Subgroups — D₇×C₁₉×AGL(1,𝔽₇) #26013911

Order 11172.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₇×C₁₉×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₉×ΓL(1,𝔽₈)⋊D₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×C₁₉×AGL(1,𝔽₇)	C₂	1	1	yes	maximal, characteristic, complemented
4	D₇²×C₁₉	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	C₇×C₁₉×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₁₉×C₇⋊D₇	C₂×C₃	1	1	yes	characteristic, complemented
7	C₇×D₇×C₁₉	C₂×C₃	1	1	yes	characteristic, complemented
8	C₇×D₇×C₁₉	C₂×C₃	1	1	yes	characteristic, complemented
9	C₂×C₃×D₇×C₁₉	—	7	1	no	maximal
10	C₂×C₁₉×AGL(1,𝔽₇)	—	7	1	no	maximal
11	C₇²×C₁₉	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
12	C₂×C₃×C₇×C₁₉	—	7	1	no	—
13	C₃×D₇×C₁₉	—	7	1	no	—
14	C₃×D₇×C₁₉	—	7	1	no	—
15	C₂×C₁₉×ΓL(1,𝔽₈)	—	7	1	no	—
16	C₁₉×AGL(1,𝔽₇)	—	7	1	no	—
17	C₁₉×AGL(1,𝔽₇)	D₇	1	1	yes	characteristic, direct product factor
18	D₇×AGL(1,𝔽₇)	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₂×D₇×C₁₉	—	7	1	no	—
20	C₂×D₇×C₁₉	—	7	1	no	—
21	C₃×C₇×C₁₉	—	7	1	no	—
22	C₁₉×ΓL(1,𝔽₈)	C₂×D₇	1	1	yes	characteristic, complemented
23	ΓL(1,𝔽₈)⋊D₇	C₂×C₁₉	1	1	yes	characteristic, complemented
24	D₇×ΓL(1,𝔽₈)	C₂×C₁₉	1	1	yes	characteristic, complemented
25	C₇×AGL(1,𝔽₇)	C₂×C₁₉	1	1	yes	characteristic, complemented
26	C₂×C₇×C₁₉	—	7	1	no	—
27	C₂×C₇×C₁₉	—	7	1	no	—
28	D₇×C₁₉	—	7	1	no	—
29	D₇×C₁₉	—	7	1	no	—
30	D₇×C₁₉	—	42	1	no	—
31	D₇×C₁₉	AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
32	D₇×C₁₉	C₃×D₇	1	1	yes	characteristic, complemented
33	C₂²×C₃×C₁₉	—	49	1	no	Hall
34	D₇²	C₃×C₁₉	1	1	yes	characteristic, complemented, Hall
35	C₇×ΓL(1,𝔽₈)	C₂²×C₁₉	1	1	yes	characteristic, complemented, Hall
36	C₇×C₁₉	—	6	1	no	—
37	C₇×C₁₉	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
38	C₇×C₁₉	C₂×C₃×D₇	1	1	yes	characteristic, complemented
39	C₂×C₃×C₁₉	—	7	1	no	—
40	C₂×C₃×C₁₉	—	49	1	no	—
41	C₂×C₃×C₁₉	—	49	1	no	—
42	C₇⋊D₇	C₂×C₃×C₁₉	1	1	yes	characteristic, complemented
43	C₇×D₇	C₂×C₃×C₁₉	1	1	yes	characteristic, complemented
44	C₇×D₇	C₂×C₃×C₁₉	1	1	yes	characteristic, complemented
45	C₂×C₃×D₇	—	7	1	no	—
46	C₂×AGL(1,𝔽₇)	—	7	1	no	—
47	C₂²×C₁₉	—	49	1	no	Hall
48	C₃×C₁₉	—	7	1	no	Hall
49	C₇²	C₂²×C₃×C₁₉	1	1	yes	characteristic, complemented, Sylow
50	C₂×C₃×C₇	—	7	1	no	—
51	C₃×D₇	—	7	1	no	—
52	C₃×D₇	—	7	1	no	—
53	C₂×ΓL(1,𝔽₈)	—	7	1	no	—
54	AGL(1,𝔽₇)	—	7	1	no	—
55	AGL(1,𝔽₇)	D₇×C₁₉	1	1	yes	characteristic, direct product factor
56	C₂×C₁₉	—	7	1	no	—
57	C₂×C₁₉	—	7	1	no	—
58	C₂×C₁₉	—	49	1	no	—
59	C₂×D₇	—	7	1	no	—
60	C₂×D₇	—	7	1	no	—
61	C₃×C₇	—	7	1	no	—
62	ΓL(1,𝔽₈)	C₂×D₇×C₁₉	1	1	yes	characteristic, complemented
63	C₁₉	D₇×AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor, Sylow
64	C₂×C₇	—	7	1	no	—
65	C₂×C₇	—	7	1	no	—
66	D₇	—	7	1	no	—
67	D₇	—	7	1	no	—
68	D₇	—	42	1	no	—
69	D₇	C₁₉×AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
70	D₇	C₃×D₇×C₁₉	1	1	yes	characteristic, complemented
71	C₂²×C₃	—	49	1	no	Hall
72	C₇	—	6	1	no	—
73	C₇	C₂×C₁₉×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
74	C₇	C₂×C₃×D₇×C₁₉	1	1	yes	characteristic, complemented
75	C₂×C₃	—	7	1	no	—
76	C₂×C₃	—	49	1	no	—
77	C₂×C₃	—	49	1	no	—
78	C₂²	—	49	1	no	Sylow
79	C₃	—	7	1	no	Sylow
80	C₂	—	7	1	no	—
81	C₂	—	7	1	no	—
82	C₂	—	49	1	no	—

	subgroup	G/H
center, hypercenter	C₁₉	D₇×AGL(1,𝔽₇)
commutator, cosocle, hypocenter	C₇²	C₂²×C₃×C₁₉
Frattini	C₁	G
Fitting, socle	C₇²×C₁₉	C₂²×C₃
LCS	G ▷ C₇²
UCS	C₁ ◁ C₁₉

Complements

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

N	H ≅ G/N	type
1. D₇×C₁₉×ΓL(1,𝔽₈)	C₂	semidirect
2. C₁₉×ΓL(1,𝔽₈)⋊D₇	C₂	semidirect
3. C₇×C₁₉×AGL(1,𝔽₇)	C₂	semidirect
4. D₇²×C₁₉	C₃	semidirect
5. C₇×C₁₉×ΓL(1,𝔽₈)	C₂²	semidirect
6. C₁₉×C₇⋊D₇	C₂×C₃	semidirect
7. C₇×D₇×C₁₉	C₂×C₃	semidirect
8. C₇×D₇×C₁₉	C₂×C₃	semidirect
11. C₇²×C₁₉	C₂²×C₃	semidirect
17. C₁₉×AGL(1,𝔽₇)	D₇	direct (N × H)
18. D₇×AGL(1,𝔽₇)	C₁₉	direct (N × H)
22. C₁₉×ΓL(1,𝔽₈)	C₂×D₇	semidirect
23. ΓL(1,𝔽₈)⋊D₇	C₂×C₁₉	semidirect
24. D₇×ΓL(1,𝔽₈)	C₂×C₁₉	semidirect
25. C₇×AGL(1,𝔽₇)	C₂×C₁₉	semidirect
31. D₇×C₁₉	AGL(1,𝔽₇)	direct (N × H)
32. D₇×C₁₉	C₃×D₇	semidirect
34. D₇²	C₃×C₁₉	semidirect
35. C₇×ΓL(1,𝔽₈)	C₂²×C₁₉	semidirect
37. C₇×C₁₉	C₂×AGL(1,𝔽₇)	semidirect
38. C₇×C₁₉	C₂×C₃×D₇	semidirect
42. C₇⋊D₇	C₂×C₃×C₁₉	semidirect
43. C₇×D₇	C₂×C₃×C₁₉	semidirect
44. C₇×D₇	C₂×C₃×C₁₉	semidirect
49. C₇²	C₂²×C₃×C₁₉	semidirect
55. AGL(1,𝔽₇)	D₇×C₁₉	direct (N × H)
62. ΓL(1,𝔽₈)	C₂×D₇×C₁₉	semidirect
63. C₁₉	D₇×AGL(1,𝔽₇)	direct (N × H)
69. D₇	C₁₉×AGL(1,𝔽₇)	direct (N × H)
70. D₇	C₃×D₇×C₁₉	semidirect
73. C₇	C₂×C₁₉×AGL(1,𝔽₇)	semidirect
74. C₇	C₂×C₃×D₇×C₁₉	semidirect

Maximal subgroups

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

subgroup	index
D₇×C₁₉×ΓL(1,𝔽₈)	2
C₁₉×ΓL(1,𝔽₈)⋊D₇	2
C₇×C₁₉×AGL(1,𝔽₇)	2
D₇²×C₁₉	3
C₂×C₃×D₇×C₁₉	7
C₂×C₁₉×AGL(1,𝔽₇)	7
D₇×AGL(1,𝔽₇)	19

Subgroups — D7×C19×AGL(1,𝔽7) #26013911

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₇×C₁₉×AGL(1,𝔽₇) #26013911