Subgroups — C₁₉×C₃⋊(ΓL(1,𝔽₈)⋊D₇) #30660840

Order 16758.

88 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	C₁₉×C₃⋊(C₇⋊D₇)	C₃	1	1	yes	maximal, characteristic, complemented
3	C₁₉×ΓL(1,𝔽₈)⋊D₇	—	3	1	no	maximal
4	C₇×C₁₉×ΓL(1,𝔽₈)	—	2	1	no	—
5	C₇×C₁₉×ΓL(1,𝔽₈)	D₃	1	1	yes	characteristic, complemented
6	C₃×C₇²×C₁₉	C₂×C₃	1	1	yes	characteristic, complemented
7	C₃×C₁₉×D₂₁	—	7	1	no	maximal
8	C₁₉×C₃⋊AGL(1,𝔽₇)	—	7	1	no	maximal
9	C₁₉×C₇⋊D₇	—	3	1	no	Hall
10	C₃²×C₇×C₁₉	—	7	1	no	—
11	C₃×C₁₉×ΓL(1,𝔽₈)	D₇	1	1	yes	characteristic, complemented
12	C₇²×C₁₉	C₃×D₃	1	1	yes	characteristic, complemented, Hall
13	C₃⋊(ΓL(1,𝔽₈)⋊D₇)	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₁₉×D₂₁	—	7	1	no	—
15	C₁₉×D₂₁	—	7	1	no	—
16	C₁₉×D₂₁	—	21	2	no	—
17	C₃×D₇×C₁₉	—	21	1	no	—
18	C₁₉×AGL(1,𝔽₇)	—	21	1	no	—
19	C₃×C₇×ΓL(1,𝔽₈)	C₂×C₁₉	1	1	yes	characteristic, complemented, Hall
20	C₃×C₇×C₁₉	—	3	2	no	—
21	C₃×C₇×C₁₉	—	7	1	no	—
22	C₃×C₇×C₁₉	—	14	1	no	—
23	C₃×C₇×C₁₉	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
24	C₃×C₇×C₁₉	C₃×D₇	1	1	yes	characteristic, complemented
25	C₁₉×ΓL(1,𝔽₈)	—	2	1	no	—
26	C₁₉×ΓL(1,𝔽₈)	D₂₁	1	1	yes	characteristic, complemented
27	C₃×D₃×C₁₉	—	49	1	no	Hall
28	C₃⋊(C₇⋊D₇)	C₃×C₁₉	1	1	yes	characteristic, complemented
29	ΓL(1,𝔽₈)⋊D₇	—	3	1	no	—
30	D₇×C₁₉	—	21	1	no	—
31	D₇×C₁₉	—	21	1	no	—
32	D₇×C₁₉	—	63	2	no	—
33	C₃²×C₁₉	—	7	1	no	Hall
34	C₃×C₇²	C₂×C₃×C₁₉	1	1	yes	characteristic, complemented
35	C₇×ΓL(1,𝔽₈)	—	2	1	no	—
36	C₇×ΓL(1,𝔽₈)	D₃×C₁₉	1	1	yes	characteristic, complemented
37	C₇×C₁₉	—	3	2	no	—
38	C₇×C₁₉	C₃⋊AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
39	C₇×C₁₉	C₃×D₂₁	1	1	yes	characteristic, complemented
40	C₃×D₂₁	—	7	1	no	—
41	C₃⋊AGL(1,𝔽₇)	—	7	1	no	—
42	C₂×C₃×C₁₉	—	147	1	no	—
43	D₃×C₁₉	—	49	1	no	—
44	C₇⋊D₇	—	3	1	no	Hall
45	C₃²×C₇	—	7	1	no	—
46	C₃×ΓL(1,𝔽₈)	D₇×C₁₉	1	1	yes	characteristic, complemented
47	C₃×C₁₉	—	7	1	no	—
48	C₃×C₁₉	—	14	1	no	—
49	C₃×C₁₉	ΓL(1,𝔽₈)⋊D₇	1	1	yes	characteristic, complemented
50	C₇²	C₃×D₃×C₁₉	1	1	yes	characteristic, complemented, Sylow
51	D₂₁	—	7	1	no	—
52	D₂₁	—	7	1	no	—
53	D₂₁	—	21	2	no	—
54	C₃×D₇	—	21	1	no	—
55	AGL(1,𝔽₇)	—	21	1	no	—
56	C₂×C₁₉	—	147	1	no	Hall
57	C₃×C₇	—	3	2	no	—
58	C₃×C₇	—	7	1	no	—
59	C₃×C₇	—	14	1	no	—
60	C₃×C₇	C₁₉×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
61	C₃×C₇	C₃×D₇×C₁₉	1	1	yes	characteristic, complemented
62	ΓL(1,𝔽₈)	—	2	1	no	—
63	ΓL(1,𝔽₈)	C₁₉×D₂₁	1	1	yes	characteristic, complemented
64	C₁₉	C₃⋊(ΓL(1,𝔽₈)⋊D₇)	1	1	yes	characteristic, direct product factor, Sylow
65	C₃×D₃	—	49	1	no	Hall
66	D₇	—	21	1	no	—
67	D₇	—	21	1	no	—
68	D₇	—	63	2	no	—
69	C₃²	—	7	1	no	Sylow
70	C₇	—	3	2	no	—
71	C₇	C₁₉×C₃⋊AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
72	C₇	C₃×C₁₉×D₂₁	1	1	yes	characteristic, complemented
73	C₂×C₃	—	147	1	no	—
74	D₃	—	49	1	no	—
75	C₃	—	7	1	no	—
76	C₃	—	14	1	no	—
77	C₃	C₁₉×ΓL(1,𝔽₈)⋊D₇	1	1	yes	characteristic, complemented
78	C₂	—	147	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₁₉	C₃⋊(ΓL(1,𝔽₈)⋊D₇)
commutator, cosocle, hypocenter	C₃×C₇²	C₂×C₃×C₁₉
Frattini	C₁	G
Fitting, socle	C₃×C₇²×C₁₉	C₂×C₃
LCS	G ▷ C₃×C₇²
UCS	C₁ ◁ 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. C₁₉×C₃⋊(C₇⋊D₇)	C₃	semidirect
5. C₇×C₁₉×ΓL(1,𝔽₈)	D₃	semidirect
6. C₃×C₇²×C₁₉	C₂×C₃	semidirect
11. C₃×C₁₉×ΓL(1,𝔽₈)	D₇	semidirect
12. C₇²×C₁₉	C₃×D₃	semidirect
13. C₃⋊(ΓL(1,𝔽₈)⋊D₇)	C₁₉	direct (N × H)
19. C₃×C₇×ΓL(1,𝔽₈)	C₂×C₁₉	semidirect
23. C₃×C₇×C₁₉	AGL(1,𝔽₇)	semidirect
24. C₃×C₇×C₁₉	C₃×D₇	semidirect
26. C₁₉×ΓL(1,𝔽₈)	D₂₁	semidirect
28. C₃⋊(C₇⋊D₇)	C₃×C₁₉	semidirect
34. C₃×C₇²	C₂×C₃×C₁₉	semidirect
36. C₇×ΓL(1,𝔽₈)	D₃×C₁₉	semidirect
38. C₇×C₁₉	C₃⋊AGL(1,𝔽₇)	semidirect
39. C₇×C₁₉	C₃×D₂₁	semidirect
46. C₃×ΓL(1,𝔽₈)	D₇×C₁₉	semidirect
49. C₃×C₁₉	ΓL(1,𝔽₈)⋊D₇	semidirect
50. C₇²	C₃×D₃×C₁₉	semidirect
60. C₃×C₇	C₁₉×AGL(1,𝔽₇)	semidirect
61. C₃×C₇	C₃×D₇×C₁₉	semidirect
63. ΓL(1,𝔽₈)	C₁₉×D₂₁	semidirect
64. C₁₉	C₃⋊(ΓL(1,𝔽₈)⋊D₇)	direct (N × H)
71. C₇	C₁₉×C₃⋊AGL(1,𝔽₇)	semidirect
72. C₇	C₃×C₁₉×D₂₁	semidirect
77. C₃	C₁₉×ΓL(1,𝔽₈)⋊D₇	semidirect

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
C₁₉×C₃⋊(C₇⋊D₇)	3
C₁₉×ΓL(1,𝔽₈)⋊D₇	3
C₃×C₁₉×D₂₁	7
C₁₉×C₃⋊AGL(1,𝔽₇)	7
C₃⋊(ΓL(1,𝔽₈)⋊D₇)	19

Subgroups — C19×C3⋊(ΓL(1,𝔽8)⋊D7) #30660840

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×C₃⋊(ΓL(1,𝔽₈)⋊D₇) #30660840