Subgroups — C₃×D₇×ΓL(1,𝔽₈)×C₃₇ #45657512

Order 32634.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇×ΓL(1,𝔽₈)×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₃×C₇×D₇×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented
3	D₇×ΓL(1,𝔽₈)×C₃₇	C₃	1	3	yes	maximal, direct product factor
4	C₇×ΓL(1,𝔽₈)×C₃₇	C₂×C₃	1	3	yes	complemented
5	C₃×C₇²×C₃₇	C₂×C₃	1	1	yes	characteristic, complemented
6	C₃²×D₇×C₃₇	—	7	1	no	maximal
7	C₂×C₃×ΓL(1,𝔽₈)×C₃₇	—	7	1	no	maximal
8	C₇×D₇×C₃₇	C₃²	1	1	yes	characteristic, complemented, Hall
9	C₃×ΓL(1,𝔽₈)×C₃₇	D₇	1	1	yes	characteristic, direct product factor
10	C₃²×C₇×C₃₇	—	7	1	no	—
11	C₇²×C₃₇	C₂×C₃²	1	1	yes	characteristic, complemented, Hall
12	C₂×C₃×C₇×C₃₇	—	7	1	no	—
13	C₃×D₇×C₃₇	—	7	3	no	—
14	C₃×D₇×C₃₇	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
15	C₂×ΓL(1,𝔽₈)×C₃₇	—	7	3	no	—
16	C₃×D₇×ΓL(1,𝔽₈)	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₃×C₇×C₃₇	—	6	1	no	—
18	C₃×C₇×C₃₇	—	7	3	no	—
19	C₃×C₇×C₃₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
20	C₃×C₇×C₃₇	C₃×D₇	1	1	yes	characteristic, complemented
21	ΓL(1,𝔽₈)×C₃₇	C₃×D₇	1	3	yes	direct product factor
22	C₂×C₃²×C₃₇	—	49	1	no	Hall
23	C₂×C₇×C₃₇	—	7	1	no	—
24	D₇×C₃₇	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
25	C₃×C₇×ΓL(1,𝔽₈)	C₂×C₃₇	1	1	yes	characteristic, complemented, Hall
26	C₃²×C₃₇	—	7	1	no	Hall
27	C₃×C₇×D₇	C₃×C₃₇	1	1	yes	characteristic, complemented
28	D₇×ΓL(1,𝔽₈)	C₃×C₃₇	1	3	yes	direct product factor
29	C₇×C₃₇	—	6	1	no	—
30	C₇×C₃₇	C₂×C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
31	C₇×C₃₇	C₃²×D₇	1	1	yes	characteristic, complemented
32	C₂×C₃×C₃₇	—	7	1	no	—
33	C₂×C₃×C₃₇	—	49	3	no	—
34	C₃×C₇²	C₂×C₃×C₃₇	1	1	yes	characteristic, complemented
35	C₇×ΓL(1,𝔽₈)	C₂×C₃×C₃₇	1	3	yes	complemented
36	C₃²×D₇	—	7	1	no	—
37	C₂×C₃×ΓL(1,𝔽₈)	—	7	1	no	—
38	C₃×C₃₇	—	7	3	no	—
39	C₃×C₃₇	D₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
40	C₇×D₇	C₃²×C₃₇	1	1	yes	characteristic, complemented, Hall
41	C₂×C₃₇	—	7	1	no	Hall
42	C₃²×C₇	—	7	1	no	—
43	C₃×ΓL(1,𝔽₈)	D₇×C₃₇	1	1	yes	characteristic, direct product factor
44	C₇²	C₂×C₃²×C₃₇	1	1	yes	characteristic, complemented, Sylow
45	C₂×C₃×C₇	—	7	1	no	—
46	C₃×D₇	—	7	3	no	—
47	C₃×D₇	ΓL(1,𝔽₈)×C₃₇	1	1	yes	characteristic, direct product factor
48	C₂×ΓL(1,𝔽₈)	—	7	3	no	—
49	C₃₇	C₃×D₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
50	C₃×C₇	—	6	1	no	—
51	C₃×C₇	—	7	3	no	—
52	C₃×C₇	C₂×ΓL(1,𝔽₈)×C₃₇	1	1	yes	characteristic, complemented
53	C₃×C₇	C₃×D₇×C₃₇	1	1	yes	characteristic, complemented
54	ΓL(1,𝔽₈)	C₃×D₇×C₃₇	1	3	yes	direct product factor
55	C₂×C₃²	—	49	1	no	Hall
56	C₂×C₇	—	7	1	no	—
57	D₇	C₃×ΓL(1,𝔽₈)×C₃₇	1	1	yes	characteristic, direct product factor
58	C₃²	—	7	1	no	Sylow
59	C₇	—	6	1	no	—
60	C₇	C₂×C₃×ΓL(1,𝔽₈)×C₃₇	1	1	yes	characteristic, complemented
61	C₇	C₃²×D₇×C₃₇	1	1	yes	characteristic, complemented
62	C₂×C₃	—	7	1	no	—
63	C₂×C₃	—	49	3	no	—
64	C₃	—	7	3	no	—
65	C₃	D₇×ΓL(1,𝔽₈)×C₃₇	1	1	yes	characteristic, direct product factor
66	C₂	—	7	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃×C₃₇	D₇×ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇²	C₂×C₃²×C₃₇
Frattini	C₁	G
Fitting, socle	C₃×C₇²×C₃₇	C₂×C₃
LCS	G ▷ C₇²
UCS	C₁ ◁ C₃×C₃₇

Complements

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

N	H ≅ G/N	type
1. C₃×C₇×ΓL(1,𝔽₈)×C₃₇	C₂	semidirect
2. C₃×C₇×D₇×C₃₇	C₃	semidirect
3. D₇×ΓL(1,𝔽₈)×C₃₇	C₃	direct (N × H)
4. C₇×ΓL(1,𝔽₈)×C₃₇	C₂×C₃	semidirect
5. C₃×C₇²×C₃₇	C₂×C₃	semidirect
8. C₇×D₇×C₃₇	C₃²	semidirect
9. C₃×ΓL(1,𝔽₈)×C₃₇	D₇	direct (N × H)
11. C₇²×C₃₇	C₂×C₃²	semidirect
14. C₃×D₇×C₃₇	ΓL(1,𝔽₈)	direct (N × H)
16. C₃×D₇×ΓL(1,𝔽₈)	C₃₇	direct (N × H)
19. C₃×C₇×C₃₇	C₂×ΓL(1,𝔽₈)	semidirect
20. C₃×C₇×C₃₇	C₃×D₇	semidirect
21. ΓL(1,𝔽₈)×C₃₇	C₃×D₇	direct (N × H)
24. D₇×C₃₇	C₃×ΓL(1,𝔽₈)	direct (N × H)
25. C₃×C₇×ΓL(1,𝔽₈)	C₂×C₃₇	semidirect
27. C₃×C₇×D₇	C₃×C₃₇	semidirect
28. D₇×ΓL(1,𝔽₈)	C₃×C₃₇	direct (N × H)
30. C₇×C₃₇	C₂×C₃×ΓL(1,𝔽₈)	semidirect
31. C₇×C₃₇	C₃²×D₇	semidirect
34. C₃×C₇²	C₂×C₃×C₃₇	semidirect
35. C₇×ΓL(1,𝔽₈)	C₂×C₃×C₃₇	semidirect
39. C₃×C₃₇	D₇×ΓL(1,𝔽₈)	direct (N × H)
40. C₇×D₇	C₃²×C₃₇	semidirect
43. C₃×ΓL(1,𝔽₈)	D₇×C₃₇	direct (N × H)
44. C₇²	C₂×C₃²×C₃₇	semidirect
47. C₃×D₇	ΓL(1,𝔽₈)×C₃₇	direct (N × H)
49. C₃₇	C₃×D₇×ΓL(1,𝔽₈)	direct (N × H)
52. C₃×C₇	C₂×ΓL(1,𝔽₈)×C₃₇	semidirect
53. C₃×C₇	C₃×D₇×C₃₇	semidirect
54. ΓL(1,𝔽₈)	C₃×D₇×C₃₇	direct (N × H)
57. D₇	C₃×ΓL(1,𝔽₈)×C₃₇	direct (N × H)
60. C₇	C₂×C₃×ΓL(1,𝔽₈)×C₃₇	semidirect
61. C₇	C₃²×D₇×C₃₇	semidirect
65. C₃	D₇×ΓL(1,𝔽₈)×C₃₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₇×ΓL(1,𝔽₈)×C₃₇	2
C₃×C₇×D₇×C₃₇	3
D₇×ΓL(1,𝔽₈)×C₃₇	3
C₃²×D₇×C₃₇	7
C₂×C₃×ΓL(1,𝔽₈)×C₃₇	7
C₃×D₇×ΓL(1,𝔽₈)	37

Subgroups — C3×D7×ΓL(1,𝔽8)×C37 #45657512

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₇×ΓL(1,𝔽₈)×C₃₇ #45657512