Subgroups — C₅×D₃×ΓL(1,𝔽₈)×C₃₇ #37552099

Order 23310.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×ΓL(1,𝔽₈)×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₅×D₃×C₇×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×ΓL(1,𝔽₈)×C₃₇	—	3	1	no	maximal
4	D₃×ΓL(1,𝔽₈)×C₃₇	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₅×ΓL(1,𝔽₈)×C₃₇	—	2	1	no	—
6	C₅×ΓL(1,𝔽₈)×C₃₇	D₃	1	1	yes	characteristic, direct product factor
7	C₃×C₅×C₇×C₃₇	C₂×C₃	1	1	yes	characteristic, complemented
8	C₃×C₅×D₃×C₃₇	—	7	1	no	maximal, Hall
9	C₂×C₅×C₇×C₃₇	—	3	1	no	Hall
10	C₃×ΓL(1,𝔽₈)×C₃₇	C₂×C₅	1	1	yes	characteristic, complemented, Hall
11	C₃²×C₅×C₃₇	—	7	1	no	Hall
12	D₃×C₇×C₃₇	C₃×C₅	1	1	yes	characteristic, complemented
13	C₂×ΓL(1,𝔽₈)×C₃₇	—	3	1	no	—
14	C₅×C₇×C₃₇	C₃×D₃	1	1	yes	characteristic, complemented, Hall
15	C₂×C₃×C₅×C₃₇	—	21	1	no	—
16	C₅×D₃×C₃₇	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
17	C₃×C₇×C₃₇	C₂×C₃×C₅	1	1	yes	characteristic, complemented
18	ΓL(1,𝔽₈)×C₃₇	—	2	1	no	—
19	ΓL(1,𝔽₈)×C₃₇	C₅×D₃	1	1	yes	characteristic, direct product factor
20	C₃×D₃×C₃₇	—	7	1	no	Hall
21	C₅×D₃×ΓL(1,𝔽₈)	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₃×C₅×C₃₇	—	7	1	no	—
23	C₃×C₅×C₃₇	—	14	1	no	—
24	C₃×C₅×C₃₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
25	C₂×C₇×C₃₇	—	3	1	no	Hall
26	C₂×C₅×C₃₇	—	3	1	no	Hall
27	C₃²×C₃₇	—	7	1	no	Hall
28	C₃×C₅×ΓL(1,𝔽₈)	C₂×C₃₇	1	1	yes	characteristic, complemented, Hall
29	C₇×C₃₇	C₃×C₅×D₃	1	1	yes	characteristic, complemented, Hall
30	C₂×C₃×C₃₇	—	21	1	no	—
31	D₃×C₃₇	C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
32	C₅×D₃×C₇	C₃×C₃₇	1	1	yes	characteristic, complemented
33	C₂×C₅×ΓL(1,𝔽₈)	—	3	1	no	—
34	C₅×C₃₇	D₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
35	D₃×ΓL(1,𝔽₈)	C₅×C₃₇	1	1	yes	characteristic, direct product factor, Hall
36	C₃×C₃₇	—	7	1	no	—
37	C₃×C₃₇	—	14	1	no	—
38	C₃×C₃₇	C₂×C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
39	C₃×C₅×C₇	C₂×C₃×C₃₇	1	1	yes	characteristic, complemented
40	C₅×ΓL(1,𝔽₈)	—	2	1	no	—
41	C₅×ΓL(1,𝔽₈)	D₃×C₃₇	1	1	yes	characteristic, direct product factor
42	C₃×C₅×D₃	—	7	1	no	Hall
43	C₂×C₃₇	—	3	1	no	Hall
44	C₂×C₅×C₇	—	3	1	no	Hall
45	C₃×ΓL(1,𝔽₈)	C₂×C₅×C₃₇	1	1	yes	characteristic, complemented, Hall
46	C₃²×C₅	—	7	1	no	Hall
47	D₃×C₇	C₃×C₅×C₃₇	1	1	yes	characteristic, complemented
48	C₂×ΓL(1,𝔽₈)	—	3	1	no	—
49	C₃₇	C₅×D₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
50	C₅×C₇	C₃×D₃×C₃₇	1	1	yes	characteristic, complemented, Hall
51	C₂×C₃×C₅	—	21	1	no	—
52	C₅×D₃	ΓL(1,𝔽₈)×C₃₇	1	1	yes	characteristic, direct product factor
53	C₃×C₇	C₂×C₃×C₅×C₃₇	1	1	yes	characteristic, complemented
54	ΓL(1,𝔽₈)	—	2	1	no	—
55	ΓL(1,𝔽₈)	C₅×D₃×C₃₇	1	1	yes	characteristic, direct product factor
56	C₃×D₃	—	7	1	no	Hall
57	C₃×C₅	—	7	1	no	—
58	C₃×C₅	—	14	1	no	—
59	C₃×C₅	C₂×ΓL(1,𝔽₈)×C₃₇	1	1	yes	characteristic, complemented
60	C₂×C₇	—	3	1	no	Hall
61	C₂×C₅	—	3	1	no	Hall
62	C₃²	—	7	1	no	Sylow
63	C₇	C₃×C₅×D₃×C₃₇	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₃	—	21	1	no	—
65	D₃	C₅×ΓL(1,𝔽₈)×C₃₇	1	1	yes	characteristic, direct product factor
66	C₅	D₃×ΓL(1,𝔽₈)×C₃₇	1	1	yes	characteristic, direct product factor, Sylow
67	C₃	—	7	1	no	—
68	C₃	—	14	1	no	—
69	C₃	C₂×C₅×ΓL(1,𝔽₈)×C₃₇	1	1	yes	characteristic, complemented
70	C₂	—	3	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₅×C₃₇	D₃×ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₃×C₇	C₂×C₃×C₅×C₃₇
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₇×C₃₇	C₂×C₃
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₅×C₃₇

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×ΓL(1,𝔽₈)×C₃₇	C₂	semidirect
2. C₅×D₃×C₇×C₃₇	C₃	semidirect
4. D₃×ΓL(1,𝔽₈)×C₃₇	C₅	direct (N × H)
6. C₅×ΓL(1,𝔽₈)×C₃₇	D₃	direct (N × H)
7. C₃×C₅×C₇×C₃₇	C₂×C₃	semidirect
10. C₃×ΓL(1,𝔽₈)×C₃₇	C₂×C₅	semidirect
12. D₃×C₇×C₃₇	C₃×C₅	semidirect
14. C₅×C₇×C₃₇	C₃×D₃	semidirect
16. C₅×D₃×C₃₇	ΓL(1,𝔽₈)	direct (N × H)
17. C₃×C₇×C₃₇	C₂×C₃×C₅	semidirect
19. ΓL(1,𝔽₈)×C₃₇	C₅×D₃	direct (N × H)
21. C₅×D₃×ΓL(1,𝔽₈)	C₃₇	direct (N × H)
24. C₃×C₅×C₃₇	C₂×ΓL(1,𝔽₈)	semidirect
28. C₃×C₅×ΓL(1,𝔽₈)	C₂×C₃₇	semidirect
29. C₇×C₃₇	C₃×C₅×D₃	semidirect
31. D₃×C₃₇	C₅×ΓL(1,𝔽₈)	direct (N × H)
32. C₅×D₃×C₇	C₃×C₃₇	semidirect
34. C₅×C₃₇	D₃×ΓL(1,𝔽₈)	direct (N × H)
35. D₃×ΓL(1,𝔽₈)	C₅×C₃₇	direct (N × H)
38. C₃×C₃₇	C₂×C₅×ΓL(1,𝔽₈)	semidirect
39. C₃×C₅×C₇	C₂×C₃×C₃₇	semidirect
41. C₅×ΓL(1,𝔽₈)	D₃×C₃₇	direct (N × H)
45. C₃×ΓL(1,𝔽₈)	C₂×C₅×C₃₇	semidirect
47. D₃×C₇	C₃×C₅×C₃₇	semidirect
49. C₃₇	C₅×D₃×ΓL(1,𝔽₈)	direct (N × H)
50. C₅×C₇	C₃×D₃×C₃₇	semidirect
52. C₅×D₃	ΓL(1,𝔽₈)×C₃₇	direct (N × H)
53. C₃×C₇	C₂×C₃×C₅×C₃₇	semidirect
55. ΓL(1,𝔽₈)	C₅×D₃×C₃₇	direct (N × H)
59. C₃×C₅	C₂×ΓL(1,𝔽₈)×C₃₇	semidirect
63. C₇	C₃×C₅×D₃×C₃₇	semidirect
65. D₃	C₅×ΓL(1,𝔽₈)×C₃₇	direct (N × H)
66. C₅	D₃×ΓL(1,𝔽₈)×C₃₇	direct (N × H)
69. C₃	C₂×C₅×ΓL(1,𝔽₈)×C₃₇	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)×C₃₇	2
C₅×D₃×C₇×C₃₇	3
C₂×C₅×ΓL(1,𝔽₈)×C₃₇	3
D₃×ΓL(1,𝔽₈)×C₃₇	5
C₃×C₅×D₃×C₃₇	7
C₅×D₃×ΓL(1,𝔽₈)	37

Subgroups — C5×D3×ΓL(1,𝔽8)×C37 #37552099

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×D₃×ΓL(1,𝔽₈)×C₃₇ #37552099