Group #10695014 — subgroups

Subgroups — #10695014

Order 768.

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

normal subgroups only

This group has 526 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×#4699	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	#69582	C₂	1	2	yes	maximal
6	#9835	—	3	1	no	maximal, Sylow
7	C₂²×Dic₁₂	—	2	2	no	—
8	C₂×#869	C₂²	1	2	yes	—
9	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₃×#384	C₂²	1	2	yes	—
11	C₂×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
13	#4699	—	2	2	no	—
14	#4699	C₄	1	4	yes	complemented
15	C₃×#340	—	2	2	no	—
16	#4529	—	2	4	no	—
17	#4529	—	2	4	no	—
18	#4454	—	2	2	no	—
19	C₂×#494	—	3	1	no	—
20	#1990	—	3	2	no	—
21	#1990	—	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	C₂×Dic₁₂	—	2	8	no	—
35	C₂×Dic₁₂	—	4	8	no	—
36	#869	—	2	2	no	—
37	#869	—	2	2	no	—
38	#869	—	2	2	no	—
39	#869	C₂×C₄	1	4	yes	—
40	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
41	C₃×C₄×C₈	D₄	1	2	yes	—
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	C₂²×Q₁₆	—	6	2	no	—
47	C₂×C₄⋊Q₈	—	3	2	no	—
48	#494	—	3	4	no	—
49	#494	—	6	2	no	—
50	C₂×C₄×C₈	—	3	1	no	—
51	C₂×C₄×C₈	—	3	1	no	—
52	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
53	#384	C₂×D₃	1	2	yes	—
54	#340	—	2	2	no	—
55	#340	—	6	4	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	8	no	—
65	C₂×Dic₆	—	4	2	no	—
66	C₂×C₃×C₈	—	2	4	no	—
67	C₂×C₃×C₈	C₂²⋊C₄	1	4	yes	—
68	C₃×C₄⋊C₄	—	4	4	no	—
69	C₃×C₄⋊C₄	—	4	4	no	—
70	C₃×C₄⋊C₄	—	4	4	no	—
71	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
72	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
73	C₃×C₄²	#41	1	1	yes	characteristic
74	C₃×C₄²	#41	1	1	yes	characteristic
75	C₄⋊Dic₃	—	2	4	no	—
76	C₄⋊Dic₃	—	2	4	no	—
77	C₄⋊Dic₃	—	4	2	no	—
78	C₄⋊Dic₃	—	4	2	no	—
79	C₂×C₃⋊C₈	—	2	2	no	—
80	C₂×C₃⋊C₈	—	2	2	no	—
81	C₂×C₃⋊C₈	—	2	2	no	—
82	C₂×C₃⋊C₈	—	2	4	no	—
83	C₂×C₃⋊C₈	Q₁₆	1	4	yes	complemented
84	Dic₁₂	—	4	16	no	—
85	C₂²×Q₈	—	6	2	no	—
86	C₂×Q₁₆	—	6	8	no	—
87	C₂×Q₁₆	—	12	8	no	—
88	C₂²×C₈	—	3	2	no	—
89	C₂²×C₈	—	3	2	no	—
90	C₂²×C₈	#67	1	2	yes	—
91	C₄⋊Q₈	—	3	4	no	—
92	C₄⋊Q₈	—	6	2	no	—
93	C₄⋊Q₈	—	6	2	no	—
94	C₄⋊Q₈	—	6	2	no	—
95	C₂×C₄⋊C₄	—	2	2	no	—
96	C₂×C₄⋊C₄	—	2	2	no	—
97	C₂×C₄⋊C₄	—	2	2	no	—
98	C₂×C₄⋊C₄	—	6	2	no	—
99	C₂×C₄⋊C₄	—	6	2	no	—
100	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
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×Q₁₆	1	2	yes	—
118	C₂×C₃×C₄	#135	1	2	yes	—
119	C₂×C₃×C₄	AGL(1,ℤ/8)	1	4	yes	—
120	C₂×C₃×C₄	#137	1	4	yes	—
121	C₂×Dic₃	—	4	4	no	—
122	C₂×Dic₃	—	8	4	no	—
123	Dic₆	—	4	8	no	—
124	Dic₆	—	8	4	no	—
125	C₃×C₈	—	2	4	no	—
126	C₃⋊C₈	—	2	4	no	—
127	C₃⋊C₈	—	4	2	no	—
128	C₂×Q₈	—	6	4	no	—
129	C₂×Q₈	—	12	2	no	—
130	C₂×Q₈	—	12	8	no	—
131	C₂²×C₄	—	4	2	no	—
132	C₂²×C₄	—	12	2	no	—
133	C₂²×C₄	D₃×D₄	1	1	yes	characteristic
134	C₂²×C₄	D₃×D₄	1	1	yes	characteristic
135	C₂²×C₄	C₂×#67	1	1	yes	characteristic
136	Q₁₆	—	12	16	no	—
137	C₂×C₈	—	2	4	no	—
138	C₂×C₈	—	3	4	no	—
139	C₂×C₈	—	6	2	no	—
140	C₂×C₈	—	6	2	no	—
141	C₂×C₈	—	6	2	no	—
142	C₂×C₈	—	6	4	no	—
143	C₂×C₈	(C₂×D₃)⋊C₄	1	4	yes	—
144	C₄⋊C₄	—	4	4	no	—
145	C₄⋊C₄	—	4	4	no	—
146	C₄⋊C₄	—	4	4	no	—
147	C₄⋊C₄	—	6	4	no	—
148	C₄⋊C₄	—	6	4	no	—
149	C₄⋊C₄	—	12	2	no	—
150	C₄⋊C₄	—	12	2	no	—
151	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
152	C₄²	C₂×D₁₂	1	1	yes	characteristic
153	C₄²	#239	1	1	yes	characteristic
154	C₄²	#239	1	1	yes	characteristic
155	C₂²×C₃	#390	1	1	yes	characteristic
156	C₂²×C₃	#494	1	1	yes	characteristic
157	C₂²×C₃	#495	1	1	yes	characteristic
158	C₂²×C₃	#496	1	2	yes	—
159	C₂²×C₃	#497	1	2	yes	—
160	C₃×C₄	—	16	2	no	—
161	C₃×C₄	#418	1	2	yes	—
162	C₃×C₄	C₄×Q₁₆	1	2	yes	—
163	C₃×C₄	#441	1	2	yes	—
164	C₃×C₄	#467	1	2	yes	—
165	C₃×C₄	#470	1	2	yes	—
166	C₃×C₄	#490	1	2	yes	—
167	Dic₃	—	8	4	no	—
168	C₂³	#940	1	1	yes	characteristic
169	Q₈	—	12	8	no	—
170	Q₈	—	24	4	no	—
171	C₂×C₄	—	8	2	no	—
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	—	12	4	no	—
175	C₂×C₄	—	24	4	no	—
176	C₂×C₄	#893	1	1	yes	characteristic
177	C₂×C₄	#893	1	1	yes	characteristic
178	C₂×C₄	#895	1	1	yes	characteristic
179	C₂×C₄	#895	1	1	yes	characteristic
180	C₂×C₄	#912	1	2	yes	—
181	C₂×C₄	#915	1	4	yes	—
182	C₂×C₄	D₃×Q₁₆	1	2	yes	—
183	C₂×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
184	C₂×C₄	#947	1	1	yes	characteristic
185	C₂×C₄	#949	1	4	yes	—
186	C₈	—	2	4	no	—
187	C₈	—	6	4	no	—
188	C₈	—	12	2	no	—
189	C₂×C₃	#1720	1	2	yes	—
190	C₂×C₃	#1723	1	2	yes	—
191	C₂×C₃	#1961	1	1	yes	characteristic
192	C₂×C₃	#1969	1	1	yes	characteristic
193	C₂×C₃	#1971	1	1	yes	characteristic
194	C₂²	#4988	1	1	yes	characteristic
195	C₂²	#5058	1	2	yes	—
196	C₂²	#5155	1	1	yes	characteristic
197	C₂²	#5167	1	2	yes	—
198	C₂²	#5183	1	1	yes	characteristic
199	C₄	—	16	2	no	—
200	C₄	—	24	4	no	—
201	C₄	#4853	1	2	yes	—
202	C₄	#4867	1	2	yes	—
203	C₄	#4873	1	2	yes	—
204	C₄	#4887	1	2	yes	—
205	C₄	#5001	1	2	yes	—
206	C₄	#5067	1	2	yes	—
207	C₃	#9835	1	1	yes	characteristic, complemented, Sylow
208	C₂	#69553	1	1	yes	characteristic
209	C₂	#70575	1	1	yes	characteristic
210	C₂	#70627	1	1	yes	characteristic
211	C₂	#71193	1	2	yes	—
212	C₂	#71226	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#4699

C₂

yes

maximal, characteristic

C₃×#1834

C₂

yes

maximal, characteristic

C₂×#4454

C₂

yes

maximal, characteristic

#69582

C₂

yes

maximal

#69582

C₂

yes

maximal

#9835

—

maximal, Sylow

C₂²×Dic₁₂

—

C₂×#869

C₂²

yes

—

C₂×C₃×C₄×C₈

C₂²

yes

characteristic

C₃×#384

C₂²

yes

—

C₂×C₄×C₃⋊C₈

C₂²

yes

characteristic

C₂×C₄×C₃⋊C₈

C₂²

yes

characteristic

#4699

—

#4699

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×Dic₁₂

—

C₂×Dic₁₂

—

—

—

—

C₂×C₄

yes

—

C₃×C₄×C₈

C₂×C₄

yes

—

C₃×C₄×C₈

D₄

yes

—

C₄×C₃⋊C₈

—

C₄×C₃⋊C₈

—

C₄×C₃⋊C₈

—

C₄×C₃⋊C₈

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

#494

—

#494

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₃

yes

characteristic

#384

C₂×D₃

yes

—

—

—

—

—

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×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

Q₁₆

yes

complemented

Dic₁₂

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

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

—

100

C₂×C₄²

C₂²×D₃

yes

characteristic

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×Q₁₆

yes

—

118

C₂×C₃×C₄

#135

yes

—

119

C₂×C₃×C₄

AGL(1,ℤ/8)

yes

—

120

C₂×C₃×C₄

#137

yes

—

121

C₂×Dic₃

—

122

C₂×Dic₃

—

123

Dic₆

—

124

Dic₆

—

125

C₃×C₈

—

126

C₃⋊C₈

—

127

C₃⋊C₈

—

128

C₂×Q₈

—

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

D₃×D₄

yes

characteristic

134

C₂²×C₄

D₃×D₄

yes

characteristic

135

C₂²×C₄

C₂×#67

yes

characteristic

136

Q₁₆

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

(C₂×D₃)⋊C₄

yes

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄²

C₂×C₄×D₃

yes

characteristic

152

C₄²

C₂×D₁₂

yes

characteristic

153

C₄²

#239

yes

characteristic

154

C₄²

#239

yes

characteristic

155

C₂²×C₃

#390

yes

characteristic

156

C₂²×C₃

#494

yes

characteristic

157

C₂²×C₃

#495

yes

characteristic

158

C₂²×C₃

#496

yes

—

159

C₂²×C₃

#497

yes

—

160

C₃×C₄

—

161

C₃×C₄

#418

yes

—

162

C₃×C₄

C₄×Q₁₆

yes

—

163

C₃×C₄

#441

yes

—

164

C₃×C₄

#467

yes

—

165

C₃×C₄

#470

yes

—

166

C₃×C₄

#490

yes

—

167

Dic₃

—

168

C₂³

#940

yes

characteristic

169

Q₈

—

170

Q₈

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

#893

yes

characteristic

177

C₂×C₄

#893

yes

characteristic

178

C₂×C₄

#895

yes

characteristic

179

C₂×C₄

#895

yes

characteristic

180

C₂×C₄

#912

yes

—

181

C₂×C₄

#915

yes

—

182

C₂×C₄

D₃×Q₁₆

yes

—

183

C₂×C₄

C₂×(C₂×D₃)⋊C₄

yes

characteristic

184

C₂×C₄

#947

yes

characteristic

185

C₂×C₄

#949

yes

—

186

C₈

—

187

C₈

—

188

C₈

—

189

C₂×C₃

#1720

yes

—

190

C₂×C₃

#1723

yes

—

191

C₂×C₃

#1961

yes

characteristic

192

C₂×C₃

#1969

yes

characteristic

193

C₂×C₃

#1971

yes

characteristic

194

C₂²

#4988

yes

characteristic

195

C₂²

#5058

yes

—

196

C₂²

#5155

yes

characteristic

197

C₂²

#5167

yes

—

198

C₂²

#5183

yes

characteristic

199

C₄

—

200

C₄

—

201

C₄

#4853

yes

—

202

C₄

#4867

yes

—

203

C₄

#4873

yes

—

204

C₄

#4887

yes

—

205

C₄

#5001

yes

—

206

C₄

#5067

yes

—

207

C₃

#9835

yes

characteristic, complemented, Sylow

208

C₂

#69553

yes

characteristic

209

C₂

#70575

yes

characteristic

210

C₂

#70627

yes

characteristic

211

C₂

#71193

yes

—

212

C₂

#71226

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	#9835
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
14. #4699	C₄	semidirect
83. C₂×C₃⋊C₈	Q₁₆	semidirect
207. C₃	#9835	semidirect

H ≅ G/N

type

14. #4699

C₄

semidirect

83. C₂×C₃⋊C₈

Q₁₆

semidirect

207. C₃

#9835

semidirect

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

subgroup	index
C₂×#4699	2
C₃×#1834	2
C₂×#4454	2
#69582	2
#9835	3

subgroup

index

Subgroups — #10695014

Conjugacy classes of subgroups

Complements

Maximal subgroups