Subgroups — C₃×C₃₇×ΓL(1,𝔽₈)⋊D₇ #45657506

Order 32634.

112 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×C₃₇×C₇⋊D₇	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃₇×ΓL(1,𝔽₈)⋊D₇	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	C₃×C₃₇×AGL(1,𝔽₇)	—	7	1	no	maximal
8	C₃₇×C₇⋊D₇	C₃²	1	1	yes	characteristic, complemented, Hall
9	C₃×ΓL(1,𝔽₈)×C₃₇	D₇	1	1	yes	characteristic, complemented
10	C₃²×C₇×C₃₇	—	7	1	no	—
11	C₇²×C₃₇	C₂×C₃²	1	1	yes	characteristic, complemented, Hall
12	C₃×D₇×C₃₇	—	7	3	no	—
13	C₃×D₇×C₃₇	—	7	1	no	—
14	C₃×D₇×C₃₇	—	7	1	no	—
15	C₃×D₇×C₃₇	—	21	2	no	—
16	C₃₇×AGL(1,𝔽₇)	—	7	3	no	—
17	C₃×ΓL(1,𝔽₈)⋊D₇	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₃×C₇×C₃₇	—	3	2	no	—
19	C₃×C₇×C₃₇	—	7	3	no	—
20	C₃×C₇×C₃₇	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
21	C₃×C₇×C₃₇	C₃×D₇	1	1	yes	characteristic, complemented
22	ΓL(1,𝔽₈)×C₃₇	C₃×D₇	1	3	yes	complemented
23	C₂×C₃²×C₃₇	—	49	1	no	Hall
24	D₇×C₃₇	—	7	1	no	—
25	D₇×C₃₇	—	7	1	no	—
26	D₇×C₃₇	—	21	2	no	—
27	C₃×C₇×ΓL(1,𝔽₈)	C₂×C₃₇	1	1	yes	characteristic, complemented, Hall
28	C₃²×C₃₇	—	7	1	no	Hall
29	C₃×C₇⋊D₇	C₃×C₃₇	1	1	yes	characteristic, complemented
30	ΓL(1,𝔽₈)⋊D₇	C₃×C₃₇	1	3	yes	direct product factor
31	C₇×C₃₇	—	3	2	no	—
32	C₇×C₃₇	C₃×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
33	C₇×C₃₇	C₃²×D₇	1	1	yes	characteristic, complemented
34	C₂×C₃×C₃₇	—	49	3	no	—
35	C₂×C₃×C₃₇	—	49	1	no	—
36	C₃×C₇²	C₂×C₃×C₃₇	1	1	yes	characteristic, complemented
37	C₇×ΓL(1,𝔽₈)	C₂×C₃×C₃₇	1	3	yes	complemented
38	C₃²×D₇	—	7	1	no	—
39	C₃×AGL(1,𝔽₇)	—	7	1	no	—
40	C₃×C₃₇	—	7	3	no	—
41	C₃×C₃₇	ΓL(1,𝔽₈)⋊D₇	1	1	yes	characteristic, direct product factor
42	C₇⋊D₇	C₃²×C₃₇	1	1	yes	characteristic, complemented, Hall
43	C₂×C₃₇	—	49	1	no	Hall
44	C₃²×C₇	—	7	1	no	—
45	C₃×ΓL(1,𝔽₈)	D₇×C₃₇	1	1	yes	characteristic, complemented
46	C₇²	C₂×C₃²×C₃₇	1	1	yes	characteristic, complemented, Sylow
47	C₃×D₇	—	7	3	no	—
48	C₃×D₇	—	7	1	no	—
49	C₃×D₇	—	7	1	no	—
50	C₃×D₇	—	21	2	no	—
51	AGL(1,𝔽₇)	—	7	3	no	—
52	C₃₇	C₃×ΓL(1,𝔽₈)⋊D₇	1	1	yes	characteristic, direct product factor, Sylow
53	C₃×C₇	—	3	2	no	—
54	C₃×C₇	—	7	3	no	—
55	C₃×C₇	C₃₇×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
56	C₃×C₇	C₃×D₇×C₃₇	1	1	yes	characteristic, complemented
57	ΓL(1,𝔽₈)	C₃×D₇×C₃₇	1	3	yes	complemented
58	C₂×C₃²	—	49	1	no	Hall
59	D₇	—	7	1	no	—
60	D₇	—	7	1	no	—
61	D₇	—	21	2	no	—
62	C₃²	—	7	1	no	Sylow
63	C₇	—	3	2	no	—
64	C₇	C₃×C₃₇×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
65	C₇	C₃²×D₇×C₃₇	1	1	yes	characteristic, complemented
66	C₂×C₃	—	49	3	no	—
67	C₂×C₃	—	49	1	no	—
68	C₃	—	7	3	no	—
69	C₃	C₃₇×ΓL(1,𝔽₈)⋊D₇	1	1	yes	characteristic, direct product factor
70	C₂	—	49	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃×C₃₇	ΓL(1,𝔽₈)⋊D₇
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×C₃₇×C₇⋊D₇	C₃	semidirect
3. C₃₇×ΓL(1,𝔽₈)⋊D₇	C₃	direct (N × H)
4. C₇×ΓL(1,𝔽₈)×C₃₇	C₂×C₃	semidirect
5. C₃×C₇²×C₃₇	C₂×C₃	semidirect
8. C₃₇×C₇⋊D₇	C₃²	semidirect
9. C₃×ΓL(1,𝔽₈)×C₃₇	D₇	semidirect
11. C₇²×C₃₇	C₂×C₃²	semidirect
17. C₃×ΓL(1,𝔽₈)⋊D₇	C₃₇	direct (N × H)
20. C₃×C₇×C₃₇	AGL(1,𝔽₇)	semidirect
21. C₃×C₇×C₃₇	C₃×D₇	semidirect
22. ΓL(1,𝔽₈)×C₃₇	C₃×D₇	semidirect
27. C₃×C₇×ΓL(1,𝔽₈)	C₂×C₃₇	semidirect
29. C₃×C₇⋊D₇	C₃×C₃₇	semidirect
30. ΓL(1,𝔽₈)⋊D₇	C₃×C₃₇	direct (N × H)
32. C₇×C₃₇	C₃×AGL(1,𝔽₇)	semidirect
33. C₇×C₃₇	C₃²×D₇	semidirect
36. C₃×C₇²	C₂×C₃×C₃₇	semidirect
37. C₇×ΓL(1,𝔽₈)	C₂×C₃×C₃₇	semidirect
41. C₃×C₃₇	ΓL(1,𝔽₈)⋊D₇	direct (N × H)
42. C₇⋊D₇	C₃²×C₃₇	semidirect
45. C₃×ΓL(1,𝔽₈)	D₇×C₃₇	semidirect
46. C₇²	C₂×C₃²×C₃₇	semidirect
52. C₃₇	C₃×ΓL(1,𝔽₈)⋊D₇	direct (N × H)
55. C₃×C₇	C₃₇×AGL(1,𝔽₇)	semidirect
56. C₃×C₇	C₃×D₇×C₃₇	semidirect
57. ΓL(1,𝔽₈)	C₃×D₇×C₃₇	semidirect
64. C₇	C₃×C₃₇×AGL(1,𝔽₇)	semidirect
65. C₇	C₃²×D₇×C₃₇	semidirect
69. C₃	C₃₇×ΓL(1,𝔽₈)⋊D₇	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×C₃₇×C₇⋊D₇	3
C₃₇×ΓL(1,𝔽₈)⋊D₇	3
C₃²×D₇×C₃₇	7
C₃×C₃₇×AGL(1,𝔽₇)	7
C₃×ΓL(1,𝔽₈)⋊D₇	37

Subgroups — C3×C37×ΓL(1,𝔽8)⋊D7 #45657506

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₃₇×ΓL(1,𝔽₈)⋊D₇ #45657506