Group #10695217 — subgroups

Subgroups — #10695217

Order 768.

486 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 155 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	C₄×#870	C₂	1	1	yes	maximal, characteristic
2	C₃×#1842	C₂	1	1	yes	maximal, characteristic
3	C₄×#803	C₂	1	1	yes	maximal, characteristic
4	#67704	C₂	1	4	yes	maximal
5	#9876	—	3	1	no	maximal, Sylow
6	C₂×#870	C₄	1	1	yes	characteristic
7	C₂×C₃×C₄⋊C₈	—	2	2	no	—
8	C₂×C₃×C₄⋊C₈	C₂²	1	2	yes	—
9	C₃×C₄³	C₂²	1	1	yes	characteristic
10	#4936	C₄	1	1	yes	characteristic
11	C₄×C₄⋊Dic₃	C₂²	1	2	yes	—
12	#4933	—	2	2	no	—
13	C₄×Dic₃⋊C₄	—	2	2	no	—
14	#4928	C₄	1	2	yes	—
15	C₂×#803	—	2	2	no	—
16	C₂×C₄×C₃⋊C₈	C₂²	1	2	yes	—
17	C₄×#125	—	3	1	no	—
18	#1842	D₃	1	1	yes	characteristic
19	C₄×#97	—	3	1	no	—
20	#1279	—	3	4	no	—
21	C₂²×C₃×C₈	—	2	2	no	—
22	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃×C₄²	D₄	1	2	yes	—
25	C₂×C₃×C₄²	D₄	1	2	yes	—
26	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
27	C₂×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
28	C₂×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
29	C₂×Dic₃⋊C₄	—	2	2	no	—
30	C₂×Dic₃⋊C₄	—	2	2	no	—
31	C₂×C₄×Dic₃	—	2	2	no	—
32	C₂²×C₃⋊C₈	—	2	2	no	—
33	#870	—	2	4	no	—
34	C₃×C₄⋊C₈	—	2	4	no	—
35	C₃×C₄⋊C₈	—	4	4	no	—
36	C₃⋊#95	—	2	2	no	—
37	C₃⋊#95	—	2	2	no	—
38	C₃⋊#95	—	2	2	no	—
39	#803	—	2	8	no	—
40	C₄×C₃⋊C₈	—	2	2	no	—
41	C₄×C₃⋊C₈	Q₈	1	4	yes	—
42	C₂×#125	—	3	1	no	—
43	C₂×C₄⋊C₈	—	2	2	no	—
44	C₂×C₄⋊C₈	C₂×D₃	1	2	yes	—
45	C₂×#97	—	6	2	no	—
46	C₂×C₄×C₈	—	3	2	no	—
47	#389	—	3	2	no	—
48	#387	—	6	2	no	—
49	#382	—	3	1	no	—
50	C₄×C₄⋊C₄	—	3	2	no	—
51	C₄×C₄⋊C₄	—	6	2	no	—
52	C₄³	C₂×D₃	1	1	yes	characteristic
53	C₂²×C₃×C₄	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₃×C₄	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
58	C₂²×Dic₃	—	2	2	no	—
59	C₂²×Dic₃	—	2	2	no	—
60	C₂×C₃×C₈	—	4	2	no	—
61	C₂×C₃×C₈	—	4	4	no	—
62	C₃×C₄²	—	2	4	no	—
63	C₃×C₄²	—	2	2	no	—
64	C₃×C₄²	—	2	2	no	—
65	C₃×C₄²	—	2	2	no	—
66	C₃×C₄²	—	2	2	no	—
67	C₃×C₄²	C₂×Q₈	1	2	yes	—
68	C₃×C₄²	#41	1	2	yes	—
69	C₄⋊Dic₃	—	2	2	no	—
70	C₄⋊Dic₃	—	2	4	no	—
71	C₄⋊Dic₃	M₁₆	1	4	yes	—
72	Dic₃⋊C₄	—	2	8	no	—
73	Dic₃⋊C₄	—	4	4	no	—
74	C₄×Dic₃	—	4	2	no	—
75	C₄×Dic₃	—	4	2	no	—
76	C₂×C₃⋊C₈	—	2	8	no	—
77	C₂×C₃⋊C₈	—	2	4	no	—
78	C₂²×C₈	—	2	2	no	—
79	C₂²×C₈	—	6	2	no	—
80	#125	—	6	4	no	—
81	C₂×C₄⋊C₄	—	3	2	no	—
82	C₂×C₄⋊C₄	—	3	2	no	—
83	C₂×C₄⋊C₄	—	6	2	no	—
84	C₂×C₄⋊C₄	—	6	2	no	—
85	C₂×C₄²	—	6	2	no	—
86	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
87	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
88	C₂×C₄²	D₁₂	1	2	yes	—
89	C₂×C₄²	#67	1	2	yes	—
90	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
91	C₄⋊C₈	—	2	4	no	—
92	C₄⋊C₈	—	4	4	no	—
93	#97	—	6	8	no	—
94	C₄×C₈	—	3	4	no	—
95	C₄×C₈	—	6	2	no	—
96	#95	—	6	2	no	—
97	#95	—	6	2	no	—
98	#95	—	6	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	2	no	—
105	C₂×C₃×C₄	—	2	2	no	—
106	C₂×C₃×C₄	#104	1	4	yes	—
107	C₂×C₃×C₄	C₄×Q₈	1	2	yes	—
108	C₂×C₃×C₄	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₃×C₄	#124	1	1	yes	characteristic
110	C₂×C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
111	C₂×C₃×C₄	C₂×M₁₆	1	2	yes	—
112	C₂×C₃×C₄	#131	1	2	yes	—
113	C₂×C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
114	C₂×C₃×C₄	#137	1	2	yes	—
115	C₂×Dic₃	—	2	4	no	—
116	C₂×Dic₃	—	4	2	no	—
117	C₂×Dic₃	—	4	2	no	—
118	C₂×Dic₃	—	4	2	no	—
119	C₂×Dic₃	—	4	2	no	—
120	C₂×Dic₃	—	4	2	no	—
121	C₃×C₈	—	8	2	no	—
122	C₃⋊C₈	—	2	8	no	—
123	C₂²×C₄	—	6	2	no	—
124	C₂²×C₄	—	6	2	no	—
125	C₂²×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
126	C₂²×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
127	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
128	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
129	C₂²×C₄	C₂×#67	1	1	yes	characteristic
130	C₂×C₈	—	4	4	no	—
131	C₂×C₈	—	4	2	no	—
132	C₂×C₈	—	6	8	no	—
133	C₂×C₈	—	6	4	no	—
134	C₄⋊C₄	—	3	4	no	—
135	C₄⋊C₄	—	6	8	no	—
136	C₄⋊C₄	—	6	2	no	—
137	C₄⋊C₄	—	6	4	no	—
138	C₄⋊C₄	—	12	4	no	—
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	—	12	2	no	—
145	C₄²	—	12	2	no	—
146	C₄²	#237	1	2	yes	—
147	C₄²	D₃×Q₈	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	#446	1	4	yes	—
157	C₃×C₄	#490	1	1	yes	characteristic
158	C₃×C₄	#491	1	1	yes	characteristic
159	C₃×C₄	#501	1	2	yes	—
160	Dic₃	—	4	4	no	—
161	Dic₃	—	8	2	no	—
162	C₂³	C₂×(C₂×D₃)⋊C₄	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	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	6	4	no	—
170	C₂×C₄	—	12	2	no	—
171	C₂×C₄	—	12	2	no	—
172	C₂×C₄	—	12	2	no	—
173	C₂×C₄	—	12	2	no	—
174	C₂×C₄	—	12	2	no	—
175	C₂×C₄	#825	1	4	yes	—
176	C₂×C₄	#887	1	2	yes	—
177	C₂×C₄	#897	1	2	yes	—
178	C₂×C₄	#901	1	2	yes	—
179	C₂×C₄	D₃×M₁₆	1	2	yes	—
180	C₂×C₄	#932	1	2	yes	—
181	C₂×C₄	#936	1	1	yes	characteristic
182	C₂×C₄	#942	1	2	yes	—
183	C₂×C₄	#944	1	1	yes	characteristic
184	C₈	—	6	8	no	—
185	C₈	—	8	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	#4966	1	2	yes	—
192	C₂²	#4973	1	1	yes	characteristic
193	C₂²	#5110	1	1	yes	characteristic
194	C₂²	#5126	1	1	yes	characteristic
195	C₂²	C₂×#825	1	2	yes	—
196	C₄	—	2	4	no	—
197	C₄	—	2	2	no	—
198	C₄	—	4	2	no	—
199	C₄	—	12	4	no	—
200	C₄	—	24	2	no	—
201	C₄	#4828	1	4	yes	—
202	C₄	#5071	1	1	yes	characteristic
203	C₄	#5081	1	2	yes	—
204	C₄	#5082	1	1	yes	characteristic
205	C₃	#9876	1	1	yes	characteristic, complemented, Sylow
206	C₂	#69583	1	1	yes	characteristic
207	C₂	#70427	1	2	yes	—
208	C₂	#70473	1	1	yes	characteristic
209	C₂	#70497	1	2	yes	—
210	C₂	#70514	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#870

C₂

yes

maximal, characteristic

C₃×#1842

C₂

yes

maximal, characteristic

C₄×#803

C₂

yes

maximal, characteristic

#67704

C₂

yes

—

C₄

yes

characteristic

C₂×C₃×C₄⋊C₈

—

C₂×C₃×C₄⋊C₈

C₂²

yes

—

C₃×C₄³

C₂²

yes

characteristic

#4936

C₄

yes

characteristic

C₄×C₄⋊Dic₃

C₂²

yes

—

#4933

—

C₄×Dic₃⋊C₄

—

#4928

C₄

yes

—

C₂×#803

—

C₂×C₄×C₃⋊C₈

C₂²

yes

—

C₄×#125

—

#1842

D₃

yes

characteristic

C₄×#97

—

#1279

—

C₂²×C₃×C₈

—

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

C₂×C₄⋊Dic₃

C₂×C₄

yes

—

C₂×C₄⋊Dic₃

C₂×C₄

yes

—

C₂×Dic₃⋊C₄

—

C₂×Dic₃⋊C₄

—

C₂×C₄×Dic₃

—

C₂²×C₃⋊C₈

—

#870

—

C₃×C₄⋊C₈

—

C₃×C₄⋊C₈

—

—

—

—

—

C₄×C₃⋊C₈

—

C₄×C₃⋊C₈

Q₈

yes

—

C₂×#125

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂×D₃

yes

—

C₂×#97

—

C₂×C₄×C₈

—

#389

—

#387

—

#382

—

C₄×C₄⋊C₄

—

C₄×C₄⋊C₄

—

C₄³

C₂×D₃

yes

characteristic

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

—

C₂²×Dic₃

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

C₂×Q₈

yes

—

C₃×C₄²

#41

yes

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

M₁₆

yes

—

Dic₃⋊C₄

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₈

—

C₂²×C₈

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₄×D₃

yes

characteristic

C₂×C₄²

C₄×D₃

yes

characteristic

C₂×C₄²

D₁₂

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

#104

yes

—

107

C₂×C₃×C₄

C₄×Q₈

yes

—

108

C₂×C₃×C₄

C₂²⋊Q₈

yes

—

109

C₂×C₃×C₄

#124

yes

characteristic

110

C₂×C₃×C₄

C₄⋊Q₈

yes

characteristic

111

C₂×C₃×C₄

C₂×M₁₆

yes

—

112

C₂×C₃×C₄

#131

yes

—

113

C₂×C₃×C₄

AGL(1,ℤ/8)

yes

—

114

C₂×C₃×C₄

#137

yes

—

115

C₂×Dic₃

—

116

C₂×Dic₃

—

117

C₂×Dic₃

—

118

C₂×Dic₃

—

119

C₂×Dic₃

—

120

C₂×Dic₃

—

121

C₃×C₈

—

122

C₃⋊C₈

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

(C₂×D₃)⋊C₄

yes

—

126

C₂²×C₄

(C₂×D₃)⋊C₄

yes

—

127

C₂²×C₄

C₂×C₄×D₃

yes

characteristic

128

C₂²×C₄

C₂×D₁₂

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

—

139

C₄²

—

140

C₄²

—

141

C₄²

—

142

C₄²

—

143

C₄²

—

144

C₄²

—

145

C₄²

—

146

C₄²

#237

yes

—

147

C₄²

D₃×Q₈

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

#446

yes

—

157

C₃×C₄

#490

yes

characteristic

158

C₃×C₄

#491

yes

characteristic

159

C₃×C₄

#501

yes

—

160

Dic₃

—

161

Dic₃

—

162

C₂³

C₂×(C₂×D₃)⋊C₄

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

#825

yes

—

176

C₂×C₄

#887

yes

—

177

C₂×C₄

#897

yes

—

178

C₂×C₄

#901

yes

—

179

C₂×C₄

D₃×M₁₆

yes

—

180

C₂×C₄

#932

yes

—

181

C₂×C₄

#936

yes

characteristic

182

C₂×C₄

#942

yes

—

183

C₂×C₄

#944

yes

characteristic

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

#4966

yes

—

192

C₂²

#4973

yes

characteristic

193

C₂²

#5110

yes

characteristic

194

C₂²

#5126

yes

characteristic

195

C₂²

C₂×#825

yes

—

196

C₄

—

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

—

201

C₄

#4828

yes

—

202

C₄

#5071

yes

characteristic

203

C₄

#5081

yes

—

204

C₄

#5082

yes

characteristic

205

C₃

#9876

yes

characteristic, complemented, Sylow

206

C₂

#69583

yes

characteristic

207

C₂

#70427

yes

—

208

C₂

#70473

yes

characteristic

209

C₂

#70497

yes

—

210

C₂

#70514

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×#67
commutator	C₂²×C₃×C₄	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×#1842	C₂
socle, prime radical	C₂³×C₃	C₂×C₂²⋊C₄
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	#1842	D₃
hypocenter	C₃	#9876
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

subgroup

G/H

center

C₂²×C₄

C₂×#67

commutator

C₂²×C₃×C₄

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₃

Fitting

C₃×#1842

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 ◁ #1842

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
C₄×#870	2
C₃×#1842	2
C₄×#803	2
#67704	2
#9876	3

subgroup

index

Subgroups — #10695217

Conjugacy classes of subgroups

Complements

Maximal subgroups