Group #11895453 — subgroups

Subgroups — #11895453

Order 1152.

504 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 182 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×#557	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₁₈	C₂	1	1	yes	maximal, characteristic
3	#10588763	C₂	1	3	yes	maximal
4	#10588759	C₂	1	3	yes	maximal
5	C₄×#3674	C₂	1	3	yes	maximal
6	#10588703	C₂	1	3	yes	maximal
7	Q₈×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic
8	#83696	—	3	1	no	maximal
9	C₉×C₄⋊Q₈	C₂²	1	3	yes	—
10	C₉×#125	—	2	3	no	—
11	C₄×Q₈×C₉	—	2	1	no	—
12	C₄×Q₈×C₉	C₂²	1	3	yes	—
13	C₄×Q₈×C₉	C₂²	1	1	yes	characteristic
14	Q₈×Dic₉	—	2	1	no	—
15	C₂×#3674	C₂²	1	3	yes	—
16	#63450	—	2	3	no	—
17	#63431	C₂²	1	3	yes	—
18	C₄×Dic₁₈	C₂²	1	3	yes	—
19	#63395	C₂²	1	6	yes	—
20	#63395	C₂²	1	3	yes	—
21	#63368	C₂²	1	1	yes	characteristic
22	#63366	C₂²	1	3	yes	—
23	C₄⋊(C₉⋊C₈)	C₂²	1	3	yes	—
24	C₄×C₉⋊C₈	C₂²	1	3	yes	—
25	C₃×#557	D₃	1	1	yes	characteristic
26	Q₈×Dic₆	—	3	1	no	—
27	#5090	—	3	3	no	—
28	#5079	—	3	3	no	—
29	C₄×#216	—	3	3	no	—
30	Q₈×C₃⋊C₈	—	3	1	no	—
31	#5021	—	3	3	no	—
32	C₂×Q₈×C₉	D₄	1	1	yes	characteristic
33	C₂×Q₈×C₉	C₂³	1	3	yes	—
34	C₂×Dic₁₈	—	2	3	no	—
35	C₂×Dic₁₈	C₂³	1	1	yes	characteristic
36	C₉×C₄⋊C₄	—	2	3	no	—
37	C₉×C₄⋊C₄	—	2	3	no	—
38	C₉×C₄⋊C₄	—	2	3	no	—
39	C₉×C₄⋊C₄	D₄	1	3	yes	—
40	C₉×C₄⋊C₄	C₂³	1	1	yes	characteristic
41	C₄²×C₉	—	2	3	no	—
42	C₄²×C₉	C₂³	1	3	yes	—
43	#3674	—	2	6	no	—
44	C₄⋊Dic₉	—	2	3	no	—
45	C₄⋊Dic₉	C₂³	1	3	yes	—
46	Dic₉⋊C₄	—	2	3	no	—
47	C₄×Dic₉	—	2	3	no	—
48	C₂×C₉⋊C₈	C₂³	1	3	yes	—
49	C₂×C₉⋊C₈	C₂³	1	1	yes	characteristic
50	#3363	—	9	1	no	Sylow
51	C₃×C₄⋊Q₈	C₂×D₃	1	3	yes	—
52	C₃×#125	—	2	3	no	—
53	C₃×C₄×Q₈	—	2	1	no	—
54	C₃×C₄×Q₈	C₂×D₃	1	3	yes	—
55	C₃×C₄×Q₈	C₂×D₃	1	1	yes	characteristic
56	Q₈×Dic₃	—	6	1	no	—
57	C₂×#216	—	3	3	no	—
58	#888	—	6	3	no	—
59	#869	—	3	3	no	—
60	C₄×Dic₆	—	3	3	no	—
61	#835	—	3	6	no	—
62	#835	—	3	3	no	—
63	#810	—	3	1	no	—
64	#807	—	3	3	no	—
65	C₄⋊(C₃⋊C₈)	—	3	3	no	—
66	C₄×C₃⋊C₈	—	3	3	no	—
67	Q₈×C₉	—	2	3	no	—
68	Q₈×C₉	—	2	3	no	—
69	Q₈×C₉	Q₁₆	1	4	yes	complemented
70	C₂×C₄×C₉	—	2	1	no	—
71	C₂×C₄×C₉	—	2	3	no	—
72	C₂×C₄×C₉	C₂×D₄	1	3	yes	—
73	C₂×C₄×C₉	C₂×D₄	1	3	yes	—
74	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
75	C₂×Dic₉	—	2	1	no	—
76	C₂×Dic₉	—	2	3	no	—
77	Dic₁₈	—	2	1	no	—
78	Dic₁₈	—	8	3	no	—
79	Dic₁₈	#41	1	2	yes	—
80	C₉⋊C₈	—	2	3	no	—
81	C₉⋊C₈	—	2	1	no	—
82	Q₈²	—	9	1	no	—
83	#557	D₉	1	1	yes	characteristic
84	#487	—	9	3	no	—
85	#477	—	9	3	no	—
86	#462	—	9	3	no	—
87	C₈×Q₈	—	9	1	no	—
88	C₄×Q₁₆	—	9	3	no	—
89	C₂×C₃×Q₈	#67	1	1	yes	characteristic
90	C₂×C₃×Q₈	C₂²×D₃	1	3	yes	—
91	C₂×Dic₆	—	3	1	no	—
92	C₂×Dic₆	—	6	3	no	—
93	C₃×C₄⋊C₄	—	2	3	no	—
94	C₃×C₄⋊C₄	—	2	3	no	—
95	C₃×C₄⋊C₄	—	2	3	no	—
96	C₃×C₄⋊C₄	#67	1	3	yes	—
97	C₃×C₄⋊C₄	C₂²×D₃	1	1	yes	characteristic
98	C₃×C₄²	—	2	3	no	—
99	C₃×C₄²	C₂²×D₃	1	3	yes	—
100	#216	—	6	6	no	—
101	C₄⋊Dic₃	—	3	3	no	—
102	C₄⋊Dic₃	—	6	3	no	—
103	Dic₃⋊C₄	—	6	3	no	—
104	C₄×Dic₃	—	6	3	no	—
105	C₂×C₃⋊C₈	—	3	1	no	—
106	C₂×C₃⋊C₈	—	3	3	no	—
107	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
108	C₄×C₉	—	2	3	no	—
109	C₄×C₉	—	4	3	no	—
110	C₄×C₉	—	4	1	no	—
111	C₄×C₉	C₂×Q₁₆	1	6	yes	—
112	C₄×C₉	C₂×#41	1	1	yes	characteristic
113	C₄×C₉	2₋¹⁺⁴	1	1	yes	characteristic
114	Dic₉	—	2	2	no	—
115	Dic₉	—	4	3	no	—
116	C₂×Q₁₆	—	9	3	no	—
117	C₄⋊Q₈	—	9	3	no	—
118	C₄⋊Q₈	—	18	3	no	—
119	C₄⋊Q₈	C₂×D₉	1	3	yes	—
120	#125	—	2	3	no	—
121	C₄×Q₈	—	2	1	no	—
122	C₄×Q₈	—	9	3	no	—
123	C₄×Q₈	—	18	1	no	—
124	C₄×Q₈	C₂×D₉	1	3	yes	—
125	C₄×Q₈	C₂×D₉	1	1	yes	characteristic
126	#107	—	9	3	no	—
127	C₄⋊C₈	—	9	3	no	—
128	#103	—	9	3	no	—
129	#103	—	9	6	no	—
130	#103	—	9	1	no	—
131	C₄×C₈	—	9	3	no	—
132	C₃×Q₈	—	2	3	no	—
133	C₃×Q₈	—	2	3	no	—
134	C₃×Q₈	#216	1	4	yes	—
135	C₂×C₃×C₄	—	2	1	no	—
136	C₂×C₃×C₄	—	2	3	no	—
137	C₂×C₃×C₄	C₂×#67	1	3	yes	—
138	C₂×C₃×C₄	C₂×#67	1	3	yes	—
139	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
140	C₂×Dic₃	—	6	1	no	—
141	C₂×Dic₃	—	6	3	no	—
142	Dic₆	—	3	2	no	—
143	Dic₆	—	6	1	no	—
144	Dic₆	—	24	3	no	—
145	C₃⋊C₈	—	6	3	no	—
146	C₃⋊C₈	—	6	1	no	—
147	C₂×C₉	#548	1	1	yes	characteristic
148	C₂×C₉	C₂²×Q₁₆	1	1	yes	characteristic
149	C₂×C₉	#577	1	1	yes	characteristic
150	C₂×Q₈	—	9	1	no	—
151	C₂×Q₈	—	18	3	no	—
152	C₂×Q₈	#611	1	1	yes	characteristic
153	C₂×Q₈	C₂²×D₉	1	3	yes	—
154	Q₁₆	—	18	6	no	—
155	C₂×C₈	—	9	1	no	—
156	C₂×C₈	—	9	3	no	—
157	C₄⋊C₄	—	2	3	no	—
158	C₄⋊C₄	—	2	3	no	—
159	C₄⋊C₄	—	2	3	no	—
160	C₄⋊C₄	—	9	3	no	—
161	C₄⋊C₄	—	18	3	no	—
162	C₄⋊C₄	—	18	3	no	—
163	C₄⋊C₄	#611	1	3	yes	—
164	C₄⋊C₄	C₂²×D₉	1	1	yes	characteristic
165	C₄²	—	2	3	no	—
166	C₄²	—	18	3	no	—
167	C₄²	C₂²×D₉	1	3	yes	—
168	C₂²×C₃	C₂²×#67	1	1	yes	characteristic
169	C₃×C₄	—	2	3	no	—
170	C₃×C₄	—	4	3	no	—
171	C₃×C₄	—	4	1	no	—
172	C₃×C₄	C₂×#216	1	6	yes	—
173	C₃×C₄	D₃×#41	1	1	yes	characteristic
174	C₃×C₄	#1010	1	1	yes	characteristic
175	Dic₃	—	6	2	no	—
176	Dic₃	—	12	3	no	—
177	C₉	#3363	1	1	yes	characteristic, complemented, Sylow
178	Q₈	—	2	3	no	—
179	Q₈	—	2	3	no	—
180	Q₈	—	9	2	no	—
181	Q₈	—	18	1	no	—
182	Q₈	—	72	3	no	—
183	Q₈	#3674	1	4	yes	complemented
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	3	no	—
186	C₂×C₄	—	18	1	no	—
187	C₂×C₄	—	18	3	no	—
188	C₂×C₄	C₂×#611	1	3	yes	—
189	C₂×C₄	C₂×#611	1	3	yes	—
190	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
191	C₈	—	18	3	no	—
192	C₈	—	18	1	no	—
193	C₂×C₃	C₂²×#216	1	1	yes	characteristic
194	C₂×C₃	#5831	1	1	yes	characteristic
195	C₂×C₃	#5837	1	1	yes	characteristic
196	C₂²	C₂²×#611	1	1	yes	characteristic
197	C₄	—	2	3	no	—
198	C₄	—	4	3	no	—
199	C₄	—	4	1	no	—
200	C₄	—	18	2	no	—
201	C₄	—	36	3	no	—
202	C₄	C₂×#3674	1	6	yes	—
203	C₄	#41×D₉	1	1	yes	characteristic
204	C₄	#63716	1	1	yes	characteristic
205	C₃	#83696	1	1	yes	characteristic
206	C₂	C₂²×#3674	1	1	yes	characteristic
207	C₂	#10589933	1	1	yes	characteristic
208	C₂	#10589939	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₉×#557

C₂

yes

maximal, characteristic

Q₈×Dic₁₈

C₂

yes

maximal, characteristic

#10588763

C₂

yes

maximal

#10588759

C₂

yes

maximal

C₄×#3674

C₂

yes

maximal

#10588703

C₂

yes

maximal

Q₈×C₉⋊C₈

C₂

yes

maximal, characteristic

#83696

—

maximal

C₉×C₄⋊Q₈

C₂²

yes

—

C₉×#125

—

C₄×Q₈×C₉

—

C₄×Q₈×C₉

C₂²

yes

—

C₄×Q₈×C₉

C₂²

yes

characteristic

Q₈×Dic₉

—

C₂×#3674

C₂²

yes

—

#63450

—

#63431

C₂²

yes

—

C₄×Dic₁₈

C₂²

yes

—

#63395

C₂²

yes

—

#63395

C₂²

yes

—

#63368

C₂²

yes

characteristic

#63366

C₂²

yes

—

C₄⋊(C₉⋊C₈)

C₂²

yes

—

C₄×C₉⋊C₈

C₂²

yes

—

C₃×#557

D₃

yes

characteristic

Q₈×Dic₆

—

#5090

—

#5079

—

C₄×#216

—

Q₈×C₃⋊C₈

—

#5021

—

C₂×Q₈×C₉

D₄

yes

characteristic

C₂×Q₈×C₉

C₂³

yes

—

C₂×Dic₁₈

—

C₂×Dic₁₈

C₂³

yes

characteristic

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

D₄

yes

—

C₉×C₄⋊C₄

C₂³

yes

characteristic

C₄²×C₉

—

C₄²×C₉

C₂³

yes

—

#3674

—

C₄⋊Dic₉

—

C₄⋊Dic₉

C₂³

yes

—

Dic₉⋊C₄

—

C₄×Dic₉

—

C₂×C₉⋊C₈

C₂³

yes

—

C₂×C₉⋊C₈

C₂³

yes

characteristic

#3363

—

Sylow

C₃×C₄⋊Q₈

C₂×D₃

yes

—

C₃×#125

—

C₃×C₄×Q₈

—

C₃×C₄×Q₈

C₂×D₃

yes

—

C₃×C₄×Q₈

C₂×D₃

yes

characteristic

Q₈×Dic₃

—

C₂×#216

—

#888

—

#869

—

C₄×Dic₆

—

—

—

—

—

C₄⋊(C₃⋊C₈)

—

C₄×C₃⋊C₈

—

Q₈×C₉

—

Q₈×C₉

—

Q₈×C₉

Q₁₆

yes

complemented

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂×D₄

yes

—

C₂×C₄×C₉

C₂×D₄

yes

—

C₂×C₄×C₉

C₂⁴

yes

characteristic

C₂×Dic₉

—

C₂×Dic₉

—

Dic₁₈

—

Dic₁₈

—

Dic₁₈

#41

yes

—

C₉⋊C₈

—

C₉⋊C₈

—

Q₈²

—

#557

D₉

yes

—

—

—

C₈×Q₈

—

C₄×Q₁₆

—

C₂×C₃×Q₈

#67

yes

characteristic

C₂×C₃×Q₈

C₂²×D₃

yes

—

C₂×Dic₆

—

C₂×Dic₆

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

#67

yes

—

C₃×C₄⋊C₄

C₂²×D₃

yes

characteristic

C₃×C₄²

—

C₃×C₄²

C₂²×D₃

yes

—

100

#216

—

101

C₄⋊Dic₃

—

102

C₄⋊Dic₃

—

103

Dic₃⋊C₄

—

104

C₄×Dic₃

—

105

C₂×C₃⋊C₈

—

106

C₂×C₃⋊C₈

—

107

C₂²×C₉

C₂²×D₄

yes

characteristic

108

C₄×C₉

—

109

C₄×C₉

—

110

C₄×C₉

—

111

C₄×C₉

C₂×Q₁₆

yes

—

112

C₄×C₉

C₂×#41

yes

characteristic

113

C₄×C₉

2₋¹⁺⁴

yes

characteristic

114

Dic₉

—

115

Dic₉

—

116

C₂×Q₁₆

—

117

C₄⋊Q₈

—

118

C₄⋊Q₈

—

119

C₄⋊Q₈

C₂×D₉

yes

—

120

#125

—

121

C₄×Q₈

—

122

C₄×Q₈

—

123

C₄×Q₈

—

124

C₄×Q₈

C₂×D₉

yes

—

125

C₄×Q₈

C₂×D₉

yes

characteristic

126

#107

—

127

C₄⋊C₈

—

128

#103

—

129

#103

—

130

#103

—

131

C₄×C₈

—

132

C₃×Q₈

—

133

C₃×Q₈

—

134

C₃×Q₈

#216

yes

—

135

C₂×C₃×C₄

—

136

C₂×C₃×C₄

—

137

C₂×C₃×C₄

C₂×#67

yes

—

138

C₂×C₃×C₄

C₂×#67

yes

—

139

C₂×C₃×C₄

C₂³×D₃

yes

characteristic

140

C₂×Dic₃

—

141

C₂×Dic₃

—

142

Dic₆

—

143

Dic₆

—

144

Dic₆

—

145

C₃⋊C₈

—

146

C₃⋊C₈

—

147

C₂×C₉

#548

yes

characteristic

148

C₂×C₉

C₂²×Q₁₆

yes

characteristic

149

C₂×C₉

#577

yes

characteristic

150

C₂×Q₈

—

151

C₂×Q₈

—

152

C₂×Q₈

#611

yes

characteristic

153

C₂×Q₈

C₂²×D₉

yes

—

154

Q₁₆

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

#611

yes

—

164

C₄⋊C₄

C₂²×D₉

yes

characteristic

165

C₄²

—

166

C₄²

—

167

C₄²

C₂²×D₉

yes

—

168

C₂²×C₃

C₂²×#67

yes

characteristic

169

C₃×C₄

—

170

C₃×C₄

—

171

C₃×C₄

—

172

C₃×C₄

C₂×#216

yes

—

173

C₃×C₄

D₃×#41

yes

characteristic

174

C₃×C₄

#1010

yes

characteristic

175

Dic₃

—

176

Dic₃

—

177

C₉

#3363

yes

characteristic, complemented, Sylow

178

Q₈

—

179

Q₈

—

180

Q₈

—

181

Q₈

—

182

Q₈

—

183

Q₈

#3674

yes

complemented

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

C₂×#611

yes

—

189

C₂×C₄

C₂×#611

yes

—

190

C₂×C₄

C₂³×D₉

yes

characteristic

191

C₈

—

192

C₈

—

193

C₂×C₃

C₂²×#216

yes

characteristic

194

C₂×C₃

#5831

yes

characteristic

195

C₂×C₃

#5837

yes

characteristic

196

C₂²

C₂²×#611

yes

characteristic

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

—

201

C₄

—

202

C₄

C₂×#3674

yes

—

203

C₄

#41×D₉

yes

characteristic

204

C₄

#63716

yes

characteristic

205

C₃

#83696

yes

characteristic

206

C₂

C₂²×#3674

yes

characteristic

207

C₂

#10589933

yes

characteristic

208

C₂

#10589939

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×#611
commutator, cosocle	C₂×C₄×C₉	C₂⁴
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	C₉×#557	C₂
socle, prime radical	C₂²×C₃	C₂²×#67
hypercenter	#557	D₉
hypocenter	C₉	#3363
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #557

subgroup

G/H

center

C₂²

C₂²×#611

commutator, cosocle

C₂×C₄×C₉

C₂⁴

Frattini

C₂×C₃×C₄

C₂³×D₃

Fitting

C₉×#557

C₂

socle, prime radical

C₂²×C₃

C₂²×#67

D₉

C₉

G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #557

N	H ≅ G/N	type
69. Q₈×C₉	Q₁₆	semidirect
177. C₉	#3363	semidirect
183. Q₈	#3674	semidirect

H ≅ G/N

type

69. Q₈×C₉

Q₁₆

semidirect

177. C₉

#3363

semidirect

183. Q₈

#3674

semidirect

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

subgroup	index
C₉×#557	2
Q₈×Dic₁₈	2
#10588763	2
#10588759	2
C₄×#3674	2
#10588703	2
Q₈×C₉⋊C₈	2
#83696	3

subgroup

index

Q₈×Dic₁₈

Q₈×C₉⋊C₈

Subgroups — #11895453

Conjugacy classes of subgroups

Complements

Maximal subgroups