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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 (29 matches)

  displayed columns for results
Label Subgroup Ambient Quotient
Name Order Sylow norm char max cent ab Name Order Name Size max ab
10080.k.5040.a1.a1 $C_2$ $2$ ✓ ✓ ✓ ✓ $C_2.S_7$ $2^{5} \cdot 3^{2} \cdot 5 \cdot 7$ $S_7$ $2^{4} \cdot 3^{2} \cdot 5 \cdot 7$ ✓
10080.l.5040.a1.a1 $C_2$ $2$ ✓ ✓ ✓ ✓ $C_2\times S_7$ $2^{5} \cdot 3^{2} \cdot 5 \cdot 7$ $S_7$ $2^{4} \cdot 3^{2} \cdot 5 \cdot 7$ ✓
10080.m.5040.a1.a1 $C_2$ $2$ ✓ ✓ ✓ ✓ $A_7:C_4$ $2^{5} \cdot 3^{2} \cdot 5 \cdot 7$ $S_7$ $2^{4} \cdot 3^{2} \cdot 5 \cdot 7$ ✓
15120.d.5040.a1.a1 $C_3$ $3$ ✓ ✓ ✓ ✓ $C_3\times S_7$ $2^{4} \cdot 3^{3} \cdot 5 \cdot 7$ $S_7$ $2^{4} \cdot 3^{2} \cdot 5 \cdot 7$ ✓
15120.f.5040.a1.a1 $C_3$ $3$ ✓ ✓ ✓ $C_3:S_7$ $2^{4} \cdot 3^{3} \cdot 5 \cdot 7$ $S_7$ $2^{4} \cdot 3^{2} \cdot 5 \cdot 7$ ✓
25200.l.5040.a1.a1 $C_5$ $5$ ✓ ✓ ✓ ✓ $C_5\times S_7$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7$ $S_7$ $2^{4} \cdot 3^{2} \cdot 5 \cdot 7$ ✓
25200.m.5040.a1.a1 $C_5$ $5$ ✓ ✓ ✓ $C_5:S_7$ $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7$ $S_7$ $2^{4} \cdot 3^{2} \cdot 5 \cdot 7$ ✓
35280.g.5040.a1.a1 $C_7$ $7$ ✓ ✓ ✓ ✓ $C_7\times S_7$ $2^{4} \cdot 3^{2} \cdot 5 \cdot 7^{2}$ $S_7$ $2^{4} \cdot 3^{2} \cdot 5 \cdot 7$ ✓
35280.h.5040.a1.a1 $C_7$ $7$ ✓ ✓ ✓ $C_7:S_7$ $2^{4} \cdot 3^{2} \cdot 5 \cdot 7^{2}$ $S_7$ $2^{4} \cdot 3^{2} \cdot 5 \cdot 7$ ✓
302400.b.5040.a1 $A_5$ $2^{2} \cdot 3 \cdot 5$ ✓ ✓ $A_5\times S_7$ $2^{6} \cdot 3^{3} \cdot 5^{2} \cdot 7$ $S_7$ $2^{4} \cdot 3^{2} \cdot 5 \cdot 7$ ✓
302400.c.5040.a1 $A_5$ $2^{2} \cdot 3 \cdot 5$ ✓ ✓ $A_5:S_7$ $2^{6} \cdot 3^{3} \cdot 5^{2} \cdot 7$ $S_7$ $2^{4} \cdot 3^{2} \cdot 5 \cdot 7$ ✓
322560.b.5040.A $C_2^6$ $2^{6}$ ✓ ✓ ✓ $C_2^6.S_7$ $2^{10} \cdot 3^{2} \cdot 5 \cdot 7$ $S_7$ $2^{4} \cdot 3^{2} \cdot 5 \cdot 7$ ✓
846720.e.5040.a1 $\PSL(2,7)$ $2^{3} \cdot 3 \cdot 7$ ✓ ✓ $\GL(3,2):S_7$ $2^{7} \cdot 3^{3} \cdot 5 \cdot 7^{2}$ $S_7$ $2^{4} \cdot 3^{2} \cdot 5 \cdot 7$ ✓
846720.f.5040.A $\PSL(2,7)$ $2^{3} \cdot 3 \cdot 7$ ✓ ✓ $S_7\times \GL(3,2)$ $2^{7} \cdot 3^{3} \cdot 5 \cdot 7^{2}$ $S_7$ $2^{4} \cdot 3^{2} \cdot 5 \cdot 7$ ✓
1814400.c.5040.A $A_6$ $2^{3} \cdot 3^{2} \cdot 5$ ✓ ✓ $A_6:S_7$ $2^{7} \cdot 3^{4} \cdot 5^{2} \cdot 7$ $S_7$ $2^{4} \cdot 3^{2} \cdot 5 \cdot 7$ ✓
1814400.d.5040._.A $A_6$ $2^{3} \cdot 3^{2} \cdot 5$ ✓ ✓ $A_6\times S_7$ $2^{7} \cdot 3^{4} \cdot 5^{2} \cdot 7$ $S_7$ $2^{4} \cdot 3^{2} \cdot 5 \cdot 7$ ✓
3674160.c.5040.A $C_3^6$ $3^{6}$ ✓ ✓ ✓ $C_3^6.S_7$ $2^{4} \cdot 3^{8} \cdot 5 \cdot 7$ $S_7$ $2^{4} \cdot 3^{2} \cdot 5 \cdot 7$ ✓
12700800.b.5040._.A $A_7$ $2^{3} \cdot 3^{2} \cdot 5 \cdot 7$ ✓ $A_7.S_7$ $2^{7} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2}$ $S_7$ $2^{4} \cdot 3^{2} \cdot 5 \cdot 7$ ✓
12700800.b.5040._.B $A_7$ $2^{3} \cdot 3^{2} \cdot 5 \cdot 7$ ✓ $A_7.S_7$ $2^{7} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2}$ $S_7$ $2^{4} \cdot 3^{2} \cdot 5 \cdot 7$ ✓
12700800.c.5040._.A $A_7$ $2^{3} \cdot 3^{2} \cdot 5 \cdot 7$ ✓ ✓ $A_7\times S_7$ $2^{7} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2}$ $S_7$ $2^{4} \cdot 3^{2} \cdot 5 \cdot 7$ ✓
78750000.b.5040._.A $C_5^6$ $5^{6}$ ✓ ✓ ✓ $C_5^6.S_7$ $2^{4} \cdot 3^{2} \cdot 5^{7} \cdot 7$ $S_7$ $2^{4} \cdot 3^{2} \cdot 5 \cdot 7$ ✓
78750000.c.5040._.A $C_5^6$ $5^{6}$ ✓ ✓ ✓ $C_5^6.S_7$ $2^{4} \cdot 3^{2} \cdot 5^{7} \cdot 7$ $S_7$ $2^{4} \cdot 3^{2} \cdot 5 \cdot 7$ ✓
82575360.q.5040._.A $C_2^{14}$ $2^{14}$ ✓ ✓ ✓ $C_2^{14}.A_7.C_2$ $2^{18} \cdot 3^{2} \cdot 5 \cdot 7$ $S_7$ $2^{4} \cdot 3^{2} \cdot 5 \cdot 7$ ✓
101606400.f.5040._.A $A_8$ $2^{6} \cdot 3^{2} \cdot 5 \cdot 7$ ✓ ✓ $A_7.S_8$ $2^{10} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2}$ $S_7$ $2^{4} \cdot 3^{2} \cdot 5 \cdot 7$ ✓
101606400.g.5040._.A $A_8$ $2^{6} \cdot 3^{2} \cdot 5 \cdot 7$ ✓ ✓ $S_7\times A_8$ $2^{10} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2}$ $S_7$ $2^{4} \cdot 3^{2} \cdot 5 \cdot 7$ ✓
14108774400000000.b.5040._.A $A_5.A_5.A_5.A_5.A_5.A_5.A_5$ $2^{14} \cdot 3^{7} \cdot 5^{7}$ ✓ ✓ $A_5^7.A_7.C_2$ $2^{18} \cdot 3^{9} \cdot 5^{8} \cdot 7$ $S_7$ $2^{4} \cdot 3^{2} \cdot 5 \cdot 7$ ✓
14108774400000000.c.5040._.A $A_5.A_5.A_5.A_5.A_5.A_5.A_5$ $2^{14} \cdot 3^{7} \cdot 5^{7}$ ✓ ✓ $A_5^7.A_7.C_2$ $2^{18} \cdot 3^{9} \cdot 5^{8} \cdot 7$ $S_7$ $2^{4} \cdot 3^{2} \cdot 5 \cdot 7$ ✓
3949553870438400000000.b.5040._.A $A_6.A_6.A_6.A_6.A_6.A_6.A_6$ $2^{21} \cdot 3^{14} \cdot 5^{7}$ ✓ ✓ $A_6^7.A_7.C_2$ $2^{25} \cdot 3^{16} \cdot 5^{8} \cdot 7$ $S_7$ $2^{4} \cdot 3^{2} \cdot 5 \cdot 7$ ✓
3949553870438400000000.c.5040._.A $A_6.A_6.A_6.A_6.A_6.A_6.A_6$ $2^{21} \cdot 3^{14} \cdot 5^{7}$ ✓ ✓ $A_6^7.A_7.C_2$ $2^{25} \cdot 3^{16} \cdot 5^{8} \cdot 7$ $S_7$ $2^{4} \cdot 3^{2} \cdot 5 \cdot 7$ ✓
  displayed columns for results