Group #35309387 — subgroups

Subgroups — #35309387

Order 21248.

372 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 99 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	#24907638	C₂	1	1	yes	maximal, characteristic, complemented
2	#24906233	C₂	1	2	yes	maximal, complemented
3	#24905888	C₂	1	2	yes	maximal
4	#24905294	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₃×#1955	C₂	1	1	yes	maximal, characteristic
6	#19127754	C₂²	1	2	yes	—
7	C₂×#14581728	—	2	2	no	—
8	#19127725	C₂²	1	2	yes	—
9	#19127451	—	2	4	no	—
10	D₄⋊(C₈₃⋊C₈)	—	2	4	no	—
11	C₂×C₄×D₄×C₈₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊C₈×C₈₃	—	2	2	no	—
14	C₂×#97×C₈₃	C₂²	1	1	yes	characteristic
15	C₂×#13050574	—	2	2	no	—
16	#14581728	—	4	4	no	—
17	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
18	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
19	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
20	C₂²⋊(C₈₃⋊C₈)	—	2	2	no	—
21	C₄×C₈₃⋊C₈	D₄	1	4	yes	—
22	C₂²×D₄×C₈₃	Q₈	1	1	yes	characteristic
23	C₂³×C₄×C₈₃	—	2	1	no	—
24	C₂²×C₈×C₈₃	—	2	1	no	—
25	C₂²×C₈×C₈₃	D₄	1	2	yes	—
26	C₄×D₄×C₈₃	—	2	2	no	—
27	C₄×D₄×C₈₃	—	2	2	no	—
28	C₂×C₄⋊C₄×C₈₃	Q₈	1	1	yes	characteristic
29	C₂×C₂²⋊C₄×C₈₃	—	2	1	no	—
30	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₈₃	—	2	2	no	—
32	C₂²⋊C₈×C₈₃	—	4	4	no	—
33	#97×C₈₃	—	2	2	no	—
34	#13050574	—	4	4	no	—
35	C₂×C₈₃⋊C₈	—	2	4	no	—
36	C₂×C₈₃⋊C₈	—	4	2	no	—
37	C₂⁴×C₈₃	—	2	1	no	—
38	C₂×D₄×C₈₃	—	2	2	no	—
39	C₂×D₄×C₈₃	—	4	2	no	—
40	C₂²×C₄×C₈₃	—	2	1	no	—
41	C₂²×C₄×C₈₃	—	4	1	no	—
42	C₂²×C₄×C₈₃	—	4	1	no	—
43	C₂²×C₄×C₈₃	—	4	2	no	—
44	C₂²×C₄×C₈₃	—	4	2	no	—
45	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
47	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
48	C₂×C₈×C₈₃	—	2	2	no	—
49	C₂×C₈×C₈₃	—	2	2	no	—
50	C₂×C₈×C₈₃	—	4	1	no	—
51	C₂×C₈×C₈₃	—	4	2	no	—
52	C₂×C₈×C₈₃	SD₁₆	1	4	yes	—
53	C₄⋊C₄×C₈₃	—	2	2	no	—
54	C₂²⋊C₄×C₈₃	—	4	2	no	—
55	C₄²×C₈₃	C₂×D₄	1	2	yes	—
56	C₄²×C₈₃	#41	1	2	yes	—
57	C₂³×C₈₃	—	2	2	no	—
58	C₂³×C₈₃	—	4	1	no	—
59	C₂³×C₈₃	—	4	2	no	—
60	C₂³×C₈₃	—	4	1	no	—
61	C₂³×C₈₃	—	4	2	no	—
62	C₂³×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
63	D₄×C₈₃	—	2	4	no	—
64	D₄×C₈₃	—	4	2	no	—
65	C₂×C₄×C₈₃	—	4	1	no	—
66	C₂×C₄×C₈₃	—	4	1	no	—
67	C₂×C₄×C₈₃	—	4	1	no	—
68	C₂×C₄×C₈₃	—	4	2	no	—
69	C₂×C₄×C₈₃	—	8	2	no	—
70	C₂×C₄×C₈₃	—	8	1	no	—
71	C₂×C₄×C₈₃	—	8	1	no	—
72	C₂×C₄×C₈₃	C₂²⋊Q₈	1	2	yes	—
73	C₂×C₄×C₈₃	#123	1	1	yes	characteristic
74	C₂×C₄×C₈₃	#123	1	2	yes	—
75	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
76	C₂×C₄×C₈₃	C₂×SD₁₆	1	2	yes	—
77	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
78	C₈×C₈₃	—	2	4	no	—
79	C₈×C₈₃	—	8	1	no	—
80	C₈₃⋊C₈	—	4	4	no	—
81	C₈₃⋊C₈	—	8	2	no	—
82	C₂²×C₈₃	—	2	1	no	—
83	C₂²×C₈₃	—	2	1	no	—
84	C₂²×C₈₃	—	4	4	no	—
85	C₂²×C₈₃	—	4	2	no	—
86	C₂²×C₈₃	—	8	2	no	—
87	C₂²×C₈₃	—	8	1	no	—
88	C₂²×C₈₃	—	8	1	no	—
89	C₂²×C₈₃	#399	1	1	yes	characteristic
90	C₂²×C₈₃	#476	1	2	yes	—
91	C₄×C₈₃	—	2	2	no	—
92	C₄×C₈₃	—	2	2	no	—
93	C₄×C₈₃	—	8	1	no	—
94	C₄×C₈₃	#481	1	2	yes	—
95	C₄×C₈₃	#492	1	1	yes	characteristic
96	C₄×C₈₃	#496	1	1	yes	characteristic
97	#10661	—	83	1	no	maximal, Sylow
98	C₂×C₈₃	—	2	2	no	—
99	C₂×C₈₃	—	8	2	no	—
100	C₂×C₈₃	#2091	1	1	yes	characteristic
101	C₂×C₈₃	#2094	1	1	yes	characteristic
102	C₂×C₈₃	#2098	1	1	yes	characteristic
103	#1955	D₈₃	1	1	yes	characteristic
104	#1849	—	83	1	no	—
105	#1570	—	83	2	no	—
106	#1486	—	83	2	no	—
107	#1275	—	83	1	no	—
108	C₈₃	#10661	1	1	yes	characteristic, complemented, Sylow
109	C₂×C₄×D₄	C₂×D₈₃	1	1	yes	characteristic
110	#431	—	83	2	no	—
111	C₂×#108	—	166	2	no	—
112	#423	—	83	2	no	—
113	C₂×C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
114	C₂×C₂²⋊C₈	—	2	2	no	—
115	C₂×#97	C₂×D₈₃	1	1	yes	characteristic
116	#334	—	166	4	no	—
117	D₄⋊C₈	—	166	4	no	—
118	C₂²×D₄	Dic₁₆₆	1	1	yes	characteristic
119	C₂³×C₄	—	2	1	no	—
120	C₂×M₁₆	—	166	2	no	—
121	C₂²×C₈	—	2	1	no	—
122	C₂²×C₈	C₂²⋊D₈₃	1	2	yes	—
123	C₄×D₄	—	2	2	no	—
124	C₄×D₄	—	2	2	no	—
125	C₂×C₄⋊C₄	Dic₁₆₆	1	1	yes	characteristic
126	C₂×C₂²⋊C₄	—	2	1	no	—
127	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
128	#108	—	332	4	no	—
129	C₄⋊C₈	—	2	2	no	—
130	C₄⋊C₈	—	166	2	no	—
131	C₄⋊C₈	—	166	2	no	—
132	C₄⋊C₈	—	166	2	no	—
133	C₂²⋊C₈	—	4	4	no	—
134	C₂²⋊C₈	—	166	2	no	—
135	#97	—	2	2	no	—
136	C₄×C₈	—	83	4	no	—
137	C₂⁴	—	2	1	no	—
138	C₂×D₄	—	2	2	no	—
139	C₂×D₄	—	4	2	no	—
140	C₂²×C₄	—	2	1	no	—
141	C₂²×C₄	—	4	1	no	—
142	C₂²×C₄	—	4	1	no	—
143	C₂²×C₄	—	4	2	no	—
144	C₂²×C₄	—	4	2	no	—
145	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
146	C₂²×C₄	C₂×Dic₁₆₆	1	1	yes	characteristic
147	C₂²×C₄	#13050590	1	1	yes	characteristic
148	M₁₆	—	332	4	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	4	1	no	—
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	—	332	2	no	—
154	C₂×C₈	—	166	4	no	—
155	C₂×C₈	#13050578	1	4	yes	—
156	C₄⋊C₄	—	2	2	no	—
157	C₂²⋊C₄	—	4	2	no	—
158	C₄²	D₄×D₈₃	1	2	yes	—
159	C₄²	#13050589	1	2	yes	—
160	C₂³	—	2	2	no	—
161	C₂³	—	4	1	no	—
162	C₂³	—	4	2	no	—
163	C₂³	—	4	1	no	—
164	C₂³	—	4	2	no	—
165	C₂³	#14581771	1	1	yes	characteristic
166	D₄	—	2	4	no	—
167	D₄	—	4	2	no	—
168	C₂×C₄	—	4	1	no	—
169	C₂×C₄	—	4	1	no	—
170	C₂×C₄	—	4	1	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	—	8	1	no	—
173	C₂×C₄	—	8	1	no	—
174	C₂×C₄	—	8	2	no	—
175	C₂×C₄	#14581769	1	2	yes	—
176	C₂×C₄	#14581773	1	1	yes	characteristic
177	C₂×C₄	#14581774	1	2	yes	—
178	C₂×C₄	#14581788	1	1	yes	characteristic
179	C₂×C₄	C₂×#13050578	1	2	yes	—
180	C₂×C₄	#14581817	1	2	yes	—
181	C₈	—	2	4	no	—
182	C₈	—	8	1	no	—
183	C₈	—	332	4	no	—
184	C₈	—	664	2	no	—
185	C₂²	—	2	1	no	—
186	C₂²	—	2	1	no	—
187	C₂²	—	4	4	no	—
188	C₂²	—	4	2	no	—
189	C₂²	—	8	2	no	—
190	C₂²	—	8	1	no	—
191	C₂²	—	8	1	no	—
192	C₂²	#19127637	1	1	yes	characteristic
193	C₂²	#19127991	1	2	yes	—
194	C₄	—	2	2	no	—
195	C₄	—	2	2	no	—
196	C₄	—	8	1	no	—
197	C₄	#19128017	1	2	yes	—
198	C₄	#19128073	1	1	yes	characteristic
199	C₄	#19128091	1	1	yes	characteristic
200	C₂	—	2	2	no	—
201	C₂	—	8	2	no	—
202	C₂	#24908979	1	1	yes	characteristic
203	C₂	#24908996	1	1	yes	characteristic
204	C₂	#24909016	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24907638

C₂

yes

maximal, characteristic, complemented

#24906233

C₂

yes

maximal, complemented

#24905888

C₂

yes

maximal

#24905294

C₂

yes

maximal, characteristic, complemented

C₈₃×#1955

C₂

yes

maximal, characteristic

#19127754

C₂²

yes

—

C₂×#14581728

—

#19127725

C₂²

yes

—

#19127451

—

D₄⋊(C₈₃⋊C₈)

—

C₂×C₄×D₄×C₈₃

C₂²

yes

characteristic

C₂×C₄⋊C₈×C₈₃

C₂²

yes

characteristic

C₂×C₂²⋊C₈×C₈₃

—

C₂×#97×C₈₃

C₂²

yes

characteristic

C₂×#13050574

—

#14581728

—

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

—

C₂²⋊(C₈₃⋊C₈)

—

C₄×C₈₃⋊C₈

D₄

yes

—

C₂²×D₄×C₈₃

Q₈

yes

characteristic

C₂³×C₄×C₈₃

—

C₂²×C₈×C₈₃

—

C₂²×C₈×C₈₃

D₄

yes

—

C₄×D₄×C₈₃

—

C₄×D₄×C₈₃

—

C₂×C₄⋊C₄×C₈₃

Q₈

yes

characteristic

C₂×C₂²⋊C₄×C₈₃

—

C₂×C₄²×C₈₃

C₂³

yes

characteristic

C₄⋊C₈×C₈₃

—

C₂²⋊C₈×C₈₃

—

#97×C₈₃

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂⁴×C₈₃

—

C₂×D₄×C₈₃

—

C₂×D₄×C₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₃

C₂×Q₈

yes

characteristic

C₂²×C₄×C₈₃

#41

yes

characteristic

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

SD₁₆

yes

—

C₄⋊C₄×C₈₃

—

C₂²⋊C₄×C₈₃

—

C₄²×C₈₃

C₂×D₄

yes

—

C₄²×C₈₃

#41

yes

—

C₂³×C₈₃

—

C₂³×C₈₃

—

C₂³×C₈₃

—

C₂³×C₈₃

—

C₂³×C₈₃

—

C₂³×C₈₃

C₂²⋊Q₈

yes

characteristic

D₄×C₈₃

—

D₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₈₃

#123

yes

characteristic

C₂×C₄×C₈₃

#123

yes

—

C₂×C₄×C₈₃

#127

yes

characteristic

C₂×C₄×C₈₃

C₂×SD₁₆

yes

—

C₂×C₄×C₈₃

AGL(1,ℤ/8)

yes

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

#399

yes

characteristic

C₂²×C₈₃

#476

yes

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#481

yes

—

C₄×C₈₃

yes

C₄×C₈₃

yes

—

C₂×C₈₃

—

C₂×C₈₃

—

100

C₂×C₈₃

#2091

yes

characteristic

101

C₂×C₈₃

#2094

yes

characteristic

102

C₂×C₈₃

#2098

yes

characteristic

103

#1955

D₈₃

yes

characteristic

104

#1849

—

105

#1570

—

106

#1486

—

107

#1275

—

108

C₈₃

#10661

yes

characteristic, complemented, Sylow

109

C₂×C₄×D₄

C₂×D₈₃

yes

characteristic

110

#431

—

111

C₂×#108

—

166

—

112

#423

—

113

C₂×C₄⋊C₈

C₂×D₈₃

yes

characteristic

114

C₂×C₂²⋊C₈

—

115

C₂×#97

C₂×D₈₃

yes

characteristic

116

#334

—

166

—

117

D₄⋊C₈

—

166

—

118

C₂²×D₄

Dic₁₆₆

yes

characteristic

119

C₂³×C₄

—

120

C₂×M₁₆

—

166

—

121

C₂²×C₈

—

122

C₂²×C₈

C₂²⋊D₈₃

yes

—

123

C₄×D₄

—

124

C₄×D₄

—

125

C₂×C₄⋊C₄

Dic₁₆₆

yes

characteristic

126

C₂×C₂²⋊C₄

—

127

C₂×C₄²

C₂²×D₈₃

yes

characteristic

128

#108

—

332

—

129

C₄⋊C₈

—

130

C₄⋊C₈

—

166

—

131

C₄⋊C₈

—

166

—

132

C₄⋊C₈

—

166

—

133

C₂²⋊C₈

—

134

C₂²⋊C₈

—

166

—

135

#97

—

136

C₄×C₈

—

137

C₂⁴

—

138

C₂×D₄

—

139

C₂×D₄

—

140

C₂²×C₄

—

141

C₂²×C₄

—

142

C₂²×C₄

—

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

146

C₂²×C₄

C₂×Dic₁₆₆

yes

characteristic

147

C₂²×C₄

#13050590

yes

characteristic

148

M₁₆

—

332

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

332

—

154

C₂×C₈

—

166

—

155

C₂×C₈

#13050578

yes

—

156

C₄⋊C₄

—

157

C₂²⋊C₄

—

158

C₄²

D₄×D₈₃

yes

—

159

C₄²

#13050589

yes

—

160

C₂³

—

161

C₂³

—

162

C₂³

—

163

C₂³

—

164

C₂³

—

165

C₂³

#14581771

yes

characteristic

166

D₄

—

167

D₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

#14581769

yes

—

176

C₂×C₄

#14581773

yes

characteristic

177

C₂×C₄

#14581774

yes

—

178

C₂×C₄

#14581788

yes

characteristic

179

C₂×C₄

C₂×#13050578

yes

—

180

C₂×C₄

#14581817

yes

—

181

C₈

—

182

C₈

—

183

C₈

—

332

—

184

C₈

—

664

—

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

—

189

C₂²

—

190

C₂²

—

191

C₂²

—

192

C₂²

#19127637

yes

characteristic

193

C₂²

#19127991

yes

—

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

#19128017

yes

—

198

C₄

#19128073

yes

characteristic

199

C₄

#19128091

yes

characteristic

200

C₂

—

201

C₂

—

202

C₂

#24908979

yes

characteristic

203

C₂

#24908996

yes

characteristic

204

C₂

#24909016

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14581788
commutator, cosocle	C₂×C₄²×C₈₃	C₂³
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1955	C₂
socle	C₂²×C₈₃	#399
prime radical	C₂²×D₄×C₈₃	Q₈
hypercenter	#1955	D₈₃
hypocenter	C₈₃	#10661
LCS	G ▷ C₂×C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1955

subgroup

G/H

center

C₂×C₄

#14581788

commutator, cosocle

C₂×C₄²×C₈₃

C₂³

Frattini

C₂×C₄²

C₂²×D₈₃

Fitting

C₈₃×#1955

C₂

socle

C₂²×C₈₃

#399

prime radical

C₂²×D₄×C₈₃

Q₈

D₈₃

C₈₃

G ▷ C₂×C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1955

N	H ≅ G/N	type
1. #24907638	C₂	semidirect
2. #24906233	C₂	semidirect
4. #24905294	C₂	semidirect
108. C₈₃	#10661	semidirect

H ≅ G/N

type

1. #24907638

C₂

semidirect

2. #24906233

C₂

semidirect

4. #24905294

C₂

semidirect

108. C₈₃

#10661

semidirect

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

subgroup	index
#24907638	2
#24906233	2
#24905888	2
#24905294	2
C₈₃×#1955	2
#10661	83

subgroup

index

C₈₃×#1955

Subgroups — #35309387

Conjugacy classes of subgroups

Complements

Maximal subgroups