Subgroups — C₂×C₇×ΓL(1,𝔽₈)⋊D₄₁ #37563111

Order 24108.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×ΓL(1,𝔽₈)×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×ΓL(1,𝔽₈)⋊D₄₁	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₇×D₂₈₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₇×ΓL(1,𝔽₈)×C₄₁	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₇×D₂₈₇	C₂×C₃	1	2	yes	complemented
6	C₂×C₇²×C₄₁	C₂×C₃	1	1	yes	characteristic, complemented
7	C₂×C₃×C₇×D₄₁	—	7	1	no	maximal
8	C₂×ΓL(1,𝔽₈)⋊D₄₁	C₇	1	1	yes	maximal, characteristic, direct product factor
9	C₇²×C₄₁	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
10	C₂×C₃×C₇×C₄₁	—	14	1	no	—
11	C₃×C₇×D₄₁	—	7	2	no	—
12	C₂×ΓL(1,𝔽₈)×C₄₁	C₂×C₇	1	1	yes	characteristic, complemented
13	ΓL(1,𝔽₈)⋊D₄₁	C₂×C₇	1	2	yes	direct product factor
14	C₂×D₂₈₇	C₃×C₇	1	1	yes	characteristic, complemented
15	C₂×C₇×D₄₁	—	7	1	no	—
16	C₃×C₇×C₄₁	—	14	1	no	—
17	ΓL(1,𝔽₈)×C₄₁	C₂²×C₇	1	1	yes	characteristic, complemented
18	C₂×C₇×AGL(1,𝔽₇)	—	41	1	no	maximal, Hall
19	C₂×C₇×C₄₁	—	6	1	no	—
20	C₂×C₇×C₄₁	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
21	C₂×C₇×C₄₁	C₂×C₃×C₇	1	1	yes	characteristic, complemented
22	D₂₈₇	C₂×C₃×C₇	1	2	yes	complemented
23	C₇×D₄₁	—	7	2	no	—
24	C₂×C₃×D₄₁	—	7	1	no	Hall
25	C₂×C₇×ΓL(1,𝔽₈)	D₄₁	1	1	yes	characteristic, complemented
26	C₇×AGL(1,𝔽₇)	—	41	2	no	—
27	C₇×C₄₁	—	6	1	no	—
28	C₇×C₄₁	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
29	C₇×C₄₁	C₂²×C₃×C₇	1	1	yes	characteristic, complemented
30	C₂×C₃×C₄₁	—	7	1	no	—
31	C₃×D₄₁	—	7	2	no	—
32	C₂×C₇×D₇	—	41	1	no	Hall
33	C₂×D₄₁	—	7	1	no	Hall
34	C₇×ΓL(1,𝔽₈)	C₂×D₄₁	1	1	yes	characteristic, complemented, Hall
35	C₃×C₄₁	—	7	1	no	Hall
36	C₂×C₇²	C₃×D₄₁	1	1	yes	characteristic, complemented
37	C₇×D₇	—	41	2	no	—
38	C₂²×C₃×C₇	—	287	1	no	—
39	C₂×AGL(1,𝔽₇)	—	41	1	no	—
40	C₂×C₄₁	C₇×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
41	D₄₁	—	7	2	no	—
42	C₇²	C₂×C₃×D₄₁	1	1	yes	characteristic, complemented, Sylow
43	C₂×C₃×C₇	—	7	1	no	—
44	C₂×C₃×C₇	—	287	2	no	—
45	C₂×ΓL(1,𝔽₈)	C₇×D₄₁	1	1	yes	characteristic, complemented
46	AGL(1,𝔽₇)	—	41	2	no	—
47	C₄₁	C₂×C₇×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Sylow
48	C₂²×C₇	—	287	1	no	—
49	C₂×D₇	—	41	1	no	—
50	C₃×C₇	—	7	1	no	—
51	ΓL(1,𝔽₈)	C₂×C₇×D₄₁	1	1	yes	characteristic, complemented
52	C₂×C₇	—	6	1	no	—
53	C₂×C₇	—	287	2	no	—
54	C₂×C₇	ΓL(1,𝔽₈)⋊D₄₁	1	1	yes	characteristic, direct product factor
55	C₂×C₇	C₃×C₇×D₄₁	1	1	yes	characteristic, complemented
56	D₇	—	41	2	no	—
57	C₂²×C₃	—	287	1	no	Hall
58	C₇	—	6	1	no	—
59	C₇	C₂×ΓL(1,𝔽₈)⋊D₄₁	1	1	yes	characteristic, direct product factor
60	C₇	C₂×C₃×C₇×D₄₁	1	1	yes	characteristic, complemented
61	C₂×C₃	—	7	1	no	—
62	C₂×C₃	—	287	2	no	—
63	C₂²	—	287	1	no	Sylow
64	C₃	—	7	1	no	Sylow
65	C₂	—	287	2	no	—
66	C₂	C₇×ΓL(1,𝔽₈)⋊D₄₁	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₇	ΓL(1,𝔽₈)⋊D₄₁
commutator, cosocle, hypocenter	C₇×C₄₁	C₂²×C₃×C₇
Frattini	C₁	G
Fitting, socle	C₂×C₇²×C₄₁	C₂×C₃
LCS	G ▷ C₇×C₄₁
UCS	C₁ ◁ C₂×C₇

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×ΓL(1,𝔽₈)×C₄₁	C₂	semidirect
2. C₇×ΓL(1,𝔽₈)⋊D₄₁	C₂	direct (N × H)
3. C₂×C₇×D₂₈₇	C₃	semidirect
4. C₇×ΓL(1,𝔽₈)×C₄₁	C₂²	semidirect
5. C₇×D₂₈₇	C₂×C₃	semidirect
6. C₂×C₇²×C₄₁	C₂×C₃	semidirect
8. C₂×ΓL(1,𝔽₈)⋊D₄₁	C₇	direct (N × H)
9. C₇²×C₄₁	C₂²×C₃	semidirect
12. C₂×ΓL(1,𝔽₈)×C₄₁	C₂×C₇	semidirect
13. ΓL(1,𝔽₈)⋊D₄₁	C₂×C₇	direct (N × H)
14. C₂×D₂₈₇	C₃×C₇	semidirect
17. ΓL(1,𝔽₈)×C₄₁	C₂²×C₇	semidirect
20. C₂×C₇×C₄₁	AGL(1,𝔽₇)	semidirect
21. C₂×C₇×C₄₁	C₂×C₃×C₇	semidirect
22. D₂₈₇	C₂×C₃×C₇	semidirect
25. C₂×C₇×ΓL(1,𝔽₈)	D₄₁	semidirect
28. C₇×C₄₁	C₂×AGL(1,𝔽₇)	semidirect
29. C₇×C₄₁	C₂²×C₃×C₇	semidirect
34. C₇×ΓL(1,𝔽₈)	C₂×D₄₁	semidirect
36. C₂×C₇²	C₃×D₄₁	semidirect
40. C₂×C₄₁	C₇×AGL(1,𝔽₇)	semidirect
42. C₇²	C₂×C₃×D₄₁	semidirect
45. C₂×ΓL(1,𝔽₈)	C₇×D₄₁	semidirect
47. C₄₁	C₂×C₇×AGL(1,𝔽₇)	semidirect
51. ΓL(1,𝔽₈)	C₂×C₇×D₄₁	semidirect
54. C₂×C₇	ΓL(1,𝔽₈)⋊D₄₁	direct (N × H)
55. C₂×C₇	C₃×C₇×D₄₁	semidirect
59. C₇	C₂×ΓL(1,𝔽₈)⋊D₄₁	direct (N × H)
60. C₇	C₂×C₃×C₇×D₄₁	semidirect
66. C₂	C₇×ΓL(1,𝔽₈)⋊D₄₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇×ΓL(1,𝔽₈)×C₄₁	2
C₇×ΓL(1,𝔽₈)⋊D₄₁	2
C₂×C₇×D₂₈₇	3
C₂×C₃×C₇×D₄₁	7
C₂×ΓL(1,𝔽₈)⋊D₄₁	7
C₂×C₇×AGL(1,𝔽₇)	41

Subgroups — C2×C7×ΓL(1,𝔽8)⋊D41 #37563111

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×ΓL(1,𝔽₈)⋊D₄₁ #37563111