Group #10694269 — subgroups

Subgroups — #10694269

Order 768.

538 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 175 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×#2868	C₂	1	1	yes	maximal, characteristic
2	C₂×#4546	C₂	1	4	yes	maximal
3	#69392	C₂	1	1	yes	maximal, characteristic
4	#69391	C₂	1	1	yes	maximal, characteristic
5	#9702	—	3	1	no	maximal, Sylow
6	C₂×C₃×C₄⋊Q₈	C₄	1	4	yes	—
7	C₂×C₃×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₃×#384	—	2	1	no	—
9	C₃×#384	—	2	2	no	—
10	C₃×#384	—	2	1	no	—
11	C₃×C₄³	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
13	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
14	C₂×#803	—	2	1	no	—
15	#4546	—	2	8	no	—
16	#4524	—	2	1	no	—
17	#4524	—	2	2	no	—
18	#2868	D₃	1	1	yes	characteristic
19	#1831	—	3	1	no	—
20	#1767	—	3	1	no	—
21	C₂×#333	—	3	4	no	—
22	C₂²×C₃×Q₈	C₂×C₄	1	4	yes	—
23	C₃×C₄⋊Q₈	—	2	4	no	—
24	C₃×C₄⋊Q₈	—	2	4	no	—
25	C₃×C₄⋊Q₈	—	4	8	no	—
26	C₂×C₃×C₄⋊C₄	—	2	4	no	—
27	C₂×C₃×C₄⋊C₄	—	2	4	no	—
28	C₂×C₃×C₄⋊C₄	—	2	2	no	—
29	C₂×C₃×C₄⋊C₄	—	2	2	no	—
30	C₂×C₃×C₄²	C₂×C₄	1	2	yes	—
31	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
32	C₂×C₃×C₄²	D₄	1	2	yes	—
33	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
34	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
35	C₂²×C₃⋊C₈	—	2	2	no	—
36	C₂²×C₃⋊C₈	—	2	2	no	—
37	C₄⋊(C₃⋊C₈)	—	2	2	no	—
38	C₄⋊(C₃⋊C₈)	—	2	4	no	—
39	C₄⋊(C₃⋊C₈)	D₄	1	4	yes	—
40	#803	—	4	2	no	—
41	C₂×C₄⋊Q₈	Dic₃	1	4	yes	—
42	C₂×C₄⋊Q₈	C₂×D₃	1	2	yes	—
43	C₂×C₄⋊C₈	—	3	2	no	—
44	C₂×C₄⋊C₈	—	3	2	no	—
45	C₂×#97	—	6	1	no	—
46	#384	—	2	1	no	—
47	#384	—	2	2	no	—
48	#384	—	2	1	no	—
49	C₄³	C₂×D₃	1	1	yes	characteristic
50	#336	—	6	2	no	—
51	#336	—	6	1	no	—
52	#333	—	6	8	no	—
53	C₂×C₃×Q₈	—	2	8	no	—
54	C₂×C₃×Q₈	—	4	8	no	—
55	C₂²×C₃×C₄	—	2	4	no	—
56	C₂²×C₃×C₄	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₃×C₄	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
61	C₃×C₄⋊C₄	—	4	8	no	—
62	C₃×C₄⋊C₄	—	4	8	no	—
63	C₃×C₄⋊C₄	—	4	4	no	—
64	C₃×C₄⋊C₄	—	4	2	no	—
65	C₃×C₄⋊C₄	—	4	2	no	—
66	C₃×C₄²	—	2	4	no	—
67	C₃×C₄²	—	2	2	no	—
68	C₃×C₄²	—	2	4	no	—
69	C₃×C₄²	SD₁₆	1	2	yes	—
70	C₃×C₄²	Q₁₆	1	2	yes	—
71	C₃×C₄²	C₂×D₄	1	2	yes	—
72	C₃×C₄²	#41	1	2	yes	—
73	C₂×C₃⋊C₈	—	2	4	no	—
74	C₂×C₃⋊C₈	—	4	2	no	—
75	C₂×C₃⋊C₈	—	4	2	no	—
76	C₂×C₃⋊C₈	—	4	2	no	—
77	C₂×C₃⋊C₈	—	4	4	no	—
78	C₂²×Q₈	C₂×Dic₃	1	4	yes	—
79	C₂²×C₈	—	6	2	no	—
80	C₂²×C₈	—	6	2	no	—
81	C₄⋊Q₈	—	2	4	no	—
82	C₄⋊Q₈	—	2	4	no	—
83	C₄⋊Q₈	—	4	8	no	—
84	C₂×C₄⋊C₄	—	2	4	no	—
85	C₂×C₄⋊C₄	—	2	4	no	—
86	C₂×C₄⋊C₄	—	2	2	no	—
87	C₂×C₄⋊C₄	—	2	2	no	—
88	C₂×C₄²	C₂×Dic₃	1	2	yes	—
89	C₂×C₄²	#67	1	1	yes	characteristic
90	C₂×C₄²	#67	1	2	yes	—
91	C₂×C₄²	#67	1	1	yes	characteristic
92	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
93	C₄⋊C₈	—	3	4	no	—
94	C₄⋊C₈	—	6	4	no	—
95	C₄⋊C₈	—	6	2	no	—
96	#97	—	12	2	no	—
97	C₂³×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₃×Q₈	—	8	8	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	—	2	2	no	—
104	C₂×C₃×C₄	—	4	4	no	—
105	C₂×C₃×C₄	—	4	8	no	—
106	C₂×C₃×C₄	#100	1	4	yes	—
107	C₂×C₃×C₄	#103	1	4	yes	—
108	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
109	C₂×C₃×C₄	#121	1	2	yes	—
110	C₂×C₃×C₄	#124	1	1	yes	characteristic
111	C₂×C₃×C₄	#127	1	1	yes	characteristic
112	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
113	C₂×C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
114	C₂×C₃×C₄	#137	1	2	yes	—
115	C₂×C₃×C₄	#137	1	4	yes	—
116	C₃⋊C₈	—	4	4	no	—
117	C₃⋊C₈	—	8	2	no	—
118	C₂×Q₈	—	2	8	no	—
119	C₂×Q₈	—	4	8	no	—
120	C₂²×C₄	—	2	4	no	—
121	C₂²×C₄	#217	1	2	yes	—
122	C₂²×C₄	#217	1	2	yes	—
123	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
124	C₂²×C₄	C₂×#67	1	1	yes	characteristic
125	C₂²×C₄	C₂×#67	1	1	yes	characteristic
126	C₂×C₈	—	6	4	no	—
127	C₂×C₈	—	12	2	no	—
128	C₂×C₈	—	12	2	no	—
129	C₂×C₈	—	12	2	no	—
130	C₂×C₈	—	12	4	no	—
131	C₄⋊C₄	—	4	8	no	—
132	C₄⋊C₄	—	4	8	no	—
133	C₄⋊C₄	—	4	2	no	—
134	C₄⋊C₄	—	4	4	no	—
135	C₄⋊C₄	—	4	2	no	—
136	C₄²	—	2	4	no	—
137	C₄²	—	2	4	no	—
138	C₄²	—	2	2	no	—
139	C₄²	#215	1	2	yes	—
140	C₄²	#216	1	2	yes	—
141	C₄²	D₃×D₄	1	2	yes	—
142	C₄²	#237	1	2	yes	—
143	C₂²×C₃	#390	1	1	yes	characteristic
144	C₂²×C₃	C₂×#100	1	2	yes	—
145	C₂²×C₃	C₂×#103	1	1	yes	characteristic
146	C₂²×C₃	#419	1	2	yes	—
147	C₂²×C₃	#419	1	1	yes	characteristic
148	C₃×C₄	—	2	4	no	—
149	C₃×C₄	—	2	4	no	—
150	C₃×C₄	—	2	2	no	—
151	C₃×C₄	—	8	4	no	—
152	C₃×C₄	#461	1	2	yes	—
153	C₃×C₄	#462	1	2	yes	—
154	C₃×C₄	#491	1	2	yes	—
155	C₃×C₄	#497	1	2	yes	—
156	C₂³	C₂×#217	1	1	yes	characteristic
157	Q₈	—	8	8	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	4	8	no	—
164	C₂×C₄	—	4	4	no	—
165	C₂×C₄	#833	1	4	yes	—
166	C₂×C₄	#835	1	4	yes	—
167	C₂×C₄	D₄×Dic₃	1	2	yes	—
168	C₂×C₄	#936	1	1	yes	characteristic
169	C₂×C₄	#938	1	2	yes	—
170	C₂×C₄	#940	1	1	yes	characteristic
171	C₂×C₄	C₂×#215	1	1	yes	characteristic
172	C₂×C₄	#942	1	4	yes	—
173	C₂×C₄	#942	1	2	yes	—
174	C₂×C₄	C₂×#216	1	1	yes	characteristic
175	C₈	—	12	4	no	—
176	C₈	—	24	2	no	—
177	C₂×C₃	#1548	1	2	yes	—
178	C₂×C₃	#1558	1	2	yes	—
179	C₂×C₃	#1960	1	1	yes	characteristic
180	C₂×C₃	#1965	1	1	yes	characteristic
181	C₂×C₃	#1981	1	1	yes	characteristic
182	C₂²	C₂×#833	1	2	yes	—
183	C₂²	#5221	1	1	yes	characteristic
184	C₂²	C₂×#835	1	1	yes	characteristic
185	C₂²	#5225	1	2	yes	—
186	C₂²	#5225	1	1	yes	characteristic
187	C₄	—	2	2	no	—
188	C₄	—	2	4	no	—
189	C₄	—	2	4	no	—
190	C₄	—	8	4	no	—
191	C₄	#5082	1	2	yes	—
192	C₄	#5083	1	2	yes	—
193	C₄	#5086	1	2	yes	—
194	C₄	#5090	1	2	yes	—
195	C₃	#9702	1	1	yes	characteristic, complemented, Sylow
196	C₂	#71766	1	1	yes	characteristic
197	C₂	#71774	1	2	yes	—
198	C₂	#71777	1	2	yes	—
199	C₂	#71778	1	1	yes	characteristic
200	C₂	#71786	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#2868

C₂

yes

maximal, characteristic

C₂×#4546

C₂

yes

maximal

#69392

C₂

yes

maximal, characteristic

#69391

C₂

yes

maximal, characteristic

#9702

—

maximal, Sylow

C₂×C₃×C₄⋊Q₈

C₄

yes

—

C₂×C₃×C₄⋊Q₈

C₂²

yes

—

C₃×#384

—

C₃×#384

—

C₃×#384

—

C₃×C₄³

C₂²

yes

characteristic

C₂×C₄⋊(C₃⋊C₈)

C₂²

yes

—

C₂×C₄⋊(C₃⋊C₈)

C₂²

yes

—

—

—

—

—

D₃

yes

—

—

—

C₂²×C₃×Q₈

C₂×C₄

yes

—

C₃×C₄⋊Q₈

—

C₃×C₄⋊Q₈

—

C₃×C₄⋊Q₈

—

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄²

C₂×C₄

yes

—

C₂×C₃×C₄²

D₄

yes

characteristic

C₂×C₃×C₄²

D₄

yes

—

C₂×C₃×C₄²

D₄

yes

characteristic

C₂×C₃×C₄²

C₂³

yes

characteristic

C₂²×C₃⋊C₈

—

C₂²×C₃⋊C₈

—

C₄⋊(C₃⋊C₈)

—

C₄⋊(C₃⋊C₈)

—

C₄⋊(C₃⋊C₈)

D₄

yes

—

#803

—

C₂×C₄⋊Q₈

Dic₃

yes

—

C₂×C₄⋊Q₈

C₂×D₃

yes

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×#97

—

#384

—

#384

—

#384

—

C₄³

C₂×D₃

yes

—

—

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

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

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

SD₁₆

yes

—

C₃×C₄²

Q₁₆

yes

—

C₃×C₄²

C₂×D₄

yes

—

C₃×C₄²

#41

yes

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×Q₈

C₂×Dic₃

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×Dic₃

yes

—

C₂×C₄²

#67

yes

characteristic

C₂×C₄²

#67

yes

—

C₂×C₄²

#67

yes

characteristic

C₂×C₄²

C₂²×D₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₂³×C₃

C₂×C₂²⋊C₄

yes

characteristic

C₃×Q₈

—

C₂×C₃×C₄

—

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

#100

yes

—

107

C₂×C₃×C₄

#103

yes

—

108

C₂×C₃×C₄

C₄×D₄

yes

—

109

C₂×C₃×C₄

#121

yes

—

110

C₂×C₃×C₄

#124

yes

characteristic

111

C₂×C₃×C₄

#127

yes

characteristic

112

C₂×C₃×C₄

C₂×SD₁₆

yes

characteristic

113

C₂×C₃×C₄

C₂×Q₁₆

yes

characteristic

114

C₂×C₃×C₄

#137

yes

—

115

C₂×C₃×C₄

#137

yes

—

116

C₃⋊C₈

—

117

C₃⋊C₈

—

118

C₂×Q₈

—

119

C₂×Q₈

—

120

C₂²×C₄

—

121

C₂²×C₄

#217

yes

—

122

C₂²×C₄

#217

yes

—

123

C₂²×C₄

C₂²×Dic₃

yes

characteristic

124

C₂²×C₄

C₂×#67

yes

characteristic

125

C₂²×C₄

C₂×#67

yes

characteristic

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄²

—

137

C₄²

—

138

C₄²

—

139

C₄²

#215

yes

—

140

C₄²

#216

yes

—

141

C₄²

D₃×D₄

yes

—

142

C₄²

#237

yes

—

143

C₂²×C₃

#390

yes

characteristic

144

C₂²×C₃

C₂×#100

yes

—

145

C₂²×C₃

C₂×#103

yes

characteristic

146

C₂²×C₃

#419

yes

—

147

C₂²×C₃

#419

yes

characteristic

148

C₃×C₄

—

149

C₃×C₄

—

150

C₃×C₄

—

151

C₃×C₄

—

152

C₃×C₄

#461

yes

—

153

C₃×C₄

#462

yes

—

154

C₃×C₄

#491

yes

—

155

C₃×C₄

#497

yes

—

156

C₂³

C₂×#217

yes

characteristic

157

Q₈

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

#833

yes

—

166

C₂×C₄

#835

yes

—

167

C₂×C₄

D₄×Dic₃

yes

—

168

C₂×C₄

#936

yes

characteristic

169

C₂×C₄

#938

yes

—

170

C₂×C₄

#940

yes

characteristic

171

C₂×C₄

C₂×#215

yes

characteristic

172

C₂×C₄

#942

yes

—

173

C₂×C₄

#942

yes

—

174

C₂×C₄

C₂×#216

yes

characteristic

175

C₈

—

176

C₈

—

177

C₂×C₃

#1548

yes

—

178

C₂×C₃

#1558

yes

—

179

C₂×C₃

#1960

yes

characteristic

180

C₂×C₃

#1965

yes

characteristic

181

C₂×C₃

#1981

yes

characteristic

182

C₂²

C₂×#833

yes

—

183

C₂²

#5221

yes

characteristic

184

C₂²

C₂×#835

yes

characteristic

185

C₂²

#5225

yes

—

186

C₂²

#5225

yes

characteristic

187

C₄

—

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

#5082

yes

—

192

C₄

#5083

yes

—

193

C₄

#5086

yes

—

194

C₄

#5090

yes

—

195

C₃

#9702

yes

characteristic, complemented, Sylow

196

C₂

#71766

yes

characteristic

197

C₂

#71774

yes

—

198

C₂

#71777

yes

—

199

C₂

#71778

yes

characteristic

200

C₂

#71786

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×#217
commutator	C₂²×C₃×C₄	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×#2868	C₂
socle, prime radical	C₂³×C₃	C₂×C₂²⋊C₄
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	#2868	D₃
hypocenter	C₃	#9702
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2868

subgroup

G/H

center

C₂³

C₂×#217

commutator

C₂²×C₃×C₄

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₃

Fitting

C₃×#2868

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 ◁ #2868

N	H ≅ G/N	type
195. C₃	#9702	semidirect

H ≅ G/N

type

195. C₃

#9702

semidirect

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

subgroup	index
C₃×#2868	2
C₂×#4546	2
#69392	2
#69391	2
#9702	3

subgroup

index

Subgroups — #10694269

Conjugacy classes of subgroups

Complements

Maximal subgroups