Group #10694992 — subgroups

Subgroups — #10694992

Order 768.

494 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 163 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×#4682	C₂	1	1	yes	maximal, characteristic
2	C₃×#1834	C₂	1	1	yes	maximal, characteristic
3	C₂×#4454	C₂	1	1	yes	maximal, characteristic
4	#69582	C₂	1	2	yes	maximal
5	#69579	C₂	1	2	yes	maximal
6	#9831	—	3	1	no	maximal, Sylow
7	C₂×#869	C₂²	1	2	yes	—
8	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₃×#384	C₂²	1	2	yes	—
10	C₂×#818	—	2	2	no	—
11	C₂×#810	—	2	4	no	—
12	C₂×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
14	#4682	—	2	2	no	—
15	#4682	C₄	1	4	yes	complemented
16	C₃×#340	—	2	2	no	—
17	#4529	—	2	4	no	—
18	#4454	—	2	2	no	—
19	C₂×#500	—	3	1	no	—
20	#1990	—	3	2	no	—
21	#1956	—	3	2	no	—
22	#1834	D₃	1	1	yes	characteristic
23	C₂×#322	—	3	1	no	—
24	C₂²×Dic₆	—	2	2	no	—
25	C₂²×C₃×C₈	D₄	1	2	yes	—
26	C₂×C₃×C₄⋊C₄	—	2	2	no	—
27	C₂×C₃×C₄⋊C₄	—	2	2	no	—
28	C₂×C₃×C₄⋊C₄	—	2	2	no	—
29	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
30	C₂×C₄⋊Dic₃	—	2	2	no	—
31	C₂×C₄⋊Dic₃	—	2	2	no	—
32	C₂²×C₃⋊C₈	D₄	1	2	yes	—
33	C₂²×C₃⋊C₈	D₄	1	2	yes	—
34	#869	—	2	2	no	—
35	#869	—	2	2	no	—
36	#869	—	2	2	no	—
37	#869	C₂×C₄	1	4	yes	—
38	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
39	C₃×C₄×C₈	D₄	1	2	yes	—
40	#818	—	2	8	no	—
41	#810	—	4	8	no	—
42	C₄×C₃⋊C₈	—	2	1	no	—
43	C₄×C₃⋊C₈	—	2	1	no	—
44	C₄×C₃⋊C₈	—	2	1	no	—
45	C₄×C₃⋊C₈	—	2	1	no	—
46	C₂×C₄⋊Q₈	—	3	2	no	—
47	#500	—	3	4	no	—
48	#500	—	6	2	no	—
49	C₂×#107	—	6	2	no	—
50	C₂×#103	—	6	4	no	—
51	C₂×C₄×C₈	—	3	1	no	—
52	C₂×C₄×C₈	—	3	1	no	—
53	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
54	#384	C₂×D₃	1	2	yes	—
55	#340	—	2	2	no	—
56	#340	—	6	4	no	—
57	#322	—	6	2	no	—
58	C₂²×C₃×C₄	—	4	2	no	—
59	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
62	C₂²×Dic₃	—	4	2	no	—
63	C₂×Dic₆	—	2	4	no	—
64	C₂×Dic₆	—	4	4	no	—
65	C₂×Dic₆	—	4	4	no	—
66	C₂×Dic₆	—	4	2	no	—
67	C₂×C₃×C₈	—	2	4	no	—
68	C₂×C₃×C₈	C₂²⋊C₄	1	4	yes	—
69	C₃×C₄⋊C₄	—	4	4	no	—
70	C₃×C₄⋊C₄	—	4	4	no	—
71	C₃×C₄⋊C₄	—	4	4	no	—
72	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
73	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
74	C₃×C₄²	#41	1	1	yes	characteristic
75	C₃×C₄²	#41	1	1	yes	characteristic
76	C₄⋊Dic₃	—	2	4	no	—
77	C₄⋊Dic₃	—	2	4	no	—
78	C₄⋊Dic₃	—	4	2	no	—
79	C₄⋊Dic₃	—	4	2	no	—
80	C₂×C₃⋊C₈	—	2	2	no	—
81	C₂×C₃⋊C₈	—	2	2	no	—
82	C₂×C₃⋊C₈	—	2	2	no	—
83	C₂×C₃⋊C₈	—	2	4	no	—
84	C₂×C₃⋊C₈	D₈	1	4	yes	—
85	C₂²×Q₈	—	6	2	no	—
86	C₂²×C₈	—	3	2	no	—
87	C₂²×C₈	—	3	2	no	—
88	C₂²×C₈	#67	1	2	yes	—
89	C₄⋊Q₈	—	3	4	no	—
90	C₄⋊Q₈	—	6	2	no	—
91	C₄⋊Q₈	—	6	2	no	—
92	C₄⋊Q₈	—	6	2	no	—
93	C₂×C₄⋊C₄	—	2	2	no	—
94	C₂×C₄⋊C₄	—	2	2	no	—
95	C₂×C₄⋊C₄	—	2	2	no	—
96	C₂×C₄⋊C₄	—	6	2	no	—
97	C₂×C₄⋊C₄	—	6	2	no	—
98	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
99	#107	—	6	8	no	—
100	#103	—	12	8	no	—
101	C₄×C₈	—	6	1	no	—
102	C₄×C₈	—	6	1	no	—
103	C₄×C₈	—	6	1	no	—
104	C₄×C₈	—	6	1	no	—
105	C₄×C₈	C₄×D₃	1	2	yes	—
106	C₄×C₈	D₁₂	1	2	yes	—
107	C₂³×C₃	#127	1	1	yes	characteristic
108	C₂×C₃×C₄	—	8	2	no	—
109	C₂×C₃×C₄	—	8	2	no	—
110	C₂×C₃×C₄	—	8	2	no	—
111	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
113	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
114	C₂×C₃×C₄	#121	1	1	yes	characteristic
115	C₂×C₃×C₄	#121	1	1	yes	characteristic
116	C₂×C₃×C₄	#124	1	1	yes	characteristic
117	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
118	C₂×C₃×C₄	#135	1	2	yes	—
119	C₂×C₃×C₄	#137	1	4	yes	—
120	C₂×C₃×C₄	#137	1	4	yes	—
121	C₂×Dic₃	—	4	4	no	—
122	C₂×Dic₃	—	8	2	no	—
123	C₂×Dic₃	—	8	2	no	—
124	Dic₆	—	4	8	no	—
125	Dic₆	—	8	4	no	—
126	C₃×C₈	—	2	4	no	—
127	C₃⋊C₈	—	2	4	no	—
128	C₃⋊C₈	—	4	2	no	—
129	C₂×Q₈	—	6	4	no	—
130	C₂×Q₈	—	12	2	no	—
131	C₂×Q₈	—	12	4	no	—
132	C₂×Q₈	—	12	4	no	—
133	C₂²×C₄	—	4	2	no	—
134	C₂²×C₄	—	12	2	no	—
135	C₂²×C₄	D₃×D₄	1	1	yes	characteristic
136	C₂²×C₄	D₃×D₄	1	1	yes	characteristic
137	C₂²×C₄	C₂×#67	1	1	yes	characteristic
138	C₂×C₈	—	2	4	no	—
139	C₂×C₈	—	3	4	no	—
140	C₂×C₈	—	6	2	no	—
141	C₂×C₈	—	6	2	no	—
142	C₂×C₈	—	6	2	no	—
143	C₂×C₈	—	6	4	no	—
144	C₂×C₈	(C₂×D₃)⋊C₄	1	4	yes	—
145	C₄⋊C₄	—	4	4	no	—
146	C₄⋊C₄	—	4	4	no	—
147	C₄⋊C₄	—	4	4	no	—
148	C₄⋊C₄	—	6	4	no	—
149	C₄⋊C₄	—	6	4	no	—
150	C₄⋊C₄	—	12	2	no	—
151	C₄⋊C₄	—	12	2	no	—
152	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
153	C₄²	C₂×D₁₂	1	1	yes	characteristic
154	C₄²	#239	1	1	yes	characteristic
155	C₄²	#239	1	1	yes	characteristic
156	C₂²×C₃	#390	1	1	yes	characteristic
157	C₂²×C₃	#493	1	1	yes	characteristic
158	C₂²×C₃	#495	1	1	yes	characteristic
159	C₂²×C₃	#497	1	2	yes	—
160	C₂²×C₃	#497	1	2	yes	—
161	C₃×C₄	—	16	2	no	—
162	C₃×C₄	#419	1	2	yes	—
163	C₃×C₄	C₄×D₈	1	2	yes	—
164	C₃×C₄	#441	1	2	yes	—
165	C₃×C₄	#466	1	2	yes	—
166	C₃×C₄	#470	1	2	yes	—
167	C₃×C₄	#491	1	2	yes	—
168	Dic₃	—	8	4	no	—
169	C₂³	#940	1	1	yes	characteristic
170	Q₈	—	12	8	no	—
171	Q₈	—	24	4	no	—
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	—	8	2	no	—
175	C₂×C₄	—	12	4	no	—
176	C₂×C₄	—	24	2	no	—
177	C₂×C₄	—	24	2	no	—
178	C₂×C₄	#893	1	1	yes	characteristic
179	C₂×C₄	#893	1	1	yes	characteristic
180	C₂×C₄	#895	1	1	yes	characteristic
181	C₂×C₄	#895	1	1	yes	characteristic
182	C₂×C₄	D₃×D₈	1	2	yes	—
183	C₂×C₄	#915	1	4	yes	—
184	C₂×C₄	#919	1	2	yes	—
185	C₂×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
186	C₂×C₄	#947	1	1	yes	characteristic
187	C₂×C₄	#951	1	4	yes	—
188	C₈	—	2	4	no	—
189	C₈	—	6	4	no	—
190	C₈	—	12	2	no	—
191	C₂×C₃	#1718	1	2	yes	—
192	C₂×C₃	#1722	1	2	yes	—
193	C₂×C₃	#1960	1	1	yes	characteristic
194	C₂×C₃	#1970	1	1	yes	characteristic
195	C₂×C₃	#1971	1	1	yes	characteristic
196	C₂²	#4988	1	1	yes	characteristic
197	C₂²	#5060	1	2	yes	—
198	C₂²	#5151	1	1	yes	characteristic
199	C₂²	#5167	1	2	yes	—
200	C₂²	#5191	1	1	yes	characteristic
201	C₄	—	16	2	no	—
202	C₄	—	24	4	no	—
203	C₄	#4853	1	2	yes	—
204	C₄	#4867	1	2	yes	—
205	C₄	#4872	1	2	yes	—
206	C₄	#4883	1	2	yes	—
207	C₄	#5004	1	2	yes	—
208	C₄	#5069	1	2	yes	—
209	C₃	#9831	1	1	yes	characteristic, complemented, Sylow
210	C₂	#69552	1	1	yes	characteristic
211	C₂	#70575	1	1	yes	characteristic
212	C₂	#70628	1	1	yes	characteristic
213	C₂	#71195	1	2	yes	—
214	C₂	#71222	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#4682

C₂

yes

maximal, characteristic

C₃×#1834

C₂

yes

maximal, characteristic

C₂×#4454

C₂

yes

maximal, characteristic

#69582

C₂

yes

maximal

#69579

C₂

yes

—

C₂²

yes

—

C₂×C₃×C₄×C₈

C₂²

yes

characteristic

C₃×#384

C₂²

yes

—

C₂×#818

—

C₂×#810

—

C₂×C₄×C₃⋊C₈

C₂²

yes

characteristic

C₂×C₄×C₃⋊C₈

C₂²

yes

characteristic

#4682

—

#4682

C₄

yes

—

—

—

—

—

—

D₃

yes

characteristic

C₂×#322

—

C₂²×Dic₆

—

C₂²×C₃×C₈

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

—

C₂²×C₃⋊C₈

D₄

yes

—

—

—

—

C₂×C₄

yes

—

C₃×C₄×C₈

C₂×C₄

yes

—

C₃×C₄×C₈

D₄

yes

—

#818

—

#810

—

C₄×C₃⋊C₈

—

C₄×C₃⋊C₈

—

C₄×C₃⋊C₈

—

C₄×C₃⋊C₈

—

C₂×C₄⋊Q₈

—

—

—

—

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₃

yes

characteristic

#384

C₂×D₃

yes

—

#340

—

#340

—

#322

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂×D₄

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

C₂²⋊C₄

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂²×C₄

yes

characteristic

C₃×C₄²

C₂×D₄

yes

C₃×C₄²

yes

C₃×C₄²

yes

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

D₈

yes

—

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

#67

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₃

yes

characteristic

#107

—

100

#103

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₄×C₈

—

104

C₄×C₈

—

105

C₄×C₈

C₄×D₃

yes

—

106

C₄×C₈

D₁₂

yes

—

107

C₂³×C₃

#127

yes

characteristic

108

C₂×C₃×C₄

—

109

C₂×C₃×C₄

—

110

C₂×C₃×C₄

—

111

C₂×C₃×C₄

C₂×C₂²⋊C₄

yes

characteristic

112

C₂×C₃×C₄

C₄×D₄

yes

characteristic

113

C₂×C₃×C₄

C₄×D₄

yes

characteristic

114

C₂×C₃×C₄

#121

yes

characteristic

115

C₂×C₃×C₄

#121

yes

characteristic

116

C₂×C₃×C₄

#124

yes

characteristic

117

C₂×C₃×C₄

C₂×D₈

yes

—

118

C₂×C₃×C₄

#135

yes

—

119

C₂×C₃×C₄

#137

yes

—

120

C₂×C₃×C₄

#137

yes

—

121

C₂×Dic₃

—

122

C₂×Dic₃

—

123

C₂×Dic₃

—

124

Dic₆

—

125

Dic₆

—

126

C₃×C₈

—

127

C₃⋊C₈

—

128

C₃⋊C₈

—

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂²×C₄

—

134

C₂²×C₄

—

135

C₂²×C₄

D₃×D₄

yes

characteristic

136

C₂²×C₄

D₃×D₄

yes

characteristic

137

C₂²×C₄

C₂×#67

yes

characteristic

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

(C₂×D₃)⋊C₄

yes

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄²

C₂×C₄×D₃

yes

characteristic

153

C₄²

C₂×D₁₂

yes

characteristic

154

C₄²

#239

yes

characteristic

155

C₄²

#239

yes

characteristic

156

C₂²×C₃

#390

yes

characteristic

157

C₂²×C₃

#493

yes

characteristic

158

C₂²×C₃

#495

yes

characteristic

159

C₂²×C₃

#497

yes

—

160

C₂²×C₃

#497

yes

—

161

C₃×C₄

—

162

C₃×C₄

#419

yes

—

163

C₃×C₄

C₄×D₈

yes

—

164

C₃×C₄

#441

yes

—

165

C₃×C₄

#466

yes

—

166

C₃×C₄

#470

yes

—

167

C₃×C₄

#491

yes

—

168

Dic₃

—

169

C₂³

#940

yes

characteristic

170

Q₈

—

171

Q₈

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

#893

yes

characteristic

179

C₂×C₄

#893

yes

characteristic

180

C₂×C₄

#895

yes

characteristic

181

C₂×C₄

#895

yes

characteristic

182

C₂×C₄

D₃×D₈

yes

—

183

C₂×C₄

#915

yes

—

184

C₂×C₄

#919

yes

—

185

C₂×C₄

C₂×(C₂×D₃)⋊C₄

yes

characteristic

186

C₂×C₄

#947

yes

characteristic

187

C₂×C₄

#951

yes

—

188

C₈

—

189

C₈

—

190

C₈

—

191

C₂×C₃

#1718

yes

—

192

C₂×C₃

#1722

yes

—

193

C₂×C₃

#1960

yes

characteristic

194

C₂×C₃

#1970

yes

characteristic

195

C₂×C₃

#1971

yes

characteristic

196

C₂²

#4988

yes

characteristic

197

C₂²

#5060

yes

—

198

C₂²

#5151

yes

characteristic

199

C₂²

#5167

yes

—

200

C₂²

#5191

yes

characteristic

201

C₄

—

202

C₄

—

203

C₄

#4853

yes

—

204

C₄

#4867

yes

—

205

C₄

#4872

yes

—

206

C₄

#4883

yes

—

207

C₄

#5004

yes

—

208

C₄

#5069

yes

—

209

C₃

#9831

yes

characteristic, complemented, Sylow

210

C₂

#69552

yes

characteristic

211

C₂

#70575

yes

characteristic

212

C₂

#70628

yes

characteristic

213

C₂

#71195

yes

—

214

C₂

#71222

yes

—

	subgroup	G/H
center	C₂³	#940
commutator	C₃×C₄²	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×#1834	C₂
socle, prime radical	C₂³×C₃	#127
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	#1834	D₃
hypocenter	C₃	#9831
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

subgroup

G/H

center

C₂³

#940

commutator

C₃×C₄²

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₃

C₂

C₂³×C₃

C₂×C₃×C₄²

C₂³

D₃

C₃

G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

N	H ≅ G/N	type
15. #4682	C₄	semidirect
209. C₃	#9831	semidirect

H ≅ G/N

type

15. #4682

C₄

semidirect

209. C₃

#9831

semidirect

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

subgroup	index
C₂×#4682	2
C₃×#1834	2
C₂×#4454	2
#69582	2
#69579	2
#9831	3

subgroup

index

Subgroups — #10694992

Conjugacy classes of subgroups

Complements

Maximal subgroups