Subgroups — C₃×ΓL(1,𝔽₈)×D₂₅₉ #45657517

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×D₂₅₉	C₃	1	1	yes	maximal, characteristic, complemented
3	ΓL(1,𝔽₈)×D₂₅₉	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×D₂₅₉	—	7	1	no	maximal
7	C₃×ΓL(1,𝔽₈)×D₃₇	—	7	1	no	maximal
8	C₇×D₂₅₉	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×D₂₅₉	—	7	3	no	—
13	C₃×D₂₅₉	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
14	C₃×C₇×D₃₇	—	7	1	no	—
15	ΓL(1,𝔽₈)×D₃₇	—	7	3	no	—
16	C₃×D₇×ΓL(1,𝔽₈)	—	37	1	no	maximal, 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	complemented
22	C₃²×D₃₇	—	49	1	no	Hall
23	D₂₅₉	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
24	C₇×D₃₇	—	7	1	no	—
25	C₃×C₇×ΓL(1,𝔽₈)	D₃₇	1	1	yes	characteristic, complemented, Hall
26	C₃²×C₃₇	—	7	1	no	Hall
27	C₃×C₇×D₇	—	37	1	no	—
28	D₇×ΓL(1,𝔽₈)	—	37	3	no	—
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	C₃×D₃₇	—	7	1	no	—
33	C₃×D₃₇	—	49	3	no	—
34	C₃×C₇²	C₃×D₃₇	1	1	yes	characteristic, complemented
35	C₇×ΓL(1,𝔽₈)	C₃×D₃₇	1	3	yes	complemented
36	C₃²×D₇	—	259	1	no	—
37	C₂×C₃×ΓL(1,𝔽₈)	—	259	1	no	—
38	C₃×C₃₇	—	7	3	no	—
39	C₃×C₃₇	D₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
40	C₇×D₇	—	37	1	no	Hall
41	D₃₇	—	7	1	no	Hall
42	C₃²×C₇	—	7	1	no	—
43	C₃×ΓL(1,𝔽₈)	D₂₅₉	1	1	yes	characteristic, direct product factor
44	C₇²	C₃²×D₃₇	1	1	yes	characteristic, complemented, Sylow
45	C₂×C₃×C₇	—	259	1	no	—
46	C₃×D₇	—	259	3	no	—
47	C₃×D₇	—	37	1	no	—
48	C₂×ΓL(1,𝔽₈)	—	259	3	no	—
49	C₃₇	C₃×D₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
50	C₃×C₇	—	6	1	no	—
51	C₃×C₇	—	7	3	no	—
52	C₃×C₇	ΓL(1,𝔽₈)×D₃₇	1	1	yes	characteristic, complemented
53	C₃×C₇	C₃×D₂₅₉	1	1	yes	characteristic, complemented
54	ΓL(1,𝔽₈)	C₃×D₂₅₉	1	3	yes	direct product factor
55	C₂×C₃²	—	1813	1	no	Hall
56	C₂×C₇	—	259	1	no	—
57	D₇	—	37	1	no	—
58	C₃²	—	7	1	no	Sylow
59	C₇	—	6	1	no	—
60	C₇	C₃×ΓL(1,𝔽₈)×D₃₇	1	1	yes	characteristic, complemented
61	C₇	C₃²×D₂₅₉	1	1	yes	characteristic, complemented
62	C₂×C₃	—	259	1	no	—
63	C₂×C₃	—	1813	3	no	—
64	C₃	—	7	3	no	—
65	C₃	ΓL(1,𝔽₈)×D₂₅₉	1	1	yes	characteristic, direct product factor
66	C₂	—	259	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃	ΓL(1,𝔽₈)×D₂₅₉
commutator, cosocle, hypocenter	C₇²×C₃₇	C₂×C₃²
Frattini	C₁	G
Fitting, socle	C₃×C₇²×C₃₇	C₂×C₃
LCS	G ▷ C₇²×C₃₇
UCS	C₁ ◁ C₃

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×D₂₅₉	C₃	semidirect
3. ΓL(1,𝔽₈)×D₂₅₉	C₃	direct (N × H)
4. C₇×ΓL(1,𝔽₈)×C₃₇	C₂×C₃	semidirect
5. C₃×C₇²×C₃₇	C₂×C₃	semidirect
8. C₇×D₂₅₉	C₃²	semidirect
9. C₃×ΓL(1,𝔽₈)×C₃₇	D₇	semidirect
11. C₇²×C₃₇	C₂×C₃²	semidirect
13. C₃×D₂₅₉	ΓL(1,𝔽₈)	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	semidirect
23. D₂₅₉	C₃×ΓL(1,𝔽₈)	direct (N × H)
25. C₃×C₇×ΓL(1,𝔽₈)	D₃₇	semidirect
30. C₇×C₃₇	C₂×C₃×ΓL(1,𝔽₈)	semidirect
31. C₇×C₃₇	C₃²×D₇	semidirect
34. C₃×C₇²	C₃×D₃₇	semidirect
35. C₇×ΓL(1,𝔽₈)	C₃×D₃₇	semidirect
39. C₃×C₃₇	D₇×ΓL(1,𝔽₈)	semidirect
43. C₃×ΓL(1,𝔽₈)	D₂₅₉	direct (N × H)
44. C₇²	C₃²×D₃₇	semidirect
49. C₃₇	C₃×D₇×ΓL(1,𝔽₈)	semidirect
52. C₃×C₇	ΓL(1,𝔽₈)×D₃₇	semidirect
53. C₃×C₇	C₃×D₂₅₉	semidirect
54. ΓL(1,𝔽₈)	C₃×D₂₅₉	direct (N × H)
60. C₇	C₃×ΓL(1,𝔽₈)×D₃₇	semidirect
61. C₇	C₃²×D₂₅₉	semidirect
65. C₃	ΓL(1,𝔽₈)×D₂₅₉	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×D₂₅₉	3
ΓL(1,𝔽₈)×D₂₅₉	3
C₃²×D₂₅₉	7
C₃×ΓL(1,𝔽₈)×D₃₇	7
C₃×D₇×ΓL(1,𝔽₈)	37

Subgroups — C3×ΓL(1,𝔽8)×D259 #45657517

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×ΓL(1,𝔽₈)×D₂₅₉ #45657517