Subgroups — C₃×ΓL(1,𝔽₈)×D₂₀₉ #39911878

Order 26334.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₁₁×C₁₉×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	ΓL(1,𝔽₈)×D₂₀₉	C₃	1	3	yes	maximal, direct product factor
3	C₃×C₇×D₂₀₉	C₃	1	1	yes	maximal, characteristic, complemented
4	C₁₁×C₁₉×ΓL(1,𝔽₈)	C₂×C₃	1	3	yes	complemented
5	C₃×C₇×C₁₁×C₁₉	C₂×C₃	1	1	yes	characteristic, complemented
6	C₃²×D₂₀₉	—	7	1	no	maximal, Hall
7	C₇×D₂₀₉	C₃²	1	1	yes	characteristic, complemented, Hall
8	C₃×ΓL(1,𝔽₈)×D₁₉	—	11	1	no	maximal, Hall
9	C₃²×C₁₁×C₁₉	—	7	1	no	Hall
10	C₇×C₁₁×C₁₉	C₂×C₃²	1	1	yes	characteristic, complemented, Hall
11	C₃×ΓL(1,𝔽₈)×D₁₁	—	19	1	no	maximal, Hall
12	C₃×D₂₀₉	—	7	3	no	—
13	C₃×D₂₀₉	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
14	C₃×C₁₉×ΓL(1,𝔽₈)	D₁₁	1	1	yes	characteristic, complemented, Hall
15	C₃×C₇×D₁₉	—	11	1	no	—
16	ΓL(1,𝔽₈)×D₁₉	—	11	3	no	—
17	C₃×C₁₁×ΓL(1,𝔽₈)	D₁₉	1	1	yes	characteristic, complemented, Hall
18	C₃×C₁₁×C₁₉	—	7	3	no	—
19	C₃×C₁₁×C₁₉	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
20	C₃×C₇×D₁₁	—	19	1	no	—
21	ΓL(1,𝔽₈)×D₁₁	—	19	3	no	—
22	D₂₀₉	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
23	C₃×C₇×C₁₉	C₃×D₁₁	1	1	yes	characteristic, complemented
24	C₁₉×ΓL(1,𝔽₈)	C₃×D₁₁	1	3	yes	complemented
25	C₃²×D₁₉	—	77	1	no	Hall
26	C₇×D₁₉	—	11	1	no	Hall
27	C₃×C₇×C₁₁	C₃×D₁₉	1	1	yes	characteristic, complemented
28	C₁₁×ΓL(1,𝔽₈)	C₃×D₁₉	1	3	yes	complemented
29	C₁₁×C₁₉	C₂×C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
30	C₃²×D₁₁	—	133	1	no	Hall
31	C₃²×C₁₉	—	7	1	no	Hall
32	C₇×D₁₁	—	19	1	no	Hall
33	C₇×C₁₉	C₃²×D₁₁	1	1	yes	characteristic, complemented, Hall
34	C₂×C₃×ΓL(1,𝔽₈)	—	209	1	no	Hall
35	C₃×D₁₉	—	11	1	no	—
36	C₃×D₁₉	—	77	3	no	—
37	C₃²×C₁₁	—	7	1	no	Hall
38	C₇×C₁₁	C₃²×D₁₉	1	1	yes	characteristic, complemented, Hall
39	C₃×D₁₁	—	19	1	no	—
40	C₃×D₁₁	—	133	3	no	—
41	C₃×ΓL(1,𝔽₈)	D₂₀₉	1	1	yes	characteristic, direct product factor, Hall
42	C₃×C₁₉	—	7	3	no	—
43	C₃×C₁₉	ΓL(1,𝔽₈)×D₁₁	1	1	yes	characteristic, complemented
44	C₂×C₃×C₇	—	209	1	no	—
45	C₂×ΓL(1,𝔽₈)	—	209	3	no	—
46	D₁₉	—	11	1	no	Hall
47	C₃×C₁₁	—	7	3	no	—
48	C₃×C₁₁	ΓL(1,𝔽₈)×D₁₉	1	1	yes	characteristic, complemented
49	D₁₁	—	19	1	no	Hall
50	C₃×C₇	C₃×D₂₀₉	1	1	yes	characteristic, complemented
51	ΓL(1,𝔽₈)	C₃×D₂₀₉	1	3	yes	direct product factor
52	C₁₉	C₃×ΓL(1,𝔽₈)×D₁₁	1	1	yes	characteristic, complemented, Sylow
53	C₂×C₃²	—	1463	1	no	Hall
54	C₂×C₇	—	209	1	no	Hall
55	C₁₁	C₃×ΓL(1,𝔽₈)×D₁₉	1	1	yes	characteristic, complemented, Sylow
56	C₃²	—	7	1	no	Sylow
57	C₇	C₃²×D₂₀₉	1	1	yes	characteristic, complemented, Sylow
58	C₂×C₃	—	1463	3	no	—
59	C₂×C₃	—	209	1	no	—
60	C₃	—	7	3	no	—
61	C₃	ΓL(1,𝔽₈)×D₂₀₉	1	1	yes	characteristic, direct product factor
62	C₂	—	209	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃	ΓL(1,𝔽₈)×D₂₀₉
commutator, cosocle, hypocenter	C₇×C₁₁×C₁₉	C₂×C₃²
Frattini	C₁	G
Fitting, socle	C₃×C₇×C₁₁×C₁₉	C₂×C₃
LCS	G ▷ C₇×C₁₁×C₁₉
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
1. C₃×C₁₁×C₁₉×ΓL(1,𝔽₈)	C₂	semidirect
2. ΓL(1,𝔽₈)×D₂₀₉	C₃	direct (N × H)
3. C₃×C₇×D₂₀₉	C₃	semidirect
4. C₁₁×C₁₉×ΓL(1,𝔽₈)	C₂×C₃	semidirect
5. C₃×C₇×C₁₁×C₁₉	C₂×C₃	semidirect
7. C₇×D₂₀₉	C₃²	semidirect
10. C₇×C₁₁×C₁₉	C₂×C₃²	semidirect
13. C₃×D₂₀₉	ΓL(1,𝔽₈)	direct (N × H)
14. C₃×C₁₉×ΓL(1,𝔽₈)	D₁₁	semidirect
17. C₃×C₁₁×ΓL(1,𝔽₈)	D₁₉	semidirect
19. C₃×C₁₁×C₁₉	C₂×ΓL(1,𝔽₈)	semidirect
22. D₂₀₉	C₃×ΓL(1,𝔽₈)	direct (N × H)
23. C₃×C₇×C₁₉	C₃×D₁₁	semidirect
24. C₁₉×ΓL(1,𝔽₈)	C₃×D₁₁	semidirect
27. C₃×C₇×C₁₁	C₃×D₁₉	semidirect
28. C₁₁×ΓL(1,𝔽₈)	C₃×D₁₉	semidirect
29. C₁₁×C₁₉	C₂×C₃×ΓL(1,𝔽₈)	semidirect
33. C₇×C₁₉	C₃²×D₁₁	semidirect
38. C₇×C₁₁	C₃²×D₁₉	semidirect
41. C₃×ΓL(1,𝔽₈)	D₂₀₉	direct (N × H)
43. C₃×C₁₉	ΓL(1,𝔽₈)×D₁₁	semidirect
48. C₃×C₁₁	ΓL(1,𝔽₈)×D₁₉	semidirect
50. C₃×C₇	C₃×D₂₀₉	semidirect
51. ΓL(1,𝔽₈)	C₃×D₂₀₉	direct (N × H)
52. C₁₉	C₃×ΓL(1,𝔽₈)×D₁₁	semidirect
55. C₁₁	C₃×ΓL(1,𝔽₈)×D₁₉	semidirect
57. C₇	C₃²×D₂₀₉	semidirect
61. C₃	ΓL(1,𝔽₈)×D₂₀₉	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×C₁₁×C₁₉×ΓL(1,𝔽₈)	2
ΓL(1,𝔽₈)×D₂₀₉	3
C₃×C₇×D₂₀₉	3
C₃²×D₂₀₉	7
C₃×ΓL(1,𝔽₈)×D₁₉	11
C₃×ΓL(1,𝔽₈)×D₁₁	19

Subgroups — C3×ΓL(1,𝔽8)×D209 #39911878

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×ΓL(1,𝔽₈)×D₂₀₉ #39911878