Subgroups — ΓL(1,𝔽₈)×D₁₁×D₃₇ #46803213

Order 34188.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)×D₄₀₇	C₂	1	1	yes	maximal, characteristic, complemented
2	ΓL(1,𝔽₈)×D₁₁×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₁×ΓL(1,𝔽₈)×D₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×D₁₁×D₃₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	C₁₁×ΓL(1,𝔽₈)×C₃₇	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₇×D₄₀₇	C₂×C₃	1	1	yes	characteristic, complemented
7	C₇×D₁₁×C₃₇	C₂×C₃	1	1	yes	characteristic, complemented
8	C₇×C₁₁×D₃₇	C₂×C₃	1	1	yes	characteristic, complemented
9	C₃×D₁₁×D₃₇	—	7	1	no	maximal, Hall
10	C₂×ΓL(1,𝔽₈)×D₃₇	—	11	1	no	maximal, Hall
11	C₇×C₁₁×C₃₇	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
12	C₃×D₄₀₇	—	7	1	no	—
13	C₃×D₁₁×C₃₇	—	7	1	no	—
14	C₃×C₁₁×D₃₇	—	7	1	no	—
15	D₁₁×D₃₇	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
16	C₂×ΓL(1,𝔽₈)×C₃₇	—	11	1	no	—
17	ΓL(1,𝔽₈)×D₃₇	—	11	1	no	—
18	ΓL(1,𝔽₈)×D₃₇	D₁₁	1	1	yes	characteristic, direct product factor
19	C₃×C₁₁×C₃₇	—	7	1	no	Hall
20	C₂×C₇×D₃₇	—	11	1	no	Hall
21	C₂×ΓL(1,𝔽₈)×D₁₁	—	37	1	no	maximal, Hall
22	D₄₀₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
23	C₁₁×D₃₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
24	D₁₁×C₃₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
25	ΓL(1,𝔽₈)×C₃₇	C₂×D₁₁	1	1	yes	characteristic, complemented, Hall
26	C₂×C₇×C₃₇	—	11	1	no	—
27	C₇×D₃₇	—	11	1	no	—
28	C₇×D₃₇	C₃×D₁₁	1	1	yes	characteristic, complemented
29	C₂×C₁₁×ΓL(1,𝔽₈)	—	37	1	no	—
30	ΓL(1,𝔽₈)×D₁₁	—	37	1	no	—
31	ΓL(1,𝔽₈)×D₁₁	D₃₇	1	1	yes	characteristic, direct product factor
32	C₂×C₃×D₃₇	—	77	1	no	Hall
33	C₁₁×C₃₇	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
34	C₂×C₇×D₁₁	—	37	1	no	Hall
35	C₇×C₃₇	C₂×C₃×D₁₁	1	1	yes	characteristic, complemented, Hall
36	C₁₁×ΓL(1,𝔽₈)	C₂×D₃₇	1	1	yes	characteristic, complemented, Hall
37	C₂×C₃×C₃₇	—	77	1	no	—
38	C₃×D₃₇	—	7	1	no	—
39	C₃×D₃₇	—	77	1	no	—
40	C₂×C₇×C₁₁	—	37	1	no	—
41	C₇×D₁₁	—	37	1	no	—
42	C₇×D₁₁	C₃×D₃₇	1	1	yes	characteristic, complemented
43	C₂×D₃₇	—	11	1	no	Hall
44	C₂×C₃×D₁₁	—	259	1	no	Hall
45	C₃×C₃₇	—	7	1	no	Hall
46	C₂²×ΓL(1,𝔽₈)	—	407	1	no	Hall
47	C₇×C₁₁	C₂×C₃×D₃₇	1	1	yes	characteristic, complemented, Hall
48	C₂×C₃₇	—	11	1	no	—
49	D₃₇	—	11	1	no	—
50	D₃₇	ΓL(1,𝔽₈)×D₁₁	1	1	yes	characteristic, direct product factor
51	C₂×C₃×C₁₁	—	259	1	no	—
52	C₃×D₁₁	—	259	1	no	—
53	C₃×D₁₁	—	7	1	no	—
54	C₂×D₁₁	—	37	1	no	Hall
55	C₂×ΓL(1,𝔽₈)	—	11	1	no	—
56	C₂×ΓL(1,𝔽₈)	—	37	1	no	—
57	C₂×ΓL(1,𝔽₈)	—	407	1	no	—
58	C₃₇	C₂×ΓL(1,𝔽₈)×D₁₁	1	1	yes	characteristic, complemented, Sylow
59	C₃×C₁₁	—	7	1	no	Hall
60	C₂²×C₇	—	407	1	no	Hall
61	C₂×C₁₁	—	37	1	no	—
62	D₁₁	—	37	1	no	—
63	D₁₁	ΓL(1,𝔽₈)×D₃₇	1	1	yes	characteristic, direct product factor
64	ΓL(1,𝔽₈)	D₁₁×D₃₇	1	1	yes	characteristic, direct product factor, Hall
65	C₂×C₇	—	11	1	no	—
66	C₂×C₇	—	37	1	no	—
67	C₂×C₇	—	407	1	no	—
68	C₂²×C₃	—	2849	1	no	Hall
69	C₁₁	C₂×ΓL(1,𝔽₈)×D₃₇	1	1	yes	characteristic, complemented, Sylow
70	C₇	C₃×D₁₁×D₃₇	1	1	yes	characteristic, complemented, Sylow
71	C₂×C₃	—	259	1	no	—
72	C₂×C₃	—	2849	1	no	—
73	C₂×C₃	—	77	1	no	—
74	C₂²	—	407	1	no	Sylow
75	C₃	—	7	1	no	Sylow
76	C₂	—	11	1	no	—
77	C₂	—	37	1	no	—
78	C₂	—	407	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₁₁×C₃₇	C₂²×C₃
LCS	G ▷ C₇×C₁₁×C₃₇
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. ΓL(1,𝔽₈)×D₄₀₇	C₂	semidirect
2. ΓL(1,𝔽₈)×D₁₁×C₃₇	C₂	semidirect
3. C₁₁×ΓL(1,𝔽₈)×D₃₇	C₂	semidirect
4. C₇×D₁₁×D₃₇	C₃	semidirect
5. C₁₁×ΓL(1,𝔽₈)×C₃₇	C₂²	semidirect
6. C₇×D₄₀₇	C₂×C₃	semidirect
7. C₇×D₁₁×C₃₇	C₂×C₃	semidirect
8. C₇×C₁₁×D₃₇	C₂×C₃	semidirect
11. C₇×C₁₁×C₃₇	C₂²×C₃	semidirect
15. D₁₁×D₃₇	ΓL(1,𝔽₈)	direct (N × H)
18. ΓL(1,𝔽₈)×D₃₇	D₁₁	direct (N × H)
22. D₄₀₇	C₂×ΓL(1,𝔽₈)	semidirect
23. C₁₁×D₃₇	C₂×ΓL(1,𝔽₈)	semidirect
24. D₁₁×C₃₇	C₂×ΓL(1,𝔽₈)	semidirect
25. ΓL(1,𝔽₈)×C₃₇	C₂×D₁₁	semidirect
28. C₇×D₃₇	C₃×D₁₁	semidirect
31. ΓL(1,𝔽₈)×D₁₁	D₃₇	direct (N × H)
33. C₁₁×C₃₇	C₂²×ΓL(1,𝔽₈)	semidirect
35. C₇×C₃₇	C₂×C₃×D₁₁	semidirect
36. C₁₁×ΓL(1,𝔽₈)	C₂×D₃₇	semidirect
42. C₇×D₁₁	C₃×D₃₇	semidirect
47. C₇×C₁₁	C₂×C₃×D₃₇	semidirect
50. D₃₇	ΓL(1,𝔽₈)×D₁₁	direct (N × H)
58. C₃₇	C₂×ΓL(1,𝔽₈)×D₁₁	semidirect
63. D₁₁	ΓL(1,𝔽₈)×D₃₇	direct (N × H)
64. ΓL(1,𝔽₈)	D₁₁×D₃₇	direct (N × H)
69. C₁₁	C₂×ΓL(1,𝔽₈)×D₃₇	semidirect
70. C₇	C₃×D₁₁×D₃₇	semidirect

Maximal subgroups

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

subgroup	index
ΓL(1,𝔽₈)×D₄₀₇	2
ΓL(1,𝔽₈)×D₁₁×C₃₇	2
C₁₁×ΓL(1,𝔽₈)×D₃₇	2
C₇×D₁₁×D₃₇	3
C₃×D₁₁×D₃₇	7
C₂×ΓL(1,𝔽₈)×D₃₇	11
C₂×ΓL(1,𝔽₈)×D₁₁	37

Subgroups — ΓL(1,𝔽8)×D11×D37 #46803213

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×D₁₁×D₃₇ #46803213