Subgroups — C₅×C₁₃×ΓL(1,𝔽₈)×D₁₁ #44473277

Order 30030.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₁₁×C₁₃×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₅×C₇×C₁₃×D₁₁	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₁₃×ΓL(1,𝔽₈)×D₁₁	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₅×C₇×C₁₁×C₁₃	C₂×C₃	1	1	yes	characteristic, complemented, Hall
5	C₃×C₅×C₁₃×D₁₁	—	7	1	no	maximal, Hall
6	C₁₁×C₁₃×ΓL(1,𝔽₈)	C₂×C₅	1	1	yes	characteristic, complemented, Hall
7	C₂×C₅×C₁₃×ΓL(1,𝔽₈)	—	11	1	no	maximal, Hall
8	C₅×ΓL(1,𝔽₈)×D₁₁	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₃×C₅×C₁₁×C₁₃	—	7	1	no	Hall
10	C₇×C₁₃×D₁₁	C₃×C₅	1	1	yes	characteristic, complemented, Hall
11	C₅×C₁₃×D₁₁	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
12	C₅×C₁₃×ΓL(1,𝔽₈)	D₁₁	1	1	yes	characteristic, direct product factor, Hall
13	C₅×C₁₁×ΓL(1,𝔽₈)	C₂×C₁₃	1	1	yes	characteristic, complemented, Hall
14	C₇×C₁₁×C₁₃	C₂×C₃×C₅	1	1	yes	characteristic, complemented, Hall
15	C₂×C₅×C₇×C₁₃	—	11	1	no	Hall
16	C₃×C₁₃×D₁₁	—	7	1	no	Hall
17	C₅×C₇×D₁₁	C₃×C₁₃	1	1	yes	characteristic, complemented, Hall
18	C₅×C₁₁×C₁₃	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
19	C₂×C₁₃×ΓL(1,𝔽₈)	—	11	1	no	Hall
20	ΓL(1,𝔽₈)×D₁₁	C₅×C₁₃	1	1	yes	characteristic, direct product factor, Hall
21	C₅×C₇×C₁₃	C₃×D₁₁	1	1	yes	characteristic, complemented, Hall
22	C₃×C₁₁×C₁₃	—	7	1	no	Hall
23	C₂×C₃×C₅×C₁₃	—	77	1	no	Hall
24	C₅×C₇×C₁₁	C₂×C₃×C₁₃	1	1	yes	characteristic, complemented, Hall
25	C₃×C₅×D₁₁	—	7	1	no	Hall
26	C₁₃×D₁₁	C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
27	C₁₃×ΓL(1,𝔽₈)	C₅×D₁₁	1	1	yes	characteristic, direct product factor, Hall
28	C₁₁×ΓL(1,𝔽₈)	C₂×C₅×C₁₃	1	1	yes	characteristic, complemented, Hall
29	C₂×C₅×ΓL(1,𝔽₈)	—	11	1	no	Hall
30	C₃×C₅×C₁₃	—	7	1	no	Hall
31	C₂×C₇×C₁₃	—	11	1	no	Hall
32	C₃×C₅×C₁₁	—	7	1	no	Hall
33	C₇×D₁₁	C₃×C₅×C₁₃	1	1	yes	characteristic, complemented, Hall
34	C₁₁×C₁₃	C₂×C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
35	C₂×C₅×C₁₃	—	11	1	no	Hall
36	C₅×D₁₁	C₁₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
37	C₅×ΓL(1,𝔽₈)	C₁₃×D₁₁	1	1	yes	characteristic, direct product factor, Hall
38	C₇×C₁₃	C₃×C₅×D₁₁	1	1	yes	characteristic, complemented, Hall
39	C₂×C₃×C₁₃	—	77	1	no	Hall
40	C₇×C₁₁	C₂×C₃×C₅×C₁₃	1	1	yes	characteristic, complemented, Hall
41	C₂×C₅×C₇	—	11	1	no	Hall
42	C₃×D₁₁	—	7	1	no	Hall
43	C₅×C₁₃	ΓL(1,𝔽₈)×D₁₁	1	1	yes	characteristic, direct product factor, Hall
44	C₅×C₁₁	C₂×C₁₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
45	C₂×ΓL(1,𝔽₈)	—	11	1	no	Hall
46	C₃×C₁₃	—	7	1	no	Hall
47	C₅×C₇	C₃×C₁₃×D₁₁	1	1	yes	characteristic, complemented, Hall
48	C₃×C₁₁	—	7	1	no	Hall
49	C₂×C₃×C₅	—	77	1	no	Hall
50	C₂×C₁₃	—	11	1	no	Hall
51	D₁₁	C₅×C₁₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
52	ΓL(1,𝔽₈)	C₅×C₁₃×D₁₁	1	1	yes	characteristic, direct product factor, Hall
53	C₃×C₅	—	7	1	no	Hall
54	C₂×C₇	—	11	1	no	Hall
55	C₁₃	C₅×ΓL(1,𝔽₈)×D₁₁	1	1	yes	characteristic, direct product factor, Sylow
56	C₁₁	C₂×C₅×C₁₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
57	C₂×C₅	—	11	1	no	Hall
58	C₇	C₃×C₅×C₁₃×D₁₁	1	1	yes	characteristic, complemented, Sylow
59	C₂×C₃	—	77	1	no	Hall
60	C₅	C₁₃×ΓL(1,𝔽₈)×D₁₁	1	1	yes	characteristic, direct product factor, Sylow
61	C₃	—	7	1	no	Sylow
62	C₂	—	11	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₅×C₁₃	ΓL(1,𝔽₈)×D₁₁
commutator, cosocle, hypocenter	C₇×C₁₁	C₂×C₃×C₅×C₁₃
Frattini	C₁	G
Fitting, socle	C₅×C₇×C₁₁×C₁₃	C₂×C₃
LCS	G ▷ C₇×C₁₁
UCS	C₁ ◁ C₅×C₁₃

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₅×C₁₁×C₁₃×ΓL(1,𝔽₈)	C₂	semidirect
2. C₅×C₇×C₁₃×D₁₁	C₃	semidirect
3. C₁₃×ΓL(1,𝔽₈)×D₁₁	C₅	direct (N × H)
4. C₅×C₇×C₁₁×C₁₃	C₂×C₃	semidirect
6. C₁₁×C₁₃×ΓL(1,𝔽₈)	C₂×C₅	semidirect
8. C₅×ΓL(1,𝔽₈)×D₁₁	C₁₃	direct (N × H)
10. C₇×C₁₃×D₁₁	C₃×C₅	semidirect
11. C₅×C₁₃×D₁₁	ΓL(1,𝔽₈)	direct (N × H)
12. C₅×C₁₃×ΓL(1,𝔽₈)	D₁₁	direct (N × H)
13. C₅×C₁₁×ΓL(1,𝔽₈)	C₂×C₁₃	semidirect
14. C₇×C₁₁×C₁₃	C₂×C₃×C₅	semidirect
17. C₅×C₇×D₁₁	C₃×C₁₃	semidirect
18. C₅×C₁₁×C₁₃	C₂×ΓL(1,𝔽₈)	semidirect
20. ΓL(1,𝔽₈)×D₁₁	C₅×C₁₃	direct (N × H)
21. C₅×C₇×C₁₃	C₃×D₁₁	semidirect
24. C₅×C₇×C₁₁	C₂×C₃×C₁₃	semidirect
26. C₁₃×D₁₁	C₅×ΓL(1,𝔽₈)	direct (N × H)
27. C₁₃×ΓL(1,𝔽₈)	C₅×D₁₁	direct (N × H)
28. C₁₁×ΓL(1,𝔽₈)	C₂×C₅×C₁₃	semidirect
33. C₇×D₁₁	C₃×C₅×C₁₃	semidirect
34. C₁₁×C₁₃	C₂×C₅×ΓL(1,𝔽₈)	semidirect
36. C₅×D₁₁	C₁₃×ΓL(1,𝔽₈)	direct (N × H)
37. C₅×ΓL(1,𝔽₈)	C₁₃×D₁₁	direct (N × H)
38. C₇×C₁₃	C₃×C₅×D₁₁	semidirect
40. C₇×C₁₁	C₂×C₃×C₅×C₁₃	semidirect
43. C₅×C₁₃	ΓL(1,𝔽₈)×D₁₁	direct (N × H)
44. C₅×C₁₁	C₂×C₁₃×ΓL(1,𝔽₈)	semidirect
47. C₅×C₇	C₃×C₁₃×D₁₁	semidirect
51. D₁₁	C₅×C₁₃×ΓL(1,𝔽₈)	direct (N × H)
52. ΓL(1,𝔽₈)	C₅×C₁₃×D₁₁	direct (N × H)
55. C₁₃	C₅×ΓL(1,𝔽₈)×D₁₁	direct (N × H)
56. C₁₁	C₂×C₅×C₁₃×ΓL(1,𝔽₈)	semidirect
58. C₇	C₃×C₅×C₁₃×D₁₁	semidirect
60. C₅	C₁₃×ΓL(1,𝔽₈)×D₁₁	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₅×C₁₁×C₁₃×ΓL(1,𝔽₈)	2
C₅×C₇×C₁₃×D₁₁	3
C₁₃×ΓL(1,𝔽₈)×D₁₁	5
C₃×C₅×C₁₃×D₁₁	7
C₂×C₅×C₁₃×ΓL(1,𝔽₈)	11
C₅×ΓL(1,𝔽₈)×D₁₁	13

Subgroups — C5×C13×ΓL(1,𝔽8)×D11 #44473277

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×C₁₃×ΓL(1,𝔽₈)×D₁₁ #44473277