Group #11795355 — subgroups

Subgroups — #11795355

Order 1152.

333 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 333 conjugacy classes of subgroups — too many to draw. Only its 86 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×#430	C₂	1	1	yes	maximal, characteristic, complemented
2	#10588808	C₂	1	2	yes	maximal
3	#10588808	C₂	1	1	yes	maximal, characteristic
4	#10588799	C₂	1	2	yes	maximal, complemented
5	C₉⋊#338	C₂	1	1	yes	maximal, characteristic, complemented
6	#70729	—	3	1	no	maximal
7	C₂²×C₄×D₉	C₂²	1	1	yes	characteristic
8	C₂×C₉×M₁₆	C₂²	1	2	yes	—
9	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
10	C₂×#3667	C₂²	1	1	yes	characteristic
11	C₂×#3667	C₂²	1	2	yes	—
12	C₉×#108	—	2	2	no	—
13	#63381	—	2	4	no	—
14	(C₂×D₉)⋊C₈	—	2	2	no	—
15	(C₂×D₉)⋊C₈	—	2	4	no	—
16	C₃×#430	D₃	1	1	yes	characteristic
17	#5126	—	3	2	no	—
18	#5126	—	3	1	no	—
19	#5124	—	3	2	no	—
20	C₃⋊#338	—	3	1	no	—
21	C₂³×D₉	Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
23	C₂²×Dic₉	Q₈	1	1	yes	characteristic
24	C₂×C₄×D₉	—	2	2	no	—
25	C₂×C₄×D₉	—	2	1	no	—
26	C₂×C₄×D₉	—	2	2	no	—
27	C₂×C₄×D₉	—	2	1	no	—
28	C₉×M₁₆	—	2	2	no	—
29	C₉×M₁₆	—	2	2	no	—
30	C₉×M₁₆	—	2	2	no	—
31	C₂×C₈×C₉	—	2	2	no	—
32	C₂×C₈×C₉	D₄	1	2	yes	—
33	#3667	—	2	2	no	—
34	#3667	—	2	2	no	—
35	#3667	—	2	2	no	—
36	C₂×C₉⋊C₈	—	2	1	no	—
37	C₂×C₉⋊C₈	D₄	1	4	yes	—
38	#2091	—	9	1	no	Sylow
39	C₂²×C₄×D₃	—	3	1	no	—
40	C₂×C₃×M₁₆	C₂×D₃	1	2	yes	—
41	C₂×C₃×M₁₆	C₂×D₃	1	1	yes	characteristic
42	C₂×#208	—	3	2	no	—
43	C₂×#208	—	3	1	no	—
44	C₃×#108	—	2	2	no	—
45	#822	—	6	4	no	—
46	(C₂×D₃)⋊C₈	—	6	4	no	—
47	(C₂×D₃)⋊C₈	—	6	2	no	—
48	C₂³×C₉	C₂×Q₈	1	1	yes	characteristic
49	C₂²×D₉	—	2	2	no	—
50	C₂²×D₉	—	2	1	no	—
51	C₂²×D₉	—	4	2	no	—
52	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₉	#41	1	2	yes	—
55	C₂×C₄×C₉	#41	1	1	yes	characteristic
56	C₂×Dic₉	—	2	2	no	—
57	C₂×Dic₉	—	2	1	no	—
58	C₄×D₉	—	4	2	no	—
59	C₄×D₉	—	4	1	no	—
60	C₄×D₉	—	4	1	no	—
61	C₈×C₉	—	2	2	no	—
62	C₈×C₉	—	4	2	no	—
63	C₉⋊C₈	—	2	4	no	—
64	C₉⋊C₈	—	4	1	no	—
65	#430	—	9	2	no	—
66	#430	D₉	1	1	yes	characteristic, complemented
67	#407	—	9	2	no	—
68	#407	—	9	1	no	—
69	#338	—	9	1	no	—
70	C₂³×D₃	—	3	1	no	—
71	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
72	C₂²×Dic₃	—	3	1	no	—
73	C₂×C₄×D₃	—	6	2	no	—
74	C₂×C₄×D₃	—	6	2	no	—
75	C₂×C₄×D₃	—	6	1	no	—
76	C₂×C₄×D₃	—	6	1	no	—
77	C₃×M₁₆	—	2	2	no	—
78	C₃×M₁₆	—	2	2	no	—
79	C₃×M₁₆	—	2	2	no	—
80	C₂×C₃×C₈	—	2	2	no	—
81	C₂×C₃×C₈	#67	1	2	yes	—
82	#208	—	6	2	no	—
83	#208	—	6	2	no	—
84	#208	—	6	2	no	—
85	C₂×C₃⋊C₈	—	3	4	no	—
86	C₂×C₃⋊C₈	—	6	1	no	—
87	C₂²×C₉	—	4	1	no	—
88	C₂²×C₉	C₂²⋊Q₈	1	2	yes	—
89	C₂²×C₉	C₂²⋊Q₈	1	1	yes	characteristic
90	C₂×D₉	—	2	4	no	—
91	C₂×D₉	—	2	2	no	—
92	C₂×D₉	—	4	2	no	—
93	C₂×D₉	—	4	1	no	—
94	C₂×D₉	—	4	1	no	—
95	C₄×C₉	#123	1	2	yes	—
96	C₄×C₉	#123	1	1	yes	characteristic
97	C₄×C₉	#127	1	1	yes	characteristic
98	Dic₉	—	4	1	no	—
99	C₂³×C₄	—	9	1	no	—
100	C₂×M₁₆	—	9	2	no	—
101	C₂×M₁₆	—	9	1	no	—
102	C₂×M₁₆	C₂×D₉	1	2	yes	—
103	C₂×M₁₆	C₂×D₉	1	1	yes	characteristic
104	#108	—	2	2	no	—
105	#108	—	18	4	no	—
106	C₂²⋊C₈	—	18	4	no	—
107	C₂²⋊C₈	—	18	2	no	—
108	C₂³×C₃	D₃×Q₈	1	1	yes	characteristic
109	C₂²×D₃	—	6	2	no	—
110	C₂²×D₃	—	6	1	no	—
111	C₂²×D₃	—	12	2	no	—
112	C₂×C₃×C₄	#235	1	2	yes	—
113	C₂×C₃×C₄	D₃×D₄	1	2	yes	—
114	C₂×C₃×C₄	#239	1	1	yes	characteristic
115	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
116	C₂×Dic₃	—	6	2	no	—
117	C₂×Dic₃	—	6	1	no	—
118	C₄×D₃	—	12	2	no	—
119	C₄×D₃	—	12	1	no	—
120	C₄×D₃	—	12	1	no	—
121	C₃×C₈	—	2	2	no	—
122	C₃×C₈	—	4	2	no	—
123	C₃⋊C₈	—	6	4	no	—
124	C₃⋊C₈	—	12	1	no	—
125	C₂×C₉	—	2	2	no	—
126	C₂×C₉	—	2	1	no	—
127	C₂×C₉	#399	1	1	yes	characteristic
128	D₉	—	4	2	no	—
129	C₂⁴	—	9	1	no	—
130	C₂²×C₄	—	9	1	no	—
131	C₂²×C₄	—	18	2	no	—
132	C₂²×C₄	—	18	2	no	—
133	C₂²×C₄	—	18	1	no	—
134	C₂²×C₄	—	18	1	no	—
135	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
136	M₁₆	—	2	2	no	—
137	M₁₆	—	2	2	no	—
138	M₁₆	—	2	2	no	—
139	M₁₆	—	18	2	no	—
140	M₁₆	—	18	2	no	—
141	M₁₆	—	18	2	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	9	4	no	—
144	C₂×C₈	—	18	1	no	—
145	C₂×C₈	#611	1	2	yes	—
146	C₂²×C₃	—	4	1	no	—
147	C₂²×C₃	#896	1	2	yes	—
148	C₂²×C₃	#946	1	1	yes	characteristic
149	C₂×D₃	—	6	4	no	—
150	C₂×D₃	—	6	2	no	—
151	C₂×D₃	—	12	2	no	—
152	C₂×D₃	—	12	1	no	—
153	C₂×D₃	—	12	1	no	—
154	C₃×C₄	#894	1	2	yes	—
155	C₃×C₄	#929	1	1	yes	characteristic
156	C₃×C₄	#940	1	1	yes	characteristic
157	Dic₃	—	12	1	no	—
158	C₉	#2091	1	1	yes	characteristic, complemented, Sylow
159	C₂³	—	18	1	no	—
160	C₂³	—	18	2	no	—
161	C₂³	—	36	2	no	—
162	C₂³	Q₈×D₉	1	1	yes	characteristic
163	C₂×C₄	—	18	1	no	—
164	C₂×C₄	—	18	2	no	—
165	C₂×C₄	—	36	1	no	—
166	C₂×C₄	—	36	1	no	—
167	C₂×C₄	—	36	2	no	—
168	C₂×C₄	#3697	1	2	yes	—
169	C₂×C₄	D₄×D₉	1	2	yes	—
170	C₂×C₄	#3701	1	1	yes	characteristic
171	C₂×C₄	C₂×#611	1	1	yes	characteristic
172	C₈	—	2	2	no	—
173	C₈	—	4	2	no	—
174	C₈	—	18	4	no	—
175	C₈	—	36	1	no	—
176	C₂×C₃	—	2	2	no	—
177	C₂×C₃	—	2	1	no	—
178	C₂×C₃	#4993	1	1	yes	characteristic
179	D₃	—	12	2	no	—
180	C₂²	—	4	1	no	—
181	C₂²	—	18	2	no	—
182	C₂²	—	18	4	no	—
183	C₂²	—	36	1	no	—
184	C₂²	—	36	2	no	—
185	C₂²	—	36	1	no	—
186	C₂²	#63458	1	2	yes	—
187	C₂²	#63508	1	1	yes	characteristic
188	C₄	—	36	1	no	—
189	C₄	#63456	1	2	yes	—
190	C₄	#63491	1	1	yes	characteristic
191	C₄	#63502	1	1	yes	characteristic
192	C₃	#70729	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	2	no	—
195	C₂	—	36	2	no	—
196	C₂	#10588673	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₉×#430

C₂

yes

maximal, characteristic, complemented

#10588808

C₂

yes

maximal

#10588808

C₂

yes

maximal, characteristic

#10588799

C₂

yes

maximal, complemented

C₉⋊#338

C₂

yes

maximal, characteristic, complemented

#70729

—

maximal

C₂²×C₄×D₉

C₂²

yes

characteristic

C₂×C₉×M₁₆

C₂²

yes

—

C₂×C₉×M₁₆

C₂²

yes

characteristic

C₂×#3667

C₂²

yes

characteristic

C₂×#3667

C₂²

yes

—

C₉×#108

—

#63381

—

(C₂×D₉)⋊C₈

—

(C₂×D₉)⋊C₈

—

C₃×#430

D₃

yes

—

—

—

—

C₂³×D₉

Q₈

yes

characteristic

C₂²×C₄×C₉

C₂³

yes

characteristic

C₂²×Dic₉

Q₈

yes

characteristic

C₂×C₄×D₉

—

C₂×C₄×D₉

—

C₂×C₄×D₉

—

C₂×C₄×D₉

—

C₉×M₁₆

—

C₉×M₁₆

—

C₉×M₁₆

—

C₂×C₈×C₉

—

C₂×C₈×C₉

D₄

yes

—

#3667

—

#3667

—

#3667

—

C₂×C₉⋊C₈

—

C₂×C₉⋊C₈

D₄

yes

—

#2091

—

Sylow

C₂²×C₄×D₃

—

C₂×C₃×M₁₆

C₂×D₃

yes

—

C₂×C₃×M₁₆

C₂×D₃

yes

—

—

—

—

(C₂×D₃)⋊C₈

—

(C₂×D₃)⋊C₈

—

C₂³×C₉

C₂×Q₈

yes

characteristic

C₂²×D₉

—

C₂²×D₉

—

C₂²×D₉

—

C₂×C₄×C₉

C₂×D₄

yes

—

C₂×C₄×C₉

C₂×D₄

yes

characteristic

C₂×C₄×C₉

#41

yes

—

C₂×C₄×C₉

#41

yes

characteristic

C₂×Dic₉

—

C₂×Dic₉

—

C₄×D₉

—

C₄×D₉

—

C₄×D₉

—

C₈×C₉

—

C₈×C₉

—

C₉⋊C₈

—

C₉⋊C₈

—

#430

—

#430

D₉

yes

characteristic, complemented

#407

—

#407

—

#338

—

C₂³×D₃

—

C₂²×C₃×C₄

C₂²×D₃

yes

characteristic

C₂²×Dic₃

—

C₂×C₄×D₃

—

C₂×C₄×D₃

—

C₂×C₄×D₃

—

C₂×C₄×D₃

—

C₃×M₁₆

—

C₃×M₁₆

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

#67

yes

—

#208

—

#208

—

#208

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₉

—

C₂²×C₉

C₂²⋊Q₈

yes

—

C₂²×C₉

C₂²⋊Q₈

yes

characteristic

C₂×D₉

—

C₂×D₉

—

C₂×D₉

—

C₂×D₉

—

C₂×D₉

—

C₄×C₉

#123

yes

—

C₄×C₉

yes

C₄×C₉

yes

Dic₉

—

C₂³×C₄

—

100

C₂×M₁₆

—

101

C₂×M₁₆

—

102

C₂×M₁₆

C₂×D₉

yes

—

103

C₂×M₁₆

C₂×D₉

yes

characteristic

104

#108

—

105

#108

—

106

C₂²⋊C₈

—

107

C₂²⋊C₈

—

108

C₂³×C₃

D₃×Q₈

yes

characteristic

109

C₂²×D₃

—

110

C₂²×D₃

—

111

C₂²×D₃

—

112

C₂×C₃×C₄

#235

yes

—

113

C₂×C₃×C₄

D₃×D₄

yes

—

114

C₂×C₃×C₄

#239

yes

characteristic

115

C₂×C₃×C₄

C₂×#67

yes

characteristic

116

C₂×Dic₃

—

117

C₂×Dic₃

—

118

C₄×D₃

—

119

C₄×D₃

—

120

C₄×D₃

—

121

C₃×C₈

—

122

C₃×C₈

—

123

C₃⋊C₈

—

124

C₃⋊C₈

—

125

C₂×C₉

—

126

C₂×C₉

—

127

C₂×C₉

#399

yes

characteristic

128

D₉

—

129

C₂⁴

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

—

134

C₂²×C₄

—

135

C₂²×C₄

C₂²×D₉

yes

characteristic

136

M₁₆

—

137

M₁₆

—

138

M₁₆

—

139

M₁₆

—

140

M₁₆

—

141

M₁₆

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

#611

yes

—

146

C₂²×C₃

—

147

C₂²×C₃

#896

yes

—

148

C₂²×C₃

#946

yes

characteristic

149

C₂×D₃

—

150

C₂×D₃

—

151

C₂×D₃

—

152

C₂×D₃

—

153

C₂×D₃

—

154

C₃×C₄

#894

yes

—

155

C₃×C₄

#929

yes

characteristic

156

C₃×C₄

#940

yes

characteristic

157

Dic₃

—

158

C₉

#2091

yes

characteristic, complemented, Sylow

159

C₂³

—

160

C₂³

—

161

C₂³

—

162

C₂³

Q₈×D₉

yes

characteristic

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

#3697

yes

—

169

C₂×C₄

D₄×D₉

yes

—

170

C₂×C₄

#3701

yes

characteristic

171

C₂×C₄

C₂×#611

yes

characteristic

172

C₈

—

173

C₈

—

174

C₈

—

175

C₈

—

176

C₂×C₃

—

177

C₂×C₃

—

178

C₂×C₃

#4993

yes

characteristic

179

D₃

—

180

C₂²

—

181

C₂²

—

182

C₂²

—

183

C₂²

—

184

C₂²

—

185

C₂²

—

186

C₂²

#63458

yes

—

187

C₂²

#63508

yes

characteristic

188

C₄

—

189

C₄

#63456

yes

—

190

C₄

#63491

yes

characteristic

191

C₄

#63502

yes

characteristic

192

C₃

#70729

yes

characteristic

193

C₂

—

194

C₂

—

195

C₂

—

196

C₂

#10588673

yes

characteristic

	subgroup	G/H
center	C₄	#63502
commutator, cosocle	C₂²×C₄×C₉	C₂³
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₉×#430	C₂
socle	C₂×C₃	#4993
prime radical	C₂³×D₉	Q₈
hypercenter	#430	D₉
hypocenter	C₉	#2091
LCS	G ▷ C₂²×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #430

subgroup

G/H

center

C₄

#63502

commutator, cosocle

C₂²×C₄×C₉

C₂³

Frattini

C₂²×C₃×C₄

C₂²×D₃

C₂

C₂×C₃

C₂³×D₉

Q₈

D₉

C₉

G ▷ C₂²×C₄×C₉ ▷ C₂×C₉ ▷ C₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #430

N	H ≅ G/N	type
1. C₉×#430	C₂	semidirect
4. #10588799	C₂	semidirect
5. C₉⋊#338	C₂	semidirect
66. #430	D₉	semidirect
158. C₉	#2091	semidirect

H ≅ G/N

type

1. C₉×#430

C₂

semidirect

4. #10588799

C₂

semidirect

5. C₉⋊#338

C₂

semidirect

66. #430

D₉

semidirect

158. C₉

#2091

semidirect

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

subgroup	index
C₉×#430	2
#10588808	2
#10588799	2
C₉⋊#338	2
#70729	3

subgroup

index

Subgroups — #11795355

Conjugacy classes of subgroups

Complements

Maximal subgroups