Subgroups — (ΓL(1,𝔽₈)⋊D₇)⋊D₆₁ #49089849

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	C₆₁×ΓL(1,𝔽₈)⋊D₇	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	C₆₁×C₇⋊D₇	C₂×C₃	1	1	yes	characteristic, complemented
7	C₇×D₄₂₇	C₂×C₃	1	1	yes	characteristic, complemented
8	C₇×D₄₂₇	C₂×C₃	1	1	yes	characteristic, complemented
9	C₃×D₇×D₆₁	—	7	1	no	maximal
10	AGL(1,𝔽₇)×D₆₁	—	7	1	no	maximal
11	C₇²×C₆₁	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
12	ΓL(1,𝔽₈)⋊D₆₁	D₇	1	1	yes	characteristic, complemented
13	AGL(1,𝔽₇)×C₆₁	—	7	1	no	—
14	ΓL(1,𝔽₈)×D₆₁	—	7	1	no	—
15	C₃×D₄₂₇	—	7	1	no	—
16	C₃×D₇×C₆₁	—	7	1	no	—
17	C₃×C₇×D₆₁	—	7	1	no	—
18	D₇×D₆₁	—	7	1	no	—
19	D₇×D₆₁	—	7	1	no	—
20	C₃×C₇×C₆₁	—	7	1	no	—
21	ΓL(1,𝔽₈)×C₆₁	C₂×D₇	1	1	yes	characteristic, complemented
22	D₄₂₇	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
23	D₄₂₇	C₃×D₇	1	1	yes	characteristic, complemented
24	C₇×D₆₁	—	7	1	no	—
25	C₇×D₆₁	—	7	1	no	—
26	D₇×C₆₁	—	7	1	no	—
27	D₇×C₆₁	—	7	1	no	—
28	D₇×C₆₁	—	42	1	no	—
29	C₂×C₃×D₆₁	—	49	1	no	Hall
30	D₇×AGL(1,𝔽₇)	—	61	1	no	maximal, Hall
31	C₇×C₆₁	—	6	1	no	—
32	C₇×C₆₁	C₂×AGL(1,𝔽₇)	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	ΓL(1,𝔽₈)⋊D₇	D₆₁	1	1	yes	characteristic, complemented
38	D₇×ΓL(1,𝔽₈)	—	61	1	no	—
39	C₇×AGL(1,𝔽₇)	—	61	1	no	—
40	C₂×D₆₁	—	49	1	no	Hall
41	D₇²	—	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	—	49	1	no	—
45	D₆₁	—	7	1	no	—
46	D₆₁	—	7	1	no	—
47	C₇⋊D₇	C₃×D₆₁	1	1	yes	characteristic, complemented
48	C₇×D₇	—	61	1	no	—
49	C₇×D₇	—	61	1	no	—
50	C₂×C₃×D₇	—	427	1	no	—
51	C₂×AGL(1,𝔽₇)	—	427	1	no	—
52	C₆₁	D₇×AGL(1,𝔽₇)	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)	—	427	1	no	—
58	AGL(1,𝔽₇)	—	7	1	no	—
59	AGL(1,𝔽₇)	—	61	1	no	—
60	C₂×D₇	—	427	1	no	—
61	C₂×D₇	—	427	1	no	—
62	C₃×C₇	—	7	1	no	—
63	ΓL(1,𝔽₈)	D₇×D₆₁	1	1	yes	characteristic, complemented
64	C₂×C₇	—	427	1	no	—
65	C₂×C₇	—	427	1	no	—
66	D₇	—	7	1	no	—
67	D₇	—	7	1	no	—
68	D₇	—	42	1	no	—
69	D₇	—	61	1	no	—
70	D₇	—	61	1	no	—
71	C₂²×C₃	—	2989	1	no	Hall
72	C₇	—	6	1	no	—
73	C₇	AGL(1,𝔽₇)×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	—	2989	1	no	Sylow
79	C₃	—	7	1	no	Sylow
80	C₂	—	49	1	no	—
81	C₂	—	427	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. C₆₁×ΓL(1,𝔽₈)⋊D₇	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. C₆₁×C₇⋊D₇	C₂×C₃	semidirect
7. C₇×D₄₂₇	C₂×C₃	semidirect
8. C₇×D₄₂₇	C₂×C₃	semidirect
11. C₇²×C₆₁	C₂²×C₃	semidirect
12. ΓL(1,𝔽₈)⋊D₆₁	D₇	semidirect
21. ΓL(1,𝔽₈)×C₆₁	C₂×D₇	semidirect
22. D₄₂₇	AGL(1,𝔽₇)	semidirect
23. D₄₂₇	C₃×D₇	semidirect
32. C₇×C₆₁	C₂×AGL(1,𝔽₇)	semidirect
33. C₇×C₆₁	C₂×C₃×D₇	semidirect
37. ΓL(1,𝔽₈)⋊D₇	D₆₁	semidirect
43. C₇×ΓL(1,𝔽₈)	C₂×D₆₁	semidirect
47. C₇⋊D₇	C₃×D₆₁	semidirect
52. C₆₁	D₇×AGL(1,𝔽₇)	semidirect
53. C₇²	C₂×C₃×D₆₁	semidirect
63. ΓL(1,𝔽₈)	D₇×D₆₁	semidirect
73. C₇	AGL(1,𝔽₇)×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
C₆₁×ΓL(1,𝔽₈)⋊D₇	2
C₇×ΓL(1,𝔽₈)⋊D₆₁	2
(C₇⋊D₇)⋊D₆₁	3
C₃×D₇×D₆₁	7
AGL(1,𝔽₇)×D₆₁	7
D₇×AGL(1,𝔽₇)	61

Subgroups — (ΓL(1,𝔽8)⋊D7)⋊D61 #49089849

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — (ΓL(1,𝔽₈)⋊D₇)⋊D₆₁ #49089849