Subgroups — C₃×ΓL(1,𝔽₈)⋊D₁₃₃ #30660786

Order 16758.

112 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⋊D₁₃₃	C₃	1	1	yes	maximal, characteristic, complemented
3	ΓL(1,𝔽₈)⋊D₁₃₃	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×D₁₃₃	—	7	1	no	maximal
7	C₃×ΓL(1,𝔽₈)⋊D₁₉	—	7	1	no	maximal
8	C₇⋊D₁₃₃	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, complemented
11	C₇²×C₁₉	C₂×C₃²	1	1	yes	characteristic, complemented, Hall
12	C₃×ΓL(1,𝔽₈)⋊D₇	—	19	1	no	maximal, Hall
13	C₃×D₁₃₃	—	7	3	no	—
14	C₃×D₁₃₃	—	7	1	no	—
15	C₃×D₁₃₃	—	21	2	no	—
16	C₃×D₁₃₃	—	21	1	no	—
17	ΓL(1,𝔽₈)⋊D₁₉	—	7	3	no	—
18	C₃×C₇×ΓL(1,𝔽₈)	D₁₉	1	1	yes	characteristic, complemented, Hall
19	C₃×C₇×C₁₉	—	3	2	no	—
20	C₃×C₇×C₁₉	—	7	3	no	—
21	C₃×C₇×C₁₉	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
22	C₃×C₇×C₁₉	C₃×D₇	1	1	yes	characteristic, complemented
23	C₁₉×ΓL(1,𝔽₈)	C₃×D₇	1	3	yes	complemented
24	C₃²×D₁₉	—	49	1	no	Hall
25	C₃×C₇⋊D₇	—	19	1	no	—
26	ΓL(1,𝔽₈)⋊D₇	—	19	3	no	—
27	D₁₃₃	—	7	1	no	—
28	D₁₃₃	—	21	2	no	—
29	D₁₃₃	—	21	1	no	—
30	C₃²×C₁₉	—	7	1	no	Hall
31	C₃×C₇²	C₃×D₁₉	1	1	yes	characteristic, complemented
32	C₇×ΓL(1,𝔽₈)	C₃×D₁₉	1	3	yes	complemented
33	C₇×C₁₉	—	3	2	no	—
34	C₇×C₁₉	C₃×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
35	C₇×C₁₉	C₃²×D₇	1	1	yes	characteristic, complemented
36	C₃²×D₇	—	133	1	no	—
37	C₃×AGL(1,𝔽₇)	—	133	1	no	—
38	C₃×D₁₉	—	49	3	no	—
39	C₃×D₁₉	—	147	1	no	—
40	C₇⋊D₇	—	19	1	no	Hall
41	C₃²×C₇	—	7	1	no	—
42	C₃×ΓL(1,𝔽₈)	D₁₃₃	1	1	yes	characteristic, complemented
43	C₃×C₁₉	—	7	3	no	—
44	C₃×C₁₉	ΓL(1,𝔽₈)⋊D₇	1	1	yes	characteristic, complemented
45	C₇²	C₃²×D₁₉	1	1	yes	characteristic, complemented, Sylow
46	C₃×D₇	—	133	3	no	—
47	C₃×D₇	—	133	1	no	—
48	C₃×D₇	—	399	2	no	—
49	C₃×D₇	—	399	1	no	—
50	AGL(1,𝔽₇)	—	133	3	no	—
51	D₁₉	—	147	1	no	Hall
52	C₃×C₇	—	3	2	no	—
53	C₃×C₇	—	7	3	no	—
54	C₃×C₇	ΓL(1,𝔽₈)⋊D₁₉	1	1	yes	characteristic, complemented
55	C₃×C₇	C₃×D₁₃₃	1	1	yes	characteristic, complemented
56	ΓL(1,𝔽₈)	C₃×D₁₃₃	1	3	yes	complemented
57	C₁₉	C₃×ΓL(1,𝔽₈)⋊D₇	1	1	yes	characteristic, complemented, Sylow
58	C₂×C₃²	—	931	1	no	Hall
59	D₇	—	133	1	no	—
60	D₇	—	399	2	no	—
61	D₇	—	399	1	no	—
62	C₃²	—	7	1	no	Sylow
63	C₇	—	3	2	no	—
64	C₇	C₃×ΓL(1,𝔽₈)⋊D₁₉	1	1	yes	characteristic, complemented
65	C₇	C₃²×D₁₃₃	1	1	yes	characteristic, complemented
66	C₂×C₃	—	931	3	no	—
67	C₂×C₃	—	2793	1	no	—
68	C₃	—	7	3	no	—
69	C₃	ΓL(1,𝔽₈)⋊D₁₃₃	1	1	yes	characteristic, direct product factor
70	C₂	—	2793	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃	ΓL(1,𝔽₈)⋊D₁₃₃
commutator, cosocle, hypocenter	C₇²×C₁₉	C₂×C₃²
Frattini	C₁	G
Fitting, socle	C₃×C₇²×C₁₉	C₂×C₃
LCS	G ▷ C₇²×C₁₉
UCS	C₁ ◁ C₃

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⋊D₁₃₃	C₃	semidirect
3. ΓL(1,𝔽₈)⋊D₁₃₃	C₃	direct (N × H)
4. C₇×C₁₉×ΓL(1,𝔽₈)	C₂×C₃	semidirect
5. C₃×C₇²×C₁₉	C₂×C₃	semidirect
8. C₇⋊D₁₃₃	C₃²	semidirect
10. C₃×C₁₉×ΓL(1,𝔽₈)	D₇	semidirect
11. C₇²×C₁₉	C₂×C₃²	semidirect
18. C₃×C₇×ΓL(1,𝔽₈)	D₁₉	semidirect
21. C₃×C₇×C₁₉	AGL(1,𝔽₇)	semidirect
22. C₃×C₇×C₁₉	C₃×D₇	semidirect
23. C₁₉×ΓL(1,𝔽₈)	C₃×D₇	semidirect
31. C₃×C₇²	C₃×D₁₉	semidirect
32. C₇×ΓL(1,𝔽₈)	C₃×D₁₉	semidirect
34. C₇×C₁₉	C₃×AGL(1,𝔽₇)	semidirect
35. C₇×C₁₉	C₃²×D₇	semidirect
42. C₃×ΓL(1,𝔽₈)	D₁₃₃	semidirect
44. C₃×C₁₉	ΓL(1,𝔽₈)⋊D₇	semidirect
45. C₇²	C₃²×D₁₉	semidirect
54. C₃×C₇	ΓL(1,𝔽₈)⋊D₁₉	semidirect
55. C₃×C₇	C₃×D₁₃₃	semidirect
56. ΓL(1,𝔽₈)	C₃×D₁₃₃	semidirect
57. C₁₉	C₃×ΓL(1,𝔽₈)⋊D₇	semidirect
64. C₇	C₃×ΓL(1,𝔽₈)⋊D₁₉	semidirect
65. C₇	C₃²×D₁₃₃	semidirect
69. C₃	ΓL(1,𝔽₈)⋊D₁₃₃	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⋊D₁₃₃	3
ΓL(1,𝔽₈)⋊D₁₃₃	3
C₃²×D₁₃₃	7
C₃×ΓL(1,𝔽₈)⋊D₁₉	7
C₃×ΓL(1,𝔽₈)⋊D₇	19

Subgroups — C3×ΓL(1,𝔽8)⋊D133 #30660786

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×ΓL(1,𝔽₈)⋊D₁₃₃ #30660786