Group #45691580 — subgroups

Subgroups — D₇×(C₇⋊C₉)⋊D₁₉ #45691580

Order 33516.

128 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 128 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₇×C₁₉×C₇⋊C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	(C₇⋊C₉)⋊D₁₃₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×(C₇⋊C₉)⋊D₁₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×D₇×D₁₃₃	C₃	1	1	yes	maximal, characteristic
5	C₇×C₁₉×C₇⋊C₉	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₃×C₇⋊D₁₃₃	C₂×C₃	1	1	yes	characteristic
7	C₃×C₇×D₁₃₃	C₂×C₃	1	1	yes	characteristic
8	C₃×C₇×D₇×C₁₉	C₂×C₃	1	1	yes	characteristic
9	C₉×D₇×D₁₉	—	7	1	no	maximal
10	C₂×(C₇⋊C₉)⋊D₁₉	—	7	1	no	maximal
11	D₇×D₁₃₃	C₉	1	1	yes	characteristic, complemented, Hall
12	C₃×C₇²×C₁₉	C₂²×C₃	1	1	yes	characteristic
13	C₉×D₁₃₃	—	7	1	no	—
14	C₉×D₇×C₁₉	—	7	1	no	—
15	C₇×C₉×D₁₉	—	7	1	no	—
16	C₂×C₁₉×C₇⋊C₉	—	7	1	no	—
17	(C₇⋊C₉)⋊D₁₉	—	7	1	no	—
18	(C₇⋊C₉)⋊D₁₉	D₇	1	1	yes	characteristic, direct product factor
19	C₇⋊D₁₃₃	C₂×C₉	1	1	yes	characteristic, complemented
20	C₇×D₁₃₃	C₂×C₉	1	1	yes	characteristic, complemented
21	C₇×D₇×C₁₉	C₂×C₉	1	1	yes	characteristic, complemented
22	D₇×D₇⋊C₉	—	19	1	no	maximal, Hall
23	C₂×C₃×D₁₃₃	—	7	1	no	—
24	C₃×D₇×D₁₉	—	7	1	no	—
25	C₇×C₉×C₁₉	—	7	1	no	—
26	C₁₉×C₇⋊C₉	C₂×D₇	1	1	yes	characteristic, complemented
27	C₇²×C₁₉	C₂²×C₉	1	1	yes	characteristic, complemented, Hall
28	(C₇⋊C₉)⋊D₇	—	19	1	no	—
29	D₇×C₇⋊C₉	D₁₉	1	1	yes	characteristic, complemented
30	C₇×D₇⋊C₉	—	19	1	no	—
31	C₂×C₃×C₇×C₁₉	—	7	1	no	—
32	C₃×D₁₃₃	—	7	1	no	—
33	C₃×D₁₃₃	—	7	1	no	—
34	C₃×D₁₃₃	—	42	1	no	—
35	C₃×D₁₃₃	C₃×D₇	1	1	yes	characteristic
36	C₃×D₇×C₁₉	AGL(1,𝔽₇)	1	1	yes	characteristic
37	C₃×C₇×D₁₉	—	7	1	no	—
38	C₂×C₉×D₁₉	—	49	1	no	Hall
39	C₃×D₇²	—	19	1	no	—
40	C₂×D₁₃₃	—	7	1	no	—
41	D₇×D₁₉	—	7	1	no	—
42	C₇×C₇⋊C₉	C₂×D₁₉	1	1	yes	characteristic, complemented, Hall
43	C₃×C₇×C₁₉	—	6	1	no	—
44	C₃×C₇×C₁₉	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
45	C₃×C₇×C₁₉	C₂×C₃×D₇	1	1	yes	characteristic
46	C₂×C₉×C₁₉	—	49	1	no	—
47	C₉×D₁₉	—	7	1	no	—
48	C₉×D₁₉	—	49	1	no	—
49	C₃×C₇⋊D₇	—	19	1	no	—
50	C₃×C₇×D₇	—	19	1	no	—
51	C₃×C₇×D₇	C₃×D₁₉	1	1	yes	characteristic
52	C₂×C₇×C₁₉	—	7	1	no	—
53	D₁₃₃	—	7	1	no	—
54	D₁₃₃	—	7	1	no	—
55	D₁₃₃	—	42	1	no	—
56	D₁₃₃	C₉×D₇	1	1	yes	characteristic, complemented
57	C₇×D₁₉	—	7	1	no	—
58	D₇×C₁₉	D₇⋊C₉	1	1	yes	characteristic, complemented
59	C₂×C₉×D₇	—	133	1	no	—
60	C₂×D₇⋊C₉	—	133	1	no	—
61	C₂×C₃×D₁₉	—	49	1	no	—
62	D₇²	—	19	1	no	Hall
63	C₉×C₁₉	—	7	1	no	Hall
64	C₃×C₇²	C₂×C₃×D₁₉	1	1	yes	characteristic
65	C₇×C₁₉	—	6	1	no	—
66	C₇×C₁₉	C₂×D₇⋊C₉	1	1	yes	characteristic, complemented
67	C₇×C₁₉	C₂×C₉×D₇	1	1	yes	characteristic, complemented
68	C₂×C₇×C₉	—	133	1	no	—
69	C₉×D₇	—	7	1	no	—
70	C₉×D₇	—	133	1	no	—
71	C₂×C₇⋊C₉	—	7	1	no	—
72	D₇⋊C₉	—	19	1	no	—
73	D₇⋊C₉	—	133	1	no	—
74	C₂×C₃×C₁₉	—	7	1	no	—
75	C₃×D₁₉	—	7	1	no	—
76	C₃×D₁₉	—	49	1	no	—
77	C₇⋊D₇	—	19	1	no	—
78	C₇×D₇	—	19	1	no	—
79	C₇×D₇	C₉×D₁₉	1	1	yes	characteristic, complemented
80	C₂×C₃×D₇	—	133	1	no	—
81	C₂×C₃×D₇	—	133	1	no	—
82	C₂×D₁₉	—	49	1	no	Hall
83	C₇×C₉	—	7	1	no	—
84	C₇⋊C₉	D₇×D₁₉	1	1	yes	characteristic, complemented
85	C₃×C₁₉	D₇×AGL(1,𝔽₇)	1	1	yes	characteristic
86	C₇²	C₂×C₉×D₁₉	1	1	yes	characteristic, complemented, Sylow
87	C₂×C₃×C₇	—	7	1	no	—
88	C₂×C₃×C₇	—	133	1	no	—
89	C₃×D₇	—	19	1	no	—
90	C₃×D₇	—	798	1	no	—
91	C₃×D₇	—	133	1	no	—
92	C₃×D₇	—	133	1	no	—
93	C₃×D₇	ΓL(1,𝔽₈)⋊D₁₉	1	1	yes	characteristic
94	C₂×C₁₉	—	7	1	no	—
95	D₁₉	—	7	1	no	—
96	D₁₉	—	49	1	no	—
97	C₂²×C₉	—	931	1	no	Hall
98	C₂×D₇	—	133	1	no	—
99	C₂×D₇	—	133	1	no	—
100	C₃×C₇	—	6	1	no	—
101	C₃×C₇	C₂×ΓL(1,𝔽₈)⋊D₁₉	1	1	yes	characteristic
102	C₃×C₇	C₃×D₇×D₁₉	1	1	yes	characteristic
103	C₁₉	D₇×D₇⋊C₉	1	1	yes	characteristic, complemented, Sylow
104	C₂×C₉	—	49	1	no	—
105	C₂×C₉	—	133	1	no	—
106	C₂×C₉	—	931	1	no	—
107	C₂×C₇	—	7	1	no	—
108	C₂×C₇	—	133	1	no	—
109	D₇	—	19	1	no	—
110	D₇	—	798	1	no	—
111	D₇	—	133	1	no	—
112	D₇	—	133	1	no	—
113	D₇	(C₇⋊C₉)⋊D₁₉	1	1	yes	characteristic, direct product factor
114	C₂²×C₃	—	931	1	no	—
115	C₉	—	7	1	no	Sylow
116	C₇	—	6	1	no	—
117	C₇	C₂×(C₇⋊C₉)⋊D₁₉	1	1	yes	characteristic, complemented
118	C₇	C₉×D₇×D₁₉	1	1	yes	characteristic, complemented
119	C₂×C₃	—	7	1	no	—
120	C₂×C₃	—	133	1	no	—
121	C₂×C₃	—	931	1	no	—
122	C₂²	—	931	1	no	Sylow
123	C₃	D₇×ΓL(1,𝔽₈)⋊D₁₉	1	1	yes	characteristic
124	C₂	—	7	1	no	—
125	C₂	—	133	1	no	—
126	C₂	—	931	1	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

D₇×C₁₉×C₇⋊C₉

C₂

yes

maximal, characteristic, complemented

(C₇⋊C₉)⋊D₁₃₃

C₂

yes

maximal, characteristic, complemented

C₇×(C₇⋊C₉)⋊D₁₉

C₂

yes

maximal, characteristic, complemented

C₃×D₇×D₁₃₃

C₃

yes

maximal, characteristic

C₇×C₁₉×C₇⋊C₉

C₂²

yes

characteristic, complemented, Hall

C₃×C₇⋊D₁₃₃

C₂×C₃

yes

characteristic

C₃×C₇×D₁₃₃

C₂×C₃

yes

characteristic

C₃×C₇×D₇×C₁₉

C₂×C₃

yes

characteristic

C₉×D₇×D₁₉

—

maximal

C₂×(C₇⋊C₉)⋊D₁₉

—

maximal

D₇×D₁₃₃

C₉

yes

characteristic, complemented, Hall

C₃×C₇²×C₁₉

C₂²×C₃

yes

characteristic

C₉×D₁₃₃

—

C₉×D₇×C₁₉

—

C₇×C₉×D₁₉

—

C₂×C₁₉×C₇⋊C₉

—

(C₇⋊C₉)⋊D₁₉

—

(C₇⋊C₉)⋊D₁₉

D₇

yes

characteristic, direct product factor

C₇⋊D₁₃₃

C₂×C₉

yes

characteristic, complemented

C₇×D₁₃₃

C₂×C₉

yes

characteristic, complemented

C₇×D₇×C₁₉

C₂×C₉

yes

characteristic, complemented

D₇×D₇⋊C₉

—

maximal, Hall

C₂×C₃×D₁₃₃

—

C₃×D₇×D₁₉

—

C₇×C₉×C₁₉

—

C₁₉×C₇⋊C₉

C₂×D₇

yes

characteristic, complemented

C₇²×C₁₉

C₂²×C₉

yes

characteristic, complemented, Hall

(C₇⋊C₉)⋊D₇

—

D₇×C₇⋊C₉

D₁₉

yes

characteristic, complemented

C₇×D₇⋊C₉

—

C₂×C₃×C₇×C₁₉

—

C₃×D₁₃₃

—

C₃×D₁₃₃

—

C₃×D₁₃₃

—

C₃×D₁₃₃

C₃×D₇

yes

characteristic

C₃×D₇×C₁₉

AGL(1,𝔽₇)

yes

characteristic

C₃×C₇×D₁₉

—

C₂×C₉×D₁₉

—

Hall

C₃×D₇²

—

C₂×D₁₃₃

—

D₇×D₁₉

—

C₇×C₇⋊C₉

C₂×D₁₉

yes

characteristic, complemented, Hall

C₃×C₇×C₁₉

—

C₃×C₇×C₁₉

C₂×AGL(1,𝔽₇)

yes

characteristic

C₃×C₇×C₁₉

C₂×C₃×D₇

yes

characteristic

C₂×C₉×C₁₉

—

C₉×D₁₉

—

C₉×D₁₉

—

C₃×C₇⋊D₇

—

C₃×C₇×D₇

—

C₃×C₇×D₇

C₃×D₁₉

yes

characteristic

C₂×C₇×C₁₉

—

D₁₃₃

—

D₁₃₃

—

D₁₃₃

—

D₁₃₃

C₉×D₇

yes

characteristic, complemented

C₇×D₁₉

—

D₇×C₁₉

D₇⋊C₉

yes

characteristic, complemented

C₂×C₉×D₇

—

133

—

C₂×D₇⋊C₉

—

133

—

C₂×C₃×D₁₉

—

D₇²

—

Hall

C₉×C₁₉

—

Hall

C₃×C₇²

C₂×C₃×D₁₉

yes

characteristic

C₇×C₁₉

—

C₇×C₁₉

C₂×D₇⋊C₉

yes

characteristic, complemented

C₇×C₁₉

C₂×C₉×D₇

yes

characteristic, complemented

C₂×C₇×C₉

—

133

—

C₉×D₇

—

C₉×D₇

—

133

—

C₂×C₇⋊C₉

—

D₇⋊C₉

—

D₇⋊C₉

—

133

—

C₂×C₃×C₁₉

—

C₃×D₁₉

—

C₃×D₁₉

—

C₇⋊D₇

—

C₇×D₇

—

C₇×D₇

C₉×D₁₉

yes

characteristic, complemented

C₂×C₃×D₇

—

133

—

C₂×C₃×D₇

—

133

—

C₂×D₁₉

—

Hall

C₇×C₉

—

C₇⋊C₉

D₇×D₁₉

yes

characteristic, complemented

C₃×C₁₉

D₇×AGL(1,𝔽₇)

yes

characteristic

C₇²

C₂×C₉×D₁₉

yes

characteristic, complemented, Sylow

C₂×C₃×C₇

—

C₂×C₃×C₇

—

133

—

C₃×D₇

—

C₃×D₇

—

798

—

C₃×D₇

—

133

—

C₃×D₇

—

133

—

C₃×D₇

ΓL(1,𝔽₈)⋊D₁₉

yes

characteristic

C₂×C₁₉

—

D₁₉

—

D₁₉

—

C₂²×C₉

—

931

Hall

C₂×D₇

—

133

—

C₂×D₇

—

133

—

100

C₃×C₇

—

101

C₃×C₇

C₂×ΓL(1,𝔽₈)⋊D₁₉

yes

characteristic

102

C₃×C₇

C₃×D₇×D₁₉

yes

characteristic

103

C₁₉

D₇×D₇⋊C₉

yes

characteristic, complemented, Sylow

104

C₂×C₉

—

105

C₂×C₉

—

133

—

106

C₂×C₉

—

931

—

107

C₂×C₇

—

108

C₂×C₇

—

133

—

109

D₇

—

110

D₇

—

798

—

111

D₇

—

133

—

112

D₇

—

133

—

113

D₇

(C₇⋊C₉)⋊D₁₉

yes

characteristic, direct product factor

114

C₂²×C₃

—

931

—

115

C₉

—

Sylow

116

C₇

—

117

C₇

C₂×(C₇⋊C₉)⋊D₁₉

yes

characteristic, complemented

118

C₇

C₉×D₇×D₁₉

yes

characteristic, complemented

119

C₂×C₃

—

120

C₂×C₃

—

133

—

121

C₂×C₃

—

931

—

122

C₂²

—

931

Sylow

123

C₃

D₇×ΓL(1,𝔽₈)⋊D₁₉

yes

characteristic

124

C₂

—

125

C₂

—

133

—

126

C₂

—

931

—

	subgroup	G/H
center, Frattini, hypercenter	C₃	D₇×ΓL(1,𝔽₈)⋊D₁₉
commutator, hypocenter	C₇²×C₁₉	C₂²×C₉
Fitting, socle, cosocle	C₃×C₇²×C₁₉	C₂²×C₃
prime radical	C₃×D₇×D₁₃₃	C₃
LCS	G ▷ C₇²×C₁₉
UCS	C₁ ◁ C₃

subgroup

G/H

center, Frattini, hypercenter

C₃

D₇×ΓL(1,𝔽₈)⋊D₁₉

commutator, hypocenter

C₇²×C₁₉

C₂²×C₉

Fitting, socle, cosocle

C₃×C₇²×C₁₉

C₂²×C₃

prime radical

C₃×D₇×D₁₃₃

C₃

LCS

G ▷ C₇²×C₁₉

UCS

C₁ ◁ C₃

N	H ≅ G/N	type
1. D₇×C₁₉×C₇⋊C₉	C₂	semidirect
2. (C₇⋊C₉)⋊D₁₃₃	C₂	semidirect
3. C₇×(C₇⋊C₉)⋊D₁₉	C₂	semidirect
5. C₇×C₁₉×C₇⋊C₉	C₂²	semidirect
11. D₇×D₁₃₃	C₉	semidirect
18. (C₇⋊C₉)⋊D₁₉	D₇	direct (N × H)
19. C₇⋊D₁₃₃	C₂×C₉	semidirect
20. C₇×D₁₃₃	C₂×C₉	semidirect
21. C₇×D₇×C₁₉	C₂×C₉	semidirect
26. C₁₉×C₇⋊C₉	C₂×D₇	semidirect
27. C₇²×C₁₉	C₂²×C₉	semidirect
29. D₇×C₇⋊C₉	D₁₉	semidirect
42. C₇×C₇⋊C₉	C₂×D₁₉	semidirect
56. D₁₃₃	C₉×D₇	semidirect
58. D₇×C₁₉	D₇⋊C₉	semidirect
66. C₇×C₁₉	C₂×D₇⋊C₉	semidirect
67. C₇×C₁₉	C₂×C₉×D₇	semidirect
79. C₇×D₇	C₉×D₁₉	semidirect
84. C₇⋊C₉	D₇×D₁₉	semidirect
86. C₇²	C₂×C₉×D₁₉	semidirect
103. C₁₉	D₇×D₇⋊C₉	semidirect
113. D₇	(C₇⋊C₉)⋊D₁₉	direct (N × H)
117. C₇	C₂×(C₇⋊C₉)⋊D₁₉	semidirect
118. C₇	C₉×D₇×D₁₉	semidirect

H ≅ G/N

type

1. D₇×C₁₉×C₇⋊C₉

C₂

semidirect

2. (C₇⋊C₉)⋊D₁₃₃

C₂

semidirect

3. C₇×(C₇⋊C₉)⋊D₁₉

C₂

semidirect

5. C₇×C₁₉×C₇⋊C₉

C₂²

semidirect

11. D₇×D₁₃₃

C₉

semidirect

18. (C₇⋊C₉)⋊D₁₉

D₇

direct (N × H)

19. C₇⋊D₁₃₃

C₂×C₉

semidirect

20. C₇×D₁₃₃

C₂×C₉

semidirect

21. C₇×D₇×C₁₉

C₂×C₉

semidirect

26. C₁₉×C₇⋊C₉

C₂×D₇

semidirect

27. C₇²×C₁₉

C₂²×C₉

semidirect

29. D₇×C₇⋊C₉

D₁₉

semidirect

42. C₇×C₇⋊C₉

C₂×D₁₉

semidirect

56. D₁₃₃

C₉×D₇

semidirect

58. D₇×C₁₉

D₇⋊C₉

semidirect

66. C₇×C₁₉

C₂×D₇⋊C₉

semidirect

67. C₇×C₁₉

C₂×C₉×D₇

semidirect

79. C₇×D₇

C₉×D₁₉

semidirect

84. C₇⋊C₉

D₇×D₁₉

semidirect

86. C₇²

C₂×C₉×D₁₉

semidirect

103. C₁₉

D₇×D₇⋊C₉

semidirect

113. D₇

(C₇⋊C₉)⋊D₁₉

direct (N × H)

117. C₇

C₂×(C₇⋊C₉)⋊D₁₉

semidirect

118. C₇

C₉×D₇×D₁₉

semidirect

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

subgroup	index
D₇×C₁₉×C₇⋊C₉	2
(C₇⋊C₉)⋊D₁₃₃	2
C₇×(C₇⋊C₉)⋊D₁₉	2
C₃×D₇×D₁₃₃	3
C₉×D₇×D₁₉	7
C₂×(C₇⋊C₉)⋊D₁₉	7
D₇×D₇⋊C₉	19

subgroup

index

D₇×C₁₉×C₇⋊C₉

(C₇⋊C₉)⋊D₁₃₃

C₇×(C₇⋊C₉)⋊D₁₉

C₃×D₇×D₁₃₃

C₉×D₇×D₁₉

C₂×(C₇⋊C₉)⋊D₁₉

D₇×D₇⋊C₉

Subgroups — D₇×(C₇⋊C₉)⋊D₁₉ #45691580

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D7×(C7⋊C9)⋊D19 #45691580

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₇×(C₇⋊C₉)⋊D₁₉ #45691580