Group #10694280 — subgroups

Subgroups — #10694280

Order 768.

538 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 167 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	#73311	C₂	1	1	yes	maximal, characteristic
2	C₃×#1831	C₂	1	1	yes	maximal, characteristic
3	C₂×#4486	C₂	1	4	yes	maximal
4	#69391	C₂	1	1	yes	maximal, characteristic
5	#9705	—	3	1	no	maximal, Sylow
6	C₂×#869	—	2	1	no	—
7	C₂×#869	—	2	1	no	—
8	C₂×#869	C₂²	1	2	yes	—
9	C₂×C₃×C₄⋊C₈	C₂²	1	2	yes	—
10	C₃×C₄³	C₂²	1	1	yes	characteristic
11	#4935	—	2	2	no	—
12	#4935	C₄	1	4	yes	—
13	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
14	C₂×#803	—	2	1	no	—
15	C₃×#336	—	2	2	no	—
16	#4524	—	2	1	no	—
17	#4486	—	2	8	no	—
18	#2868	—	3	1	no	—
19	#1831	D₃	1	1	yes	characteristic
20	#1767	—	3	1	no	—
21	C₂×#332	—	3	4	no	—
22	C₂²×Dic₆	—	2	2	no	—
23	C₂²×C₃×C₈	—	2	2	no	—
24	C₂×C₃×C₄²	C₂×C₄	1	2	yes	—
25	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
26	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
27	C₂×C₃×C₄²	D₄	1	2	yes	—
28	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊Dic₃	—	2	4	no	—
30	C₂×C₄⋊Dic₃	—	2	4	no	—
31	C₂×C₄⋊Dic₃	—	2	2	no	—
32	C₂×C₄⋊Dic₃	C₂×C₄	1	4	yes	—
33	C₂²×C₃⋊C₈	—	2	2	no	—
34	#869	—	2	4	no	—
35	#869	—	2	4	no	—
36	#869	—	4	4	no	—
37	#869	—	4	2	no	—
38	#869	—	4	2	no	—
39	C₃×C₄⋊C₈	—	2	4	no	—
40	C₄⋊(C₃⋊C₈)	—	2	2	no	—
41	C₄⋊(C₃⋊C₈)	D₄	1	4	yes	—
42	#803	—	4	2	no	—
43	C₂×C₄⋊Q₈	—	3	2	no	—
44	C₂×C₄⋊Q₈	—	6	1	no	—
45	C₂×C₄⋊Q₈	—	6	1	no	—
46	C₂×C₄⋊C₈	—	3	2	no	—
47	C₂×C₄⋊C₈	C₂×D₃	1	2	yes	—
48	C₂×#97	—	6	1	no	—
49	#384	—	3	4	no	—
50	#384	—	6	2	no	—
51	C₄³	C₂×D₃	1	1	yes	characteristic
52	#336	—	2	2	no	—
53	#336	—	6	1	no	—
54	#332	—	6	8	no	—
55	C₂²×C₃×C₄	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₃×C₄	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
60	C₂²×Dic₃	—	2	4	no	—
61	C₂×Dic₆	—	4	4	no	—
62	C₂×Dic₆	—	4	2	no	—
63	C₂×Dic₆	—	4	2	no	—
64	C₂×Dic₆	—	4	4	no	—
65	C₂×C₃×C₈	—	4	2	no	—
66	C₂×C₃×C₈	—	4	4	no	—
67	C₃×C₄²	—	2	4	no	—
68	C₃×C₄²	—	2	2	no	—
69	C₃×C₄²	—	2	4	no	—
70	C₃×C₄²	D₈	1	2	yes	—
71	C₃×C₄²	SD₁₆	1	2	yes	—
72	C₃×C₄²	C₂×D₄	1	2	yes	—
73	C₃×C₄²	#41	1	2	yes	—
74	C₄⋊Dic₃	—	2	8	no	—
75	C₄⋊Dic₃	—	4	8	no	—
76	C₄⋊Dic₃	—	4	8	no	—
77	C₄⋊Dic₃	—	4	2	no	—
78	C₄⋊Dic₃	—	4	2	no	—
79	C₂×C₃⋊C₈	—	2	4	no	—
80	C₂×C₃⋊C₈	—	4	2	no	—
81	C₂×C₃⋊C₈	—	4	2	no	—
82	C₂²×Q₈	—	6	2	no	—
83	C₂²×C₈	—	2	2	no	—
84	C₂²×C₈	—	6	2	no	—
85	C₄⋊Q₈	—	6	4	no	—
86	C₄⋊Q₈	—	6	4	no	—
87	C₄⋊Q₈	—	12	4	no	—
88	C₄⋊Q₈	—	12	2	no	—
89	C₄⋊Q₈	—	12	2	no	—
90	C₂×C₄⋊C₄	—	3	4	no	—
91	C₂×C₄⋊C₄	—	6	4	no	—
92	C₂×C₄⋊C₄	—	6	4	no	—
93	C₂×C₄⋊C₄	—	6	2	no	—
94	C₂×C₄²	C₄×D₃	1	2	yes	—
95	C₂×C₄²	D₁₂	1	1	yes	characteristic
96	C₂×C₄²	D₁₂	1	1	yes	characteristic
97	C₂×C₄²	#67	1	2	yes	—
98	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
99	C₄⋊C₈	—	2	4	no	—
100	C₄⋊C₈	—	3	4	no	—
101	C₄⋊C₈	—	6	2	no	—
102	#97	—	12	2	no	—
103	C₂³×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₃×C₄	—	2	2	no	—
105	C₂×C₃×C₄	—	2	2	no	—
106	C₂×C₃×C₄	—	2	2	no	—
107	C₂×C₃×C₄	—	2	2	no	—
108	C₂×C₃×C₄	—	2	2	no	—
109	C₂×C₃×C₄	#101	1	4	yes	—
110	C₂×C₃×C₄	#102	1	4	yes	—
111	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
112	C₂×C₃×C₄	#121	1	2	yes	—
113	C₂×C₃×C₄	#124	1	1	yes	characteristic
114	C₂×C₃×C₄	#127	1	1	yes	characteristic
115	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
116	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
117	C₂×C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
118	C₂×C₃×C₄	#137	1	4	yes	—
119	C₂×Dic₃	—	4	4	no	—
120	C₂×Dic₃	—	4	4	no	—
121	C₂×Dic₃	—	4	4	no	—
122	Dic₆	—	8	4	no	—
123	Dic₆	—	8	4	no	—
124	C₃×C₈	—	8	2	no	—
125	C₃⋊C₈	—	4	4	no	—
126	C₂×Q₈	—	12	4	no	—
127	C₂×Q₈	—	12	2	no	—
128	C₂×Q₈	—	12	2	no	—
129	C₂×Q₈	—	12	4	no	—
130	C₂²×C₄	—	6	4	no	—
131	C₂²×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
132	C₂²×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
133	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
134	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
135	C₂²×C₄	C₂×#67	1	1	yes	characteristic
136	C₂×C₈	—	4	2	no	—
137	C₂×C₈	—	4	4	no	—
138	C₂×C₈	—	6	4	no	—
139	C₂×C₈	—	12	2	no	—
140	C₂×C₈	—	12	2	no	—
141	C₄⋊C₄	—	6	8	no	—
142	C₄⋊C₄	—	12	8	no	—
143	C₄⋊C₄	—	12	8	no	—
144	C₄⋊C₄	—	12	2	no	—
145	C₄⋊C₄	—	12	2	no	—
146	C₄²	—	2	4	no	—
147	C₄²	—	2	2	no	—
148	C₄²	—	2	4	no	—
149	C₄²	#204	1	2	yes	—
150	C₄²	D₂₄	1	2	yes	—
151	C₄²	D₃×D₄	1	2	yes	—
152	C₄²	#239	1	2	yes	—
153	C₂²×C₃	#390	1	1	yes	characteristic
154	C₂²×C₃	C₂×#101	1	2	yes	—
155	C₂²×C₃	C₂×#102	1	1	yes	characteristic
156	C₂²×C₃	#418	1	1	yes	characteristic
157	C₂²×C₃	#419	1	2	yes	—
158	C₃×C₄	—	2	4	no	—
159	C₃×C₄	—	2	4	no	—
160	C₃×C₄	—	2	2	no	—
161	C₃×C₄	#459	1	2	yes	—
162	C₃×C₄	#460	1	2	yes	—
163	C₃×C₄	#491	1	2	yes	—
164	C₃×C₄	#497	1	2	yes	—
165	Dic₃	—	8	4	no	—
166	C₂³	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
167	Q₈	—	24	4	no	—
168	Q₈	—	24	4	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	12	4	no	—
175	C₂×C₄	—	12	4	no	—
176	C₂×C₄	—	12	4	no	—
177	C₂×C₄	#821	1	4	yes	—
178	C₂×C₄	#824	1	4	yes	—
179	C₂×C₄	#893	1	2	yes	—
180	C₂×C₄	#895	1	2	yes	—
181	C₂×C₄	C₂×#204	1	1	yes	characteristic
182	C₂×C₄	C₂×D₂₄	1	1	yes	characteristic
183	C₂×C₄	#908	1	2	yes	—
184	C₂×C₄	#940	1	1	yes	characteristic
185	C₂×C₄	#947	1	1	yes	characteristic
186	C₂×C₄	#951	1	4	yes	—
187	C₈	—	8	2	no	—
188	C₈	—	12	4	no	—
189	C₂×C₃	#1546	1	2	yes	—
190	C₂×C₃	#1556	1	2	yes	—
191	C₂×C₃	#1960	1	1	yes	characteristic
192	C₂×C₃	#1966	1	1	yes	characteristic
193	C₂×C₃	#1982	1	1	yes	characteristic
194	C₂²	#4988	1	1	yes	characteristic
195	C₂²	#5004	1	2	yes	—
196	C₂²	C₂×#821	1	1	yes	characteristic
197	C₂²	#5131	1	1	yes	characteristic
198	C₂²	C₂×#824	1	2	yes	—
199	C₄	—	2	2	no	—
200	C₄	—	2	4	no	—
201	C₄	—	2	4	no	—
202	C₄	—	24	4	no	—
203	C₄	#4843	1	2	yes	—
204	C₄	#4848	1	2	yes	—
205	C₄	#5060	1	2	yes	—
206	C₄	#5069	1	2	yes	—
207	C₃	#9705	1	1	yes	characteristic, complemented, Sylow
208	C₂	#69552	1	1	yes	characteristic
209	C₂	#70457	1	1	yes	characteristic
210	C₂	#70522	1	1	yes	characteristic
211	C₂	#70778	1	2	yes	—
212	C₂	#70822	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#73311

C₂

yes

maximal, characteristic

C₃×#1831

C₂

yes

maximal, characteristic

C₂×#4486

C₂

yes

maximal

#69391

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

—

C₂×C₃×C₄⋊C₈

C₂²

yes

—

C₃×C₄³

C₂²

yes

characteristic

#4935

—

#4935

C₄

yes

—

C₂×C₄⋊(C₃⋊C₈)

C₂²

yes

—

—

—

—

—

—

D₃

yes

characteristic

#1767

—

C₂×#332

—

C₂²×Dic₆

—

C₂²×C₃×C₈

—

C₂×C₃×C₄²

C₂×C₄

yes

—

C₂×C₃×C₄²

D₄

yes

characteristic

C₂×C₃×C₄²

D₄

yes

characteristic

C₂×C₃×C₄²

D₄

yes

—

C₂×C₃×C₄²

C₂³

yes

characteristic

C₂×C₄⋊Dic₃

—

C₂×C₄⋊Dic₃

—

C₂×C₄⋊Dic₃

—

C₂×C₄⋊Dic₃

C₂×C₄

yes

—

C₂²×C₃⋊C₈

—

—

—

—

—

—

C₃×C₄⋊C₈

—

C₄⋊(C₃⋊C₈)

—

C₄⋊(C₃⋊C₈)

D₄

yes

—

#803

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂×D₃

yes

—

C₂×#97

—

#384

—

#384

—

C₄³

C₂×D₃

yes

—

—

—

C₂²×C₃×C₄

C₂²⋊C₄

yes

—

C₂²×C₃×C₄

C₂²⋊C₄

yes

—

C₂²×C₃×C₄

C₂²×C₄

yes

characteristic

C₂²×C₃×C₄

C₂×D₄

yes

characteristic

C₂²×C₃×C₄

C₂×D₄

yes

characteristic

C₂²×Dic₃

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

D₈

yes

—

C₃×C₄²

SD₁₆

yes

—

C₃×C₄²

C₂×D₄

yes

—

C₃×C₄²

#41

yes

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₄×D₃

yes

—

C₂×C₄²

D₁₂

yes

characteristic

C₂×C₄²

D₁₂

yes

characteristic

C₂×C₄²

#67

yes

—

C₂×C₄²

C₂²×D₃

yes

characteristic

C₄⋊C₈

—

100

C₄⋊C₈

—

101

C₄⋊C₈

—

102

#97

—

103

C₂³×C₃

C₂×C₂²⋊C₄

yes

characteristic

104

C₂×C₃×C₄

—

105

C₂×C₃×C₄

—

106

C₂×C₃×C₄

—

107

C₂×C₃×C₄

—

108

C₂×C₃×C₄

—

109

C₂×C₃×C₄

#101

yes

—

110

C₂×C₃×C₄

#102

yes

—

111

C₂×C₃×C₄

C₄×D₄

yes

—

112

C₂×C₃×C₄

#121

yes

—

113

C₂×C₃×C₄

#124

yes

characteristic

114

C₂×C₃×C₄

#127

yes

characteristic

115

C₂×C₃×C₄

C₂×D₈

yes

characteristic

116

C₂×C₃×C₄

C₂×SD₁₆

yes

characteristic

117

C₂×C₃×C₄

AGL(1,ℤ/8)

yes

—

118

C₂×C₃×C₄

#137

yes

—

119

C₂×Dic₃

—

120

C₂×Dic₃

—

121

C₂×Dic₃

—

122

Dic₆

—

123

Dic₆

—

124

C₃×C₈

—

125

C₃⋊C₈

—

126

C₂×Q₈

—

127

C₂×Q₈

—

128

C₂×Q₈

—

129

C₂×Q₈

—

130

C₂²×C₄

—

131

C₂²×C₄

(C₂×D₃)⋊C₄

yes

—

132

C₂²×C₄

(C₂×D₃)⋊C₄

yes

—

133

C₂²×C₄

C₂×C₄×D₃

yes

characteristic

134

C₂²×C₄

C₂×D₁₂

yes

characteristic

135

C₂²×C₄

C₂×#67

yes

characteristic

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄²

—

147

C₄²

—

148

C₄²

—

149

C₄²

#204

yes

—

150

C₄²

D₂₄

yes

—

151

C₄²

D₃×D₄

yes

—

152

C₄²

#239

yes

—

153

C₂²×C₃

#390

yes

characteristic

154

C₂²×C₃

C₂×#101

yes

—

155

C₂²×C₃

C₂×#102

yes

characteristic

156

C₂²×C₃

#418

yes

characteristic

157

C₂²×C₃

#419

yes

—

158

C₃×C₄

—

159

C₃×C₄

—

160

C₃×C₄

—

161

C₃×C₄

#459

yes

—

162

C₃×C₄

#460

yes

—

163

C₃×C₄

#491

yes

—

164

C₃×C₄

#497

yes

—

165

Dic₃

—

166

C₂³

C₂×(C₂×D₃)⋊C₄

yes

characteristic

167

Q₈

—

168

Q₈

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

#821

yes

—

178

C₂×C₄

#824

yes

—

179

C₂×C₄

#893

yes

—

180

C₂×C₄

#895

yes

—

181

C₂×C₄

C₂×#204

yes

characteristic

182

C₂×C₄

C₂×D₂₄

yes

characteristic

183

C₂×C₄

#908

yes

—

184

C₂×C₄

#940

yes

characteristic

185

C₂×C₄

#947

yes

characteristic

186

C₂×C₄

#951

yes

—

187

C₈

—

188

C₈

—

189

C₂×C₃

#1546

yes

—

190

C₂×C₃

#1556

yes

—

191

C₂×C₃

#1960

yes

characteristic

192

C₂×C₃

#1966

yes

characteristic

193

C₂×C₃

#1982

yes

characteristic

194

C₂²

#4988

yes

characteristic

195

C₂²

#5004

yes

—

196

C₂²

C₂×#821

yes

characteristic

197

C₂²

#5131

yes

characteristic

198

C₂²

C₂×#824

yes

—

199

C₄

—

200

C₄

—

201

C₄

—

202

C₄

—

203

C₄

#4843

yes

—

204

C₄

#4848

yes

—

205

C₄

#5060

yes

—

206

C₄

#5069

yes

—

207

C₃

#9705

yes

characteristic, complemented, Sylow

208

C₂

#69552

yes

characteristic

209

C₂

#70457

yes

characteristic

210

C₂

#70522

yes

characteristic

211

C₂

#70778

yes

—

212

C₂

#70822

yes

—

	subgroup	G/H
center	C₂³	C₂×(C₂×D₃)⋊C₄
commutator	C₂²×C₃×C₄	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×#1831	C₂
socle, prime radical	C₂³×C₃	C₂×C₂²⋊C₄
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	#1831	D₃
hypocenter	C₃	#9705
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

subgroup

G/H

center

C₂³

C₂×(C₂×D₃)⋊C₄

commutator

C₂²×C₃×C₄

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₃

Fitting

C₃×#1831

C₂

socle, prime radical

C₂³×C₃

C₂×C₂²⋊C₄

cosocle

C₂×C₃×C₄²

C₂³

D₃

C₃

G ▷ C₂²×C₃×C₄ ▷ C₂²×C₃ ▷ C₃

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

N	H ≅ G/N	type
207. C₃	#9705	semidirect

H ≅ G/N

type

207. C₃

#9705

semidirect

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

subgroup	index
#73311	2
C₃×#1831	2
C₂×#4486	2
#69391	2
#9705	3

subgroup

index

Subgroups — #10694280

Conjugacy classes of subgroups

Complements

Maximal subgroups