Group #11766811 — subgroups

Subgroups — C₉×#2090 #11766811

Order 1152.

264 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 126 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×#428	C₂	1	2	yes	maximal
2	C₂×C₉×#106	C₂	1	1	yes	maximal, characteristic
3	C₉×#424	C₂	1	1	yes	maximal, characteristic
4	C₉×#343	C₂	1	2	yes	maximal
5	C₉×#341	C₂	1	1	yes	maximal, characteristic
6	C₃×#2090	C₃	1	1	yes	maximal, characteristic
7	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₉×M₁₆	C₂²	1	2	yes	—
9	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
10	C₉×#117	C₂²	1	1	yes	characteristic
11	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
12	C₉×#107	—	2	2	no	—
13	C₉×#106	—	2	2	no	—
14	C₉×#106	—	2	2	no	—
15	C₉×C₄⋊C₈	—	2	2	no	—
16	C₃×#428	C₂×C₃	1	2	yes	—
17	C₂×C₃×#106	C₂×C₃	1	1	yes	characteristic
18	C₃×#424	C₂×C₃	1	1	yes	characteristic
19	C₃×#343	C₂×C₃	1	2	yes	—
20	C₃×#341	C₂×C₃	1	1	yes	characteristic
21	C₂²×C₄×C₉	—	2	2	no	—
22	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
23	C₉×M₁₆	—	2	2	no	—
24	C₉×M₁₆	—	2	2	no	—
25	C₉×M₁₆	Q₈	1	4	yes	—
26	C₂×C₈×C₉	—	2	2	no	—
27	C₂×C₈×C₉	—	2	2	no	—
28	C₂×C₈×C₉	—	2	1	no	—
29	C₂×C₈×C₉	D₄	1	2	yes	—
30	C₉×C₄⋊C₄	—	2	2	no	—
31	C₉×C₄⋊C₄	—	2	2	no	—
32	C₉×C₄⋊C₄	D₄	1	2	yes	—
33	C₉×C₂²⋊C₄	—	2	1	no	—
34	C₄²×C₉	—	2	1	no	—
35	#2090	C₉	1	1	yes	characteristic, direct product factor, Sylow
36	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
37	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
38	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
39	C₃×#117	C₂²×C₃	1	1	yes	characteristic
40	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
41	C₃×#107	—	2	2	no	—
42	C₃×#106	—	2	2	no	—
43	C₃×#106	—	2	2	no	—
44	C₃×C₄⋊C₈	—	2	2	no	—
45	C₂³×C₉	#41	1	1	yes	characteristic
46	C₂×C₄×C₉	—	2	2	no	—
47	C₂×C₄×C₉	—	4	2	no	—
48	C₂×C₄×C₉	—	8	2	no	—
49	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₉	C₂×Q₈	1	2	yes	—
52	C₂×C₄×C₉	#41	1	2	yes	—
53	C₈×C₉	—	2	4	no	—
54	C₈×C₉	—	2	2	no	—
55	C₈×C₉	—	4	1	no	—
56	#428	C₂×C₉	1	2	yes	—
57	C₂×#106	C₂×C₉	1	1	yes	characteristic
58	#424	C₂×C₉	1	1	yes	characteristic
59	#343	C₂×C₉	1	2	yes	—
60	#341	C₂×C₉	1	1	yes	characteristic
61	C₂²×C₃×C₄	—	2	2	no	—
62	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
63	C₃×M₁₆	—	2	2	no	—
64	C₃×M₁₆	—	2	2	no	—
65	C₃×M₁₆	C₃×Q₈	1	4	yes	—
66	C₂×C₃×C₈	—	2	2	no	—
67	C₂×C₃×C₈	—	2	2	no	—
68	C₂×C₃×C₈	—	2	1	no	—
69	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
70	C₃×C₄⋊C₄	—	2	2	no	—
71	C₃×C₄⋊C₄	—	2	2	no	—
72	C₃×C₄⋊C₄	C₃×D₄	1	2	yes	—
73	C₃×C₂²⋊C₄	—	2	1	no	—
74	C₃×C₄²	—	2	1	no	—
75	C₂²×C₉	—	2	2	no	—
76	C₂²×C₉	#121	1	1	yes	characteristic
77	C₂²×C₉	#125	1	2	yes	—
78	C₄×C₉	—	4	2	no	—
79	C₄×C₉	—	8	2	no	—
80	C₄×C₉	C₂²⋊Q₈	1	2	yes	—
81	C₄×C₉	#123	1	1	yes	characteristic
82	C₄×C₉	C₄⋊Q₈	1	1	yes	characteristic
83	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
84	C₂×M₁₆	C₂²×C₉	1	2	yes	—
85	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
86	#117	C₂²×C₉	1	1	yes	characteristic
87	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
88	#107	—	2	2	no	—
89	#106	—	2	2	no	—
90	#106	—	2	2	no	—
91	C₄⋊C₈	—	2	2	no	—
92	C₂³×C₃	C₃×#41	1	1	yes	characteristic
93	C₂×C₃×C₄	—	2	2	no	—
94	C₂×C₃×C₄	—	4	2	no	—
95	C₂×C₃×C₄	—	8	2	no	—
96	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
97	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
98	C₂×C₃×C₄	C₂×C₃×Q₈	1	2	yes	—
99	C₂×C₃×C₄	C₃×#41	1	2	yes	—
100	C₃×C₈	—	2	4	no	—
101	C₃×C₈	—	2	2	no	—
102	C₃×C₈	—	4	1	no	—
103	C₂×C₉	—	2	2	no	—
104	C₂×C₉	#398	1	1	yes	characteristic
105	C₂×C₉	#471	1	2	yes	—
106	C₂²×C₄	—	2	2	no	—
107	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
108	M₁₆	—	2	2	no	—
109	M₁₆	—	2	2	no	—
110	M₁₆	Q₈×C₉	1	4	yes	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	D₄×C₉	1	2	yes	—
115	C₄⋊C₄	—	2	2	no	—
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	D₄×C₉	1	2	yes	—
118	C₂²⋊C₄	—	2	1	no	—
119	C₄²	—	2	1	no	—
120	C₂²×C₃	—	2	2	no	—
121	C₂²×C₃	C₃×#121	1	1	yes	characteristic
122	C₂²×C₃	C₃×#125	1	2	yes	—
123	C₃×C₄	—	4	2	no	—
124	C₃×C₄	—	8	2	no	—
125	C₃×C₄	C₃×C₂²⋊Q₈	1	2	yes	—
126	C₃×C₄	C₃×#123	1	1	yes	characteristic
127	C₃×C₄	C₃×C₄⋊Q₈	1	1	yes	characteristic
128	C₉	#2090	1	1	yes	characteristic, direct product factor, Sylow
129	C₂³	C₉×#41	1	1	yes	characteristic
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	8	2	no	—
133	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
134	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
135	C₂×C₄	C₂×Q₈×C₉	1	2	yes	—
136	C₂×C₄	C₉×#41	1	2	yes	—
137	C₈	—	2	4	no	—
138	C₈	—	2	2	no	—
139	C₈	—	4	1	no	—
140	C₂×C₃	—	2	2	no	—
141	C₂×C₃	C₃×#398	1	1	yes	characteristic
142	C₂×C₃	C₃×#471	1	2	yes	—
143	C₂²	—	2	2	no	—
144	C₂²	C₉×#121	1	1	yes	characteristic
145	C₂²	C₉×#125	1	2	yes	—
146	C₄	—	4	2	no	—
147	C₄	—	8	2	no	—
148	C₄	C₉×C₂²⋊Q₈	1	2	yes	—
149	C₄	C₉×#123	1	1	yes	characteristic
150	C₄	C₉×C₄⋊Q₈	1	1	yes	characteristic
151	C₃	C₃×#2090	1	1	yes	characteristic
152	C₂	—	2	2	no	—
153	C₂	C₉×#398	1	1	yes	characteristic
154	C₂	C₉×#471	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₉×#428

C₂

yes

maximal

C₂×C₉×#106

C₂

yes

maximal, characteristic

C₉×#424

C₂

yes

maximal, characteristic

C₉×#343

C₂

yes

maximal

C₉×#341

C₂

yes

maximal, characteristic

C₃×#2090

C₃

yes

maximal, characteristic

C₂×C₉×M₁₆

C₂²

yes

characteristic

C₂×C₉×M₁₆

C₂²

yes

—

C₂²×C₈×C₉

C₂²

yes

characteristic

C₉×#117

C₂²

yes

characteristic

C₂×C₉×C₄⋊C₄

C₂²

yes

—

C₉×#107

—

C₉×#106

—

C₉×#106

—

C₉×C₄⋊C₈

—

C₃×#428

C₂×C₃

yes

—

C₂×C₃×#106

C₂×C₃

yes

characteristic

C₃×#424

C₂×C₃

yes

characteristic

C₃×#343

C₂×C₃

yes

—

C₃×#341

C₂×C₃

yes

characteristic

C₂²×C₄×C₉

—

C₂²×C₄×C₉

C₂³

yes

characteristic

C₉×M₁₆

—

C₉×M₁₆

—

C₉×M₁₆

Q₈

yes

—

C₂×C₈×C₉

—

C₂×C₈×C₉

—

C₂×C₈×C₉

—

C₂×C₈×C₉

D₄

yes

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

D₄

yes

—

C₉×C₂²⋊C₄

—

C₄²×C₉

—

#2090

C₉

yes

characteristic, direct product factor, Sylow

C₂×C₃×M₁₆

C₂²×C₃

yes

characteristic

C₂×C₃×M₁₆

C₂²×C₃

yes

—

C₂²×C₃×C₈

C₂²×C₃

yes

characteristic

C₃×#117

C₂²×C₃

yes

characteristic

C₂×C₃×C₄⋊C₄

C₂²×C₃

yes

—

C₃×#107

—

C₃×#106

—

C₃×#106

—

C₃×C₄⋊C₈

—

C₂³×C₉

#41

yes

characteristic

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂×D₄

yes

characteristic

C₂×C₄×C₉

C₂×D₄

yes

characteristic

C₂×C₄×C₉

C₂×Q₈

yes

—

C₂×C₄×C₉

#41

yes

—

C₈×C₉

—

C₈×C₉

—

C₈×C₉

—

#428

C₂×C₉

yes

—

C₂×#106

C₂×C₉

yes

characteristic

#424

C₂×C₉

yes

characteristic

#343

C₂×C₉

yes

—

#341

C₂×C₉

yes

characteristic

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂³×C₃

yes

characteristic

C₃×M₁₆

—

C₃×M₁₆

—

C₃×M₁₆

C₃×Q₈

yes

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₃×D₄

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

C₃×D₄

yes

—

C₃×C₂²⋊C₄

—

C₃×C₄²

—

C₂²×C₉

—

C₂²×C₉

#121

yes

characteristic

C₂²×C₉

#125

yes

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

C₂²⋊Q₈

yes

—

C₄×C₉

#123

yes

characteristic

C₄×C₉

C₄⋊Q₈

yes

characteristic

C₂×M₁₆

C₂²×C₉

yes

characteristic

C₂×M₁₆

C₂²×C₉

yes

—

C₂²×C₈

C₂²×C₉

yes

characteristic

#117

C₂²×C₉

yes

characteristic

C₂×C₄⋊C₄

C₂²×C₉

yes

—

#107

—

#106

—

#106

—

C₄⋊C₈

—

C₂³×C₃

C₃×#41

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×C₃×D₄

yes

characteristic

C₂×C₃×C₄

C₂×C₃×D₄

yes

characteristic

C₂×C₃×C₄

C₂×C₃×Q₈

yes

—

C₂×C₃×C₄

C₃×#41

yes

—

100

C₃×C₈

—

101

C₃×C₈

—

102

C₃×C₈

—

103

C₂×C₉

—

104

C₂×C₉

#398

yes

characteristic

105

C₂×C₉

#471

yes

—

106

C₂²×C₄

—

107

C₂²×C₄

C₂³×C₉

yes

characteristic

108

M₁₆

—

109

M₁₆

—

110

M₁₆

Q₈×C₉

yes

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

D₄×C₉

yes

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

D₄×C₉

yes

—

118

C₂²⋊C₄

—

119

C₄²

—

120

C₂²×C₃

—

121

C₂²×C₃

C₃×#121

yes

characteristic

122

C₂²×C₃

C₃×#125

yes

—

123

C₃×C₄

—

124

C₃×C₄

—

125

C₃×C₄

C₃×C₂²⋊Q₈

yes

—

126

C₃×C₄

C₃×#123

yes

characteristic

127

C₃×C₄

C₃×C₄⋊Q₈

yes

characteristic

128

C₉

#2090

yes

characteristic, direct product factor, Sylow

129

C₂³

C₉×#41

yes

characteristic

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

C₂×D₄×C₉

yes

characteristic

134

C₂×C₄

C₂×D₄×C₉

yes

characteristic

135

C₂×C₄

C₂×Q₈×C₉

yes

—

136

C₂×C₄

C₉×#41

yes

—

137

C₈

—

138

C₈

—

139

C₈

—

140

C₂×C₃

—

141

C₂×C₃

C₃×#398

yes

characteristic

142

C₂×C₃

C₃×#471

yes

—

143

C₂²

—

144

C₂²

C₉×#121

yes

characteristic

145

C₂²

C₉×#125

yes

—

146

C₄

—

147

C₄

—

148

C₄

C₉×C₂²⋊Q₈

yes

—

149

C₄

C₉×#123

yes

characteristic

150

C₄

C₉×C₄⋊Q₈

yes

characteristic

151

C₃

C₃×#2090

yes

characteristic

152

C₂

—

153

C₂

C₉×#398

yes

characteristic

154

C₂

C₉×#471

yes

—

	subgroup	G/H
center	C₂²×C₉	#121
commutator	C₂²×C₄	C₂³×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂²×C₃	C₃×#121
prime radical	C₂³×C₃	C₃×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₂²×C₄×C₉ ◁ G

subgroup

G/H

center

C₂²×C₉

#121

commutator

C₂²×C₄

C₂³×C₉

Frattini, cosocle

C₂²×C₃×C₄

C₂³×C₃

socle

C₂²×C₃

C₃×#121

prime radical

C₂³×C₃

C₃×#41

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₉ ◁ C₂²×C₄×C₉ ◁ G

N	H ≅ G/N	type
35. #2090	C₉	direct (N × H)
128. C₉	#2090	direct (N × H)

H ≅ G/N

type

35. #2090

C₉

direct (N × H)

128. C₉

#2090

direct (N × H)

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

subgroup	index
C₉×#428	2
C₂×C₉×#106	2
C₉×#424	2
C₉×#343	2
C₉×#341	2
C₃×#2090	3

subgroup

index

C₂×C₉×#106

Subgroups — C₉×#2090 #11766811

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C9×#2090 #11766811

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2090 #11766811