Group #10698030 — subgroups

Subgroups — #10698030

Order 768.

554 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 151 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	C₂×#5093	C₂	1	2	yes	maximal
2	C₂×#4680	C₂	1	1	yes	maximal, characteristic
3	C₃×#1956	C₂	1	1	yes	maximal, characteristic
4	#69582	C₂	1	2	yes	maximal
5	C₃⋊#1273	C₂	1	1	yes	maximal, characteristic
6	#10317	—	3	1	no	maximal, Sylow
7	C₂×C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
8	C₂×#869	C₂²	1	2	yes	—
9	C₂×C₃×#103	—	2	2	no	—
10	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
11	C₃×#384	C₂²	1	1	yes	characteristic
12	C₂×#817	—	2	2	no	—
13	#5093	—	2	4	no	—
14	#5093	—	2	4	no	—
15	C₂×#810	—	2	4	no	—
16	C₂×C₄×C₃⋊C₈	C₂²	1	2	yes	—
17	#4680	—	2	4	no	—
18	#4529	—	2	4	no	—
19	C₂×#498	—	3	1	no	—
20	C₂×#487	—	3	2	no	—
21	#1990	—	3	2	no	—
22	#1956	D₃	1	1	yes	characteristic
23	#1273	—	3	1	no	—
24	C₂²×C₃×Q₈	—	2	1	no	—
25	C₂²×Dic₆	—	2	2	no	—
26	C₂²×C₃×C₈	D₄	1	2	yes	—
27	C₃×C₄⋊Q₈	—	2	2	no	—
28	C₃×C₄⋊Q₈	—	2	1	no	—
29	C₃×C₄⋊Q₈	D₄	1	2	yes	—
30	C₂×C₃×C₄⋊C₄	—	2	2	no	—
31	C₂×C₃×C₄⋊C₄	—	2	1	no	—
32	C₂×C₃×C₄⋊C₄	—	2	2	no	—
33	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
34	C₂×C₄⋊Dic₃	—	2	2	no	—
35	C₂×C₄⋊Dic₃	—	2	2	no	—
36	C₂²×C₃⋊C₈	D₄	1	4	yes	—
37	#869	—	2	2	no	—
38	#869	—	2	2	no	—
39	#869	—	2	2	no	—
40	#869	D₄	1	4	yes	—
41	C₃×#103	—	4	4	no	—
42	C₃×C₄×C₈	—	2	2	no	—
43	#817	—	2	8	no	—
44	#810	—	2	16	no	—
45	C₄×C₃⋊C₈	—	2	2	no	—
46	C₄×C₃⋊C₈	—	2	2	no	—
47	C₂×C₄⋊Q₈	—	3	2	no	—
48	C₂×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
49	#498	—	6	4	no	—
50	#487	—	6	4	no	—
51	#487	—	6	4	no	—
52	C₂×#106	—	6	2	no	—
53	C₂×#103	—	2	2	no	—
54	C₂×#103	—	6	4	no	—
55	C₂×C₄×C₈	—	3	2	no	—
56	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
57	#384	C₂×D₃	1	1	yes	characteristic
58	#340	—	6	4	no	—
59	C₂×C₃×Q₈	—	2	2	no	—
60	C₂×C₃×Q₈	—	4	1	no	—
61	C₂×C₃×Q₈	—	4	2	no	—
62	C₂×C₃×Q₈	—	4	2	no	—
63	C₂²×C₃×C₄	—	4	1	no	—
64	C₂²×C₃×C₄	—	4	1	no	—
65	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
66	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
67	C₂²×Dic₃	—	4	2	no	—
68	C₂×Dic₆	—	2	4	no	—
69	C₂×Dic₆	—	4	4	no	—
70	C₂×Dic₆	—	4	4	no	—
71	C₂×Dic₆	—	4	2	no	—
72	C₂×C₃×C₈	—	2	2	no	—
73	C₂×C₃×C₈	—	2	2	no	—
74	C₂×C₃×C₈	SD₁₆	1	4	yes	—
75	C₃×C₄⋊C₄	—	2	4	no	—
76	C₃×C₄⋊C₄	—	4	2	no	—
77	C₃×C₄⋊C₄	—	4	2	no	—
78	C₃×C₄⋊C₄	—	4	4	no	—
79	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
80	C₃×C₄²	C₂×D₄	1	2	yes	—
81	C₃×C₄²	#41	1	1	yes	characteristic
82	C₄⋊Dic₃	—	2	4	no	—
83	C₄⋊Dic₃	—	2	4	no	—
84	C₄⋊Dic₃	—	4	2	no	—
85	C₄⋊Dic₃	—	4	2	no	—
86	C₂×C₃⋊C₈	—	2	4	no	—
87	C₂×C₃⋊C₈	—	2	4	no	—
88	C₂×C₃⋊C₈	—	2	4	no	—
89	C₂²×Q₈	—	2	1	no	—
90	C₂²×Q₈	—	6	2	no	—
91	C₂²×C₈	—	3	4	no	—
92	C₂²×C₈	#67	1	2	yes	—
93	C₄⋊Q₈	—	2	1	no	—
94	C₄⋊Q₈	—	2	2	no	—
95	C₄⋊Q₈	—	3	4	no	—
96	C₄⋊Q₈	—	6	2	no	—
97	C₄⋊Q₈	—	6	2	no	—
98	C₄⋊Q₈	—	6	2	no	—
99	C₄⋊Q₈	D₁₂	1	2	yes	—
100	C₂×C₄⋊C₄	—	2	2	no	—
101	C₂×C₄⋊C₄	—	2	2	no	—
102	C₂×C₄⋊C₄	—	2	1	no	—
103	C₂×C₄⋊C₄	—	6	2	no	—
104	C₂×C₄⋊C₄	—	6	2	no	—
105	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
106	#106	—	6	8	no	—
107	#103	—	4	4	no	—
108	#103	—	6	16	no	—
109	C₄×C₈	—	2	2	no	—
110	C₄×C₈	—	6	2	no	—
111	C₄×C₈	—	6	2	no	—
112	C₂³×C₃	#127	1	1	yes	characteristic
113	C₃×Q₈	—	4	4	no	—
114	C₃×Q₈	—	8	2	no	—
115	C₂×C₃×C₄	—	4	2	no	—
116	C₂×C₃×C₄	—	8	1	no	—
117	C₂×C₃×C₄	—	8	1	no	—
118	C₂×C₃×C₄	—	8	1	no	—
119	C₂×C₃×C₄	—	8	2	no	—
120	C₂×C₃×C₄	#120	1	2	yes	—
121	C₂×C₃×C₄	#120	1	1	yes	characteristic
122	C₂×C₃×C₄	#121	1	2	yes	—
123	C₂×C₃×C₄	#121	1	1	yes	characteristic
124	C₂×C₃×C₄	C₂×SD₁₆	1	2	yes	—
125	C₂×C₃×C₄	#135	1	4	yes	—
126	C₂×C₃×C₄	#137	1	4	yes	—
127	C₂×C₃×C₄	#137	1	2	yes	—
128	C₂×Dic₃	—	4	4	no	—
129	C₂×Dic₃	—	8	2	no	—
130	C₂×Dic₃	—	8	2	no	—
131	Dic₆	—	4	8	no	—
132	Dic₆	—	8	4	no	—
133	C₃×C₈	—	2	4	no	—
134	C₃⋊C₈	—	4	4	no	—
135	C₂×Q₈	—	2	2	no	—
136	C₂×Q₈	—	4	1	no	—
137	C₂×Q₈	—	4	2	no	—
138	C₂×Q₈	—	4	2	no	—
139	C₂×Q₈	—	6	4	no	—
140	C₂×Q₈	—	12	2	no	—
141	C₂×Q₈	—	12	4	no	—
142	C₂×Q₈	—	12	4	no	—
143	C₂²×C₄	—	4	1	no	—
144	C₂²×C₄	—	4	1	no	—
145	C₂²×C₄	—	12	2	no	—
146	C₂²×C₄	D₃×D₄	1	2	yes	—
147	C₂²×C₄	C₂×#67	1	1	yes	characteristic
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	6	4	no	—
151	C₂×C₈	—	6	4	no	—
152	C₂×C₈	—	6	4	no	—
153	C₂×C₈	#214	1	4	yes	—
154	C₄⋊C₄	—	2	4	no	—
155	C₄⋊C₄	—	4	2	no	—
156	C₄⋊C₄	—	4	4	no	—
157	C₄⋊C₄	—	4	2	no	—
158	C₄⋊C₄	—	6	4	no	—
159	C₄⋊C₄	—	6	4	no	—
160	C₄⋊C₄	—	12	2	no	—
161	C₄⋊C₄	—	12	2	no	—
162	C₄²	C₂×D₁₂	1	1	yes	characteristic
163	C₄²	#235	1	1	yes	characteristic
164	C₄²	D₃×D₄	1	2	yes	—
165	C₂²×C₃	#392	1	1	yes	characteristic
166	C₂²×C₃	#492	1	1	yes	characteristic
167	C₂²×C₃	#495	1	2	yes	—
168	C₂²×C₃	#497	1	2	yes	—
169	C₂²×C₃	#497	1	1	yes	characteristic
170	C₃×C₄	—	8	2	no	—
171	C₃×C₄	—	16	1	no	—
172	C₃×C₄	#448	1	2	yes	—
173	C₃×C₄	#452	1	4	yes	—
174	C₃×C₄	#461	1	2	yes	—
175	C₃×C₄	#464	1	4	yes	—
176	Dic₃	—	8	4	no	—
177	C₂³	#940	1	1	yes	characteristic
178	Q₈	—	4	4	no	—
179	Q₈	—	8	2	no	—
180	Q₈	—	12	8	no	—
181	Q₈	—	24	4	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	8	1	no	—
184	C₂×C₄	—	8	1	no	—
185	C₂×C₄	—	8	2	no	—
186	C₂×C₄	—	8	1	no	—
187	C₂×C₄	—	12	4	no	—
188	C₂×C₄	—	24	2	no	—
189	C₂×C₄	—	24	2	no	—
190	C₂×C₄	#882	1	2	yes	—
191	C₂×C₄	#884	1	2	yes	—
192	C₂×C₄	#916	1	4	yes	—
193	C₂×C₄	#918	1	4	yes	—
194	C₂×C₄	#930	1	1	yes	characteristic
195	C₂×C₄	C₂×#214	1	2	yes	—
196	C₂×C₄	#937	1	1	yes	characteristic
197	C₂×C₄	#951	1	2	yes	—
198	C₈	—	2	4	no	—
199	C₈	—	12	4	no	—
200	C₂×C₃	#1708	1	1	yes	characteristic
201	C₂×C₃	#1716	1	1	yes	characteristic
202	C₂×C₃	#1718	1	2	yes	—
203	C₂×C₃	#2013	1	1	yes	characteristic
204	C₂×C₃	#2015	1	2	yes	—
205	C₂²	#4960	1	1	yes	characteristic
206	C₂²	#5060	1	1	yes	characteristic
207	C₂²	#5076	1	1	yes	characteristic
208	C₂²	#5168	1	2	yes	—
209	C₂²	#5190	1	2	yes	—
210	C₄	—	8	2	no	—
211	C₄	—	16	1	no	—
212	C₄	—	24	4	no	—
213	C₄	#4798	1	4	yes	—
214	C₄	#4808	1	4	yes	—
215	C₄	#5019	1	2	yes	—
216	C₄	#5040	1	2	yes	—
217	C₃	#10317	1	1	yes	characteristic, complemented, Sylow
218	C₂	#69661	1	1	yes	characteristic
219	C₂	#70278	1	2	yes	—
220	C₂	#71219	1	2	yes	—
221	C₂	#71256	1	1	yes	characteristic
222	C₂	#71299	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#5093

C₂

yes

maximal

C₂×#4680

C₂

yes

maximal, characteristic

C₃×#1956

C₂

yes

maximal, characteristic

#69582

C₂

yes

maximal

C₃⋊#1273

C₂

yes

maximal, characteristic

#10317

—

maximal, Sylow

C₂×C₃×C₄⋊Q₈

C₂²

yes

characteristic

C₂×#869

C₂²

yes

—

C₂×C₃×#103

—

C₂×C₃×C₄×C₈

C₂²

yes

characteristic

C₃×#384

C₂²

yes

—

—

—

—

C₂×C₄×C₃⋊C₈

C₂²

yes

—

—

—

—

—

—

D₃

yes

characteristic

#1273

—

C₂²×C₃×Q₈

—

C₂²×Dic₆

—

C₂²×C₃×C₈

D₄

yes

—

C₃×C₄⋊Q₈

—

C₃×C₄⋊Q₈

—

C₃×C₄⋊Q₈

D₄

yes

—

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄²

C₂³

yes

characteristic

C₂×C₄⋊Dic₃

—

C₂×C₄⋊Dic₃

—

C₂²×C₃⋊C₈

D₄

yes

—

—

—

—

D₄

yes

—

C₃×#103

—

C₃×C₄×C₈

—

#817

—

#810

—

C₄×C₃⋊C₈

—

C₄×C₃⋊C₈

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

C₂×D₃

yes

—

—

—

—

—

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₃

yes

characteristic

#384

C₂×D₃

yes

characteristic

#340

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂×D₄

yes

—

C₂²×C₃×C₄

C₂×D₄

yes

characteristic

C₂²×Dic₃

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

SD₁₆

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂×D₄

yes

characteristic

C₃×C₄²

C₂×D₄

yes

—

C₃×C₄²

#41

yes

characteristic

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

#67

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

D₁₂

yes

—

100

C₂×C₄⋊C₄

—

101

C₂×C₄⋊C₄

—

102

C₂×C₄⋊C₄

—

103

C₂×C₄⋊C₄

—

104

C₂×C₄⋊C₄

—

105

C₂×C₄²

C₂²×D₃

yes

characteristic

106

#106

—

107

#103

—

108

#103

—

109

C₄×C₈

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₂³×C₃

#127

yes

characteristic

113

C₃×Q₈

—

114

C₃×Q₈

—

115

C₂×C₃×C₄

—

116

C₂×C₃×C₄

—

117

C₂×C₃×C₄

—

118

C₂×C₃×C₄

—

119

C₂×C₃×C₄

—

120

C₂×C₃×C₄

#120

yes

—

121

C₂×C₃×C₄

#120

yes

characteristic

122

C₂×C₃×C₄

#121

yes

—

123

C₂×C₃×C₄

#121

yes

characteristic

124

C₂×C₃×C₄

C₂×SD₁₆

yes

—

125

C₂×C₃×C₄

#135

yes

—

126

C₂×C₃×C₄

#137

yes

—

127

C₂×C₃×C₄

#137

yes

—

128

C₂×Dic₃

—

129

C₂×Dic₃

—

130

C₂×Dic₃

—

131

Dic₆

—

132

Dic₆

—

133

C₃×C₈

—

134

C₃⋊C₈

—

135

C₂×Q₈

—

136

C₂×Q₈

—

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂×Q₈

—

142

C₂×Q₈

—

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

—

146

C₂²×C₄

D₃×D₄

yes

—

147

C₂²×C₄

C₂×#67

yes

characteristic

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

#214

yes

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄²

C₂×D₁₂

yes

characteristic

163

C₄²

#235

yes

characteristic

164

C₄²

D₃×D₄

yes

—

165

C₂²×C₃

#392

yes

characteristic

166

C₂²×C₃

#492

yes

characteristic

167

C₂²×C₃

#495

yes

—

168

C₂²×C₃

#497

yes

—

169

C₂²×C₃

#497

yes

characteristic

170

C₃×C₄

—

171

C₃×C₄

—

172

C₃×C₄

#448

yes

—

173

C₃×C₄

#452

yes

—

174

C₃×C₄

#461

yes

—

175

C₃×C₄

#464

yes

—

176

Dic₃

—

177

C₂³

#940

yes

characteristic

178

Q₈

—

179

Q₈

—

180

Q₈

—

181

Q₈

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

#882

yes

—

191

C₂×C₄

#884

yes

—

192

C₂×C₄

#916

yes

—

193

C₂×C₄

#918

yes

—

194

C₂×C₄

#930

yes

characteristic

195

C₂×C₄

C₂×#214

yes

—

196

C₂×C₄

#937

yes

characteristic

197

C₂×C₄

#951

yes

—

198

C₈

—

199

C₈

—

200

C₂×C₃

#1708

yes

characteristic

201

C₂×C₃

#1716

yes

characteristic

202

C₂×C₃

#1718

yes

—

203

C₂×C₃

#2013

yes

characteristic

204

C₂×C₃

#2015

yes

—

205

C₂²

#4960

yes

characteristic

206

C₂²

#5060

yes

characteristic

207

C₂²

#5076

yes

characteristic

208

C₂²

#5168

yes

—

209

C₂²

#5190

yes

—

210

C₄

—

211

C₄

—

212

C₄

—

213

C₄

#4798

yes

—

214

C₄

#4808

yes

—

215

C₄

#5019

yes

—

216

C₄

#5040

yes

—

217

C₃

#10317

yes

characteristic, complemented, Sylow

218

C₂

#69661

yes

characteristic

219

C₂

#70278

yes

—

220

C₂

#71219

yes

—

221

C₂

#71256

yes

characteristic

222

C₂

#71299

yes

characteristic

	subgroup	G/H
center	C₂³	#940
commutator, cosocle	C₂×C₃×C₄²	C₂³
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×#1956	C₂
socle, prime radical	C₂³×C₃	#127
hypercenter	#1956	D₃
hypocenter	C₃	#10317
LCS	G ▷ C₂×C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1956

subgroup

G/H

center

C₂³

#940

commutator, cosocle

C₂×C₃×C₄²

C₂³

Frattini

C₂×C₄²

C₂²×D₃

C₂

C₂³×C₃

D₃

C₃

G ▷ C₂×C₃×C₄² ▷ C₂²×C₃ ▷ C₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1956

N	H ≅ G/N	type
217. C₃	#10317	semidirect

H ≅ G/N

type

217. C₃

#10317

semidirect

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

subgroup	index
C₂×#5093	2
C₂×#4680	2
C₃×#1956	2
#69582	2
C₃⋊#1273	2
#10317	3

subgroup

index

Subgroups — #10698030

Conjugacy classes of subgroups

Complements

Maximal subgroups