Group #35304074 — subgroups

Subgroups — #35304074

Order 21248.

434 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 127 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×#19128106	C₂	1	1	yes	maximal, characteristic
2	#24908306	C₂	1	2	yes	maximal
3	#24906136	C₂	1	2	yes	maximal
4	#24905752	C₂	1	1	yes	maximal, characteristic
5	C₈₃×#1566	C₂	1	1	yes	maximal, characteristic
6	C₂×#14581792	C₂²	1	2	yes	—
7	#19128106	—	2	1	no	—
8	#19128106	—	2	1	no	—
9	#19128106	—	2	2	no	—
10	#19127754	C₂²	1	1	yes	characteristic
11	C₂×#14581726	C₄	1	4	yes	—
12	C₂×#14581710	—	2	4	no	—
13	C₄×#13050574	C₂²	1	1	yes	characteristic
14	#19127448	—	2	4	no	—
15	#19127423	—	2	1	no	—
16	#19127422	—	2	1	no	—
17	#423×C₈₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
19	#334×C₈₃	—	2	2	no	—
20	C₂²×Dic₁₆₆	—	2	2	no	—
21	C₂×#13050574	D₄	1	2	yes	—
22	#14581792	—	2	2	no	—
23	#14581792	—	2	2	no	—
24	#14581792	—	2	2	no	—
25	#14581792	—	2	2	no	—
26	C₂×C₄⋊Dic₈₃	—	2	2	no	—
27	C₂×C₄⋊Dic₈₃	C₂×C₄	1	4	yes	—
28	#14581726	—	2	4	no	—
29	#14581726	—	2	4	no	—
30	C₄⋊(C₈₃⋊C₈)	D₄	1	2	yes	—
31	#14581710	—	4	8	no	—
32	C₂²⋊(C₈₃⋊C₈)	—	2	1	no	—
33	#14581703	—	2	1	no	—
34	C₄×C₈₃⋊C₈	—	2	1	no	—
35	C₄×C₈₃⋊C₈	—	2	1	no	—
36	C₂×M₁₆×C₈₃	—	2	2	no	—
37	C₂²×C₈×C₈₃	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₈₃	—	2	2	no	—
40	C₄⋊C₈×C₈₃	—	2	2	no	—
41	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
42	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
43	C₄×C₈×C₈₃	D₄	1	2	yes	—
44	C₂×Dic₁₆₆	—	4	2	no	—
45	C₂×Dic₁₆₆	—	4	2	no	—
46	C₂×Dic₁₆₆	—	4	4	no	—
47	C₂×Dic₁₆₆	—	4	4	no	—
48	C₂²×Dic₈₃	—	2	4	no	—
49	#13050574	—	2	4	no	—
50	C₂×C₈₃⋊C₈	—	2	2	no	—
51	C₂×C₈₃⋊C₈	—	2	2	no	—
52	C₄⋊Dic₈₃	—	2	4	no	—
53	C₄⋊Dic₈₃	—	2	4	no	—
54	C₄⋊Dic₈₃	—	4	2	no	—
55	C₄⋊Dic₈₃	—	4	2	no	—
56	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
59	M₁₆×C₈₃	—	4	4	no	—
60	C₂×C₈×C₈₃	—	2	4	no	—
61	C₂×C₈×C₈₃	—	4	2	no	—
62	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
64	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₈₃	#41	1	1	yes	characteristic
68	C₂³×C₈₃	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₈₃	—	2	2	no	—
70	C₂×C₄×C₈₃	—	2	2	no	—
71	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
73	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
74	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
75	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
76	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
77	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	4	yes	—
78	C₂×Dic₈₃	—	4	4	no	—
79	C₂×Dic₈₃	—	4	8	no	—
80	Dic₁₆₆	—	8	4	no	—
81	Dic₁₆₆	—	8	2	no	—
82	Dic₁₆₆	—	8	2	no	—
83	C₈×C₈₃	—	2	2	no	—
84	C₈×C₈₃	—	8	2	no	—
85	C₈×C₈₃	#101	1	4	yes	complemented
86	C₈₃⋊C₈	—	4	2	no	—
87	C₈₃⋊C₈	—	4	2	no	—
88	C₂²×C₈₃	—	2	1	no	—
89	C₂²×C₈₃	—	2	1	no	—
90	C₂²×C₈₃	#390	1	1	yes	characteristic
91	C₂²×C₈₃	#442	1	2	yes	—
92	C₄×C₈₃	—	2	1	no	—
93	C₄×C₈₃	—	2	1	no	—
94	C₄×C₈₃	C₂×#101	1	2	yes	—
95	C₄×C₈₃	#469	1	2	yes	—
96	C₄×C₈₃	#473	1	2	yes	—
97	C₄×C₈₃	#496	1	2	yes	—
98	Dic₈₃	—	8	4	no	—
99	#9808	—	83	1	no	maximal, Sylow
100	C₂×C₈₃	—	2	2	no	—
101	C₂×C₈₃	#1966	1	1	yes	characteristic
102	C₂×C₈₃	#1972	1	1	yes	characteristic
103	C₂×C₈₃	#1979	1	1	yes	characteristic
104	C₂×#500	—	83	1	no	—
105	#1966	—	83	2	no	—
106	#1566	D₈₃	1	1	yes	characteristic
107	#1546	—	83	2	no	—
108	#1451	—	83	1	no	—
109	C₈₃	#9808	1	1	yes	characteristic, complemented, Sylow
110	C₂×C₄⋊Q₈	—	83	2	no	—
111	#500	—	166	1	no	—
112	#500	—	166	1	no	—
113	#500	—	166	2	no	—
114	#431	—	83	1	no	—
115	C₂×#107	—	83	4	no	—
116	#423	C₂×D₈₃	1	2	yes	—
117	C₂×#101	—	166	4	no	—
118	C₄×M₁₆	—	83	1	no	—
119	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
120	#334	—	2	2	no	—
121	#332	—	166	4	no	—
122	#322	—	166	1	no	—
123	#322	—	166	1	no	—
124	C₂²×Q₈	—	166	2	no	—
125	C₂×M₁₆	—	2	2	no	—
126	C₂×M₁₆	—	83	2	no	—
127	C₂²×C₈	C₄×D₈₃	1	2	yes	—
128	C₄⋊Q₈	—	166	2	no	—
129	C₄⋊Q₈	—	166	2	no	—
130	C₄⋊Q₈	—	166	2	no	—
131	C₄⋊Q₈	—	166	2	no	—
132	C₂×C₄⋊C₄	—	83	4	no	—
133	C₂×C₄⋊C₄	—	166	2	no	—
134	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
135	#107	—	166	4	no	—
136	#107	—	166	4	no	—
137	C₄⋊C₈	—	2	2	no	—
138	C₄⋊C₈	—	2	2	no	—
139	C₄⋊C₈	—	83	2	no	—
140	#101	—	332	8	no	—
141	C₂²⋊C₈	—	166	1	no	—
142	#97	—	166	1	no	—
143	C₄×C₈	—	166	1	no	—
144	C₄×C₈	—	166	1	no	—
145	C₄×C₈	D₃₃₂	1	1	yes	characteristic
146	C₄×C₈	D₃₃₂	1	1	yes	characteristic
147	C₄×C₈	C₂²⋊D₈₃	1	2	yes	—
148	C₂×Q₈	—	332	2	no	—
149	C₂×Q₈	—	332	2	no	—
150	C₂×Q₈	—	332	4	no	—
151	C₂×Q₈	—	332	4	no	—
152	C₂²×C₄	—	166	4	no	—
153	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
154	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
155	C₂²×C₄	#13050591	1	1	yes	characteristic
156	M₁₆	—	4	4	no	—
157	M₁₆	—	166	4	no	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	—	4	2	no	—
160	C₂×C₈	—	166	2	no	—
161	C₂×C₈	—	166	2	no	—
162	C₂×C₈	(C₂×D₈₃)⋊C₄	1	2	yes	—
163	C₂×C₈	(C₂×D₈₃)⋊C₄	1	2	yes	—
164	C₄⋊C₄	—	332	2	no	—
165	C₄⋊C₄	—	332	2	no	—
166	C₄⋊C₄	—	166	4	no	—
167	C₄⋊C₄	—	166	4	no	—
168	C₄²	D₄×D₈₃	1	1	yes	characteristic
169	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
170	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
171	C₄²	#13050591	1	1	yes	characteristic
172	C₂³	#14581754	1	1	yes	characteristic
173	Q₈	—	664	2	no	—
174	Q₈	—	664	2	no	—
175	Q₈	—	664	4	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	332	8	no	—
179	C₂×C₄	—	332	4	no	—
180	C₂×C₄	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
181	C₂×C₄	#14581754	1	1	yes	characteristic
182	C₂×C₄	#14581767	1	1	yes	characteristic
183	C₂×C₄	#14581767	1	1	yes	characteristic
184	C₂×C₄	#14581781	1	1	yes	characteristic
185	C₂×C₄	#14581788	1	1	yes	characteristic
186	C₂×C₄	#14581819	1	4	yes	—
187	C₈	—	2	2	no	—
188	C₈	—	8	2	no	—
189	C₈	—	332	2	no	—
190	C₈	—	332	2	no	—
191	C₈	#14581710	1	4	yes	complemented
192	C₂²	—	2	1	no	—
193	C₂²	—	2	1	no	—
194	C₂²	#19127602	1	1	yes	characteristic
195	C₂²	#19127812	1	2	yes	—
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	—	664	4	no	—
199	C₄	C₂×#14581710	1	2	yes	—
200	C₄	#19127955	1	2	yes	—
201	C₄	#19127977	1	2	yes	—
202	C₄	#19128087	1	2	yes	—
203	C₂	—	2	2	no	—
204	C₂	#24908306	1	1	yes	characteristic
205	C₂	#24908337	1	1	yes	characteristic
206	C₂	#24908376	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19128106

C₂

yes

maximal, characteristic

#24908306

C₂

yes

maximal

#24906136

C₂

yes

maximal

#24905752

C₂

yes

maximal, characteristic

C₈₃×#1566

C₂

yes

maximal, characteristic

C₂×#14581792

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₂×#14581726

C₄

yes

—

C₂×#14581710

—

C₄×#13050574

C₂²

yes

—

—

—

#423×C₈₃

C₂²

yes

—

C₂×C₄×C₈×C₈₃

C₂²

yes

characteristic

#334×C₈₃

—

C₂²×Dic₁₆₆

—

C₂×#13050574

D₄

yes

—

—

—

—

—

C₂×C₄⋊Dic₈₃

—

C₂×C₄⋊Dic₈₃

C₂×C₄

yes

—

#14581726

—

#14581726

—

C₄⋊(C₈₃⋊C₈)

D₄

yes

—

#14581710

—

C₂²⋊(C₈₃⋊C₈)

—

#14581703

—

C₄×C₈₃⋊C₈

—

C₄×C₈₃⋊C₈

—

C₂×M₁₆×C₈₃

—

C₂²×C₈×C₈₃

C₂×C₄

yes

—

C₂×C₄²×C₈₃

C₂³

yes

characteristic

C₄⋊C₈×C₈₃

—

C₄⋊C₈×C₈₃

—

C₄×C₈×C₈₃

D₄

yes

characteristic

C₄×C₈×C₈₃

D₄

yes

characteristic

C₄×C₈×C₈₃

D₄

yes

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂²×Dic₈₃

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₂²×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₃

#41

yes

characteristic

M₁₆×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₂²⋊C₄

yes

—

C₂×C₈×C₈₃

C₂²⋊C₄

yes

—

C₄²×C₈₃

C₂×D₄

yes

characteristic

C₄²×C₈₃

C₂×D₄

yes

characteristic

C₄²×C₈₃

C₂×D₄

yes

characteristic

C₄²×C₈₃

#41

yes

characteristic

C₂³×C₈₃

C₄×D₄

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₄×D₄

yes

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

#101

yes

complemented

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

#390

yes

characteristic

C₂²×C₈₃

#442

yes

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₂×#101

yes

—

C₄×C₈₃

#469

yes

—

C₄×C₈₃

#473

yes

—

C₄×C₈₃

#496

yes

—

Dic₈₃

—

#9808

—

maximal, Sylow

100

C₂×C₈₃

—

101

C₂×C₈₃

#1966

yes

characteristic

102

C₂×C₈₃

#1972

yes

characteristic

103

C₂×C₈₃

#1979

yes

characteristic

104

C₂×#500

—

105

#1966

—

106

#1566

D₈₃

yes

characteristic

107

#1546

—

108

#1451

—

109

C₈₃

#9808

yes

characteristic, complemented, Sylow

110

C₂×C₄⋊Q₈

—

111

#500

—

166

—

112

#500

—

166

—

113

#500

—

166

—

114

#431

—

115

C₂×#107

—

116

#423

C₂×D₈₃

yes

—

117

C₂×#101

—

166

—

118

C₄×M₁₆

—

119

C₂×C₄×C₈

C₂×D₈₃

yes

characteristic

120

#334

—

121

#332

—

166

—

122

#322

—

166

—

123

#322

—

166

—

124

C₂²×Q₈

—

166

—

125

C₂×M₁₆

—

126

C₂×M₁₆

—

127

C₂²×C₈

C₄×D₈₃

yes

—

128

C₄⋊Q₈

—

166

—

129

C₄⋊Q₈

—

166

—

130

C₄⋊Q₈

—

166

—

131

C₄⋊Q₈

—

166

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄⋊C₄

—

166

—

134

C₂×C₄²

C₂²×D₈₃

yes

characteristic

135

#107

—

166

—

136

#107

—

166

—

137

C₄⋊C₈

—

138

C₄⋊C₈

—

139

C₄⋊C₈

—

140

#101

—

332

—

141

C₂²⋊C₈

—

166

—

142

#97

—

166

—

143

C₄×C₈

—

166

—

144

C₄×C₈

—

166

—

145

C₄×C₈

D₃₃₂

yes

characteristic

146

C₄×C₈

D₃₃₂

yes

characteristic

147

C₄×C₈

C₂²⋊D₈₃

yes

—

148

C₂×Q₈

—

332

—

149

C₂×Q₈

—

332

—

150

C₂×Q₈

—

332

—

151

C₂×Q₈

—

332

—

152

C₂²×C₄

—

166

—

153

C₂²×C₄

C₂×C₄×D₈₃

yes

characteristic

154

C₂²×C₄

D₄×D₈₃

yes

characteristic

155

C₂²×C₄

#13050591

yes

characteristic

156

M₁₆

—

157

M₁₆

—

166

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

166

—

161

C₂×C₈

—

166

—

162

C₂×C₈

(C₂×D₈₃)⋊C₄

yes

—

163

C₂×C₈

(C₂×D₈₃)⋊C₄

yes

—

164

C₄⋊C₄

—

332

—

165

C₄⋊C₄

—

332

—

166

C₄⋊C₄

—

166

—

167

C₄⋊C₄

—

166

—

168

C₄²

D₄×D₈₃

yes

characteristic

169

C₄²

C₂×C₂²⋊D₈₃

yes

characteristic

170

C₄²

C₂×D₃₃₂

yes

characteristic

171

C₄²

#13050591

yes

characteristic

172

C₂³

#14581754

yes

characteristic

173

Q₈

—

664

—

174

Q₈

—

664

—

175

Q₈

—

664

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

332

—

179

C₂×C₄

—

332

—

180

C₂×C₄

C₂×(C₂×D₈₃)⋊C₄

yes

characteristic

181

C₂×C₄

#14581754

yes

characteristic

182

C₂×C₄

#14581767

yes

characteristic

183

C₂×C₄

#14581767

yes

characteristic

184

C₂×C₄

#14581781

yes

characteristic

185

C₂×C₄

#14581788

yes

characteristic

186

C₂×C₄

#14581819

yes

—

187

C₈

—

188

C₈

—

189

C₈

—

332

—

190

C₈

—

332

—

191

C₈

#14581710

yes

complemented

192

C₂²

—

193

C₂²

—

194

C₂²

#19127602

yes

characteristic

195

C₂²

#19127812

yes

—

196

C₄

—

197

C₄

—

198

C₄

—

664

—

199

C₄

C₂×#14581710

yes

—

200

C₄

#19127955

yes

—

201

C₄

#19127977

yes

—

202

C₄

#19128087

yes

—

203

C₂

—

204

C₂

#24908306

yes

characteristic

205

C₂

#24908337

yes

characteristic

206

C₂

#24908376

yes

characteristic

	subgroup	G/H
center	C₂²	#19127602
commutator	C₂²×C₄×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1566	C₂
socle	C₂²×C₈₃	#390
prime radical	C₂³×C₈₃	C₄×D₄
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	#1566	D₈₃
hypocenter	C₈₃	#9808
LCS	G ▷ C₂²×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1566

subgroup

G/H

center

C₂²

#19127602

commutator

C₂²×C₄×C₈₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₃

Fitting

C₈₃×#1566

C₂

socle

C₂²×C₈₃

#390

prime radical

C₂³×C₈₃

C₄×D₄

cosocle

C₂×C₄²×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂²×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1566

N	H ≅ G/N	type
85. C₈×C₈₃	#101	semidirect
109. C₈₃	#9808	semidirect
191. C₈	#14581710	semidirect

H ≅ G/N

type

85. C₈×C₈₃

#101

semidirect

109. C₈₃

#9808

semidirect

191. C₈

#14581710

semidirect

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

subgroup	index
C₂×#19128106	2
#24908306	2
#24906136	2
#24905752	2
C₈₃×#1566	2
#9808	83

subgroup

index

C₈₃×#1566

Subgroups — #35304074

Conjugacy classes of subgroups

Complements

Maximal subgroups