Subgroups — D₇×ΓL(1,𝔽₈)×D₆₁ #49089855

Order 35868.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)×D₄₂₇	C₂	1	1	yes	maximal, characteristic, complemented
2	D₇×ΓL(1,𝔽₈)×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×ΓL(1,𝔽₈)×D₆₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×D₇×D₆₁	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	C₇×ΓL(1,𝔽₈)×C₆₁	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₇×D₄₂₇	C₂×C₃	1	1	yes	characteristic, complemented
7	C₇×D₇×C₆₁	C₂×C₃	1	1	yes	characteristic, complemented
8	C₇²×D₆₁	C₂×C₃	1	1	yes	characteristic, complemented
9	C₃×D₇×D₆₁	—	7	1	no	maximal
10	C₂×ΓL(1,𝔽₈)×D₆₁	—	7	1	no	maximal
11	C₇²×C₆₁	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
12	ΓL(1,𝔽₈)×D₆₁	—	7	1	no	—
13	ΓL(1,𝔽₈)×D₆₁	D₇	1	1	yes	characteristic, direct product factor
14	C₃×D₄₂₇	—	7	1	no	—
15	C₃×D₇×C₆₁	—	7	1	no	—
16	C₃×C₇×D₆₁	—	7	1	no	—
17	C₂×ΓL(1,𝔽₈)×C₆₁	—	7	1	no	—
18	C₂×C₇×D₆₁	—	7	1	no	—
19	D₇×D₆₁	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
20	C₃×C₇×C₆₁	—	7	1	no	—
21	ΓL(1,𝔽₈)×C₆₁	C₂×D₇	1	1	yes	characteristic, complemented
22	C₂×C₇×C₆₁	—	7	1	no	—
23	D₄₂₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
24	C₇×D₆₁	—	6	1	no	—
25	C₇×D₆₁	—	7	1	no	—
26	C₇×D₆₁	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
27	C₇×D₆₁	C₃×D₇	1	1	yes	characteristic, complemented
28	D₇×C₆₁	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
29	C₂×C₃×D₆₁	—	49	1	no	Hall
30	C₂×D₇×ΓL(1,𝔽₈)	—	61	1	no	maximal, Hall
31	C₇×C₆₁	—	6	1	no	—
32	C₇×C₆₁	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
33	C₇×C₆₁	C₂×C₃×D₇	1	1	yes	characteristic, complemented
34	C₂×C₃×C₆₁	—	49	1	no	—
35	C₃×D₆₁	—	7	1	no	—
36	C₃×D₆₁	—	49	1	no	—
37	C₂×C₇×ΓL(1,𝔽₈)	—	61	1	no	—
38	D₇×ΓL(1,𝔽₈)	—	61	1	no	—
39	D₇×ΓL(1,𝔽₈)	D₆₁	1	1	yes	characteristic, direct product factor
40	C₂×D₆₁	—	7	1	no	Hall
41	C₂×C₇×D₇	—	61	1	no	Hall
42	C₃×C₆₁	—	7	1	no	Hall
43	C₇×ΓL(1,𝔽₈)	C₂×D₆₁	1	1	yes	characteristic, complemented, Hall
44	C₂×C₆₁	—	7	1	no	—
45	D₆₁	—	7	1	no	—
46	D₆₁	D₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
47	C₂×C₇²	—	61	1	no	—
48	C₇×D₇	—	61	1	no	—
49	C₇×D₇	C₃×D₆₁	1	1	yes	characteristic, complemented
50	C₂×C₃×D₇	—	427	1	no	—
51	C₂²×ΓL(1,𝔽₈)	—	427	1	no	—
52	C₆₁	C₂×D₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
53	C₇²	C₂×C₃×D₆₁	1	1	yes	characteristic, complemented, Sylow
54	C₂×C₃×C₇	—	427	1	no	—
55	C₃×D₇	—	7	1	no	—
56	C₃×D₇	—	427	1	no	—
57	C₂×ΓL(1,𝔽₈)	—	7	1	no	—
58	C₂×ΓL(1,𝔽₈)	—	61	1	no	—
59	C₂×ΓL(1,𝔽₈)	—	427	1	no	—
60	C₂²×C₇	—	427	1	no	—
61	C₂×D₇	—	61	1	no	—
62	C₃×C₇	—	7	1	no	—
63	ΓL(1,𝔽₈)	D₇×D₆₁	1	1	yes	characteristic, direct product factor
64	C₂×C₇	—	7	1	no	—
65	C₂×C₇	—	61	1	no	—
66	C₂×C₇	—	61	1	no	—
67	C₂×C₇	—	366	1	no	—
68	C₂×C₇	—	427	1	no	—
69	D₇	—	61	1	no	—
70	D₇	ΓL(1,𝔽₈)×D₆₁	1	1	yes	characteristic, direct product factor
71	C₂²×C₃	—	2989	1	no	Hall
72	C₇	—	6	1	no	—
73	C₇	C₂×ΓL(1,𝔽₈)×D₆₁	1	1	yes	characteristic, complemented
74	C₇	C₃×D₇×D₆₁	1	1	yes	characteristic, complemented
75	C₂×C₃	—	49	1	no	—
76	C₂×C₃	—	427	1	no	—
77	C₂×C₃	—	2989	1	no	—
78	C₂²	—	427	1	no	Sylow
79	C₃	—	7	1	no	Sylow
80	C₂	—	7	1	no	—
81	C₂	—	61	1	no	—
82	C₂	—	427	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇²×C₆₁	C₂²×C₃
LCS	G ▷ C₇²×C₆₁
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. ΓL(1,𝔽₈)×D₄₂₇	C₂	semidirect
2. D₇×ΓL(1,𝔽₈)×C₆₁	C₂	semidirect
3. C₇×ΓL(1,𝔽₈)×D₆₁	C₂	semidirect
4. C₇×D₇×D₆₁	C₃	semidirect
5. C₇×ΓL(1,𝔽₈)×C₆₁	C₂²	semidirect
6. C₇×D₄₂₇	C₂×C₃	semidirect
7. C₇×D₇×C₆₁	C₂×C₃	semidirect
8. C₇²×D₆₁	C₂×C₃	semidirect
11. C₇²×C₆₁	C₂²×C₃	semidirect
13. ΓL(1,𝔽₈)×D₆₁	D₇	direct (N × H)
19. D₇×D₆₁	ΓL(1,𝔽₈)	direct (N × H)
21. ΓL(1,𝔽₈)×C₆₁	C₂×D₇	semidirect
23. D₄₂₇	C₂×ΓL(1,𝔽₈)	semidirect
26. C₇×D₆₁	C₂×ΓL(1,𝔽₈)	semidirect
27. C₇×D₆₁	C₃×D₇	semidirect
28. D₇×C₆₁	C₂×ΓL(1,𝔽₈)	semidirect
32. C₇×C₆₁	C₂²×ΓL(1,𝔽₈)	semidirect
33. C₇×C₆₁	C₂×C₃×D₇	semidirect
39. D₇×ΓL(1,𝔽₈)	D₆₁	direct (N × H)
43. C₇×ΓL(1,𝔽₈)	C₂×D₆₁	semidirect
46. D₆₁	D₇×ΓL(1,𝔽₈)	direct (N × H)
49. C₇×D₇	C₃×D₆₁	semidirect
52. C₆₁	C₂×D₇×ΓL(1,𝔽₈)	semidirect
53. C₇²	C₂×C₃×D₆₁	semidirect
63. ΓL(1,𝔽₈)	D₇×D₆₁	direct (N × H)
70. D₇	ΓL(1,𝔽₈)×D₆₁	direct (N × H)
73. C₇	C₂×ΓL(1,𝔽₈)×D₆₁	semidirect
74. C₇	C₃×D₇×D₆₁	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)×D₄₂₇	2
D₇×ΓL(1,𝔽₈)×C₆₁	2
C₇×ΓL(1,𝔽₈)×D₆₁	2
C₇×D₇×D₆₁	3
C₃×D₇×D₆₁	7
C₂×ΓL(1,𝔽₈)×D₆₁	7
C₂×D₇×ΓL(1,𝔽₈)	61

Subgroups — D7×ΓL(1,𝔽8)×D61 #49089855

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₇×ΓL(1,𝔽₈)×D₆₁ #49089855