Group #10714580 — subgroups

Subgroups — #10714580

Order 768.

436 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 97 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	C₂×#4517	C₂	1	2	yes	maximal
3	#71445	C₂	1	2	yes	maximal
4	#69794	C₂	1	1	yes	maximal, characteristic
5	C₃×#1376	C₂	1	1	yes	maximal, characteristic
6	#13415	—	3	1	no	maximal, Sylow
7	C₂²×Dic₁₂	—	2	1	no	—
8	C₂²×Dic₁₂	C₂²	1	2	yes	—
9	C₂×#869	—	2	1	no	—
10	C₂×C₃×M₃₂	C₂²	1	2	yes	—
11	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₂×#819	C₂²	1	1	yes	characteristic
13	#5106	C₂²	1	1	yes	characteristic
14	#5102	—	2	1	no	—
15	#4699	—	2	4	no	—
16	#4523	—	2	1	no	—
17	#4517	—	2	4	no	—
18	#4503	—	2	4	no	—
19	C₂×#494	—	3	1	no	—
20	C₂×#362	—	3	2	no	—
21	#2141	—	3	2	no	—
22	#1945	—	3	1	no	—
23	#1376	D₃	1	1	yes	characteristic
24	C₂²×Dic₆	—	2	2	no	—
25	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
26	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
27	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
28	#926	—	2	1	no	—
29	C₂×C₄⋊Dic₃	—	4	2	no	—
30	C₂×#208	—	2	1	no	—
31	C₂×Dic₁₂	—	2	2	no	—
32	C₂×Dic₁₂	—	2	2	no	—
33	C₂×Dic₁₂	—	2	2	no	—
34	C₂×Dic₁₂	—	2	2	no	—
35	C₂×Dic₁₂	—	2	4	no	—
36	C₂×Dic₁₂	—	2	2	no	—
37	C₂×Dic₁₂	—	4	2	no	—
38	C₂×Dic₁₂	—	4	2	no	—
39	C₂×Dic₁₂	D₄	1	4	yes	—
40	#869	—	2	4	no	—
41	#869	—	4	2	no	—
42	C₃×M₃₂	—	2	2	no	—
43	C₃×M₃₂	—	2	2	no	—
44	C₂×C₃×C₁₆	—	2	2	no	—
45	C₃×C₄×C₈	—	2	2	no	—
46	#819	—	2	2	no	—
47	#818	D₄	1	2	yes	—
48	#817	—	2	1	no	—
49	Dic₃⋊C₈	—	4	2	no	—
50	C₂²×Q₁₆	—	3	2	no	—
51	C₂²×Q₁₆	—	6	1	no	—
52	C₂×C₄⋊Q₈	—	6	1	no	—
53	C₂×M₃₂	C₂×D₃	1	2	yes	—
54	#494	—	6	4	no	—
55	C₂×#108	—	3	1	no	—
56	#427	—	3	1	no	—
57	#424	—	6	1	no	—
58	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
59	#362	—	6	4	no	—
60	#358	—	6	4	no	—
61	#339	—	6	1	no	—
62	C₂²×C₃×C₄	—	2	1	no	—
63	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
64	C₂²×Dic₃	—	4	2	no	—
65	C₂×Dic₆	—	2	4	no	—
66	C₂×Dic₆	—	4	2	no	—
67	C₂×Dic₆	—	4	2	no	—
68	C₂×Dic₆	—	4	2	no	—
69	C₂×Dic₆	—	4	2	no	—
70	C₂×Dic₆	—	4	2	no	—
71	C₂×C₃×C₈	—	2	1	no	—
72	C₂×C₃×C₈	—	2	1	no	—
73	C₂×C₃×C₈	D₈	1	2	yes	—
74	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
75	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
76	C₂×C₃×C₈	#41	1	2	yes	—
77	C₂×C₃×C₈	#41	1	1	yes	characteristic
78	C₃×C₄²	—	2	1	no	—
79	C₃×C₄²	SD₁₆	1	2	yes	—
80	#217	—	4	1	no	—
81	C₄⋊Dic₃	—	2	2	no	—
82	C₄⋊Dic₃	—	4	8	no	—
83	C₄×Dic₃	—	4	1	no	—
84	#208	—	4	1	no	—
85	#208	—	4	1	no	—
86	C₂×C₃⋊C₈	—	4	1	no	—
87	Dic₁₂	—	2	4	no	—
88	Dic₁₂	—	2	4	no	—
89	Dic₁₂	—	4	8	no	—
90	Dic₁₂	—	4	2	no	—
91	Dic₁₂	—	4	2	no	—
92	C₃×C₁₆	—	4	2	no	—
93	C₂²×Q₈	—	6	2	no	—
94	C₂×Q₁₆	—	3	4	no	—
95	C₂×Q₁₆	—	6	4	no	—
96	C₂×Q₁₆	—	6	2	no	—
97	C₂×Q₁₆	—	6	2	no	—
98	C₂×Q₁₆	—	6	2	no	—
99	C₂×Q₁₆	—	6	2	no	—
100	C₂×Q₁₆	—	6	2	no	—
101	C₂×Q₁₆	—	12	2	no	—
102	C₂×Q₁₆	—	12	2	no	—
103	C₂×M₁₆	—	6	1	no	—
104	C₂²×C₈	D₁₂	1	1	yes	characteristic
105	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
106	C₄⋊Q₈	—	6	4	no	—
107	C₄⋊Q₈	—	12	2	no	—
108	#117	—	6	1	no	—
109	C₂×C₄⋊C₄	—	12	2	no	—
110	C₂×C₄²	D₁₂	1	1	yes	characteristic
111	M₃₂	—	2	2	no	—
112	M₃₂	—	2	2	no	—
113	C₂×C₁₆	—	2	2	no	—
114	#108	—	6	2	no	—
115	#107	—	3	2	no	—
116	#106	—	6	1	no	—
117	C₄⋊C₈	—	12	2	no	—
118	C₄×C₈	—	2	2	no	—
119	C₂³×C₃	AGL(1,ℤ/8)	1	1	yes	characteristic
120	C₂×C₃×C₄	—	2	4	no	—
121	C₂×C₃×C₄	—	2	2	no	—
122	C₂×C₃×C₄	#120	1	1	yes	characteristic
123	C₂×C₃×C₄	#121	1	1	yes	characteristic
124	C₂×C₃×C₄	#124	1	1	yes	characteristic
125	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
126	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
127	C₂×C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
128	C₂×Dic₃	—	4	4	no	—
129	C₂×Dic₃	—	4	2	no	—
130	C₂×Dic₃	—	8	2	no	—
131	C₂×Dic₃	—	8	2	no	—
132	Dic₆	—	4	4	no	—
133	Dic₆	—	4	4	no	—
134	Dic₆	—	8	2	no	—
135	Dic₆	—	8	2	no	—
136	C₃×C₈	—	2	2	no	—
137	C₃×C₈	#121	1	2	yes	—
138	C₃×C₈	#123	1	2	yes	—
139	C₃⋊C₈	—	8	1	no	—
140	C₂×Q₈	—	6	4	no	—
141	C₂×Q₈	—	12	2	no	—
142	C₂×Q₈	—	12	2	no	—
143	C₂×Q₈	—	12	2	no	—
144	C₂×Q₈	—	12	2	no	—
145	C₂×Q₈	—	12	2	no	—
146	C₂²×C₄	—	2	1	no	—
147	C₂²×C₄	—	12	2	no	—
148	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
149	Q₁₆	—	6	4	no	—
150	Q₁₆	—	6	4	no	—
151	Q₁₆	—	12	8	no	—
152	Q₁₆	—	12	2	no	—
153	Q₁₆	—	12	2	no	—
154	M₁₆	—	12	1	no	—
155	M₁₆	—	12	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	2	1	no	—
158	C₂×C₈	—	12	1	no	—
159	C₂×C₈	D₂₄	1	2	yes	—
160	C₂×C₈	#235	1	2	yes	—
161	C₂×C₈	D₃×D₄	1	1	yes	characteristic
162	C₂×C₈	D₃×D₄	1	2	yes	—
163	C₂×C₈	#239	1	1	yes	characteristic
164	C₄⋊C₄	—	6	2	no	—
165	C₄⋊C₄	—	12	8	no	—
166	C₂²⋊C₄	—	12	1	no	—
167	C₄²	—	2	1	no	—
168	C₄²	—	12	1	no	—
169	C₄²	#204	1	2	yes	—
170	C₁₆	—	4	2	no	—
171	C₂²×C₃	—	2	2	no	—
172	C₂²×C₃	#450	1	1	yes	characteristic
173	C₂²×C₃	#459	1	1	yes	characteristic
174	C₂²×C₃	#490	1	1	yes	characteristic
175	C₃×C₄	—	2	4	no	—
176	C₃×C₄	#394	1	1	yes	characteristic
177	C₃×C₄	#447	1	1	yes	characteristic
178	C₃×C₄	#460	1	1	yes	characteristic
179	C₃×C₄	#486	1	1	yes	characteristic
180	Dic₃	—	8	2	no	—
181	Dic₃	—	8	4	no	—
182	C₂³	#908	1	1	yes	characteristic
183	Q₈	—	12	4	no	—
184	Q₈	—	12	4	no	—
185	Q₈	—	24	2	no	—
186	Q₈	—	24	2	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	12	4	no	—
190	C₂×C₄	—	12	2	no	—
191	C₂×C₄	—	24	2	no	—
192	C₂×C₄	—	24	2	no	—
193	C₂×C₄	#875	1	1	yes	characteristic
194	C₂×C₄	#882	1	1	yes	characteristic
195	C₂×C₄	#895	1	1	yes	characteristic
196	C₂×C₄	C₂×#204	1	1	yes	characteristic
197	C₂×C₄	C₂×D₂₄	1	1	yes	characteristic
198	C₂×C₄	#908	1	1	yes	characteristic
199	C₈	—	2	2	no	—
200	C₈	—	24	1	no	—
201	C₈	#884	1	2	yes	—
202	C₈	#894	1	2	yes	—
203	C₂×C₃	—	2	1	no	—
204	C₂×C₃	—	2	1	no	—
205	C₂×C₃	#2041	1	1	yes	characteristic
206	C₂×C₃	#2214	1	2	yes	—
207	C₂²	—	2	2	no	—
208	C₂²	#4713	1	1	yes	characteristic
209	C₂²	#4731	1	1	yes	characteristic
210	C₂²	#4843	1	1	yes	characteristic
211	C₄	—	2	4	no	—
212	C₄	—	24	4	no	—
213	C₄	—	24	2	no	—
214	C₄	#4672	1	1	yes	characteristic
215	C₄	#4694	1	1	yes	characteristic
216	C₄	#4732	1	1	yes	characteristic
217	C₄	#4848	1	1	yes	characteristic
218	C₃	#13415	1	1	yes	characteristic, complemented, Sylow
219	C₂	—	2	1	no	—
220	C₂	—	2	1	no	—
221	C₂	#68629	1	1	yes	characteristic
222	C₂	#69306	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#4699

C₂

yes

maximal, characteristic

C₂×#4517

C₂

yes

maximal

#71445

C₂

yes

maximal

#69794

C₂

yes

maximal, characteristic

C₃×#1376

C₂

yes

maximal, characteristic

#13415

—

maximal, Sylow

C₂²×Dic₁₂

—

C₂²×Dic₁₂

C₂²

yes

—

C₂×#869

—

C₂×C₃×M₃₂

C₂²

yes

—

C₂×C₃×C₄×C₈

C₂²

yes

characteristic

C₂×#819

C₂²

yes

characteristic

#5106

C₂²

yes

—

—

—

—

—

—

—

—

—

D₃

yes

characteristic

C₂²×Dic₆

—

C₂²×C₃×C₈

D₄

yes

characteristic

C₂²×C₃×C₈

C₂³

yes

characteristic

C₂×C₃×C₄²

D₄

yes

characteristic

#926

—

C₂×C₄⋊Dic₃

—

C₂×#208

—

C₂×Dic₁₂

—

C₂×Dic₁₂

—

C₂×Dic₁₂

—

C₂×Dic₁₂

—

C₂×Dic₁₂

—

C₂×Dic₁₂

—

C₂×Dic₁₂

—

C₂×Dic₁₂

—

C₂×Dic₁₂

D₄

yes

—

#869

—

#869

—

C₃×M₃₂

—

C₃×M₃₂

—

C₂×C₃×C₁₆

—

C₃×C₄×C₈

—

#819

—

#818

D₄

yes

—

#817

—

Dic₃⋊C₈

—

C₂²×Q₁₆

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

C₂×M₃₂

C₂×D₃

yes

—

—

—

—

—

C₂×C₄×C₈

C₂×D₃

yes

—

—

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂×D₄

yes

characteristic

C₂²×Dic₃

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

D₈

yes

—

C₂×C₃×C₈

C₂×D₄

yes

characteristic

C₂×C₃×C₈

C₂×D₄

yes

—

C₂×C₃×C₈

#41

yes

—

C₂×C₃×C₈

#41

yes

characteristic

C₃×C₄²

—

C₃×C₄²

SD₁₆

yes

—

#217

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄×Dic₃

—

#208

—

#208

—

C₂×C₃⋊C₈

—

Dic₁₂

—

Dic₁₂

—

Dic₁₂

—

Dic₁₂

—

Dic₁₂

—

C₃×C₁₆

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

100

C₂×Q₁₆

—

101

C₂×Q₁₆

—

102

C₂×Q₁₆

—

103

C₂×M₁₆

—

104

C₂²×C₈

D₁₂

yes

characteristic

105

C₂²×C₈

C₂²×D₃

yes

characteristic

106

C₄⋊Q₈

—

107

C₄⋊Q₈

—

108

#117

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄²

D₁₂

yes

characteristic

111

M₃₂

—

112

M₃₂

—

113

C₂×C₁₆

—

114

#108

—

115

#107

—

116

#106

—

117

C₄⋊C₈

—

118

C₄×C₈

—

119

C₂³×C₃

AGL(1,ℤ/8)

yes

characteristic

120

C₂×C₃×C₄

—

121

C₂×C₃×C₄

—

122

C₂×C₃×C₄

#120

yes

characteristic

123

C₂×C₃×C₄

#121

yes

characteristic

124

C₂×C₃×C₄

#124

yes

characteristic

125

C₂×C₃×C₄

C₂×D₈

yes

characteristic

126

C₂×C₃×C₄

C₂×SD₁₆

yes

characteristic

127

C₂×C₃×C₄

AGL(1,ℤ/8)

yes

characteristic

128

C₂×Dic₃

—

129

C₂×Dic₃

—

130

C₂×Dic₃

—

131

C₂×Dic₃

—

132

Dic₆

—

133

Dic₆

—

134

Dic₆

—

135

Dic₆

—

136

C₃×C₈

—

137

C₃×C₈

#121

yes

—

138

C₃×C₈

#123

yes

—

139

C₃⋊C₈

—

140

C₂×Q₈

—

141

C₂×Q₈

—

142

C₂×Q₈

—

143

C₂×Q₈

—

144

C₂×Q₈

—

145

C₂×Q₈

—

146

C₂²×C₄

—

147

C₂²×C₄

—

148

C₂²×C₄

C₂×D₁₂

yes

characteristic

149

Q₁₆

—

150

Q₁₆

—

151

Q₁₆

—

152

Q₁₆

—

153

Q₁₆

—

154

M₁₆

—

155

M₁₆

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

D₂₄

yes

—

160

C₂×C₈

#235

yes

—

161

C₂×C₈

D₃×D₄

yes

characteristic

162

C₂×C₈

D₃×D₄

yes

—

163

C₂×C₈

#239

yes

characteristic

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₂²⋊C₄

—

167

C₄²

—

168

C₄²

—

169

C₄²

#204

yes

—

170

C₁₆

—

171

C₂²×C₃

—

172

C₂²×C₃

#450

yes

characteristic

173

C₂²×C₃

#459

yes

characteristic

174

C₂²×C₃

#490

yes

characteristic

175

C₃×C₄

—

176

C₃×C₄

#394

yes

characteristic

177

C₃×C₄

#447

yes

characteristic

178

C₃×C₄

#460

yes

characteristic

179

C₃×C₄

#486

yes

characteristic

180

Dic₃

—

181

Dic₃

—

182

C₂³

#908

yes

characteristic

183

Q₈

—

184

Q₈

—

185

Q₈

—

186

Q₈

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

#875

yes

characteristic

194

C₂×C₄

#882

yes

characteristic

195

C₂×C₄

#895

yes

characteristic

196

C₂×C₄

C₂×#204

yes

characteristic

197

C₂×C₄

C₂×D₂₄

yes

characteristic

198

C₂×C₄

#908

yes

characteristic

199

C₈

—

200

C₈

—

201

C₈

#884

yes

—

202

C₈

#894

yes

—

203

C₂×C₃

—

204

C₂×C₃

—

205

C₂×C₃

#2041

yes

characteristic

206

C₂×C₃

#2214

yes

—

207

C₂²

—

208

C₂²

#4713

yes

characteristic

209

C₂²

#4731

yes

characteristic

210

C₂²

#4843

yes

characteristic

211

C₄

—

212

C₄

—

213

C₄

—

214

C₄

#4672

yes

characteristic

215

C₄

#4694

yes

characteristic

216

C₄

#4732

yes

characteristic

217

C₄

#4848

yes

characteristic

218

C₃

#13415

yes

characteristic, complemented, Sylow

219

C₂

—

220

C₂

—

221

C₂

#68629

yes

characteristic

222

C₂

#69306

yes

—

	subgroup	G/H
center	C₂²	#4843
commutator, cosocle	C₂²×C₃×C₈	C₂³
Frattini	C₂²×C₈	C₂²×D₃
Fitting	C₃×#1376	C₂
socle	C₂²×C₃	#459
prime radical	C₂³×C₃	AGL(1,ℤ/8)
hypercenter	#1376	D₃
hypocenter	C₃	#13415
LCS	G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1376

subgroup

G/H

center

C₂²

#4843

commutator, cosocle

C₂²×C₃×C₈

C₂³

Frattini

C₂²×C₈

C₂²×D₃

C₂

C₂²×C₃

C₂³×C₃

D₃

C₃

G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1376

N	H ≅ G/N	type
218. C₃	#13415	semidirect

H ≅ G/N

type

218. C₃

#13415

semidirect

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

subgroup	index
C₂×#4699	2
C₂×#4517	2
#71445	2
#69794	2
C₃×#1376	2
#13415	3

subgroup

index

Subgroups — #10714580

Conjugacy classes of subgroups

Complements

Maximal subgroups