Subgroups — C₃×D₇×C₁₉×ΓL(1,𝔽₈) #30660792

Order 16758.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇×C₁₉×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₃×C₇×D₇×C₁₉	C₃	1	1	yes	maximal, characteristic, complemented
3	D₇×C₁₉×ΓL(1,𝔽₈)	C₃	1	3	yes	maximal, direct product factor
4	C₇×C₁₉×ΓL(1,𝔽₈)	C₂×C₃	1	3	yes	complemented
5	C₃×C₇²×C₁₉	C₂×C₃	1	1	yes	characteristic, complemented
6	C₃²×D₇×C₁₉	—	7	1	no	maximal
7	C₂×C₃×C₁₉×ΓL(1,𝔽₈)	—	7	1	no	maximal
8	C₇×D₇×C₁₉	C₃²	1	1	yes	characteristic, complemented, Hall
9	C₃²×C₇×C₁₉	—	7	1	no	—
10	C₃×C₁₉×ΓL(1,𝔽₈)	D₇	1	1	yes	characteristic, direct product factor
11	C₇²×C₁₉	C₂×C₃²	1	1	yes	characteristic, complemented, Hall
12	C₃×D₇×ΓL(1,𝔽₈)	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₃×C₇×C₁₉	—	7	1	no	—
14	C₃×D₇×C₁₉	—	7	3	no	—
15	C₃×D₇×C₁₉	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
16	C₂×C₁₉×ΓL(1,𝔽₈)	—	7	3	no	—
17	C₃×C₇×ΓL(1,𝔽₈)	C₂×C₁₉	1	1	yes	characteristic, complemented, Hall
18	C₃×C₇×C₁₉	—	6	1	no	—
19	C₃×C₇×C₁₉	—	7	3	no	—
20	C₃×C₇×C₁₉	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
21	C₃×C₇×C₁₉	C₃×D₇	1	1	yes	characteristic, complemented
22	C₁₉×ΓL(1,𝔽₈)	C₃×D₇	1	3	yes	direct product factor
23	C₂×C₃²×C₁₉	—	49	1	no	Hall
24	C₃×C₇×D₇	C₃×C₁₉	1	1	yes	characteristic, complemented
25	D₇×ΓL(1,𝔽₈)	C₃×C₁₉	1	3	yes	direct product factor
26	C₂×C₇×C₁₉	—	7	1	no	—
27	D₇×C₁₉	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
28	C₃²×C₁₉	—	7	1	no	Hall
29	C₃×C₇²	C₂×C₃×C₁₉	1	1	yes	characteristic, complemented
30	C₇×ΓL(1,𝔽₈)	C₂×C₃×C₁₉	1	3	yes	complemented
31	C₇×C₁₉	—	6	1	no	—
32	C₇×C₁₉	C₂×C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
33	C₇×C₁₉	C₃²×D₇	1	1	yes	characteristic, complemented
34	C₃²×D₇	—	7	1	no	—
35	C₂×C₃×ΓL(1,𝔽₈)	—	7	1	no	—
36	C₂×C₃×C₁₉	—	7	1	no	—
37	C₂×C₃×C₁₉	—	49	3	no	—
38	C₇×D₇	C₃²×C₁₉	1	1	yes	characteristic, complemented, Hall
39	C₃²×C₇	—	7	1	no	—
40	C₃×ΓL(1,𝔽₈)	D₇×C₁₉	1	1	yes	characteristic, direct product factor
41	C₃×C₁₉	—	7	3	no	—
42	C₃×C₁₉	D₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
43	C₇²	C₂×C₃²×C₁₉	1	1	yes	characteristic, complemented, Sylow
44	C₂×C₃×C₇	—	7	1	no	—
45	C₃×D₇	—	7	3	no	—
46	C₃×D₇	C₁₉×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
47	C₂×ΓL(1,𝔽₈)	—	7	3	no	—
48	C₂×C₁₉	—	7	1	no	Hall
49	C₃×C₇	—	6	1	no	—
50	C₃×C₇	—	7	3	no	—
51	C₃×C₇	C₂×C₁₉×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
52	C₃×C₇	C₃×D₇×C₁₉	1	1	yes	characteristic, complemented
53	ΓL(1,𝔽₈)	C₃×D₇×C₁₉	1	3	yes	direct product factor
54	C₁₉	C₃×D₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
55	C₂×C₃²	—	49	1	no	Hall
56	C₂×C₇	—	7	1	no	—
57	D₇	C₃×C₁₉×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
58	C₃²	—	7	1	no	Sylow
59	C₇	—	6	1	no	—
60	C₇	C₂×C₃×C₁₉×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
61	C₇	C₃²×D₇×C₁₉	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×C₁₉×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
66	C₂	—	7	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃×C₁₉	D₇×ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇²	C₂×C₃²×C₁₉
Frattini	C₁	G
Fitting, socle	C₃×C₇²×C₁₉	C₂×C₃
LCS	G ▷ C₇²
UCS	C₁ ◁ C₃×C₁₉

Complements

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

N	H ≅ G/N	type
1. C₃×C₇×C₁₉×ΓL(1,𝔽₈)	C₂	semidirect
2. C₃×C₇×D₇×C₁₉	C₃	semidirect
3. D₇×C₁₉×ΓL(1,𝔽₈)	C₃	direct (N × H)
4. C₇×C₁₉×ΓL(1,𝔽₈)	C₂×C₃	semidirect
5. C₃×C₇²×C₁₉	C₂×C₃	semidirect
8. C₇×D₇×C₁₉	C₃²	semidirect
10. C₃×C₁₉×ΓL(1,𝔽₈)	D₇	direct (N × H)
11. C₇²×C₁₉	C₂×C₃²	semidirect
12. C₃×D₇×ΓL(1,𝔽₈)	C₁₉	direct (N × H)
15. C₃×D₇×C₁₉	ΓL(1,𝔽₈)	direct (N × H)
17. C₃×C₇×ΓL(1,𝔽₈)	C₂×C₁₉	semidirect
20. C₃×C₇×C₁₉	C₂×ΓL(1,𝔽₈)	semidirect
21. C₃×C₇×C₁₉	C₃×D₇	semidirect
22. C₁₉×ΓL(1,𝔽₈)	C₃×D₇	direct (N × H)
24. C₃×C₇×D₇	C₃×C₁₉	semidirect
25. D₇×ΓL(1,𝔽₈)	C₃×C₁₉	direct (N × H)
27. D₇×C₁₉	C₃×ΓL(1,𝔽₈)	direct (N × H)
29. C₃×C₇²	C₂×C₃×C₁₉	semidirect
30. C₇×ΓL(1,𝔽₈)	C₂×C₃×C₁₉	semidirect
32. C₇×C₁₉	C₂×C₃×ΓL(1,𝔽₈)	semidirect
33. C₇×C₁₉	C₃²×D₇	semidirect
38. C₇×D₇	C₃²×C₁₉	semidirect
40. C₃×ΓL(1,𝔽₈)	D₇×C₁₉	direct (N × H)
42. C₃×C₁₉	D₇×ΓL(1,𝔽₈)	direct (N × H)
43. C₇²	C₂×C₃²×C₁₉	semidirect
46. C₃×D₇	C₁₉×ΓL(1,𝔽₈)	direct (N × H)
51. C₃×C₇	C₂×C₁₉×ΓL(1,𝔽₈)	semidirect
52. C₃×C₇	C₃×D₇×C₁₉	semidirect
53. ΓL(1,𝔽₈)	C₃×D₇×C₁₉	direct (N × H)
54. C₁₉	C₃×D₇×ΓL(1,𝔽₈)	direct (N × H)
57. D₇	C₃×C₁₉×ΓL(1,𝔽₈)	direct (N × H)
60. C₇	C₂×C₃×C₁₉×ΓL(1,𝔽₈)	semidirect
61. C₇	C₃²×D₇×C₁₉	semidirect
65. C₃	D₇×C₁₉×ΓL(1,𝔽₈)	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×C₁₉×ΓL(1,𝔽₈)	2
C₃×C₇×D₇×C₁₉	3
D₇×C₁₉×ΓL(1,𝔽₈)	3
C₃²×D₇×C₁₉	7
C₂×C₃×C₁₉×ΓL(1,𝔽₈)	7
C₃×D₇×ΓL(1,𝔽₈)	19

Subgroups — C3×D7×C19×ΓL(1,𝔽8) #30660792

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₇×C₁₉×ΓL(1,𝔽₈) #30660792