na$_5$Co$_{15.5}$Te$_6$O$_{36}$. Upon cooling, susceptibility measurement reveals three successive at $T_1$ 45.2 K, $T_2$ 35.4 $T_3 \sim$ 20 K. When field is applied along $c$ axis, strikingly anomalous initial magnetization lying outside hysteresis loop observed below $T_3$. Through detailed characterizations, $vs$ temperature phase diagram established. A partially disordered antiferromagnetic proposed $T_1$, which becomes frozen Na$_5$Co$_{15.5}$Te$_6$O$_{36}$ therefore promising candidate for KAFs, unique composite structure may shed light on exploratory path KAFs.