| Label |
Subgroup |
|
Ambient |
|
Quotient |
| Name |
Order |
Sylow |
norm |
char |
max |
cent |
cyc |
ab |
nilp |
sup solv |
solv |
perf |
simp |
Agp |
Zgp |
metab |
metacyc |
Name |
Order |
Name |
Size |
max |
cyc |
ab |
nilp |
sup solv |
solv |
simp |
Agp |
metab |
| 691200.g.1.a1 |
$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
|
✓ |
✓ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
|
$C_1$ |
$1$ |
|
✓ |
✓ |
✓ |
✓ |
✓ |
|
✓ |
✓ |
| 691200.g.2.A |
$S_4\times S_5^2$ |
$2^{9} \cdot 3^{3} \cdot 5^{2}$ |
|
✓ |
✓ |
✓ |
|
|
|
|
|
|
|
|
|
|
|
|
|
$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
|
$C_2$ |
$2$ |
|
✓ |
✓ |
✓ |
✓ |
✓ |
✓ |
✓ |
✓ |
| 691200.g.2.B |
$S_4\times \POPlus(4,5)$ |
$2^{9} \cdot 3^{3} \cdot 5^{2}$ |
|
✓ |
✓ |
✓ |
|
|
|
|
|
|
|
|
|
|
|
|
|
$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
|
$C_2$ |
$2$ |
|
✓ |
✓ |
✓ |
✓ |
✓ |
✓ |
✓ |
✓ |
| 691200.g.2.C |
$S_4\times A_5^2:C_4$ |
$2^{9} \cdot 3^{3} \cdot 5^{2}$ |
|
✓ |
✓ |
✓ |
|
|
|
|
|
|
|
|
|
|
|
|
|
$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
|
$C_2$ |
$2$ |
|
✓ |
✓ |
✓ |
✓ |
✓ |
✓ |
✓ |
✓ |
| 691200.g.2.D |
$A_4\times S_5\wr C_2$ |
$2^{9} \cdot 3^{3} \cdot 5^{2}$ |
|
✓ |
✓ |
✓ |
|
|
|
|
|
|
|
|
|
|
|
|
|
$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
|
$C_2$ |
$2$ |
|
✓ |
✓ |
✓ |
✓ |
✓ |
✓ |
✓ |
✓ |
| 691200.g.2.E |
$S_5^2.S_4$ |
$2^{9} \cdot 3^{3} \cdot 5^{2}$ |
|
✓ |
✓ |
✓ |
|
|
|
|
|
|
|
|
|
|
|
|
|
$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
|
$C_2$ |
$2$ |
|
✓ |
✓ |
✓ |
✓ |
✓ |
✓ |
✓ |
✓ |
| 691200.g.2.F |
$A_5^2.\GL(2,\mathbb{Z}/4)$ |
$2^{9} \cdot 3^{3} \cdot 5^{2}$ |
|
✓ |
✓ |
✓ |
|
|
|
|
|
|
|
|
|
|
|
|
|
$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
|
$C_2$ |
$2$ |
|
✓ |
✓ |
✓ |
✓ |
✓ |
✓ |
✓ |
✓ |
| 691200.g.2.G |
$(A_4\times A_5^2).D_4$ |
$2^{9} \cdot 3^{3} \cdot 5^{2}$ |
|
✓ |
✓ |
✓ |
|
|
|
|
|
|
|
|
|
|
|
|
|
$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
|
$C_2$ |
$2$ |
|
✓ |
✓ |
✓ |
✓ |
✓ |
✓ |
✓ |
✓ |
| 691200.g.3.a1 |
$A_5^2.D_4^2$ |
$2^{10} \cdot 3^{2} \cdot 5^{2}$ |
|
|
|
✓ |
|
|
|
|
|
|
|
|
|
|
|
|
|
$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
|
|
$3$ |
|
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
| 691200.g.4.A |
$S_4\times \PSOPlus(4,5)$ |
$2^{8} \cdot 3^{3} \cdot 5^{2}$ |
|
✓ |
✓ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
|
$C_2^2$ |
$2^{2}$ |
|
|
✓ |
✓ |
✓ |
✓ |
|
✓ |
✓ |
| 691200.g.4.B |
$A_4\times S_5^2$ |
$2^{8} \cdot 3^{3} \cdot 5^{2}$ |
|
✓ |
✓ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
|
$C_2^2$ |
$2^{2}$ |
|
|
✓ |
✓ |
✓ |
✓ |
|
✓ |
✓ |
| 691200.g.4.C |
$A_4\times \POPlus(4,5)$ |
$2^{8} \cdot 3^{3} \cdot 5^{2}$ |
|
✓ |
✓ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
|
$C_2^2$ |
$2^{2}$ |
|
|
✓ |
✓ |
✓ |
✓ |
|
✓ |
✓ |
| 691200.g.4.D |
$A_4:S_5^2$ |
$2^{8} \cdot 3^{3} \cdot 5^{2}$ |
|
✓ |
✓ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
|
$C_2^2$ |
$2^{2}$ |
|
|
✓ |
✓ |
✓ |
✓ |
|
✓ |
✓ |
| 691200.g.4.E |
$A_5^2:(C_2\times S_4)$ |
$2^{8} \cdot 3^{3} \cdot 5^{2}$ |
|
✓ |
✓ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
|
$C_2^2$ |
$2^{2}$ |
|
|
✓ |
✓ |
✓ |
✓ |
|
✓ |
✓ |
| 691200.g.4.F |
$A_4\times A_5^2:C_4$ |
$2^{8} \cdot 3^{3} \cdot 5^{2}$ |
|
✓ |
✓ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
|
$C_2^2$ |
$2^{2}$ |
|
|
✓ |
✓ |
✓ |
✓ |
|
✓ |
✓ |
| 691200.g.4.G |
$(A_4\times A_5^2):C_4$ |
$2^{8} \cdot 3^{3} \cdot 5^{2}$ |
|
✓ |
✓ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
|
$C_2^2$ |
$2^{2}$ |
|
|
✓ |
✓ |
✓ |
✓ |
|
✓ |
✓ |
| 691200.g.4.H |
$S_4\times A_5\times S_5$ |
$2^{8} \cdot 3^{3} \cdot 5^{2}$ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
|
|
$2^{2}$ |
|
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
| 691200.g.4.I |
$S_4\times \SOPlus(4,4)$ |
$2^{8} \cdot 3^{3} \cdot 5^{2}$ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
|
|
$2^{2}$ |
|
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
| 691200.g.4.J |
$A_4:S_5^2$ |
$2^{8} \cdot 3^{3} \cdot 5^{2}$ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
|
|
$2^{2}$ |
|
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
| 691200.g.4.K |
$\SOPlus(4,4):S_4$ |
$2^{8} \cdot 3^{3} \cdot 5^{2}$ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
|
|
$2^{2}$ |
|
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
| 691200.g.4.L |
$S_5^2:D_6$ |
$2^{8} \cdot 3^{3} \cdot 5^{2}$ |
|
|
|
✓ |
|
|
|
|
|
|
|
|
|
|
|
|
|
$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
|
|
$2^{2}$ |
|
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
| 691200.g.6.A |
$C_2^2\times S_5\wr C_2$ |
$2^{9} \cdot 3^{2} \cdot 5^{2}$ |
|
✓ |
✓ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
|
$S_3$ |
$2 \cdot 3$ |
|
|
|
|
✓ |
✓ |
|
✓ |
✓ |
| 691200.g.6.B |
$C_2^2\times S_5\wr C_2$ |
$2^{9} \cdot 3^{2} \cdot 5^{2}$ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
|
|
$2 \cdot 3$ |
|
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
| 691200.g.6.C |
$S_5^2:D_4$ |
$2^{9} \cdot 3^{2} \cdot 5^{2}$ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
|
|
$2 \cdot 3$ |
|
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
| 691200.g.6.D |
$C_2^2.A_5^2.D_4$ |
$2^{9} \cdot 3^{2} \cdot 5^{2}$ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
|
|
$2 \cdot 3$ |
|
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
| 691200.g.6.E |
$(C_2^2\times A_5^2):D_4$ |
$2^{9} \cdot 3^{2} \cdot 5^{2}$ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
|
|
$2 \cdot 3$ |
|
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
| 691200.g.6.F |
$C_4\times S_5\wr C_2$ |
$2^{9} \cdot 3^{2} \cdot 5^{2}$ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
|
|
$2 \cdot 3$ |
|
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
| 691200.g.6.G |
$C_4.A_5^2.D_4$ |
$2^{9} \cdot 3^{2} \cdot 5^{2}$ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
|
|
$2 \cdot 3$ |
|
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
| 691200.g.6.H |
$C_4.A_5^2.D_4$ |
$2^{9} \cdot 3^{2} \cdot 5^{2}$ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
|
|
$2 \cdot 3$ |
|
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
| 691200.g.6.I |
$C_4.A_5^2.D_4$ |
$2^{9} \cdot 3^{2} \cdot 5^{2}$ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
|
|
$2 \cdot 3$ |
|
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
| 691200.g.6.J |
$S_5^2\times D_4$ |
$2^{9} \cdot 3^{2} \cdot 5^{2}$ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
|
|
$2 \cdot 3$ |
|
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
| 691200.g.6.K |
$D_4\times \POPlus(4,5)$ |
$2^{9} \cdot 3^{2} \cdot 5^{2}$ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
|
|
$2 \cdot 3$ |
|
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
| 691200.g.6.L |
$D_4\times A_5^2:C_4$ |
$2^{9} \cdot 3^{2} \cdot 5^{2}$ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
|
|
$2 \cdot 3$ |
|
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
| 691200.g.6.M |
$S_5^2:D_4$ |
$2^{9} \cdot 3^{2} \cdot 5^{2}$ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
|
|
$2 \cdot 3$ |
|
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
| 691200.g.6.N |
$C_2^2.A_5^2.D_4$ |
$2^{9} \cdot 3^{2} \cdot 5^{2}$ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
|
|
$2 \cdot 3$ |
|
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
| 691200.g.6.O |
$(C_2^2\times A_5^2):D_4$ |
$2^{9} \cdot 3^{2} \cdot 5^{2}$ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
|
|
$2 \cdot 3$ |
|
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
| 691200.g.8.A |
$S_4\times A_5^2$ |
$2^{7} \cdot 3^{3} \cdot 5^{2}$ |
|
✓ |
✓ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
|
$D_4$ |
$2^{3}$ |
|
|
|
✓ |
✓ |
✓ |
|
|
✓ |
| 691200.g.8.B |
$A_4\times \PSOPlus(4,5)$ |
$2^{7} \cdot 3^{3} \cdot 5^{2}$ |
|
✓ |
✓ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
|
$C_2^3$ |
$2^{3}$ |
|
|
✓ |
✓ |
✓ |
✓ |
|
✓ |
✓ |
| 691200.g.8.C |
$A_5^2:S_4$ |
$2^{7} \cdot 3^{3} \cdot 5^{2}$ |
|
✓ |
✓ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
|
$D_4$ |
$2^{3}$ |
|
|
|
✓ |
✓ |
✓ |
|
|
✓ |
| 691200.g.8.D |
$A_4\times A_5\times S_5$ |
$2^{7} \cdot 3^{3} \cdot 5^{2}$ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
|
|
$2^{3}$ |
|
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
| 691200.g.8.E |
$A_4\times \SOPlus(4,4)$ |
$2^{7} \cdot 3^{3} \cdot 5^{2}$ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
|
|
$2^{3}$ |
|
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
| 691200.g.8.F |
$A_5^2:S_4$ |
$2^{7} \cdot 3^{3} \cdot 5^{2}$ |
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$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
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$2^{3}$ |
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$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
| 691200.g.8.G |
$A_5^2:S_4$ |
$2^{7} \cdot 3^{3} \cdot 5^{2}$ |
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$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
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$2^{3}$ |
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$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
| 691200.g.8.H |
$S_3\times S_5^2$ |
$2^{7} \cdot 3^{3} \cdot 5^{2}$ |
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$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
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$2^{3}$ |
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$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
| 691200.g.8.I |
$S_3\times \POPlus(4,5)$ |
$2^{7} \cdot 3^{3} \cdot 5^{2}$ |
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$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
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$2^{3}$ |
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$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
| 691200.g.8.J |
$A_5^2:C_4\times S_3$ |
$2^{7} \cdot 3^{3} \cdot 5^{2}$ |
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$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
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$2^{3}$ |
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$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
| 691200.g.8.K |
$S_5^2:C_6$ |
$2^{7} \cdot 3^{3} \cdot 5^{2}$ |
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$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
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$2^{3}$ |
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$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
| 691200.g.8.L |
$S_5^2:S_3$ |
$2^{7} \cdot 3^{3} \cdot 5^{2}$ |
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$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
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$2^{3}$ |
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$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
| 691200.g.8.M |
$(A_5\times \GL(2,4)):D_4$ |
$2^{7} \cdot 3^{3} \cdot 5^{2}$ |
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$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
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$2^{3}$ |
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$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
| 691200.g.8.N |
$A_5^2:D_{12}$ |
$2^{7} \cdot 3^{3} \cdot 5^{2}$ |
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$S_4\times S_5\wr C_2$ |
$2^{10} \cdot 3^{3} \cdot 5^{2}$ |
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$2^{3}$ |
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$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
$-$ |
