Subgroups — ΓL(1,𝔽₈)×D₁₅×C₄₇ #44444102

Order 29610.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×ΓL(1,𝔽₈)×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	D₅×ΓL(1,𝔽₈)×C₄₇	—	3	1	no	maximal
3	C₇×D₁₅×C₄₇	C₃	1	1	yes	maximal, characteristic, complemented
4	D₃×ΓL(1,𝔽₈)×C₄₇	—	5	1	no	maximal, Hall
5	C₅×ΓL(1,𝔽₈)×C₄₇	—	2	1	no	—
6	C₅×ΓL(1,𝔽₈)×C₄₇	D₃	1	1	yes	characteristic, complemented
7	C₃×C₅×C₇×C₄₇	C₂×C₃	1	1	yes	characteristic, complemented
8	C₃×D₁₅×C₄₇	—	7	1	no	maximal, Hall
9	C₇×D₅×C₄₇	—	3	1	no	Hall
10	C₃×ΓL(1,𝔽₈)×C₄₇	D₅	1	1	yes	characteristic, complemented, Hall
11	C₃²×C₅×C₄₇	—	7	1	no	Hall
12	D₃×C₇×C₄₇	—	5	1	no	—
13	C₂×ΓL(1,𝔽₈)×C₄₇	—	15	1	no	—
14	C₅×C₇×C₄₇	C₃×D₃	1	1	yes	characteristic, complemented, Hall
15	D₁₅×C₄₇	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
16	C₃×D₅×C₄₇	—	21	1	no	—
17	C₃×C₇×C₄₇	C₃×D₅	1	1	yes	characteristic, complemented
18	ΓL(1,𝔽₈)×C₄₇	—	2	1	no	—
19	ΓL(1,𝔽₈)×C₄₇	D₁₅	1	1	yes	characteristic, direct product factor
20	C₃×D₃×C₄₇	—	35	1	no	Hall
21	C₃×C₅×C₄₇	—	7	1	no	—
22	C₃×C₅×C₄₇	—	14	1	no	—
23	C₃×C₅×C₄₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
24	C₂×C₇×C₄₇	—	15	1	no	Hall
25	ΓL(1,𝔽₈)×D₁₅	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Hall
26	D₅×C₄₇	—	3	1	no	Hall
27	C₃²×C₄₇	—	7	1	no	Hall
28	C₇×C₄₇	C₃×D₁₅	1	1	yes	characteristic, complemented, Hall
29	C₃×C₅×ΓL(1,𝔽₈)	C₂×C₄₇	1	1	yes	characteristic, complemented, Hall
30	C₂×C₃×C₄₇	—	105	1	no	—
31	D₃×C₄₇	—	5	1	no	—
32	C₅×C₄₇	D₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
33	C₇×D₁₅	C₃×C₄₇	1	1	yes	characteristic, complemented
34	D₅×ΓL(1,𝔽₈)	—	3	1	no	—
35	C₃×C₄₇	—	7	1	no	—
36	C₃×C₄₇	—	14	1	no	—
37	C₃×C₄₇	D₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
38	D₃×ΓL(1,𝔽₈)	—	5	1	no	Hall
39	C₃×C₅×C₇	C₂×C₃×C₄₇	1	1	yes	characteristic, complemented
40	C₅×ΓL(1,𝔽₈)	—	2	1	no	—
41	C₅×ΓL(1,𝔽₈)	D₃×C₄₇	1	1	yes	characteristic, complemented
42	C₂×C₄₇	—	15	1	no	Hall
43	C₃×D₁₅	—	7	1	no	Hall
44	C₇×D₅	—	3	1	no	Hall
45	C₃×ΓL(1,𝔽₈)	D₅×C₄₇	1	1	yes	characteristic, complemented, Hall
46	C₄₇	ΓL(1,𝔽₈)×D₁₅	1	1	yes	characteristic, direct product factor, Sylow
47	C₃²×C₅	—	7	1	no	Hall
48	D₃×C₇	—	5	1	no	—
49	C₂×ΓL(1,𝔽₈)	—	15	1	no	—
50	C₅×C₇	C₃×D₃×C₄₇	1	1	yes	characteristic, complemented, Hall
51	D₁₅	ΓL(1,𝔽₈)×C₄₇	1	1	yes	characteristic, direct product factor
52	C₃×D₅	—	21	1	no	—
53	C₃×C₇	C₃×D₅×C₄₇	1	1	yes	characteristic, complemented
54	ΓL(1,𝔽₈)	—	2	1	no	—
55	ΓL(1,𝔽₈)	D₁₅×C₄₇	1	1	yes	characteristic, direct product factor
56	C₃×D₃	—	35	1	no	Hall
57	C₃×C₅	—	7	1	no	—
58	C₃×C₅	—	14	1	no	—
59	C₃×C₅	C₂×ΓL(1,𝔽₈)×C₄₇	1	1	yes	characteristic, complemented
60	C₂×C₇	—	15	1	no	Hall
61	D₅	—	3	1	no	Hall
62	C₃²	—	7	1	no	Sylow
63	C₇	C₃×D₁₅×C₄₇	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₃	—	105	1	no	—
65	D₃	—	5	1	no	—
66	C₅	D₃×ΓL(1,𝔽₈)×C₄₇	1	1	yes	characteristic, complemented, Sylow
67	C₃	—	7	1	no	—
68	C₃	—	14	1	no	—
69	C₃	D₅×ΓL(1,𝔽₈)×C₄₇	1	1	yes	characteristic, complemented
70	C₂	—	15	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₄₇	ΓL(1,𝔽₈)×D₁₅
commutator, cosocle, hypocenter	C₃×C₅×C₇	C₂×C₃×C₄₇
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₇×C₄₇	C₂×C₃
LCS	G ▷ C₃×C₅×C₇
UCS	C₁ ◁ C₄₇

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×ΓL(1,𝔽₈)×C₄₇	C₂	semidirect
3. C₇×D₁₅×C₄₇	C₃	semidirect
6. C₅×ΓL(1,𝔽₈)×C₄₇	D₃	semidirect
7. C₃×C₅×C₇×C₄₇	C₂×C₃	semidirect
10. C₃×ΓL(1,𝔽₈)×C₄₇	D₅	semidirect
14. C₅×C₇×C₄₇	C₃×D₃	semidirect
15. D₁₅×C₄₇	ΓL(1,𝔽₈)	direct (N × H)
17. C₃×C₇×C₄₇	C₃×D₅	semidirect
19. ΓL(1,𝔽₈)×C₄₇	D₁₅	direct (N × H)
23. C₃×C₅×C₄₇	C₂×ΓL(1,𝔽₈)	semidirect
25. ΓL(1,𝔽₈)×D₁₅	C₄₇	direct (N × H)
28. C₇×C₄₇	C₃×D₁₅	semidirect
29. C₃×C₅×ΓL(1,𝔽₈)	C₂×C₄₇	semidirect
32. C₅×C₄₇	D₃×ΓL(1,𝔽₈)	semidirect
33. C₇×D₁₅	C₃×C₄₇	semidirect
37. C₃×C₄₇	D₅×ΓL(1,𝔽₈)	semidirect
39. C₃×C₅×C₇	C₂×C₃×C₄₇	semidirect
41. C₅×ΓL(1,𝔽₈)	D₃×C₄₇	semidirect
45. C₃×ΓL(1,𝔽₈)	D₅×C₄₇	semidirect
46. C₄₇	ΓL(1,𝔽₈)×D₁₅	direct (N × H)
50. C₅×C₇	C₃×D₃×C₄₇	semidirect
51. D₁₅	ΓL(1,𝔽₈)×C₄₇	direct (N × H)
53. C₃×C₇	C₃×D₅×C₄₇	semidirect
55. ΓL(1,𝔽₈)	D₁₅×C₄₇	direct (N × H)
59. C₃×C₅	C₂×ΓL(1,𝔽₈)×C₄₇	semidirect
63. C₇	C₃×D₁₅×C₄₇	semidirect
66. C₅	D₃×ΓL(1,𝔽₈)×C₄₇	semidirect
69. C₃	D₅×ΓL(1,𝔽₈)×C₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×ΓL(1,𝔽₈)×C₄₇	2
D₅×ΓL(1,𝔽₈)×C₄₇	3
C₇×D₁₅×C₄₇	3
D₃×ΓL(1,𝔽₈)×C₄₇	5
C₃×D₁₅×C₄₇	7
ΓL(1,𝔽₈)×D₁₅	47

Subgroups — ΓL(1,𝔽8)×D15×C47 #44444102

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×D₁₅×C₄₇ #44444102