Group #10694272 — subgroups

Subgroups — #10694272

Order 768.

538 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 171 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	#73311	C₂	1	1	yes	maximal, characteristic
2	C₃×#1767	C₂	1	1	yes	maximal, characteristic
3	C₂×#4486	C₂	1	4	yes	maximal
4	#69392	C₂	1	1	yes	maximal, characteristic
5	#9703	—	3	1	no	maximal, Sylow
6	C₂×#869	—	2	2	no	—
7	C₂×#869	C₂²	1	2	yes	—
8	C₂×C₃×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₃×#97	—	2	1	no	—
10	C₃×C₄³	C₂²	1	1	yes	characteristic
11	#4935	—	2	1	no	—
12	#4935	—	2	1	no	—
13	#4935	C₄	1	4	yes	—
14	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
15	C₃×#336	—	2	1	no	—
16	#4524	—	2	2	no	—
17	#4486	—	2	8	no	—
18	#2868	—	3	1	no	—
19	#1831	—	3	1	no	—
20	#1767	D₃	1	1	yes	characteristic
21	C₂×#332	—	3	4	no	—
22	C₂²×Dic₆	—	2	2	no	—
23	C₂²×C₃×C₈	—	2	2	no	—
24	C₂×C₃×C₄²	C₂×C₄	1	2	yes	—
25	C₂×C₃×C₄²	D₄	1	2	yes	—
26	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
27	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
28	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊Dic₃	—	2	4	no	—
30	C₂×C₄⋊Dic₃	—	2	4	no	—
31	C₂×C₄⋊Dic₃	—	2	2	no	—
32	C₂×C₄⋊Dic₃	C₂×C₄	1	4	yes	—
33	C₂²×C₃⋊C₈	—	2	2	no	—
34	#869	—	2	4	no	—
35	#869	—	2	4	no	—
36	#869	—	4	8	no	—
37	C₃×C₄⋊C₈	—	2	2	no	—
38	C₃×C₄⋊C₈	D₄	1	4	yes	—
39	C₃×#97	—	4	2	no	—
40	C₄⋊(C₃⋊C₈)	—	2	4	no	—
41	C₂×C₄⋊Q₈	—	3	2	no	—
42	C₂×C₄⋊Q₈	—	6	2	no	—
43	C₂×C₄⋊C₈	—	3	2	no	—
44	C₂×C₄⋊C₈	C₂×D₃	1	2	yes	—
45	C₂×#97	—	2	1	no	—
46	#384	—	3	4	no	—
47	#384	—	6	1	no	—
48	#384	—	6	1	no	—
49	C₄³	C₂×D₃	1	1	yes	characteristic
50	#336	—	2	1	no	—
51	#336	—	6	2	no	—
52	#332	—	6	8	no	—
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	4	no	—
59	C₂×Dic₆	—	4	2	no	—
60	C₂×Dic₆	—	4	2	no	—
61	C₂×Dic₆	—	4	8	no	—
62	C₂×C₃×C₈	—	2	4	no	—
63	C₂×C₃×C₈	—	4	2	no	—
64	C₂×C₃×C₈	—	4	2	no	—
65	C₃×C₄²	—	2	4	no	—
66	C₃×C₄²	—	2	4	no	—
67	C₃×C₄²	—	2	2	no	—
68	C₃×C₄²	SD₁₆	1	2	yes	—
69	C₃×C₄²	Q₁₆	1	2	yes	—
70	C₃×C₄²	C₂×D₄	1	2	yes	—
71	C₃×C₄²	#41	1	2	yes	—
72	C₄⋊Dic₃	—	2	8	no	—
73	C₄⋊Dic₃	—	4	8	no	—
74	C₄⋊Dic₃	—	4	8	no	—
75	C₄⋊Dic₃	—	4	4	no	—
76	C₂×C₃⋊C₈	—	4	2	no	—
77	C₂×C₃⋊C₈	—	4	4	no	—
78	C₂²×Q₈	—	6	2	no	—
79	C₂²×C₈	—	2	2	no	—
80	C₂²×C₈	—	6	2	no	—
81	C₄⋊Q₈	—	6	4	no	—
82	C₄⋊Q₈	—	6	4	no	—
83	C₄⋊Q₈	—	12	8	no	—
84	C₂×C₄⋊C₄	—	3	4	no	—
85	C₂×C₄⋊C₄	—	6	4	no	—
86	C₂×C₄⋊C₄	—	6	4	no	—
87	C₂×C₄⋊C₄	—	6	2	no	—
88	C₂×C₄²	C₄×D₃	1	2	yes	—
89	C₂×C₄²	D₁₂	1	2	yes	—
90	C₂×C₄²	#67	1	1	yes	characteristic
91	C₂×C₄²	#67	1	1	yes	characteristic
92	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
93	C₄⋊C₈	—	2	2	no	—
94	C₄⋊C₈	—	6	4	no	—
95	C₄⋊C₈	D₁₂	1	4	yes	—
96	#97	—	4	2	no	—
97	C₂³×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₃×C₄	—	2	2	no	—
99	C₂×C₃×C₄	—	2	2	no	—
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	#101	1	4	yes	—
104	C₂×C₃×C₄	#103	1	4	yes	—
105	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
106	C₂×C₃×C₄	#121	1	2	yes	—
107	C₂×C₃×C₄	#124	1	1	yes	characteristic
108	C₂×C₃×C₄	#127	1	1	yes	characteristic
109	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
110	C₂×C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
111	C₂×C₃×C₄	AGL(1,ℤ/8)	1	4	yes	—
112	C₂×C₃×C₄	#137	1	2	yes	—
113	C₂×Dic₃	—	4	4	no	—
114	C₂×Dic₃	—	4	8	no	—
115	Dic₆	—	8	4	no	—
116	Dic₆	—	8	4	no	—
117	C₃×C₈	—	4	4	no	—
118	C₃⋊C₈	—	8	2	no	—
119	C₂×Q₈	—	12	2	no	—
120	C₂×Q₈	—	12	2	no	—
121	C₂×Q₈	—	12	8	no	—
122	C₂²×C₄	—	6	4	no	—
123	C₂²×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
124	C₂²×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
125	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
126	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
127	C₂²×C₄	C₂×#67	1	1	yes	characteristic
128	C₂×C₈	—	2	4	no	—
129	C₂×C₈	—	4	2	no	—
130	C₂×C₈	—	4	2	no	—
131	C₂×C₈	—	12	2	no	—
132	C₂×C₈	—	12	4	no	—
133	C₄⋊C₄	—	6	8	no	—
134	C₄⋊C₄	—	12	8	no	—
135	C₄⋊C₄	—	12	8	no	—
136	C₄⋊C₄	—	12	4	no	—
137	C₄²	—	2	4	no	—
138	C₄²	—	2	4	no	—
139	C₄²	—	2	2	no	—
140	C₄²	#214	1	2	yes	—
141	C₄²	#216	1	2	yes	—
142	C₄²	C₂×D₁₂	1	2	yes	—
143	C₄²	#235	1	2	yes	—
144	C₂²×C₃	#390	1	1	yes	characteristic
145	C₂²×C₃	C₂×#101	1	2	yes	—
146	C₂²×C₃	C₂×#103	1	1	yes	characteristic
147	C₂²×C₃	#418	1	2	yes	—
148	C₂²×C₃	#419	1	1	yes	characteristic
149	C₃×C₄	—	2	4	no	—
150	C₃×C₄	—	2	4	no	—
151	C₃×C₄	—	2	2	no	—
152	C₃×C₄	#461	1	2	yes	—
153	C₃×C₄	#462	1	2	yes	—
154	C₃×C₄	#490	1	2	yes	—
155	C₃×C₄	#496	1	2	yes	—
156	Dic₃	—	8	4	no	—
157	C₂³	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
158	Q₈	—	24	4	no	—
159	Q₈	—	24	4	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	12	8	no	—
166	C₂×C₄	—	12	4	no	—
167	C₂×C₄	#810	1	4	yes	—
168	C₂×C₄	#824	1	4	yes	—
169	C₂×C₄	C₄×D₁₂	1	2	yes	—
170	C₂×C₄	#874	1	1	yes	characteristic
171	C₂×C₄	#875	1	1	yes	characteristic
172	C₂×C₄	#908	1	4	yes	—
173	C₂×C₄	#930	1	2	yes	—
174	C₂×C₄	C₂×#214	1	1	yes	characteristic
175	C₂×C₄	C₂×#216	1	1	yes	characteristic
176	C₂×C₄	#951	1	2	yes	—
177	C₈	—	4	4	no	—
178	C₈	—	24	2	no	—
179	C₂×C₃	#1549	1	2	yes	—
180	C₂×C₃	#1556	1	2	yes	—
181	C₂×C₃	#1961	1	1	yes	characteristic
182	C₂×C₃	#1966	1	1	yes	characteristic
183	C₂×C₃	#1981	1	1	yes	characteristic
184	C₂²	#4939	1	1	yes	characteristic
185	C₂²	C₂×#810	1	1	yes	characteristic
186	C₂²	#5004	1	1	yes	characteristic
187	C₂²	#5131	1	2	yes	—
188	C₂²	C₂×#824	1	2	yes	—
189	C₄	—	2	4	no	—
190	C₄	—	2	4	no	—
191	C₄	—	2	2	no	—
192	C₄	—	24	4	no	—
193	C₄	#4712	1	2	yes	—
194	C₄	#4713	1	2	yes	—
195	C₄	#5019	1	2	yes	—
196	C₄	#5031	1	2	yes	—
197	C₃	#9703	1	1	yes	characteristic, complemented, Sylow
198	C₂	#69486	1	1	yes	characteristic
199	C₂	#69823	1	1	yes	characteristic
200	C₂	#69877	1	1	yes	characteristic
201	C₂	#70508	1	2	yes	—
202	C₂	#70822	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#73311

C₂

yes

maximal, characteristic

C₃×#1767

C₂

yes

maximal, characteristic

C₂×#4486

C₂

yes

maximal

#69392

C₂

yes

maximal, characteristic

—

—

C₂²

yes

—

C₂×C₃×C₄⋊C₈

C₂²

yes

—

C₂×C₃×#97

—

C₃×C₄³

C₂²

yes

—

—

C₄

yes

—

C₂×C₄⋊(C₃⋊C₈)

C₂²

yes

—

—

—

—

—

—

D₃

yes

characteristic

C₂×#332

—

C₂²×Dic₆

—

C₂²×C₃×C₈

—

C₂×C₃×C₄²

C₂×C₄

yes

—

C₂×C₃×C₄²

D₄

yes

—

C₂×C₃×C₄²

D₄

yes

characteristic

C₂×C₃×C₄²

D₄

yes

characteristic

C₂×C₃×C₄²

C₂³

yes

characteristic

C₂×C₄⋊Dic₃

—

C₂×C₄⋊Dic₃

—

C₂×C₄⋊Dic₃

—

C₂×C₄⋊Dic₃

C₂×C₄

yes

—

C₂²×C₃⋊C₈

—

#869

—

#869

—

#869

—

C₃×C₄⋊C₈

—

C₃×C₄⋊C₈

D₄

yes

—

C₃×#97

—

C₄⋊(C₃⋊C₈)

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂×D₃

yes

—

C₂×#97

—

#384

—

#384

—

#384

—

C₄³

C₂×D₃

yes

—

—

—

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

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

SD₁₆

yes

—

C₃×C₄²

Q₁₆

yes

—

C₃×C₄²

C₂×D₄

yes

—

C₃×C₄²

#41

yes

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₄×D₃

yes

—

C₂×C₄²

D₁₂

yes

—

C₂×C₄²

yes

C₂×C₄²

yes

C₂×C₄²

C₂²×D₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

D₁₂

yes

—

#97

—

C₂³×C₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

100

C₂×C₃×C₄

—

101

C₂×C₃×C₄

—

102

C₂×C₃×C₄

—

103

C₂×C₃×C₄

#101

yes

—

104

C₂×C₃×C₄

#103

yes

—

105

C₂×C₃×C₄

C₄×D₄

yes

—

106

C₂×C₃×C₄

#121

yes

—

107

C₂×C₃×C₄

#124

yes

characteristic

108

C₂×C₃×C₄

#127

yes

characteristic

109

C₂×C₃×C₄

C₂×SD₁₆

yes

characteristic

110

C₂×C₃×C₄

C₂×Q₁₆

yes

characteristic

111

C₂×C₃×C₄

AGL(1,ℤ/8)

yes

—

112

C₂×C₃×C₄

#137

yes

—

113

C₂×Dic₃

—

114

C₂×Dic₃

—

115

Dic₆

—

116

Dic₆

—

117

C₃×C₈

—

118

C₃⋊C₈

—

119

C₂×Q₈

—

120

C₂×Q₈

—

121

C₂×Q₈

—

122

C₂²×C₄

—

123

C₂²×C₄

(C₂×D₃)⋊C₄

yes

—

124

C₂²×C₄

(C₂×D₃)⋊C₄

yes

—

125

C₂²×C₄

C₂×C₄×D₃

yes

characteristic

126

C₂²×C₄

C₂×D₁₂

yes

characteristic

127

C₂²×C₄

C₂×#67

yes

characteristic

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄²

—

138

C₄²

—

139

C₄²

—

140

C₄²

#214

yes

—

141

C₄²

#216

yes

—

142

C₄²

C₂×D₁₂

yes

—

143

C₄²

#235

yes

—

144

C₂²×C₃

#390

yes

characteristic

145

C₂²×C₃

C₂×#101

yes

—

146

C₂²×C₃

C₂×#103

yes

characteristic

147

C₂²×C₃

#418

yes

—

148

C₂²×C₃

#419

yes

characteristic

149

C₃×C₄

—

150

C₃×C₄

—

151

C₃×C₄

—

152

C₃×C₄

#461

yes

—

153

C₃×C₄

#462

yes

—

154

C₃×C₄

#490

yes

—

155

C₃×C₄

#496

yes

—

156

Dic₃

—

157

C₂³

C₂×(C₂×D₃)⋊C₄

yes

characteristic

158

Q₈

—

159

Q₈

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

#810

yes

—

168

C₂×C₄

#824

yes

—

169

C₂×C₄

C₄×D₁₂

yes

—

170

C₂×C₄

#874

yes

characteristic

171

C₂×C₄

#875

yes

characteristic

172

C₂×C₄

#908

yes

—

173

C₂×C₄

#930

yes

—

174

C₂×C₄

C₂×#214

yes

characteristic

175

C₂×C₄

C₂×#216

yes

characteristic

176

C₂×C₄

#951

yes

—

177

C₈

—

178

C₈

—

179

C₂×C₃

#1549

yes

—

180

C₂×C₃

#1556

yes

—

181

C₂×C₃

#1961

yes

characteristic

182

C₂×C₃

#1966

yes

characteristic

183

C₂×C₃

#1981

yes

characteristic

184

C₂²

#4939

yes

characteristic

185

C₂²

C₂×#810

yes

characteristic

186

C₂²

#5004

yes

characteristic

187

C₂²

#5131

yes

—

188

C₂²

C₂×#824

yes

—

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

#4712

yes

—

194

C₄

#4713

yes

—

195

C₄

#5019

yes

—

196

C₄

#5031

yes

—

197

C₃

#9703

yes

characteristic, complemented, Sylow

198

C₂

#69486

yes

characteristic

199

C₂

#69823

yes

characteristic

200

C₂

#69877

yes

characteristic

201

C₂

#70508

yes

—

202

C₂

#70822

yes

—

	subgroup	G/H
center	C₂³	C₂×(C₂×D₃)⋊C₄
commutator	C₂²×C₃×C₄	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×#1767	C₂
socle, prime radical	C₂³×C₃	C₂×C₂²⋊C₄
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	#1767	D₃
hypocenter	C₃	#9703
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1767

subgroup

G/H

center

C₂³

C₂×(C₂×D₃)⋊C₄

commutator

C₂²×C₃×C₄

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₃

Fitting

C₃×#1767

C₂

socle, prime radical

C₂³×C₃

C₂×C₂²⋊C₄

cosocle

C₂×C₃×C₄²

C₂³

D₃

C₃

G ▷ C₂²×C₃×C₄ ▷ C₂²×C₃ ▷ C₃

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #1767

N	H ≅ G/N	type
197. C₃	#9703	semidirect

H ≅ G/N

type

197. C₃

#9703

semidirect

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

subgroup	index
#73311	2
C₃×#1767	2
C₂×#4486	2
#69392	2
#9703	3

subgroup

index

Subgroups — #10694272

Conjugacy classes of subgroups

Complements

Maximal subgroups