Subgroups — C₂×ΓL(1,𝔽₈)⋊D₂₈₇ #37563113

Order 24108.

100 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	ΓL(1,𝔽₈)⋊D₂₈₇	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×D₂₈₇	—	7	1	no	maximal
8	C₂×ΓL(1,𝔽₈)⋊D₄₁	—	7	1	no	maximal
9	C₇²×C₄₁	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
10	C₂×C₃×C₇×C₄₁	—	14	1	no	—
11	C₃×D₂₈₇	—	7	2	no	—
12	C₂×ΓL(1,𝔽₈)×C₄₁	D₇	1	1	yes	characteristic, complemented
13	ΓL(1,𝔽₈)⋊D₄₁	—	7	2	no	—
14	C₂×D₂₈₇	—	7	1	no	—
15	C₂×D₂₈₇	—	7	1	no	—
16	C₂×D₂₈₇	—	21	2	no	—
17	C₃×C₇×C₄₁	—	14	1	no	—
18	ΓL(1,𝔽₈)×C₄₁	C₂×D₇	1	1	yes	characteristic, complemented
19	C₂×ΓL(1,𝔽₈)⋊D₇	—	41	1	no	maximal, Hall
20	C₂×C₇×C₄₁	—	3	2	no	—
21	C₂×C₇×C₄₁	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
22	C₂×C₇×C₄₁	C₃×D₇	1	1	yes	characteristic, complemented
23	D₂₈₇	—	7	2	no	—
24	D₂₈₇	—	7	2	no	—
25	D₂₈₇	—	21	4	no	—
26	C₂×C₃×D₄₁	—	49	1	no	Hall
27	C₂×C₇×ΓL(1,𝔽₈)	D₄₁	1	1	yes	characteristic, complemented
28	ΓL(1,𝔽₈)⋊D₇	—	41	2	no	—
29	C₇×C₄₁	—	3	2	no	—
30	C₇×C₄₁	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
31	C₇×C₄₁	C₂×C₃×D₇	1	1	yes	characteristic, complemented
32	C₂×C₃×C₄₁	—	7	1	no	—
33	C₃×D₄₁	—	49	2	no	—
34	C₂×C₇⋊D₇	—	41	1	no	Hall
35	C₂×D₄₁	—	49	1	no	Hall
36	C₇×ΓL(1,𝔽₈)	C₂×D₄₁	1	1	yes	characteristic, complemented, Hall
37	C₃×C₄₁	—	7	1	no	Hall
38	C₂×C₇²	C₃×D₄₁	1	1	yes	characteristic, complemented
39	C₇⋊D₇	—	41	2	no	—
40	C₂×C₃×D₇	—	287	1	no	—
41	C₂×AGL(1,𝔽₇)	—	287	1	no	—
42	C₂×C₄₁	ΓL(1,𝔽₈)⋊D₇	1	1	yes	characteristic, complemented
43	D₄₁	—	49	2	no	—
44	C₇²	C₂×C₃×D₄₁	1	1	yes	characteristic, complemented, Sylow
45	C₂×C₃×C₇	—	14	1	no	—
46	C₃×D₇	—	287	2	no	—
47	C₂×ΓL(1,𝔽₈)	D₂₈₇	1	1	yes	characteristic, complemented
48	AGL(1,𝔽₇)	—	287	2	no	—
49	C₄₁	C₂×ΓL(1,𝔽₈)⋊D₇	1	1	yes	characteristic, complemented, Sylow
50	C₂×D₇	—	287	1	no	—
51	C₂×D₇	—	287	1	no	—
52	C₂×D₇	—	861	2	no	—
53	C₃×C₇	—	14	1	no	—
54	ΓL(1,𝔽₈)	C₂×D₂₈₇	1	1	yes	characteristic, complemented
55	C₂×C₇	—	3	2	no	—
56	C₂×C₇	ΓL(1,𝔽₈)⋊D₄₁	1	1	yes	characteristic, complemented
57	C₂×C₇	C₃×D₂₈₇	1	1	yes	characteristic, complemented
58	D₇	—	287	2	no	—
59	D₇	—	287	2	no	—
60	D₇	—	861	4	no	—
61	C₂²×C₃	—	2009	1	no	Hall
62	C₇	—	3	2	no	—
63	C₇	C₂×ΓL(1,𝔽₈)⋊D₄₁	1	1	yes	characteristic, complemented
64	C₇	C₂×C₃×D₂₈₇	1	1	yes	characteristic, complemented
65	C₂×C₃	—	14	1	no	—
66	C₂×C₃	—	2009	2	no	—
67	C₂²	—	2009	1	no	Sylow
68	C₃	—	14	1	no	Sylow
69	C₂	—	2009	2	no	—
70	C₂	ΓL(1,𝔽₈)⋊D₂₈₇	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂	ΓL(1,𝔽₈)⋊D₂₈₇
commutator, cosocle, hypocenter	C₇²×C₄₁	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₇²×C₄₁	C₂×C₃
LCS	G ▷ C₇²×C₄₁
UCS	C₁ ◁ C₂

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. ΓL(1,𝔽₈)⋊D₂₈₇	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
9. C₇²×C₄₁	C₂²×C₃	semidirect
12. C₂×ΓL(1,𝔽₈)×C₄₁	D₇	semidirect
18. ΓL(1,𝔽₈)×C₄₁	C₂×D₇	semidirect
21. C₂×C₇×C₄₁	AGL(1,𝔽₇)	semidirect
22. C₂×C₇×C₄₁	C₃×D₇	semidirect
27. C₂×C₇×ΓL(1,𝔽₈)	D₄₁	semidirect
30. C₇×C₄₁	C₂×AGL(1,𝔽₇)	semidirect
31. C₇×C₄₁	C₂×C₃×D₇	semidirect
36. C₇×ΓL(1,𝔽₈)	C₂×D₄₁	semidirect
38. C₂×C₇²	C₃×D₄₁	semidirect
42. C₂×C₄₁	ΓL(1,𝔽₈)⋊D₇	semidirect
44. C₇²	C₂×C₃×D₄₁	semidirect
47. C₂×ΓL(1,𝔽₈)	D₂₈₇	semidirect
49. C₄₁	C₂×ΓL(1,𝔽₈)⋊D₇	semidirect
54. ΓL(1,𝔽₈)	C₂×D₂₈₇	semidirect
56. C₂×C₇	ΓL(1,𝔽₈)⋊D₄₁	semidirect
57. C₂×C₇	C₃×D₂₈₇	semidirect
63. C₇	C₂×ΓL(1,𝔽₈)⋊D₄₁	semidirect
64. C₇	C₂×C₃×D₂₈₇	semidirect
70. C₂	ΓL(1,𝔽₈)⋊D₂₈₇	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
ΓL(1,𝔽₈)⋊D₂₈₇	2
C₂×C₇⋊D₂₈₇	3
C₂×C₃×D₂₈₇	7
C₂×ΓL(1,𝔽₈)⋊D₄₁	7
C₂×ΓL(1,𝔽₈)⋊D₇	41

Subgroups — C2×ΓL(1,𝔽8)⋊D287 #37563113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×ΓL(1,𝔽₈)⋊D₂₈₇ #37563113