Abstract group 283500000000.b: $C_5^8.A_9.C_2^2$

• Label: 283500000000.b
• Order: \( 2^{8} \cdot 3^{4} \cdot 5^{9} \cdot 7 \)
• Exponent: \( 2^{3} \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{4} \cdot 5^{9} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $45$
• Trans deg.: $45$
• Rank: $2$

Group information

Description:	$C_5^8.A_9.C_2^2$
Order:	\(283500000000\)\(\medspace = 2^{8} \cdot 3^{4} \cdot 5^{9} \cdot 7 \)
Exponent:	\(12600\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Automorphism group:	Group of order \(567000000000\)\(\medspace = 2^{9} \cdot 3^{4} \cdot 5^{9} \cdot 7 \)
Composition factors:	$C_2$ x 2, $C_5$ x 8, $A_9$
Derived length:	$1$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	7	8	9	10	12	14	15	18	20	25	30	35	40	50	60	70	75	100
Elements	1	14395255	37104200	607257000	236640624	8321915000	405000000	7087500000	15750000000	15970839120	1701000000	14175000000	4106020800	15750000000	32113368000	945000000	68349960000	9720000000	28350000000	8505000000	21924000000	16200000000	7560000000	5670000000	283500000000
Conjugacy classes	1	9	3	8	88	17	1	2	1	581	4	3	172	1	235	28	412	8	4	36	60	4	10	2	1690
Divisions	1	9	3	8	46	17	1	2	1	297	4	3	87	1	120	14	208	4	2	18	30	2	5	1	884
Autjugacy classes	1	9	3	8	46	17	1	2	1	297	4	3	87	1	120	14	208	4	2	18	30	2	5	1	884

Minimal presentations

Permutation degree:	$45$
Transitive degree:	$45$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of linear representations for this group have not been computed

Constructions

Permutation group:	Degree $45$ $\langle(1,8,24,20,13,44,30,32,40,5,9,23,16,12,45,29,33,39)(2,7,25,19,14,43,26,31,36,4,10,22,17,11,41,28,34,38) \!\cdots\! \rangle$
Transitive group:	45T7968				more information
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_5^8.C_2)$ . $S_9$	$C_5^8$ . $(C_2\times S_9)$	$(C_5^8.A_9)$ . $C_2^2$	$(C_5^8.C_2.A_9)$ . $C_2$	all 6

Elements of the group are displayed as permutations of degree 45.

Homology

Abelianization:	$C_{2}^{2} $
Schur multiplier:	not computed
Commutator length:	not computed

Subgroups

There are 8 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	a subgroup isomorphic to $C_1$
Commutator:	a subgroup isomorphic to $C_5^8.A_9$
Frattini:	a subgroup isomorphic to $C_1$
Fitting:	not computed
Radical:	not computed
Socle:	not computed

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

All subgroups of index up to 7500 (order at least 37800000) are shown, as well as all normal subgroups of any index.

Order 283500000000: $C_5^8.A_9.C_2^2$

Order 141750000000: $C_5^8.A_9.C_2$, $C_5^8.A_9.C_2$, $C_5^8.C_2.A_9$

Order 70875000000: $C_5^8.A_9$

Order 31500000000: $C_5^7.(D_5\times S_8)$

Order 15750000000: $C_5^7.C_5:S_8$, $C_5^7.(C_5\times S_8)$, $C_5^7.(D_5\times A_8)$

Order 7875000000: $C_5^8.C_2^2.A_7.C_2$, $C_5^8.A_8$

Order 6300000000: $C_5^7.(C_2\times S_8)$

Order 3937500000: $C_5^8.C_2.A_7.C_2$ x 6, $C_5^8.C_2^2.A_7$

Order 3375000000: $C_5^8.D_6.A_6.C_2$

Order 3150000000: $C_5^7.S_8$, $C_5^7.S_8$, $C_5^7.(C_2\times A_8)$

Order 2250000000: $C_5^3:C_2^2.C_5.C_5^3.C_5.D_6.S_5$

Order 1968750000: $C_5^8.A_7.C_2$ x 4, $C_5^8.C_2.A_7$ x 3

Order 1687500000: $C_5^8.S_3.A_6.C_2$ x 2, $C_5^8.C_6.A_6.C_2$ x 2, $C_5^2:C_3.C_5^5.C_{10}.A_6.C_2$ x 2, $C_5^8.D_6.A_6$

Order 1575000000: $C_5^6.D_{10}.A_7.C_2$ x 2, $C_5^7.A_8$

Order 1181250000: $C_5^8.\SL(2,8).C_6$

Order 1125000000: $C_5^8.A_4.C_2.S_5$ x 2, $C_5^3:C_2^2.C_5.C_5^3.D_{15}.S_5$ x 2, $C_5^3.A_4.C_5.C_5^3.C_{10}.S_5$ x 2, $C_5^7.D_{10}.A_6.C_2$, $C_5^3:C_2^2.C_5.C_5^3.C_5.D_6.A_5$

Order 1050000000: $C_5^4.D_5^4.\GL(3,2)$

Order 1012500000: $C_5^6.C_5^2:(S_3^3:D_6)$

Order 984375000: $C_5^8.A_7$

Order 900000000: $C_5^7.S_4^2:D_{10}$

Order 843750000: $C_5^8.C_3.A_6.C_2$ x 4, $C_5^8.S_3.A_6$, $C_5^8.C_6.A_6$, $C_5^2:C_3.C_5^5.C_{10}.A_6$

Order 787500000: $C_5^6.(D_5\times S_7)$ x 7, $C_5^7.C_2.A_7.C_2$ x 5, $C_5^6.D_{10}.A_7$, $C_2\times C_5^7.C_2.A_7$, $C_2\times C_5^7.A_7.C_2$, $(C_5^6\times C_{10}).A_7.C_2$

Order 750000000: $C_5^8.D_4.C_2.S_5$

Order 675000000: $C_5^7.D_6.A_6.C_2$

Order 590625000: $C_5^8.\SL(2,8).C_3$

Order 562500000: $C_5^8.C_2.A_6.C_2$ x 6, $C_5^3:C_2^2.C_5.C_5^3.C_{15}.S_5$ x 4, $C_5^8.D_6.S_5$ x 2, $C_5^8.A_4.C_2.A_5$, $C_5^7.D_{10}.A_6$, $C_5^3:C_2^2.C_5.C_5^3.D_{15}.A_5$, $C_5^3.A_4.C_5.C_5^3.C_{10}.A_5$

Order 525000000: $C_5^5.D_5^3.\PSL(2,7)$

Order 506250000: $C_5^2.C_5^6:S_3\wr S_3$, $C_5^2.C_5^6:S_3\wr S_3$, $C_5^2.C_5^6:S_3\wr S_3$, $C_5^2.C_5^6:S_3\wr S_3$, $C_5^6.C_{15}^2:C_6:S_4$, $C_5^6.C_{15}^2:C_6:S_4$, $C_5^2.C_5^6:(S_3^3:C_6)$

Order 450000000: $C_5^7.A_4^2:D_{20}$ x 2, $C_5^7.S_4^2:C_{10}$ x 2, $C_5^6.C_2^4.C_{15}^2.C_2^3$, $C_5^3:C_2^2.C_5.C_5^3.D_6.S_5$, $C_5^7.(A_4^2:C_4\times D_5)$, $C_5^7.(D_5\times \POPlus(4,3))$

Order 421875000: $C_5^8.C_3.A_6$

Order 393750000: $C_5^7.S_7$ x 5, $C_5^7.S_7$ x 5, $C_5^6.(D_5\times A_7)$ x 5, $C_5^7.C_2.A_7$ x 2, $C_5^7.A_7.C_2$ x 2, $(C_5^6\times C_{10}).A_7$, $C_5^8.C_2.\SL(2,8)$

Order 375000000: $C_5^8.D_4.S_5$ x 4, $C_5^8.C_2^3.S_5$ x 4, $C_5^8.C_4.C_2.S_5$ x 2, $C_5^7.(C_5\times D_4).S_5$ x 2, $C_5^3.C_{10}.C_5^3.C_{10}.C_2.S_5$ x 2, $C_5^8.D_4.C_2.A_5$, $C_5^7.C_{10}^2.C_6.C_2^3$

Order 337500000: $C_5^7.S_3.A_6.C_2$ x 2, $C_5^7.C_6.A_6.C_2$ x 2, $C_5^2:C_3.C_5^5.C_2.A_6.C_2$ x 2, $C_5^7.D_6.A_6$, $C_5^6.C_{15}^2.D_6.C_2^3$, $C_5^8.C_3^2.(C_2\times \GL(2,3))$

Order 315000000: $C_5^6.C_2^2.A_7.C_2$

Order 300000000: $C_5^4.(C_2\times D_5\wr S_4)$

Order 281250000: $C_5^8.S_3.S_5$ x 4, $C_5^8.C_6.S_5$ x 4, $C_5^8.A_6.C_2$ x 4, $C_5^8.C_2.A_6$ x 3, $C_5^8.D_6.A_5$ x 2, $(C_5\times C_5^2:C_3).C_5^5.C_2.S_5$ x 2, $(C_5\times C_5^2:C_3).C_5^3.C_5.C_{10}.S_5$ x 2, $C_5^3:C_2^2.C_5.C_5^3.C_{15}.A_5$

Order 262500000: $C_5^7.D_{10}.\PSL(2,7)$, $C_5^7.(D_5\times \PGL(2,7))$

Order 253125000: $C_5^6.(C_3\times C_{15}^2):S_4$, $C_5^6.(C_3\times C_{15}^2):S_4$, $C_5^6.(C_3\times C_{15}^2):S_4$, $C_5^6.(C_3\times C_{15}^2):S_4$, $C_5^6.C_3^3.C_{10}^2.C_6$, $C_5^6.C_3^3.C_{10}^2.C_6$, $C_5^6.C_3^3.C_{10}^2.C_6$, $C_5^6.(C_{15}^2.(S_3\times D_6))$

Order 225000000: $C_5^7.C_2^2.A_6.C_2$ x 3, $C_5^6.D_{10}.A_6.C_2$ x 3, $C_5^6.C_2^4.C_5.C_{15}.D_6$ x 3, $C_5^7.A_4.C_2.S_5$ x 2, $C_5^3:C_2^2.C_5.C_5^3.S_3.S_5$ x 2, $C_5^3.A_4.C_5.C_5^3.C_2.S_5$ x 2, $C_5^6.C_2^4.C_5.C_3^2.C_{20}$ x 2, $C_5^7.A_4^2:D_{10}$ x 2, $C_5^7.C_2^2.C_5.D_6^2$, $C_5^6.C_{10}^2.(S_3\times A_4).C_2$, $C_5^6.A_4^2.C_{10}^2$, $C_5^5.C_{10}^2.C_5.D_6^2$, $C_5^3:C_2^2.C_5.C_5^3.D_6.A_5$, $C_2\times C_5^7.C_2.A_6.C_2$, $C_5^7.A_4^2:D_{10}$, $C_5^7.A_4^2:D_{10}$

Order 210000000: $C_5^7:(C_2^4:\GL(3,2))$

Order 196875000: $C_5^7.A_7$ x 4, $C_5^8.\SL(2,8)$

Order 187500000: $C_5^8.C_2^2.S_5$ x 15, $C_5^7.D_{10}.S_5$ x 10, $C_5^8.C_4.S_5$ x 8, $C_5^8.D_4.A_5$ x 2, $C_5^8.C_2^3.A_5$ x 2, $C_5^7.C_{10}^2.C_{12}.C_2$ x 2, $C_5^8.C_4.C_2.A_5$, $C_5^8.(A_4\times F_5).C_2$, $C_5^7.C_{10}^2.D_6.C_2$, $C_5^7.C_{10}^2.C_6.C_2^2$, $C_5^7.(C_5\times D_4).A_5$, $C_5^6.C_{10}^2.D_{15}:C_4$, $C_5^6.C_{10}^2.C_{60}.C_2$, $C_5^3.C_{10}.C_5^3.C_{10}.C_2.A_5$

Order 180000000: $C_5^6.S_4^2:D_{10}$ x 2

Order 168750000: $C_5^6.C_{15}^2.D_6.C_2^2$ x 6, $C_5^7.C_3.A_6.C_2$ x 4, $C_5^6.C_{15}^2.C_6.C_2^3$ x 3, $C_5^6.C_3^2.C_{30}.C_5.C_2^3$ x 2, $C_5^8.S_3^3.C_2$, $C_5^8.C_3^3.C_2^3.C_2$, $C_5^8.C_3.D_6^2$, $C_5^7.S_3.A_6$, $C_5^7.C_6.A_6$, $C_5^4.C_{15}^2.C_5^2.C_6.C_2^3$, $C_5^2:C_3.C_5^5.C_2.A_6$, $C_5^8.C_3^2.\GL(2,3)$, $C_5^8.C_3^2.\GL(2,3)$, $C_5^8.C_3^2.(C_2\times \SL(2,3))$

Order 157500000: $C_5^6.(C_2\times S_7)$ x 4, $C_2\times C_5^6.A_7.C_2$ x 2, $C_5^6.C_2^2.A_7$

Order 150000000: $C_5^7.D_4.C_2.S_5$ x 2, $C_5^5.C_{10}^2.C_{30}.C_2^4$, $C_5^4.D_5\wr S_4$, $C_5^4.D_5\wr S_4$, $C_5^4.D_5\wr S_4$, $C_5^4.C_{10}\wr S_4$, $C_5^6.D_{10}^2:S_4$, $C_5^7.(D_5\times Q_8:S_4)$, $C_5^4.D_5^4.S_4$, $C_5^7.C_{10}.C_2^4.C_6.C_2$

Order 140625000: $C_5^8.C_3.S_5$ x 8, $C_5^8.S_3.A_5$ x 2, $C_5^8.C_6.A_5$ x 2, $C_5^8.A_6$, $(C_5\times C_5^2:C_3).C_5^5.C_2.A_5$, $(C_5\times C_5^2:C_3).C_5^3.C_5.C_{10}.A_5$

Order 135000000: $C_5^6.D_6.A_6.C_2$

Order 131250000: $C_5^8.C_2.\PSL(2,7)$ x 3, $C_5^8.\SO(3,7)$, $C_5^8.\SO(3,7)$, $C_5^7.(D_5\times \GL(3,2))$, $C_5\wr F_8.C_6$

Order 126562500: $C_5^6.C_3^3.C_5^2.A_4$, $C_5^8.C_3^3.D_6$, $C_5^8.C_3^3.D_6$, $C_5^8.C_3^3.D_6$, $C_5^8.C_3^3.D_6$, $C_5^8.C_3^3.C_6.C_2$, $C_5^8.C_3^3.D_6$, $C_5^6.(C_3\times C_{15}^2):D_6$

Order 125000000: $C_5^8.D_{10}.C_2^4$

Order 112500000: $C_5^7.C_2.A_6.C_2$ x 42, $C_5^5.C_{10}^2.C_{15}.D_6.C_2$ x 10, $C_5^7.D_6.S_5$ x 7, $C_5^7.(C_{15}\times A_4).C_2^3$ x 5, $C_5^6.D_{10}.A_6$ x 5, $C_5^7.(C_5\times A_4).D_6.C_2$ x 4, $C_5^3:C_2^2.C_5.C_5^3.C_3.S_5$ x 4, $C_5^7.(C_{30}\times A_4).C_2^2$ x 3, $C_5^8.C_3.A_4.C_2^3$ x 2, $C_5^7.C_2^2.A_6$ x 2, $C_5\wr (A_4\wr C_2)$ x 2, $C_5^8.A_4^2.C_2$, $C_5^7.A_4.C_{15}.C_2^3$, $C_5^7.A_4.C_2.A_5$, $C_5^6.C_{10}^2.(S_3\times A_4)$, $C_5^6.C_2^4.C_{15}:D_{15}$, $C_5^6.A_4^2.C_5.C_{10}$, $C_5^6.A_4.C_{10}^2.C_6$, $C_5^5.C_{10}^2.(D_5\times S_3^2)$, $C_5^5.C_{10}^2.(C_{10}\times S_3^2)$, $C_5^5.C_{10}^2.(C_6\times S_3\times D_5)$, $C_5^5.(C_{15}\times A_4).C_{10}^2.C_2$, $C_5^4.C_{15}^2.C_{10}^2.C_2^3$, $C_5^3:C_2^2.C_5^3.C_{15}^2.C_2^3$, $C_5^3:C_2^2.C_5.C_5^3.S_3.A_5$, $C_5^3.A_4.C_5.C_5^3.C_2.A_5$, $C_2\times C_5^7.A_6.C_2$, $(C_5^6\times C_{10}).A_6.C_2$, $C_5^8.F_9.C_2^2$

Order 105000000: $C_5^7:(C_2^3:\GL(3,2))$

Order 100000000: $C_5^7.D_4^2:D_{10}$

Order 93750000: $C_5^8.C_2.S_5$ x 29, $C_5^8.C_2^2.A_5$ x 5, $C_5^7.C_{10}^2.D_6$ x 4, $C_5^7.D_{10}.A_5$ x 3, $C_5^8.C_4.A_5$ x 2, $C_5^6.C_{10}^2.C_{30}.C_2$ x 2, $C_5^8.(D_5\times A_4).C_2$, $C_5^7.C_{10}^2.C_{12}$, $C_5^7.C_{10}^2.C_6.C_2$, $C_5^7.C_5.C_{30}.C_2^3$, $C_5^6.\He_5.A_4.C_4$, $C_5^3.C_{10}^2.C_5^3.C_{15}.C_2^2$

Order 90000000: $C_5^6.C_2^4.C_{15}.D_6.C_2$ x 2, $C_5^6.C_2^4.(D_5\times S_3^2)$ x 2, $C_5^7:S_4\wr C_2$ x 2, $C_5^7:S_4\wr C_2$ x 2, $C_5^7:S_4\wr C_2$ x 2, $C_5^7:S_4\wr C_2$ x 2, $C_5^7:(C_2\times A_4^2:C_4)$ x 2, $C_5^7:(A_4^2:C_2^3)$ x 2

Order 84375000: $C_5^8.S_3^3$ x 7, $C_5^8.C_3^3.C_2^3$ x 7, $C_5^6.C_3^2.C_{30}.C_{10}.C_2$ x 6, $C_5^8.C_3^3.C_2^2.C_2$ x 4, $C_5^4.C_3^2.C_5^3.C_{30}.C_2^2$ x 3, $C_5^7.C_3^3.C_{10}.C_2^2$ x 2, $C_5^7.C_3^3.C_5.C_2^3$ x 2, $C_5^6.C_{15}^2.D_{12}$ x 2, $C_5^6.C_{15}^2.D_6.C_2$ x 2, $C_5^6.C_{15}^2.C_3.D_4$ x 2, $C_5^7.C_3.A_6$, $C_5^6.C_3^3.C_{10}^2.C_2$, $C_5^4.C_{15}^2.C_5^2.C_6.C_2^2$, $C_5^8.C_3^2.\SL(2,3)$, $(C_5^8.\He_3):C_2^3$

Order 78750000: $C_5^6.S_7$ x 2, $C_5^6.S_7$ x 2, $C_5^6.(C_2\times A_7)$ x 2, $(C_5^5\times C_{10}).A_7$

Order 75000000: $C_5^7.C_2^3.S_5$ x 11, $C_5^7.D_4.S_5$ x 10, $C_5^5.C_{10}^2.C_{30}.C_2^3$ x 6, $C_5^7.C_4.C_2.S_5$ x 5, $C_5^7.D_4.C_2.A_5$ x 2, $C_5^6.(C_5\times D_4).S_5$ x 2, $C_5^5.C_{10}^2.D_{15}:D_4$ x 2, $C_5^5.C_{10}^2.C_{60}.C_2^2$ x 2, $C_5^3.C_{10}^2.C_5^3.C_6.C_2^3$ x 2, $C_5^3.C_{10}.C_5^3.C_2^2.S_5$ x 2, $(C_5\times C_5:D_5).C_5^3.C_{10}.C_2.S_5$ x 2, $C_5^7.C_2^4:D_{30}$ x 2, $C_5^5.D_5^3.S_4$ x 2, $C_5^8.D_6.C_2^4$, $C_5^8.(D_4\times A_4).C_2$, $C_5^6.C_{10}^2.C_6.C_2^3$, $C_5^6.C_2^4.C_5^2.D_6$, $C_5^5.C_{10}^2.C_{20}:D_6$, $C_5^5.C_2^2.C_{30}.C_{10}^2.C_2$, $C_5^3.C_{10}^2.C_5^3.D_6.C_2^2$, $C_5^5.(C_{10}^3.S_4)$, $C_5^5.(C_{10}^3.S_4)$, $C_5^5.C_{10}^3:S_4$, $C_5^5.C_{10}^3:S_4$, $C_5^4.D_5\wr A_4$, $C_5\wr (C_2\wr A_4)$, $C_5^6.C_{10}^2:(C_2\times S_4)$, $C_5^6.D_{10}^2:A_4$, $C_5^7.C_{10}.C_2^4.C_6$

Order 70312500: $C_5^8.C_3.A_5$ x 2

Order 67500000: $C_5^6.S_3.A_6.C_2$ x 2, $C_5^6.C_6.A_6.C_2$ x 2, $C_5^6.C_3^2.C_{30}.C_2^4$ x 2, $(C_5^5\times C_{15}).C_2.A_6.C_2$ x 2, $C_5^6.D_6.A_6$

Order 65625000: $C_5^8.\PSL(2,7)$, $C_5^6.C_{35}.C_{30}.C_2^2$, $C_5^8.\PSL(2,7)$, $C_5\wr F_8:C_3$

Order 63281250: $C_5^8.C_3^3.S_3$ x 2, $C_5^8.C_3^3.C_6$, $C_5^8.C_3^3.C_6$, $C_5^8.\He_3.C_6$, $C_5^8.\He_3.C_6$, $C_5^8.C_3^3.C_6$

Order 62500000: $C_5^8.(D_4\times F_5)$ x 3, $C_5^8.D_{10}.C_2^3$ x 2, $C_5^7.C_5^2.(C_4\times D_4)$ x 2, $C_5^7.C_5.(C_{20}\times D_4)$ x 2, $C_5^7.D_{10}^2.C_2$, $C_5^7.D_5^2.C_2^3$, $C_5^6.C_5^3.C_2^4.C_2$, $C_5^6.C_5^3.(C_4\times D_4)$, $C_5^5.C_5^3.(C_4^2\times D_5)$, $C_5^4.C_5^3.C_5.D_{10}.C_2^3$

Order 60000000: $D_5^4.D_5\wr S_3$

Order 56250000: $C_5^7.A_6.C_2$ x 32, $C_5^7.C_2.A_6$ x 22, $C_5^7.S_3.S_5$ x 14, $C_5^7.C_6.S_5$ x 14, $C_5^5.C_{10}^2.C_{15}.D_6$ x 13, $(C_5\times C_5^2:C_3).C_5^3.C_{10}.S_5$ x 12, $C_5^7.C_2^2.C_{15}.D_6$ x 9, $C_5^8.(S_3\times A_4).C_2$ x 8, $C_5^4.C_{15}^2.C_{10}^2.C_2^2$ x 8, $C_5^7.D_6.A_5$ x 7, $C_5^7.(C_5\times A_4).D_6$ x 6, $C_5^5.(C_{15}\times A_4).C_{10}^2$ x 5, $C_5^5.C_2^2.C_5^3.(C_6\times S_3)$ x 4, $C_5^8.D_6^2$ x 3, $C_5^7.A_4.C_{15}.C_2^2$ x 3, $C_5^7.(C_5\times S_3\times A_4).C_2$ x 3, $C_5^3.C_{10}^2.C_5^3.(C_6\times S_3)$ x 3, $C_5^8.C_2^2.C_6^2$ x 2, $C_5^7.(C_{30}\times A_4).C_2$ x 2, $C_5^5.C_{10}^2.C_3.C_{30}.C_2$ x 2, $C_5^4.C_{15}^2.C_5.(D_4\times D_5)$ x 2, $C_5^2:C_3.C_5^5.C_2.S_5$ x 2, $C_5^8.F_9.C_2$ x 2, $C_5^8.F_9.C_2$ x 2, $C_5^8.C_3:S_3.C_2^3$, $C_5^8.(C_6\times A_4).C_2$, $C_5^7.(C_{15}\times S_4).C_2$, $C_5^6.C_3.C_{10}^2.C_6.C_2$, $C_5^6.C_2^4.C_{15}^2$, $C_5^5.C_{10}^2.C_{15}.C_6.C_2$, $C_5^5.C_{10}^2.C_3^2.C_{10}.C_2$, $C_5^5.C_{10}^2.(C_{30}\times S_3)$, $C_5^5.C_{10}^2.(C_5\times S_3^2)$, $C_5^4.C_{15}^2.C_5^2.D_4.C_2$, $C_5^4.C_{15}^2.(D_5\times D_{20})$, $C_5^3:C_2^2.C_5^3.C_{15}^2.C_2^2$, $C_5^3:C_2^2.C_5^3.C_{15}.C_{30}.C_2$, $C_5^3:C_2^2.C_5.C_5^3.C_3.A_5$, $C_5^3.C_{10}^2.C_{15}.C_5.C_{30}.C_2$, $(C_5^6\times C_{10}).A_6$, $C_5^8.S_3^2.C_2^2$, $C_5^8.(C_2\times F_9)$, $(C_5^8.C_3^2):(C_2\times Q_8)$

Order 52500000: $C_5^7.C_2^2.\PSL(2,7)$, $C_5^6.D_{10}.\PSL(2,7)$, $C_5^7:(C_2\times \PGL(2,7))$

Order 50000000: $C_5^4.D_5\wr D_4$ x 2, $C_5^4.D_5\wr D_4$ x 2, $C_5^4.D_5\wr D_4$ x 2, $C_5\wr (C_2\wr D_4)$ x 2, $C_5^8.C_2^4.C_2^3$, $D_5^4.C_5:D_5^3$, $C_5^6.C_{20}^2:(C_2\times C_4)$, $C_5^6.D_{10}^2:D_4$, $C_5^4.(C_2\times D_5\wr C_4)$, $C_5^6.(D_{10}^2.D_4)$, $C_5^6.C_{20}^2:C_2^3$

Order 46875000: $C_5^8.S_5$ x 10, $C_5^8.C_2.A_5$ x 8, $C_5^7.C_{10}^2.C_6$ x 3, $C_5^7.C_{10}^2.S_3$, $C_5^7.C_5^2.D_6.C_2$, $C_5^7.C_5.C_{60}.C_2$, $C_5^6.C_{15}.C_{10}^2.C_2$, $C_5^6.C_{10}^2.C_{15}.C_2$, $C_5^6.C_5^3.D_6.C_2$, $C_5^6.C_5^3.C_6.C_2^2$, $C_5^5.\He_5.D_{15}:C_4$, $C_5^3.\He_5.C_5^3.C_{12}.C_2$, $C_5^3.C_{10}^2.C_5^3.C_{15}.C_2$, $C_5^3.C_{10}^2.C_5^2.C_{15}.C_{10}$

Order 45000000: $C_5^6.C_2^4.C_{15}.D_6$ x 14, $C_5^5.C_{10}^2.D_6^2$ x 7, $C_5^6.(C_5\times A_4).A_4.C_2^2$ x 6, $C_5^5.D_{10}.A_6.C_2$ x 5, $C_5^6.A_4^2.C_{10}.C_2$ x 4, $C_5^6.C_2^2.A_6.C_2$ x 2, $C_5^7:(A_4^2:C_4)$ x 2, $C_5^6.A_4^2:C_{20}$ x 2, $C_5^7:\POPlus(4,3)$ x 2, $C_5^7:\POPlus(4,3)$ x 2, $C_5^7:\POPlus(4,3)$ x 2, $C_5^6.A_4^2:D_{10}$ x 2, $C_5^7.C_2^2.D_6^2$

Order 43750000: $C_5^8.F_8.C_2$

Order 42187500: $C_5^8.C_3^2.D_6$ x 8, $C_5^8.C_3^3.C_2^2$ x 7, $C_5^7.C_3^3.C_{10}.C_2$ x 4, $C_5^6.C_{15}^2.C_6.C_2$ x 3, $C_5^8.C_3.C_6^2$ x 2, $C_5^7.C_3^3.C_{20}$ x 2, $C_5^7.C_3^2.C_{60}$ x 2, $C_5^7.C_3^2.C_{30}.C_2$ x 2, $C_5^6.C_3^3.C_{10}^2$ x 2, $C_5^6.C_{15}^2:D_6$, $C_5^6.C_{15}^2:D_6$, $C_5^6.C_{15}^2:D_6$, $C_5^6.C_{15}^2:D_6$, $C_5^8.\He_3.C_2^2$, $C_5^6.C_{15}^2:D_6$, $C_5^6.C_{15}^2:D_6$, $C_5^8.C_{18}.C_6$

Order 40500000: $C_5^6:S_3\wr S_3.C_2$

Order 39375000: $C_5^6.A_7$

Order 781250: $C_5^8.C_2$

Order 390625: $C_5^8$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 283500000000: $C_5^8.A_9.C_2^2$ ($C_1$)

Order 141750000000: $C_5^8.A_9.C_2$ ($C_2$), $C_5^8.A_9.C_2$ ($C_2$), $C_5^8.C_2.A_9$ ($C_2$)

Order 70875000000: $C_5^8.A_9$ ($C_2^2$)

Order 781250: $C_5^8.C_2$ ($S_9$)

Order 390625: $C_5^8$ ($C_2\times S_9$)

Order 1: $C_1$ ($C_5^8.A_9.C_2^2$)

Normal subgroups up to automorphism (quotient in parentheses)

not computed

Series

Derived series	not computed
Chief series	not computed
Lower central series	not computed
Upper central series	not computed

Supergroups

This group is a maximal subgroup of 2 larger groups in the database.

This group is a maximal quotient of 1 larger groups in the database.

Character theory

Complex character table

The $1690 \times 1690$ character table is not available for this group.

Rational character table

The $884 \times 884$ rational character table is not available for this group.