Subgroups — D₃×C₁₃×C₁₉×ΓL(1,𝔽₈) #44527958

Order 31122.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₁₃×C₁₉×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	D₃×C₇×C₁₃×C₁₉	C₃	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×C₁₉×ΓL(1,𝔽₈)	—	3	1	no	maximal
4	C₁₃×C₁₉×ΓL(1,𝔽₈)	—	2	1	no	—
5	C₁₃×C₁₉×ΓL(1,𝔽₈)	D₃	1	1	yes	characteristic, direct product factor
6	C₃×C₇×C₁₃×C₁₉	C₂×C₃	1	1	yes	characteristic, complemented
7	C₃×D₃×C₁₃×C₁₉	—	7	1	no	maximal, Hall
8	C₂×C₇×C₁₃×C₁₉	—	3	1	no	Hall
9	D₃×C₁₉×ΓL(1,𝔽₈)	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₃²×C₁₃×C₁₉	—	7	1	no	Hall
11	C₇×C₁₃×C₁₉	C₃×D₃	1	1	yes	characteristic, complemented, Hall
12	D₃×C₁₃×ΓL(1,𝔽₈)	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₃×C₁₃×C₁₉	—	21	1	no	—
14	D₃×C₁₃×C₁₉	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
15	C₃×C₁₉×ΓL(1,𝔽₈)	C₂×C₁₃	1	1	yes	characteristic, complemented, Hall
16	C₃×C₁₃×ΓL(1,𝔽₈)	C₂×C₁₉	1	1	yes	characteristic, complemented, Hall
17	D₃×C₇×C₁₉	C₃×C₁₃	1	1	yes	characteristic, complemented
18	C₂×C₁₉×ΓL(1,𝔽₈)	—	3	1	no	—
19	C₃×C₁₃×C₁₉	—	7	1	no	—
20	C₃×C₁₃×C₁₉	—	14	1	no	—
21	C₃×C₁₃×C₁₉	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
22	D₃×C₇×C₁₃	C₃×C₁₉	1	1	yes	characteristic, complemented
23	C₂×C₁₃×ΓL(1,𝔽₈)	—	3	1	no	—
24	C₂×C₁₃×C₁₉	—	3	1	no	Hall
25	C₃×C₇×C₁₉	C₂×C₃×C₁₃	1	1	yes	characteristic, complemented
26	C₁₉×ΓL(1,𝔽₈)	—	2	1	no	—
27	C₁₉×ΓL(1,𝔽₈)	D₃×C₁₃	1	1	yes	characteristic, direct product factor
28	C₃×D₃×C₁₉	—	7	1	no	Hall
29	C₃×C₇×C₁₃	C₂×C₃×C₁₉	1	1	yes	characteristic, complemented
30	C₁₃×ΓL(1,𝔽₈)	—	2	1	no	—
31	C₁₃×ΓL(1,𝔽₈)	D₃×C₁₉	1	1	yes	characteristic, direct product factor
32	C₂×C₇×C₁₉	—	3	1	no	Hall
33	C₁₃×C₁₉	D₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
34	C₃×D₃×C₁₃	—	7	1	no	Hall
35	C₂×C₇×C₁₃	—	3	1	no	Hall
36	C₃²×C₁₉	—	7	1	no	Hall
37	C₇×C₁₉	C₃×D₃×C₁₃	1	1	yes	characteristic, complemented, Hall
38	D₃×ΓL(1,𝔽₈)	C₁₃×C₁₉	1	1	yes	characteristic, direct product factor, Hall
39	C₃²×C₁₃	—	7	1	no	Hall
40	C₂×C₃×C₁₉	—	21	1	no	—
41	D₃×C₁₉	C₁₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
42	C₇×C₁₃	C₃×D₃×C₁₉	1	1	yes	characteristic, complemented, Hall
43	C₂×C₃×C₁₃	—	21	1	no	—
44	D₃×C₁₃	C₁₉×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
45	C₃×ΓL(1,𝔽₈)	C₂×C₁₃×C₁₉	1	1	yes	characteristic, complemented, Hall
46	C₃×C₁₉	—	7	1	no	—
47	C₃×C₁₉	—	14	1	no	—
48	C₃×C₁₉	C₂×C₁₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
49	D₃×C₇	C₃×C₁₃×C₁₉	1	1	yes	characteristic, complemented
50	C₂×ΓL(1,𝔽₈)	—	3	1	no	—
51	C₃×C₁₃	—	7	1	no	—
52	C₃×C₁₃	—	14	1	no	—
53	C₃×C₁₃	C₂×C₁₉×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
54	C₂×C₁₉	—	3	1	no	Hall
55	C₂×C₁₃	—	3	1	no	Hall
56	C₃×C₇	C₂×C₃×C₁₃×C₁₉	1	1	yes	characteristic, complemented
57	ΓL(1,𝔽₈)	—	2	1	no	—
58	ΓL(1,𝔽₈)	D₃×C₁₃×C₁₉	1	1	yes	characteristic, direct product factor
59	C₁₉	D₃×C₁₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
60	C₃×D₃	—	7	1	no	Hall
61	C₂×C₇	—	3	1	no	Hall
62	C₁₃	D₃×C₁₉×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
63	C₃²	—	7	1	no	Sylow
64	C₇	C₃×D₃×C₁₃×C₁₉	1	1	yes	characteristic, complemented, Sylow
65	C₂×C₃	—	21	1	no	—
66	D₃	C₁₃×C₁₉×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
67	C₃	—	7	1	no	—
68	C₃	—	14	1	no	—
69	C₃	C₂×C₁₃×C₁₉×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
70	C₂	—	3	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₁₃×C₁₉	D₃×ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₃×C₇	C₂×C₃×C₁₃×C₁₉
Frattini	C₁	G
Fitting, socle	C₃×C₇×C₁₃×C₁₉	C₂×C₃
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₁₃×C₁₉

Complements

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

N	H ≅ G/N	type
1. C₃×C₁₃×C₁₉×ΓL(1,𝔽₈)	C₂	semidirect
2. D₃×C₇×C₁₃×C₁₉	C₃	semidirect
5. C₁₃×C₁₉×ΓL(1,𝔽₈)	D₃	direct (N × H)
6. C₃×C₇×C₁₃×C₁₉	C₂×C₃	semidirect
9. D₃×C₁₉×ΓL(1,𝔽₈)	C₁₃	direct (N × H)
11. C₇×C₁₃×C₁₉	C₃×D₃	semidirect
12. D₃×C₁₃×ΓL(1,𝔽₈)	C₁₉	direct (N × H)
14. D₃×C₁₃×C₁₉	ΓL(1,𝔽₈)	direct (N × H)
15. C₃×C₁₉×ΓL(1,𝔽₈)	C₂×C₁₃	semidirect
16. C₃×C₁₃×ΓL(1,𝔽₈)	C₂×C₁₉	semidirect
17. D₃×C₇×C₁₉	C₃×C₁₃	semidirect
21. C₃×C₁₃×C₁₉	C₂×ΓL(1,𝔽₈)	semidirect
22. D₃×C₇×C₁₃	C₃×C₁₉	semidirect
25. C₃×C₇×C₁₉	C₂×C₃×C₁₃	semidirect
27. C₁₉×ΓL(1,𝔽₈)	D₃×C₁₃	direct (N × H)
29. C₃×C₇×C₁₃	C₂×C₃×C₁₉	semidirect
31. C₁₃×ΓL(1,𝔽₈)	D₃×C₁₉	direct (N × H)
33. C₁₃×C₁₉	D₃×ΓL(1,𝔽₈)	direct (N × H)
37. C₇×C₁₉	C₃×D₃×C₁₃	semidirect
38. D₃×ΓL(1,𝔽₈)	C₁₃×C₁₉	direct (N × H)
41. D₃×C₁₉	C₁₃×ΓL(1,𝔽₈)	direct (N × H)
42. C₇×C₁₃	C₃×D₃×C₁₉	semidirect
44. D₃×C₁₃	C₁₉×ΓL(1,𝔽₈)	direct (N × H)
45. C₃×ΓL(1,𝔽₈)	C₂×C₁₃×C₁₉	semidirect
48. C₃×C₁₉	C₂×C₁₃×ΓL(1,𝔽₈)	semidirect
49. D₃×C₇	C₃×C₁₃×C₁₉	semidirect
53. C₃×C₁₃	C₂×C₁₉×ΓL(1,𝔽₈)	semidirect
56. C₃×C₇	C₂×C₃×C₁₃×C₁₉	semidirect
58. ΓL(1,𝔽₈)	D₃×C₁₃×C₁₉	direct (N × H)
59. C₁₉	D₃×C₁₃×ΓL(1,𝔽₈)	direct (N × H)
62. C₁₃	D₃×C₁₉×ΓL(1,𝔽₈)	direct (N × H)
64. C₇	C₃×D₃×C₁₃×C₁₉	semidirect
66. D₃	C₁₃×C₁₉×ΓL(1,𝔽₈)	direct (N × H)
69. C₃	C₂×C₁₃×C₁₉×ΓL(1,𝔽₈)	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₁₃×C₁₉×ΓL(1,𝔽₈)	2
D₃×C₇×C₁₃×C₁₉	3
C₂×C₁₃×C₁₉×ΓL(1,𝔽₈)	3
C₃×D₃×C₁₃×C₁₉	7
D₃×C₁₉×ΓL(1,𝔽₈)	13
D₃×C₁₃×ΓL(1,𝔽₈)	19

Subgroups — D3×C13×C19×ΓL(1,𝔽8) #44527958

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃×C₁₃×C₁₉×ΓL(1,𝔽₈) #44527958