Group #10695219 — subgroups

Subgroups — #10695219

Order 768.

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

normal subgroups only

This group has 486 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	C₃×C₄×#125	C₂	1	1	yes	maximal, characteristic
2	#69394	C₂	1	1	yes	maximal, characteristic
3	C₄×#803	C₂	1	1	yes	maximal, characteristic
4	#67619	C₂	1	4	yes	maximal
5	#9876	—	3	1	no	maximal, Sylow
6	C₂×C₃×#125	C₄	1	1	yes	characteristic
7	C₃×#389	C₄	1	2	yes	—
8	C₃×#387	—	2	2	no	—
9	C₃×#382	C₄	1	1	yes	characteristic
10	C₃×C₄×C₄⋊C₄	—	2	2	no	—
11	C₃×C₄×C₄⋊C₄	C₂²	1	2	yes	—
12	C₃×C₄³	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊(C₃⋊C₈)	—	2	2	no	—
14	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
15	C₂×#803	—	2	2	no	—
16	C₂×C₄×C₃⋊C₈	C₂²	1	2	yes	—
17	C₄×#125	D₃	1	1	yes	characteristic
18	#1842	—	3	1	no	—
19	C₄×#97	—	3	1	no	—
20	#1279	—	3	4	no	—
21	C₃×#125	—	2	4	no	—
22	C₂×C₃×C₄⋊C₄	—	2	2	no	—
23	C₂×C₃×C₄⋊C₄	—	2	2	no	—
24	C₂×C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
25	C₂×C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
26	C₂×C₃×C₄²	—	2	2	no	—
27	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₄²	D₄	1	2	yes	—
30	C₂×C₃×C₄²	D₄	1	2	yes	—
31	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
32	C₂²×C₃⋊C₈	—	2	2	no	—
33	C₂²×C₃⋊C₈	—	2	2	no	—
34	C₃×#95	—	2	2	no	—
35	C₃×#95	—	2	2	no	—
36	C₃×#95	—	2	2	no	—
37	C₄⋊(C₃⋊C₈)	—	2	4	no	—
38	C₄⋊(C₃⋊C₈)	—	4	4	no	—
39	#803	—	2	8	no	—
40	C₄×C₃⋊C₈	—	2	2	no	—
41	C₄×C₃⋊C₈	Q₈	1	4	yes	—
42	C₂×#125	Dic₃	1	1	yes	characteristic
43	C₂×C₄⋊C₈	—	3	2	no	—
44	C₂×C₄⋊C₈	—	6	2	no	—
45	C₂×#97	—	6	2	no	—
46	C₂×C₄×C₈	—	3	2	no	—
47	#389	Dic₃	1	2	yes	—
48	#387	—	2	2	no	—
49	#382	Dic₃	1	1	yes	characteristic
50	C₄×C₄⋊C₄	—	2	2	no	—
51	C₄×C₄⋊C₄	C₂×D₃	1	2	yes	—
52	C₄³	C₂×D₃	1	1	yes	characteristic
53	C₂²×C₃×C₄	—	2	2	no	—
54	C₂²×C₃×C₄	—	2	2	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	C₃×C₄⋊C₄	—	2	8	no	—
61	C₃×C₄⋊C₄	—	2	2	no	—
62	C₃×C₄⋊C₄	—	2	4	no	—
63	C₃×C₄⋊C₄	—	4	4	no	—
64	C₃×C₄⋊C₄	M₁₆	1	4	yes	—
65	C₃×C₄²	—	2	4	no	—
66	C₃×C₄²	—	2	2	no	—
67	C₃×C₄²	—	2	2	no	—
68	C₃×C₄²	—	2	2	no	—
69	C₃×C₄²	—	2	2	no	—
70	C₃×C₄²	—	4	2	no	—
71	C₃×C₄²	—	4	2	no	—
72	C₃×C₄²	C₂×Q₈	1	2	yes	—
73	C₃×C₄²	#41	1	2	yes	—
74	C₂×C₃⋊C₈	—	2	8	no	—
75	C₂×C₃⋊C₈	—	2	4	no	—
76	C₂×C₃⋊C₈	—	4	2	no	—
77	C₂×C₃⋊C₈	—	4	4	no	—
78	C₂²×C₈	—	6	2	no	—
79	C₂²×C₈	—	6	2	no	—
80	#125	—	2	4	no	—
81	C₂×C₄⋊C₄	—	2	2	no	—
82	C₂×C₄⋊C₄	—	2	2	no	—
83	C₂×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
84	C₂×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
85	C₂×C₄²	—	2	2	no	—
86	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
87	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
88	C₂×C₄²	#67	1	2	yes	—
89	C₂×C₄²	#67	1	2	yes	—
90	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
91	C₄⋊C₈	—	6	4	no	—
92	C₄⋊C₈	—	12	4	no	—
93	#97	—	6	8	no	—
94	C₄×C₈	—	3	4	no	—
95	C₄×C₈	—	6	2	no	—
96	#95	—	2	2	no	—
97	#95	—	2	2	no	—
98	#95	—	2	2	no	—
99	C₂³×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₃×C₄	—	2	2	no	—
101	C₂×C₃×C₄	—	2	2	no	—
102	C₂×C₃×C₄	—	2	2	no	—
103	C₂×C₃×C₄	—	2	2	no	—
104	C₂×C₃×C₄	—	2	4	no	—
105	C₂×C₃×C₄	—	2	2	no	—
106	C₂×C₃×C₄	—	2	2	no	—
107	C₂×C₃×C₄	—	4	2	no	—
108	C₂×C₃×C₄	—	4	2	no	—
109	C₂×C₃×C₄	—	4	2	no	—
110	C₂×C₃×C₄	—	4	2	no	—
111	C₂×C₃×C₄	—	4	2	no	—
112	C₂×C₃×C₄	#104	1	4	yes	—
113	C₂×C₃×C₄	C₄×Q₈	1	2	yes	—
114	C₂×C₃×C₄	C₂²⋊Q₈	1	2	yes	—
115	C₂×C₃×C₄	#124	1	1	yes	characteristic
116	C₂×C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
117	C₂×C₃×C₄	C₂×M₁₆	1	2	yes	—
118	C₂×C₃×C₄	#131	1	2	yes	—
119	C₂×C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
120	C₂×C₃×C₄	#137	1	2	yes	—
121	C₃⋊C₈	—	2	8	no	—
122	C₃⋊C₈	—	8	2	no	—
123	C₂²×C₄	—	2	2	no	—
124	C₂²×C₄	—	2	2	no	—
125	C₂²×C₄	#217	1	2	yes	—
126	C₂²×C₄	#217	1	2	yes	—
127	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
128	C₂²×C₄	C₂×#67	1	1	yes	characteristic
129	C₂²×C₄	C₂×#67	1	1	yes	characteristic
130	C₂×C₈	—	6	8	no	—
131	C₂×C₈	—	6	4	no	—
132	C₂×C₈	—	12	2	no	—
133	C₂×C₈	—	12	4	no	—
134	C₄⋊C₄	—	2	8	no	—
135	C₄⋊C₄	—	2	2	no	—
136	C₄⋊C₄	—	2	4	no	—
137	C₄⋊C₄	—	4	4	no	—
138	C₄⋊C₄	#208	1	4	yes	—
139	C₄²	—	2	4	no	—
140	C₄²	—	2	2	no	—
141	C₄²	—	2	2	no	—
142	C₄²	—	2	2	no	—
143	C₄²	—	2	2	no	—
144	C₄²	—	4	2	no	—
145	C₄²	—	4	2	no	—
146	C₄²	D₃×Q₈	1	2	yes	—
147	C₄²	#239	1	2	yes	—
148	C₂²×C₃	#391	1	1	yes	characteristic
149	C₂²×C₃	#407	1	1	yes	characteristic
150	C₂²×C₃	#408	1	1	yes	characteristic
151	C₂²×C₃	#417	1	2	yes	—
152	C₂²×C₃	C₂×#104	1	2	yes	—
153	C₃×C₄	—	2	2	no	—
154	C₃×C₄	—	2	4	no	—
155	C₃×C₄	—	4	2	no	—
156	C₃×C₄	—	4	4	no	—
157	C₃×C₄	—	8	2	no	—
158	C₃×C₄	#446	1	4	yes	—
159	C₃×C₄	#490	1	1	yes	characteristic
160	C₃×C₄	#491	1	1	yes	characteristic
161	C₃×C₄	#501	1	2	yes	—
162	C₂³	C₂×#217	1	1	yes	characteristic
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	#837	1	4	yes	—
176	C₂×C₄	C₂×#208	1	2	yes	—
177	C₂×C₄	#944	1	1	yes	characteristic
178	C₂×C₄	Q₈×Dic₃	1	2	yes	—
179	C₂×C₄	#946	1	2	yes	—
180	C₂×C₄	#947	1	1	yes	characteristic
181	C₂×C₄	#948	1	2	yes	—
182	C₂×C₄	#949	1	2	yes	—
183	C₂×C₄	#951	1	2	yes	—
184	C₈	—	6	8	no	—
185	C₈	—	24	2	no	—
186	C₂×C₃	#1483	1	2	yes	—
187	C₂×C₃	#1489	1	2	yes	—
188	C₂×C₃	#1985	1	1	yes	characteristic
189	C₂×C₃	#1992	1	1	yes	characteristic
190	C₂×C₃	#1994	1	1	yes	characteristic
191	C₂²	#5207	1	1	yes	characteristic
192	C₂²	#5229	1	1	yes	characteristic
193	C₂²	#5232	1	2	yes	—
194	C₂²	#5234	1	1	yes	characteristic
195	C₂²	C₂×#837	1	2	yes	—
196	C₄	—	2	4	no	—
197	C₄	—	2	2	no	—
198	C₄	—	4	2	no	—
199	C₄	—	4	4	no	—
200	C₄	—	8	2	no	—
201	C₄	#5024	1	4	yes	—
202	C₄	#5064	1	2	yes	—
203	C₄	#5067	1	1	yes	characteristic
204	C₄	#5069	1	1	yes	characteristic
205	C₃	#9876	1	1	yes	characteristic, complemented, Sylow
206	C₂	#71635	1	1	yes	characteristic
207	C₂	#71641	1	2	yes	—
208	C₂	#71741	1	1	yes	characteristic
209	C₂	#71751	1	2	yes	—
210	C₂	#71839	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₄×#125

C₂

yes

maximal, characteristic

#69394

C₂

yes

maximal, characteristic

C₄×#803

C₂

yes

maximal, characteristic

#67619

C₂

yes

maximal

#9876

—

maximal, Sylow

C₂×C₃×#125

C₄

yes

characteristic

C₃×#389

C₄

yes

—

C₃×#387

—

C₃×#382

C₄

yes

characteristic

C₃×C₄×C₄⋊C₄

—

C₃×C₄×C₄⋊C₄

C₂²

yes

—

C₃×C₄³

C₂²

yes

characteristic

C₂×C₄⋊(C₃⋊C₈)

—

C₂×C₄⋊(C₃⋊C₈)

C₂²

yes

—

C₂×#803

—

C₂×C₄×C₃⋊C₈

C₂²

yes

—

C₄×#125

D₃

yes

characteristic

#1842

—

C₄×#97

—

#1279

—

C₃×#125

—

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₃×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₃×C₄²

—

C₂×C₃×C₄²

C₂×C₄

yes

characteristic

C₂×C₃×C₄²

C₂×C₄

yes

characteristic

C₂×C₃×C₄²

D₄

yes

—

C₂×C₃×C₄²

D₄

yes

—

C₂×C₃×C₄²

C₂³

yes

characteristic

C₂²×C₃⋊C₈

—

C₂²×C₃⋊C₈

—

C₃×#95

—

C₃×#95

—

C₃×#95

—

C₄⋊(C₃⋊C₈)

—

C₄⋊(C₃⋊C₈)

—

#803

—

C₄×C₃⋊C₈

—

C₄×C₃⋊C₈

Q₈

yes

—

C₂×#125

Dic₃

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×#97

—

C₂×C₄×C₈

—

#389

Dic₃

yes

—

#387

—

#382

Dic₃

yes

characteristic

C₄×C₄⋊C₄

—

C₄×C₄⋊C₄

C₂×D₃

yes

—

C₄³

C₂×D₃

yes

characteristic

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

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

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

M₁₆

yes

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

C₂×Q₈

yes

—

C₃×C₄²

#41

yes

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₈

—

C₂²×C₈

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×Dic₃

yes

—

C₂×C₄⋊C₄

C₂×Dic₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×Dic₃

yes

characteristic

C₂×C₄²

C₂×Dic₃

yes

characteristic

C₂×C₄²

#67

yes

—

C₂×C₄²

#67

yes

—

C₂×C₄²

C₂²×D₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

#95

—

#95

—

#95

—

C₂³×C₃

C₂×C₂²⋊C₄

yes

characteristic

100

C₂×C₃×C₄

—

101

C₂×C₃×C₄

—

102

C₂×C₃×C₄

—

103

C₂×C₃×C₄

—

104

C₂×C₃×C₄

—

105

C₂×C₃×C₄

—

106

C₂×C₃×C₄

—

107

C₂×C₃×C₄

—

108

C₂×C₃×C₄

—

109

C₂×C₃×C₄

—

110

C₂×C₃×C₄

—

111

C₂×C₃×C₄

—

112

C₂×C₃×C₄

#104

yes

—

113

C₂×C₃×C₄

C₄×Q₈

yes

—

114

C₂×C₃×C₄

C₂²⋊Q₈

yes

—

115

C₂×C₃×C₄

#124

yes

characteristic

116

C₂×C₃×C₄

C₄⋊Q₈

yes

characteristic

117

C₂×C₃×C₄

C₂×M₁₆

yes

—

118

C₂×C₃×C₄

#131

yes

—

119

C₂×C₃×C₄

AGL(1,ℤ/8)

yes

—

120

C₂×C₃×C₄

#137

yes

—

121

C₃⋊C₈

—

122

C₃⋊C₈

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

#217

yes

—

126

C₂²×C₄

#217

yes

—

127

C₂²×C₄

C₂²×Dic₃

yes

characteristic

128

C₂²×C₄

C₂×#67

yes

characteristic

129

C₂²×C₄

C₂×#67

yes

characteristic

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

#208

yes

—

139

C₄²

—

140

C₄²

—

141

C₄²

—

142

C₄²

—

143

C₄²

—

144

C₄²

—

145

C₄²

—

146

C₄²

D₃×Q₈

yes

—

147

C₄²

#239

yes

—

148

C₂²×C₃

#391

yes

characteristic

149

C₂²×C₃

#407

yes

characteristic

150

C₂²×C₃

#408

yes

characteristic

151

C₂²×C₃

#417

yes

—

152

C₂²×C₃

C₂×#104

yes

—

153

C₃×C₄

—

154

C₃×C₄

—

155

C₃×C₄

—

156

C₃×C₄

—

157

C₃×C₄

—

158

C₃×C₄

#446

yes

—

159

C₃×C₄

#490

yes

characteristic

160

C₃×C₄

#491

yes

characteristic

161

C₃×C₄

#501

yes

—

162

C₂³

C₂×#217

yes

characteristic

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

#837

yes

—

176

C₂×C₄

C₂×#208

yes

—

177

C₂×C₄

#944

yes

characteristic

178

C₂×C₄

Q₈×Dic₃

yes

—

179

C₂×C₄

#946

yes

—

180

C₂×C₄

#947

yes

characteristic

181

C₂×C₄

#948

yes

—

182

C₂×C₄

#949

yes

—

183

C₂×C₄

#951

yes

—

184

C₈

—

185

C₈

—

186

C₂×C₃

#1483

yes

—

187

C₂×C₃

#1489

yes

—

188

C₂×C₃

#1985

yes

characteristic

189

C₂×C₃

#1992

yes

characteristic

190

C₂×C₃

#1994

yes

characteristic

191

C₂²

#5207

yes

characteristic

192

C₂²

#5229

yes

characteristic

193

C₂²

#5232

yes

—

194

C₂²

#5234

yes

characteristic

195

C₂²

C₂×#837

yes

—

196

C₄

—

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

—

201

C₄

#5024

yes

—

202

C₄

#5064

yes

—

203

C₄

#5067

yes

characteristic

204

C₄

#5069

yes

characteristic

205

C₃

#9876

yes

characteristic, complemented, Sylow

206

C₂

#71635

yes

characteristic

207

C₂

#71641

yes

—

208

C₂

#71741

yes

characteristic

209

C₂

#71751

yes

—

210

C₂

#71839

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×#67
commutator	C₂²×C₃×C₄	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×C₄×#125	C₂
socle, prime radical	C₂³×C₃	C₂×C₂²⋊C₄
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₄×#125	D₃
hypocenter	C₃	#9876
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#125

subgroup

G/H

center

C₂²×C₄

C₂×#67

commutator

C₂²×C₃×C₄

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₃

Fitting

C₃×C₄×#125

C₂

socle, prime radical

C₂³×C₃

C₂×C₂²⋊C₄

cosocle

C₂×C₃×C₄²

C₂³

D₃

C₃

G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#125

N	H ≅ G/N	type
205. C₃	#9876	semidirect

H ≅ G/N

type

205. C₃

#9876

semidirect

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

subgroup	index
C₃×C₄×#125	2
#69394	2
C₄×#803	2
#67619	2
#9876	3

subgroup

index

C₃×C₄×#125

Subgroups — #10695219

Conjugacy classes of subgroups

Complements

Maximal subgroups