Subgroups — ΓL(1,𝔽₈)×D₁₀₅ #17984394

Order 4410.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×C₇×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₇×D₁₀₅	C₃	1	1	yes	maximal, characteristic, complemented
3	ΓL(1,𝔽₈)×D₃₅	—	3	1	no	maximal
4	ΓL(1,𝔽₈)×D₂₁	—	5	1	no	maximal, Hall
5	C₃×C₅×C₇²	C₂×C₃	1	1	yes	characteristic, complemented
6	C₅×C₇×ΓL(1,𝔽₈)	—	2	1	no	—
7	C₅×C₇×ΓL(1,𝔽₈)	D₃	1	1	yes	characteristic, complemented
8	C₃×D₁₀₅	—	7	1	no	maximal
9	ΓL(1,𝔽₈)×D₁₅	—	7	1	no	maximal
10	C₇×D₃₅	—	3	1	no	Hall
11	C₃×C₇×ΓL(1,𝔽₈)	D₅	1	1	yes	characteristic, complemented, Hall
12	C₃²×C₅×C₇	—	7	1	no	—
13	C₃×C₅×ΓL(1,𝔽₈)	D₇	1	1	yes	characteristic, complemented
14	C₇×D₂₁	—	5	1	no	—
15	D₇×ΓL(1,𝔽₈)	—	15	1	no	—
16	C₅×C₇²	C₃×D₃	1	1	yes	characteristic, complemented, Hall
17	D₁₀₅	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
18	C₇×D₁₅	—	7	1	no	—
19	C₃×D₃₅	—	21	1	no	—
20	D₅×ΓL(1,𝔽₈)	—	21	1	no	—
21	C₃×C₇²	C₃×D₅	1	1	yes	characteristic, complemented
22	C₇×ΓL(1,𝔽₈)	—	2	1	no	—
23	C₇×ΓL(1,𝔽₈)	D₁₅	1	1	yes	characteristic, complemented
24	C₃×D₂₁	—	35	1	no	—
25	D₃×ΓL(1,𝔽₈)	—	35	1	no	—
26	C₃×C₅×C₇	—	6	1	no	—
27	C₃×C₅×C₇	—	7	1	no	—
28	C₃×C₅×C₇	—	14	1	no	—
29	C₃×C₅×C₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
30	C₃×C₅×C₇	C₃×D₇	1	1	yes	characteristic, complemented
31	C₅×ΓL(1,𝔽₈)	—	2	1	no	—
32	C₅×ΓL(1,𝔽₈)	D₂₁	1	1	yes	characteristic, complemented
33	C₇×D₇	—	15	1	no	Hall
34	C₃×D₁₅	—	49	1	no	Hall
35	D₃₅	—	3	1	no	—
36	C₇×D₅	—	21	1	no	—
37	C₃²×C₇	—	7	1	no	—
38	C₃×ΓL(1,𝔽₈)	D₃₅	1	1	yes	characteristic, complemented
39	C₇²	C₃×D₁₅	1	1	yes	characteristic, complemented, Sylow
40	C₃²×C₅	—	7	1	no	Hall
41	D₂₁	—	5	1	no	—
42	C₃×D₇	—	105	1	no	—
43	D₃×C₇	—	35	1	no	—
44	C₂×ΓL(1,𝔽₈)	—	105	1	no	—
45	C₅×C₇	—	6	1	no	—
46	C₅×C₇	D₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
47	C₅×C₇	C₃×D₂₁	1	1	yes	characteristic, complemented
48	D₁₅	—	7	1	no	—
49	C₃×D₅	—	147	1	no	—
50	C₃×C₇	—	6	1	no	—
51	C₃×C₇	—	7	1	no	—
52	C₃×C₇	—	14	1	no	—
53	C₃×C₇	D₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
54	C₃×C₇	C₃×D₃₅	1	1	yes	characteristic, complemented
55	ΓL(1,𝔽₈)	—	2	1	no	—
56	ΓL(1,𝔽₈)	D₁₀₅	1	1	yes	characteristic, direct product factor
57	C₃×D₃	—	245	1	no	Hall
58	C₃×C₅	—	7	1	no	—
59	C₃×C₅	—	14	1	no	—
60	C₃×C₅	D₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
61	C₂×C₇	—	105	1	no	—
62	D₇	—	15	1	no	—
63	D₅	—	21	1	no	Hall
64	C₃²	—	7	1	no	Sylow
65	C₇	—	6	1	no	—
66	C₇	ΓL(1,𝔽₈)×D₁₅	1	1	yes	characteristic, complemented
67	C₇	C₃×D₁₀₅	1	1	yes	characteristic, complemented
68	C₂×C₃	—	735	1	no	—
69	D₃	—	35	1	no	—
70	C₅	ΓL(1,𝔽₈)×D₂₁	1	1	yes	characteristic, complemented, Sylow
71	C₃	—	7	1	no	—
72	C₃	—	14	1	no	—
73	C₃	ΓL(1,𝔽₈)×D₃₅	1	1	yes	characteristic, complemented
74	C₂	—	105	1	no	Sylow

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₅×C₇²	C₂×C₃
LCS	G ▷ C₃×C₅×C₇²
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×C₇×ΓL(1,𝔽₈)	C₂	semidirect
2. C₇×D₁₀₅	C₃	semidirect
5. C₃×C₅×C₇²	C₂×C₃	semidirect
7. C₅×C₇×ΓL(1,𝔽₈)	D₃	semidirect
11. C₃×C₇×ΓL(1,𝔽₈)	D₅	semidirect
13. C₃×C₅×ΓL(1,𝔽₈)	D₇	semidirect
16. C₅×C₇²	C₃×D₃	semidirect
17. D₁₀₅	ΓL(1,𝔽₈)	direct (N × H)
21. C₃×C₇²	C₃×D₅	semidirect
23. C₇×ΓL(1,𝔽₈)	D₁₅	semidirect
29. C₃×C₅×C₇	C₂×ΓL(1,𝔽₈)	semidirect
30. C₃×C₅×C₇	C₃×D₇	semidirect
32. C₅×ΓL(1,𝔽₈)	D₂₁	semidirect
38. C₃×ΓL(1,𝔽₈)	D₃₅	semidirect
39. C₇²	C₃×D₁₅	semidirect
46. C₅×C₇	D₃×ΓL(1,𝔽₈)	semidirect
47. C₅×C₇	C₃×D₂₁	semidirect
53. C₃×C₇	D₅×ΓL(1,𝔽₈)	semidirect
54. C₃×C₇	C₃×D₃₅	semidirect
56. ΓL(1,𝔽₈)	D₁₀₅	direct (N × H)
60. C₃×C₅	D₇×ΓL(1,𝔽₈)	semidirect
66. C₇	ΓL(1,𝔽₈)×D₁₅	semidirect
67. C₇	C₃×D₁₀₅	semidirect
70. C₅	ΓL(1,𝔽₈)×D₂₁	semidirect
73. C₃	ΓL(1,𝔽₈)×D₃₅	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×C₇×ΓL(1,𝔽₈)	2
C₇×D₁₀₅	3
ΓL(1,𝔽₈)×D₃₅	3
ΓL(1,𝔽₈)×D₂₁	5
C₃×D₁₀₅	7
ΓL(1,𝔽₈)×D₁₅	7

Subgroups — ΓL(1,𝔽8)×D105 #17984394

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×D₁₀₅ #17984394