Group #45691664 — subgroups

Subgroups — C₉×D₇×D₁₃₃ #45691664

Order 33516.

156 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 48 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	C₉×C₇⋊D₁₃₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×C₉×D₁₃₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×C₉×D₇×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×D₇×D₁₃₃	C₃	1	1	yes	maximal, characteristic
5	C₇²×C₉×C₁₉	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	C₂×C₉×D₁₃₃	—	7	1	no	maximal
10	C₉×D₇×D₁₉	—	7	1	no	maximal
11	D₇×D₁₃₃	C₉	1	1	yes	characteristic, direct product factor, Hall
12	C₃×C₇²×C₁₉	C₂²×C₃	1	1	yes	characteristic
13	C₉×D₁₃₃	—	7	1	no	—
14	C₉×D₁₃₃	—	7	1	no	—
15	C₉×D₁₃₃	—	14	3	no	—
16	C₉×D₁₃₃	D₇	1	1	yes	characteristic, direct product factor
17	C₉×D₇×C₁₉	D₇	1	1	yes	characteristic, complemented
18	C₇×C₉×D₁₉	—	7	1	no	—
19	C₂×C₇×C₉×C₁₉	—	7	1	no	—
20	C₇⋊D₁₃₃	C₂×C₉	1	1	yes	characteristic, complemented
21	C₇×D₁₃₃	C₂×C₉	1	1	yes	characteristic, complemented
22	C₇×D₇×C₁₉	C₂×C₉	1	1	yes	characteristic, complemented
23	C₉×D₇²	—	19	1	no	maximal, Hall
24	C₂×C₃×D₁₃₃	—	7	1	no	—
25	C₃×D₇×D₁₉	—	7	1	no	—
26	C₇×C₉×C₁₉	—	2	3	no	—
27	C₇×C₉×C₁₉	C₂×D₇	1	1	yes	characteristic, complemented
28	C₇×C₉×C₁₉	C₂×D₇	1	1	yes	characteristic, complemented
29	C₇²×C₁₉	C₂²×C₉	1	1	yes	characteristic, complemented, Hall
30	C₉×C₇⋊D₇	—	19	1	no	—
31	C₇×C₉×D₇	—	19	1	no	—
32	C₇×C₉×D₇	D₁₉	1	1	yes	characteristic, complemented
33	C₂×C₃×C₇×C₁₉	—	7	1	no	—
34	C₃×D₁₃₃	—	7	1	no	—
35	C₃×D₁₃₃	—	7	1	no	—
36	C₃×D₁₃₃	—	14	3	no	—
37	C₃×D₁₃₃	C₃×D₇	1	1	yes	characteristic
38	C₃×D₇×C₁₉	C₃×D₇	1	1	yes	characteristic
39	C₃×C₇×D₁₉	—	7	1	no	—
40	C₂×C₉×D₁₉	—	49	1	no	Hall
41	C₃×D₇²	—	19	1	no	—
42	C₂×D₁₃₃	—	7	1	no	—
43	D₇×D₁₉	—	7	1	no	—
44	C₇²×C₉	C₂×D₁₉	1	1	yes	characteristic, complemented, Hall
45	C₃×C₇×C₁₉	—	2	3	no	—
46	C₃×C₇×C₁₉	C₂×C₃×D₇	1	1	yes	characteristic
47	C₃×C₇×C₁₉	C₂×C₃×D₇	1	1	yes	characteristic
48	C₂×C₉×C₁₉	—	7	1	no	—
49	C₉×D₁₉	—	7	1	no	—
50	C₉×D₁₉	—	49	1	no	—
51	C₃×C₇⋊D₇	—	19	1	no	—
52	C₃×C₇×D₇	—	19	1	no	—
53	C₃×C₇×D₇	C₃×D₁₉	1	1	yes	characteristic
54	C₂×C₇×C₁₉	—	7	1	no	—
55	D₁₃₃	—	7	1	no	—
56	D₁₃₃	—	7	1	no	—
57	D₁₃₃	—	14	3	no	—
58	D₁₃₃	C₉×D₇	1	1	yes	characteristic, direct product factor
59	C₇×D₁₉	—	7	1	no	—
60	D₇×C₁₉	C₉×D₇	1	1	yes	characteristic, complemented
61	C₂×C₉×D₇	—	133	1	no	—
62	C₂×C₉×D₇	—	133	1	no	—
63	C₂×C₃×D₁₉	—	49	1	no	—
64	D₇²	—	19	1	no	Hall
65	C₉×C₁₉	D₇²	1	1	yes	characteristic, complemented, Hall
66	C₃×C₇²	C₂×C₃×D₁₉	1	1	yes	characteristic
67	C₇×C₁₉	—	2	3	no	—
68	C₇×C₁₉	C₂×C₉×D₇	1	1	yes	characteristic, complemented
69	C₇×C₁₉	C₂×C₉×D₇	1	1	yes	characteristic, complemented
70	C₂×C₇×C₉	—	7	1	no	—
71	C₂×C₇×C₉	—	133	1	no	—
72	C₉×D₇	—	266	3	no	—
73	C₉×D₇	—	19	1	no	—
74	C₉×D₇	—	133	1	no	—
75	C₉×D₇	—	133	1	no	—
76	C₉×D₇	D₁₃₃	1	1	yes	characteristic, direct product factor
77	C₂×C₃×C₁₉	—	7	1	no	—
78	C₃×D₁₉	—	7	1	no	—
79	C₃×D₁₉	—	49	1	no	—
80	C₇⋊D₇	—	19	1	no	—
81	C₇×D₇	—	19	1	no	—
82	C₇×D₇	C₉×D₁₉	1	1	yes	characteristic, complemented
83	C₂×C₃×D₇	—	133	1	no	—
84	C₂×C₃×D₇	—	133	1	no	—
85	C₂×D₁₉	—	49	1	no	Hall
86	C₇×C₉	—	2	3	no	—
87	C₇×C₉	D₇×D₁₉	1	1	yes	characteristic, complemented
88	C₇×C₉	C₂×D₁₃₃	1	1	yes	characteristic, complemented
89	C₃×C₁₉	C₃×D₇²	1	1	yes	characteristic
90	C₇²	C₂×C₉×D₁₉	1	1	yes	characteristic, complemented, Sylow
91	C₂×C₃×C₇	—	7	1	no	—
92	C₂×C₃×C₇	—	133	1	no	—
93	C₃×D₇	—	266	3	no	—
94	C₃×D₇	—	19	1	no	—
95	C₃×D₇	—	133	1	no	—
96	C₃×D₇	—	133	1	no	—
97	C₃×D₇	C₃×D₁₃₃	1	1	yes	characteristic
98	C₂×C₁₉	—	7	1	no	—
99	D₁₉	—	7	1	no	—
100	D₁₉	—	49	1	no	—
101	C₂²×C₉	—	931	1	no	Hall
102	C₂×D₇	—	133	1	no	—
103	C₂×D₇	—	133	1	no	—
104	C₃×C₇	—	2	3	no	—
105	C₃×C₇	C₃×D₇×D₁₉	1	1	yes	characteristic
106	C₃×C₇	C₂×C₃×D₁₃₃	1	1	yes	characteristic
107	C₁₉	C₉×D₇²	1	1	yes	characteristic, complemented, Sylow
108	C₂×C₉	—	7	1	no	—
109	C₂×C₉	—	133	1	no	—
110	C₂×C₉	—	931	1	no	—
111	C₂×C₇	—	7	1	no	—
112	C₂×C₇	—	133	1	no	—
113	D₇	—	266	3	no	—
114	D₇	—	19	1	no	—
115	D₇	—	133	1	no	—
116	D₇	—	133	1	no	—
117	D₇	C₉×D₁₃₃	1	1	yes	characteristic, direct product factor
118	C₂²×C₃	—	931	1	no	—
119	C₉	D₇×D₁₃₃	1	1	yes	characteristic, direct product factor, Sylow
120	C₇	—	2	3	no	—
121	C₇	C₉×D₇×D₁₉	1	1	yes	characteristic, complemented
122	C₇	C₂×C₉×D₁₃₃	1	1	yes	characteristic, complemented
123	C₂×C₃	—	7	1	no	—
124	C₂×C₃	—	133	1	no	—
125	C₂×C₃	—	931	1	no	—
126	C₂²	—	931	1	no	Sylow
127	C₃	C₃×D₇×D₁₃₃	1	1	yes	characteristic
128	C₂	—	7	1	no	—
129	C₂	—	133	1	no	—
130	C₂	—	931	1	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₉×C₇⋊D₁₃₃

C₂

yes

maximal, characteristic, complemented

C₇×C₉×D₁₃₃

C₂

yes

maximal, characteristic, complemented

C₇×C₉×D₇×C₁₉

C₂

yes

maximal, characteristic, complemented

C₃×D₇×D₁₃₃

C₃

yes

maximal, characteristic

C₇²×C₉×C₁₉

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

C₂×C₉×D₁₃₃

—

maximal

C₉×D₇×D₁₉

—

maximal

D₇×D₁₃₃

C₉

yes

characteristic, direct product factor, Hall

C₃×C₇²×C₁₉

C₂²×C₃

yes

characteristic

C₉×D₁₃₃

—

C₉×D₁₃₃

—

C₉×D₁₃₃

—

C₉×D₁₃₃

D₇

yes

characteristic, direct product factor

C₉×D₇×C₁₉

D₇

yes

characteristic, complemented

C₇×C₉×D₁₉

—

C₂×C₇×C₉×C₁₉

—

C₇⋊D₁₃₃

C₂×C₉

yes

characteristic, complemented

C₇×D₁₃₃

C₂×C₉

yes

characteristic, complemented

C₇×D₇×C₁₉

C₂×C₉

yes

characteristic, complemented

C₉×D₇²

—

maximal, Hall

C₂×C₃×D₁₃₃

—

C₃×D₇×D₁₉

—

C₇×C₉×C₁₉

—

C₇×C₉×C₁₉

C₂×D₇

yes

characteristic, complemented

C₇×C₉×C₁₉

C₂×D₇

yes

characteristic, complemented

C₇²×C₁₉

C₂²×C₉

yes

characteristic, complemented, Hall

C₉×C₇⋊D₇

—

C₇×C₉×D₇

—

C₇×C₉×D₇

D₁₉

yes

characteristic, complemented

C₂×C₃×C₇×C₁₉

—

C₃×D₁₃₃

—

C₃×D₁₃₃

—

C₃×D₁₃₃

—

C₃×D₁₃₃

C₃×D₇

yes

characteristic

C₃×D₇×C₁₉

C₃×D₇

yes

characteristic

C₃×C₇×D₁₉

—

C₂×C₉×D₁₉

—

Hall

C₃×D₇²

—

C₂×D₁₃₃

—

D₇×D₁₉

—

C₇²×C₉

C₂×D₁₉

yes

characteristic, complemented, Hall

C₃×C₇×C₁₉

—

C₃×C₇×C₁₉

C₂×C₃×D₇

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, direct product factor

C₇×D₁₉

—

D₇×C₁₉

C₉×D₇

yes

characteristic, complemented

C₂×C₉×D₇

—

133

—

C₂×C₉×D₇

—

133

—

C₂×C₃×D₁₉

—

D₇²

—

Hall

C₉×C₁₉

D₇²

yes

characteristic, complemented, Hall

C₃×C₇²

C₂×C₃×D₁₉

yes

characteristic

C₇×C₁₉

—

C₇×C₁₉

C₂×C₉×D₇

yes

characteristic, complemented

C₇×C₁₉

C₂×C₉×D₇

yes

characteristic, complemented

C₂×C₇×C₉

—

C₂×C₇×C₉

—

133

—

C₉×D₇

—

266

—

C₉×D₇

—

C₉×D₇

—

133

—

C₉×D₇

—

133

—

C₉×D₇

D₁₃₃

yes

characteristic, direct product factor

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

C₇×C₉

C₂×D₁₃₃

yes

characteristic, complemented

C₃×C₁₉

C₃×D₇²

yes

characteristic

C₇²

C₂×C₉×D₁₉

yes

characteristic, complemented, Sylow

C₂×C₃×C₇

—

C₂×C₃×C₇

—

133

—

C₃×D₇

—

266

—

C₃×D₇

—

C₃×D₇

—

133

—

C₃×D₇

—

133

—

C₃×D₇

C₃×D₁₃₃

yes

characteristic

C₂×C₁₉

—

D₁₉

—

100

D₁₉

—

101

C₂²×C₉

—

931

Hall

102

C₂×D₇

—

133

—

103

C₂×D₇

—

133

—

104

C₃×C₇

—

105

C₃×C₇

C₃×D₇×D₁₉

yes

characteristic

106

C₃×C₇

C₂×C₃×D₁₃₃

yes

characteristic

107

C₁₉

C₉×D₇²

yes

characteristic, complemented, Sylow

108

C₂×C₉

—

109

C₂×C₉

—

133

—

110

C₂×C₉

—

931

—

111

C₂×C₇

—

112

C₂×C₇

—

133

—

113

D₇

—

266

—

114

D₇

—

115

D₇

—

133

—

116

D₇

—

133

—

117

D₇

C₉×D₁₃₃

yes

characteristic, direct product factor

118

C₂²×C₃

—

931

—

119

C₉

D₇×D₁₃₃

yes

characteristic, direct product factor, Sylow

120

C₇

—

121

C₇

C₉×D₇×D₁₉

yes

characteristic, complemented

122

C₇

C₂×C₉×D₁₃₃

yes

characteristic, complemented

123

C₂×C₃

—

124

C₂×C₃

—

133

—

125

C₂×C₃

—

931

—

126

C₂²

—

931

Sylow

127

C₃

C₃×D₇×D₁₃₃

yes

characteristic

128

C₂

—

129

C₂

—

133

—

130

C₂

—

931

—

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

subgroup

G/H

center, hypercenter

C₉

D₇×D₁₃₃

commutator, hypocenter

C₇²×C₁₉

C₂²×C₉

Frattini

C₃

C₃×D₇×D₁₃₃

Fitting

C₇²×C₉×C₁₉

C₂²

socle, cosocle

C₃×C₇²×C₁₉

C₂²×C₃

prime radical

C₃×D₇×D₁₃₃

C₃

LCS

G ▷ C₇²×C₁₉

UCS

C₁ ◁ C₉

N	H ≅ G/N	type
1. C₉×C₇⋊D₁₃₃	C₂	semidirect
2. C₇×C₉×D₁₃₃	C₂	semidirect
3. C₇×C₉×D₇×C₁₉	C₂	semidirect
5. C₇²×C₉×C₁₉	C₂²	semidirect
11. D₇×D₁₃₃	C₉	direct (N × H)
16. C₉×D₁₃₃	D₇	direct (N × H)
17. C₉×D₇×C₁₉	D₇	semidirect
20. C₇⋊D₁₃₃	C₂×C₉	semidirect
21. C₇×D₁₃₃	C₂×C₉	semidirect
22. C₇×D₇×C₁₉	C₂×C₉	semidirect
27. C₇×C₉×C₁₉	C₂×D₇	semidirect
28. C₇×C₉×C₁₉	C₂×D₇	semidirect
29. C₇²×C₁₉	C₂²×C₉	semidirect
32. C₇×C₉×D₇	D₁₉	semidirect
44. C₇²×C₉	C₂×D₁₉	semidirect
58. D₁₃₃	C₉×D₇	direct (N × H)
60. D₇×C₁₉	C₉×D₇	semidirect
65. C₉×C₁₉	D₇²	semidirect
68. C₇×C₁₉	C₂×C₉×D₇	semidirect
69. C₇×C₁₉	C₂×C₉×D₇	semidirect
76. C₉×D₇	D₁₃₃	direct (N × H)
82. C₇×D₇	C₉×D₁₉	semidirect
87. C₇×C₉	D₇×D₁₉	semidirect
88. C₇×C₉	C₂×D₁₃₃	semidirect
90. C₇²	C₂×C₉×D₁₉	semidirect
107. C₁₉	C₉×D₇²	semidirect
117. D₇	C₉×D₁₃₃	direct (N × H)
119. C₉	D₇×D₁₃₃	direct (N × H)
121. C₇	C₉×D₇×D₁₉	semidirect
122. C₇	C₂×C₉×D₁₃₃	semidirect

H ≅ G/N

type

1. C₉×C₇⋊D₁₃₃

C₂

semidirect

2. C₇×C₉×D₁₃₃

C₂

semidirect

3. C₇×C₉×D₇×C₁₉

C₂

semidirect

5. C₇²×C₉×C₁₉

C₂²

semidirect

11. D₇×D₁₃₃

C₉

direct (N × H)

16. C₉×D₁₃₃

D₇

direct (N × H)

17. C₉×D₇×C₁₉

D₇

semidirect

20. C₇⋊D₁₃₃

C₂×C₉

semidirect

21. C₇×D₁₃₃

C₂×C₉

semidirect

22. C₇×D₇×C₁₉

C₂×C₉

semidirect

27. C₇×C₉×C₁₉

C₂×D₇

semidirect

28. C₇×C₉×C₁₉

C₂×D₇

semidirect

29. C₇²×C₁₉

C₂²×C₉

semidirect

32. C₇×C₉×D₇

D₁₉

semidirect

44. C₇²×C₉

C₂×D₁₉

semidirect

58. D₁₃₃

C₉×D₇

direct (N × H)

60. D₇×C₁₉

C₉×D₇

semidirect

65. C₉×C₁₉

D₇²

semidirect

68. C₇×C₁₉

C₂×C₉×D₇

semidirect

69. C₇×C₁₉

C₂×C₉×D₇

semidirect

76. C₉×D₇

D₁₃₃

direct (N × H)

82. C₇×D₇

C₉×D₁₉

semidirect

87. C₇×C₉

D₇×D₁₉

semidirect

88. C₇×C₉

C₂×D₁₃₃

semidirect

90. C₇²

C₂×C₉×D₁₉

semidirect

107. C₁₉

C₉×D₇²

semidirect

117. D₇

C₉×D₁₃₃

direct (N × H)

119. C₉

D₇×D₁₃₃

direct (N × H)

121. C₇

C₉×D₇×D₁₉

semidirect

122. C₇

C₂×C₉×D₁₃₃

semidirect

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

subgroup	index
C₉×C₇⋊D₁₃₃	2
C₇×C₉×D₁₃₃	2
C₇×C₉×D₇×C₁₉	2
C₃×D₇×D₁₃₃	3
C₂×C₉×D₁₃₃	7
C₉×D₇×D₁₉	7
C₉×D₇²	19

subgroup

index

C₉×C₇⋊D₁₃₃

C₇×C₉×D₁₃₃

C₇×C₉×D₇×C₁₉

C₃×D₇×D₁₃₃

C₂×C₉×D₁₃₃

C₉×D₇×D₁₉

C₉×D₇²

Subgroups — C₉×D₇×D₁₃₃ #45691664

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C9×D7×D133 #45691664

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×D₇×D₁₃₃ #45691664