Group #35309519 — subgroups

Subgroups — #35309519

Order 21248.

442 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 155 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	#24908425	C₂	1	2	yes	maximal
2	#24907620	C₂	1	1	yes	maximal, characteristic
3	#24907619	C₂	1	2	yes	maximal
4	C₈₃⋊#1273	C₂	1	1	yes	maximal, characteristic
5	C₈₃×#1956	C₂	1	1	yes	maximal, characteristic
6	C₂×#14581724	—	2	2	no	—
7	C₂×#14581723	—	2	4	no	—
8	C₂×C₄×C₈₃⋊C₈	C₂²	1	2	yes	—
9	#19127571	C₂²	1	2	yes	—
10	#19127463	—	2	4	no	—
11	C₂×C₄⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
12	C₂×#103×C₈₃	—	2	2	no	—
13	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
14	#384×C₈₃	C₂²	1	1	yes	characteristic
15	C₂²×C₈₃⋊C₈	D₄	1	4	yes	—
16	C₂×C₄⋊Dic₈₃	—	2	2	no	—
17	C₂×C₄⋊Dic₈₃	—	2	2	no	—
18	C₂×C₄⋊Dic₈₃	—	2	2	no	—
19	#14581724	—	4	4	no	—
20	#14581723	—	4	8	no	—
21	C₄×C₈₃⋊C₈	—	2	2	no	—
22	C₄×C₈₃⋊C₈	—	2	2	no	—
23	C₂²×Q₈×C₈₃	—	2	1	no	—
24	C₂²×C₈×C₈₃	D₄	1	2	yes	—
25	C₄⋊Q₈×C₈₃	—	2	2	no	—
26	C₄⋊Q₈×C₈₃	—	2	1	no	—
27	C₄⋊Q₈×C₈₃	Q₈	1	2	yes	—
28	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
29	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
30	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
31	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
32	#103×C₈₃	—	4	4	no	—
33	C₄×C₈×C₈₃	—	2	2	no	—
34	C₂²×Dic₈₃	—	4	2	no	—
35	C₂×C₈₃⋊C₈	—	2	4	no	—
36	C₂×C₈₃⋊C₈	—	2	4	no	—
37	C₂×C₈₃⋊C₈	SD₁₆	1	8	yes	—
38	C₄⋊Dic₈₃	—	4	4	no	—
39	C₄⋊Dic₈₃	—	4	4	no	—
40	C₄⋊Dic₈₃	—	4	4	no	—
41	C₂×Q₈×C₈₃	—	2	2	no	—
42	C₂×Q₈×C₈₃	—	4	1	no	—
43	C₂×Q₈×C₈₃	—	4	2	no	—
44	C₂×Q₈×C₈₃	—	4	2	no	—
45	C₂²×C₄×C₈₃	—	4	1	no	—
46	C₂²×C₄×C₈₃	—	4	1	no	—
47	C₂²×C₄×C₈₃	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₈₃	—	2	2	no	—
50	C₂×C₈×C₈₃	—	2	2	no	—
51	C₂×C₈×C₈₃	SD₁₆	1	4	yes	—
52	C₄⋊C₄×C₈₃	—	2	4	no	—
53	C₄⋊C₄×C₈₃	—	4	2	no	—
54	C₄⋊C₄×C₈₃	—	4	2	no	—
55	C₄⋊C₄×C₈₃	—	4	4	no	—
56	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
57	C₄²×C₈₃	#41	1	2	yes	—
58	C₄²×C₈₃	#41	1	1	yes	characteristic
59	C₂³×C₈₃	#127	1	1	yes	characteristic
60	Q₈×C₈₃	—	4	4	no	—
61	Q₈×C₈₃	—	8	2	no	—
62	C₂×C₄×C₈₃	—	4	2	no	—
63	C₂×C₄×C₈₃	—	8	1	no	—
64	C₂×C₄×C₈₃	—	8	1	no	—
65	C₂×C₄×C₈₃	—	8	2	no	—
66	C₂×C₄×C₈₃	—	8	1	no	—
67	C₂×C₄×C₈₃	C₂²⋊Q₈	1	2	yes	—
68	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
69	C₂×C₄×C₈₃	#123	1	1	yes	characteristic
70	C₂×C₄×C₈₃	#123	1	2	yes	—
71	C₂×C₄×C₈₃	C₂×SD₁₆	1	4	yes	—
72	C₂×C₄×C₈₃	C₂×SD₁₆	1	2	yes	—
73	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
74	C₂×C₄×C₈₃	#137	1	4	yes	—
75	C₂×Dic₈₃	—	8	2	no	—
76	C₂×Dic₈₃	—	8	2	no	—
77	C₂×Dic₈₃	—	8	2	no	—
78	C₈×C₈₃	—	2	4	no	—
79	C₈₃⋊C₈	—	2	8	no	—
80	C₂²×C₈₃	#399	1	1	yes	characteristic
81	C₂²×C₈₃	#492	1	2	yes	—
82	C₂²×C₈₃	#492	1	1	yes	characteristic
83	C₂²×C₈₃	#496	1	1	yes	characteristic
84	C₂²×C₈₃	#497	1	2	yes	—
85	C₄×C₈₃	—	8	2	no	—
86	C₄×C₈₃	—	16	1	no	—
87	C₄×C₈₃	#475	1	4	yes	—
88	C₄×C₈₃	#476	1	2	yes	—
89	C₄×C₈₃	#481	1	2	yes	—
90	C₄×C₈₃	#483	1	4	yes	—
91	Dic₈₃	—	16	2	no	—
92	#10685	—	83	1	no	maximal, Sylow
93	C₂×C₈₃	#1705	1	1	yes	characteristic
94	C₂×C₈₃	#1714	1	2	yes	—
95	C₂×C₈₃	#1716	1	1	yes	characteristic
96	C₂×C₈₃	#2094	1	1	yes	characteristic
97	C₂×C₈₃	#2096	1	2	yes	—
98	#1990	—	83	2	no	—
99	#1956	D₈₃	1	1	yes	characteristic
100	#1844	—	83	1	no	—
101	#1844	—	83	2	no	—
102	#1273	—	83	1	no	—
103	C₈₃	#10685	1	1	yes	characteristic, complemented, Sylow
104	C₂×C₄⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
105	C₂×#106	—	166	4	no	—
106	C₂×#106	—	166	2	no	—
107	C₂×#103	—	2	2	no	—
108	C₂×C₄×C₈	—	83	2	no	—
109	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
110	#384	—	83	2	no	—
111	#384	C₂×D₈₃	1	1	yes	characteristic
112	#340	—	166	4	no	—
113	C₂²×Q₈	—	2	1	no	—
114	C₂²×C₈	—	83	4	no	—
115	C₂²×C₈	C₂²⋊D₈₃	1	2	yes	—
116	C₄⋊Q₈	—	2	2	no	—
117	C₄⋊Q₈	—	2	1	no	—
118	C₄⋊Q₈	Dic₁₆₆	1	2	yes	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	—	2	1	no	—
122	C₂×C₄⋊C₄	—	166	2	no	—
123	C₂×C₄⋊C₄	—	166	2	no	—
124	C₂×C₄⋊C₄	—	166	2	no	—
125	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
126	#106	—	332	8	no	—
127	#106	—	332	4	no	—
128	#103	—	4	4	no	—
129	C₄×C₈	—	2	2	no	—
130	C₄×C₈	—	166	2	no	—
131	C₄×C₈	—	166	2	no	—
132	C₂×Q₈	—	2	2	no	—
133	C₂×Q₈	—	4	1	no	—
134	C₂×Q₈	—	4	2	no	—
135	C₂×Q₈	—	4	2	no	—
136	C₂²×C₄	—	4	1	no	—
137	C₂²×C₄	—	4	1	no	—
138	C₂²×C₄	—	332	2	no	—
139	C₂²×C₄	D₄×D₈₃	1	2	yes	—
140	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	83	8	no	—
144	C₂×C₈	—	166	4	no	—
145	C₂×C₈	—	166	4	no	—
146	C₂×C₈	#13050578	1	4	yes	—
147	C₄⋊C₄	—	2	4	no	—
148	C₄⋊C₄	—	4	2	no	—
149	C₄⋊C₄	—	4	4	no	—
150	C₄⋊C₄	—	4	2	no	—
151	C₄⋊C₄	—	332	4	no	—
152	C₄⋊C₄	—	332	4	no	—
153	C₄⋊C₄	—	332	4	no	—
154	C₄²	C₂×Dic₁₆₆	1	1	yes	characteristic
155	C₄²	#13050589	1	2	yes	—
156	C₄²	#13050590	1	1	yes	characteristic
157	C₂³	#14581788	1	1	yes	characteristic
158	Q₈	—	4	4	no	—
159	Q₈	—	8	2	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	8	1	no	—
162	C₂×C₄	—	8	1	no	—
163	C₂×C₄	—	8	2	no	—
164	C₂×C₄	—	8	1	no	—
165	C₂×C₄	—	664	2	no	—
166	C₂×C₄	—	664	2	no	—
167	C₂×C₄	—	664	2	no	—
168	C₂×C₄	#14581769	1	2	yes	—
169	C₂×C₄	#14581771	1	1	yes	characteristic
170	C₂×C₄	#14581773	1	1	yes	characteristic
171	C₂×C₄	#14581774	1	2	yes	—
172	C₂×C₄	SD₁₆×D₈₃	1	4	yes	—
173	C₂×C₄	C₂×#13050578	1	2	yes	—
174	C₂×C₄	#14581817	1	2	yes	—
175	C₂×C₄	#14581824	1	4	yes	—
176	C₈	—	2	4	no	—
177	C₈	—	166	8	no	—
178	C₂²	#19127637	1	1	yes	characteristic
179	C₂²	#19128072	1	2	yes	—
180	C₂²	#19128073	1	1	yes	characteristic
181	C₂²	#19128091	1	1	yes	characteristic
182	C₂²	#19128094	1	2	yes	—
183	C₄	—	8	2	no	—
184	C₄	—	16	1	no	—
185	C₄	—	1328	2	no	—
186	C₄	#19127984	1	4	yes	—
187	C₄	#19127991	1	2	yes	—
188	C₄	#19128017	1	2	yes	—
189	C₄	#19128030	1	4	yes	—
190	C₂	#24907049	1	1	yes	characteristic
191	C₂	#24907087	1	2	yes	—
192	C₂	#24907094	1	1	yes	characteristic
193	C₂	#24908996	1	1	yes	characteristic
194	C₂	#24909004	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#24908425

C₂

yes

maximal

#24907620

C₂

yes

maximal, characteristic

#24907619

C₂

yes

maximal

C₈₃⋊#1273

C₂

yes

maximal, characteristic

C₈₃×#1956

C₂

yes

maximal, characteristic

C₂×#14581724

—

C₂×#14581723

—

C₂×C₄×C₈₃⋊C₈

C₂²

yes

—

#19127571

C₂²

yes

—

#19127463

—

C₂×C₄⋊Q₈×C₈₃

C₂²

yes

characteristic

C₂×#103×C₈₃

—

C₂×C₄×C₈×C₈₃

C₂²

yes

characteristic

#384×C₈₃

C₂²

yes

characteristic

C₂²×C₈₃⋊C₈

D₄

yes

—

C₂×C₄⋊Dic₈₃

—

C₂×C₄⋊Dic₈₃

—

C₂×C₄⋊Dic₈₃

—

#14581724

—

#14581723

—

C₄×C₈₃⋊C₈

—

C₄×C₈₃⋊C₈

—

C₂²×Q₈×C₈₃

—

C₂²×C₈×C₈₃

D₄

yes

—

C₄⋊Q₈×C₈₃

—

C₄⋊Q₈×C₈₃

—

C₄⋊Q₈×C₈₃

Q₈

yes

—

C₂×C₄⋊C₄×C₈₃

—

C₂×C₄⋊C₄×C₈₃

—

C₂×C₄⋊C₄×C₈₃

—

C₂×C₄²×C₈₃

C₂³

yes

characteristic

#103×C₈₃

—

C₄×C₈×C₈₃

—

C₂²×Dic₈₃

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

SD₁₆

yes

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

C₂×D₄

yes

—

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

SD₁₆

yes

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄²×C₈₃

C₂×Q₈

yes

characteristic

C₄²×C₈₃

#41

yes

—

C₄²×C₈₃

yes

C₂³×C₈₃

yes

Q₈×C₈₃

—

Q₈×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

C₂²⋊Q₈

yes

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

—

C₂×C₄×C₈₃

C₂×SD₁₆

yes

—

C₂×C₄×C₈₃

C₂×SD₁₆

yes

—

C₂×C₄×C₈₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₈₃

#137

yes

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₂²×C₈₃

#399

yes

characteristic

C₂²×C₈₃

#492

yes

—

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#475

yes

—

C₄×C₈₃

#476

yes

—

C₄×C₈₃

#481

yes

—

C₄×C₈₃

#483

yes

—

Dic₈₃

—

—

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

#1990

—

#1956

D₈₃

yes

characteristic

100

#1844

—

101

#1844

—

102

#1273

—

103

C₈₃

#10685

yes

characteristic, complemented, Sylow

104

C₂×C₄⋊Q₈

C₂×D₈₃

yes

characteristic

105

C₂×#106

—

166

—

106

C₂×#106

—

166

—

107

C₂×#103

—

108

C₂×C₄×C₈

—

109

C₂×C₄×C₈

C₂×D₈₃

yes

characteristic

110

#384

—

111

#384

C₂×D₈₃

yes

characteristic

112

#340

—

166

—

113

C₂²×Q₈

—

114

C₂²×C₈

—

115

C₂²×C₈

C₂²⋊D₈₃

yes

—

116

C₄⋊Q₈

—

117

C₄⋊Q₈

—

118

C₄⋊Q₈

Dic₁₆₆

yes

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄⋊C₄

—

166

—

123

C₂×C₄⋊C₄

—

166

—

124

C₂×C₄⋊C₄

—

166

—

125

C₂×C₄²

C₂²×D₈₃

yes

characteristic

126

#106

—

332

—

127

#106

—

332

—

128

#103

—

129

C₄×C₈

—

130

C₄×C₈

—

166

—

131

C₄×C₈

—

166

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

332

—

139

C₂²×C₄

D₄×D₈₃

yes

—

140

C₂²×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

166

—

145

C₂×C₈

—

166

—

146

C₂×C₈

#13050578

yes

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

332

—

152

C₄⋊C₄

—

332

—

153

C₄⋊C₄

—

332

—

154

C₄²

C₂×Dic₁₆₆

yes

characteristic

155

C₄²

#13050589

yes

—

156

C₄²

#13050590

yes

characteristic

157

C₂³

#14581788

yes

characteristic

158

Q₈

—

159

Q₈

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

664

—

166

C₂×C₄

—

664

—

167

C₂×C₄

—

664

—

168

C₂×C₄

#14581769

yes

—

169

C₂×C₄

#14581771

yes

characteristic

170

C₂×C₄

#14581773

yes

characteristic

171

C₂×C₄

#14581774

yes

—

172

C₂×C₄

SD₁₆×D₈₃

yes

—

173

C₂×C₄

C₂×#13050578

yes

—

174

C₂×C₄

#14581817

yes

—

175

C₂×C₄

#14581824

yes

—

176

C₈

—

177

C₈

—

166

—

178

C₂²

#19127637

yes

characteristic

179

C₂²

#19128072

yes

—

180

C₂²

#19128073

yes

characteristic

181

C₂²

#19128091

yes

characteristic

182

C₂²

#19128094

yes

—

183

C₄

—

184

C₄

—

185

C₄

—

1328

—

186

C₄

#19127984

yes

—

187

C₄

#19127991

yes

—

188

C₄

#19128017

yes

—

189

C₄

#19128030

yes

—

190

C₂

#24907049

yes

characteristic

191

C₂

#24907087

yes

—

192

C₂

#24907094

yes

characteristic

193

C₂

#24908996

yes

characteristic

194

C₂

#24909004

yes

—

	subgroup	G/H
center	C₂³	#14581788
commutator, cosocle	C₂×C₄²×C₈₃	C₂³
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1956	C₂
socle, prime radical	C₂³×C₈₃	#127
hypercenter	#1956	D₈₃
hypocenter	C₈₃	#10685
LCS	G ▷ C₂×C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1956

subgroup

G/H

center

C₂³

#14581788

commutator, cosocle

C₂×C₄²×C₈₃

C₂³

Frattini

C₂×C₄²

C₂²×D₈₃

Fitting

C₈₃×#1956

C₂

C₂³×C₈₃

D₈₃

C₈₃

G ▷ C₂×C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1956

N	H ≅ G/N	type
103. C₈₃	#10685	semidirect

H ≅ G/N

type

103. C₈₃

#10685

semidirect

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

subgroup	index
#24908425	2
#24907620	2
#24907619	2
C₈₃⋊#1273	2
C₈₃×#1956	2
#10685	83

subgroup

index

#24908425

#24907620

#24907619

C₈₃⋊#1273

C₈₃×#1956

#10685

Subgroups — #35309519

Conjugacy classes of subgroups

Complements

Maximal subgroups