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Subgroup label Subgroup order Cyclic Abelian Solvable
Normal Characteristic Perfect Maximal Central
Ambient label Ambient order Direct product Semidirect product
Quotient label Subgroup index Cyclic quotient Abelian quotient Solvable quotient
Maximal quotient Non-trivial proper Hall subgroup Sylow subgroup
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Results (1-50 of at least 1000)

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Label Subgroup Ambient Quotient
Name Order Sylow norm char max cent ab Name Order Name Size max ab
176400.a.1.a1.a1 $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ ✓ ✓ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $C_1$ $1$ ✓
176400.a.2._.A $C_{420}.C_{210}$ $2^{3} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ ✓ ✓ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $C_2$ $2$ ✓
176400.a.2._.B $C_{105}^2:D_4$ $2^{3} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ ✓ ✓ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $C_2$ $2$ ✓
176400.a.2._.C $C_{105}^2:Q_8$ $2^{3} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ ✓ ✓ ✓ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $C_2$ $2$ ✓
176400.a.2._.D $C_{210}\times C_{420}$ $2^{3} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ ✓ ✓ ✓ ✓ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $C_2$ $2$ ✓
176400.a.2._.E $C_{420}.C_{210}$ $2^{3} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ ✓ ✓ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $C_2$ $2$ ✓
176400.a.2._.F $C_{105}^2:D_4$ $2^{3} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ ✓ ✓ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $C_2$ $2$ ✓
176400.a.2._.G $C_{105}\times D_{420}$ $2^{3} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ ✓ ✓ ✓ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $C_2$ $2$ ✓
176400.a.3._.A $C_{140}.D_{210}$ $2^{4} \cdot 3 \cdot 5^{2} \cdot 7^{2}$ ✓ ✓ ✓ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $C_3$ $3$ ✓
176400.a.3._.B $C_{420}.D_{70}$ $2^{4} \cdot 3 \cdot 5^{2} \cdot 7^{2}$ ✓ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $3$ $-$
176400.a.4._.A $C_{210}^2$ $2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ ✓ ✓ ✓ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $C_2^2$ $2^{2}$ ✓
176400.a.4._.B $C_{105}\times C_{420}$ $2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ ✓ ✓ ✓ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $C_2^2$ $2^{2}$ ✓
176400.a.4._.C $C_{105}:C_{420}$ $2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ ✓ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $C_2^2$ $2^{2}$ ✓
176400.a.4._.D $C_{105}:C_{420}$ $2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ ✓ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $C_2^2$ $2^{2}$ ✓
176400.a.4._.E $C_{105}\times C_{420}$ $2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ ✓ ✓ ✓ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $C_2^2$ $2^{2}$ ✓
176400.a.4._.F $C_{210}.C_{210}$ $2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ ✓ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $C_2^2$ $2^{2}$ ✓
176400.a.4._.G $C_{210}.C_{210}$ $2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ ✓ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $C_2^2$ $2^{2}$ ✓
176400.a.5._.A $C_{84}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5 \cdot 7^{2}$ ✓ ✓ ✓ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $C_5$ $5$ ✓
176400.a.5._.B $D_{84}.C_{210}$ $2^{4} \cdot 3^{2} \cdot 5 \cdot 7^{2}$ ✓ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $5$ $-$
176400.a.6._.A $C_{70}\times C_{420}$ $2^{3} \cdot 3 \cdot 5^{2} \cdot 7^{2}$ ✓ ✓ ✓ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $C_6$ $2 \cdot 3$ ✓
176400.a.6._.B $C_{70}\times C_{420}$ $2^{3} \cdot 3 \cdot 5^{2} \cdot 7^{2}$ ✓ ✓ ✓ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $S_3$ $2 \cdot 3$
176400.a.6._.C $C_{420}.C_{70}$ $2^{3} \cdot 3 \cdot 5^{2} \cdot 7^{2}$ ✓ ✓ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $C_6$ $2 \cdot 3$ ✓
176400.a.6._.D $C_{70}.D_{210}$ $2^{3} \cdot 3 \cdot 5^{2} \cdot 7^{2}$ ✓ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $C_6$ $2 \cdot 3$ ✓
176400.a.6._.E $C_{420}.C_{70}$ $2^{3} \cdot 3 \cdot 5^{2} \cdot 7^{2}$ ✓ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $C_6$ $2 \cdot 3$ ✓
176400.a.6._.F $C_{70}.D_{210}$ $2^{3} \cdot 3 \cdot 5^{2} \cdot 7^{2}$ ✓ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $C_6$ $2 \cdot 3$ ✓
176400.a.6._.G $C_{420}.C_{70}$ $2^{3} \cdot 3 \cdot 5^{2} \cdot 7^{2}$ ✓ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $C_6$ $2 \cdot 3$ ✓
176400.a.6._.H $C_{35}^2:D_{12}$ $2^{3} \cdot 3 \cdot 5^{2} \cdot 7^{2}$ ✓ ✓ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $C_6$ $2 \cdot 3$ ✓
176400.a.6._.I $C_{70}\times C_{420}$ $2^{3} \cdot 3 \cdot 5^{2} \cdot 7^{2}$ ✓ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $2 \cdot 3$ $-$
176400.a.6._.J $C_{140}.C_{210}$ $2^{3} \cdot 3 \cdot 5^{2} \cdot 7^{2}$ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $2 \cdot 3$ $-$
176400.a.6._.K $D_{70}.C_{210}$ $2^{3} \cdot 3 \cdot 5^{2} \cdot 7^{2}$ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $2 \cdot 3$ $-$
176400.a.6._.L $D_{70}.C_{210}$ $2^{3} \cdot 3 \cdot 5^{2} \cdot 7^{2}$ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $2 \cdot 3$ $-$
176400.a.6._.M $C_{140}.C_{210}$ $2^{3} \cdot 3 \cdot 5^{2} \cdot 7^{2}$ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $2 \cdot 3$ $-$
176400.a.6._.N $C_{140}.C_{210}$ $2^{3} \cdot 3 \cdot 5^{2} \cdot 7^{2}$ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $2 \cdot 3$ $-$
176400.a.6._.O $C_{140}.C_{210}$ $2^{3} \cdot 3 \cdot 5^{2} \cdot 7^{2}$ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $2 \cdot 3$ $-$
176400.a.7._.A $C_{60}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7$ ✓ ✓ ✓ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $C_7$ $7$ ✓
176400.a.7._.B $D_{60}.C_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7$ ✓ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $7$ $-$
176400.a.8._.A $C_{105}\times C_{210}$ $2 \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ ✓ ✓ ✓ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $C_2^3$ $2^{3}$ ✓
176400.a.8._.B $C_{105}\times C_{210}$ $2 \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ ✓ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $2^{3}$ $-$
176400.a.8._.C $C_{105}\times D_{105}$ $2 \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $2^{3}$ $-$
176400.a.8._.D $C_{105}\times D_{105}$ $2 \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $2^{3}$ $-$
176400.a.9.a1.a1 $C_{140}.D_{70}$ $2^{4} \cdot 5^{2} \cdot 7^{2}$ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $3^{2}$ $-$
176400.a.10._.A $C_{42}\times C_{420}$ $2^{3} \cdot 3^{2} \cdot 5 \cdot 7^{2}$ ✓ ✓ ✓ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $C_{10}$ $2 \cdot 5$ ✓
176400.a.10._.B $C_{42}\times C_{420}$ $2^{3} \cdot 3^{2} \cdot 5 \cdot 7^{2}$ ✓ ✓ ✓ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $D_5$ $2 \cdot 5$
176400.a.10._.C $C_{420}.C_{42}$ $2^{3} \cdot 3^{2} \cdot 5 \cdot 7^{2}$ ✓ ✓ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $C_{10}$ $2 \cdot 5$ ✓
176400.a.10._.D $C_{42}.D_{210}$ $2^{3} \cdot 3^{2} \cdot 5 \cdot 7^{2}$ ✓ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $C_{10}$ $2 \cdot 5$ ✓
176400.a.10._.E $C_{84}\times D_{105}$ $2^{3} \cdot 3^{2} \cdot 5 \cdot 7^{2}$ ✓ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $C_{10}$ $2 \cdot 5$ ✓
176400.a.10._.F $C_{42}.D_{210}$ $2^{3} \cdot 3^{2} \cdot 5 \cdot 7^{2}$ ✓ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $C_{10}$ $2 \cdot 5$ ✓
176400.a.10._.G $C_{84}\times D_{105}$ $2^{3} \cdot 3^{2} \cdot 5 \cdot 7^{2}$ ✓ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $C_{10}$ $2 \cdot 5$ ✓
176400.a.10._.H $C_{21}\times D_{420}$ $2^{3} \cdot 3^{2} \cdot 5 \cdot 7^{2}$ ✓ ✓ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $C_{10}$ $2 \cdot 5$ ✓
176400.a.10._.I $C_{42}\times C_{420}$ $2^{3} \cdot 3^{2} \cdot 5 \cdot 7^{2}$ ✓ $C_{420}.D_{210}$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$ $2 \cdot 5$ $-$
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