Group #35323944 — subgroups

Subgroups — #35323944

Order 21248.

418 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 95 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	C₂²×D₁₃₂₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#19127515	C₂	1	2	yes	maximal, complemented
3	#24908373	C₂	1	2	yes	maximal, complemented
4	#24905557	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₃×#2151	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₆₆₄	C₂²	1	2	yes	—
7	C₂×D₁₃₂₈	—	2	2	no	—
8	C₂×D₁₃₂₈	—	2	4	no	—
9	C₂×#14581728	C₂²	1	2	yes	—
10	#19127705	—	2	2	no	—
11	C₂×#14581708	—	2	2	no	—
12	#19127515	—	2	4	no	—
13	C₂×M₃₂×C₈₃	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
15	#405×C₈₃	C₂²	1	1	yes	characteristic
16	C₄⋊C₁₆×C₈₃	—	2	2	no	—
17	C₂²×D₃₃₂	—	2	2	no	—
18	C₂×D₆₆₄	—	2	2	no	—
19	C₂×D₆₆₄	—	2	2	no	—
20	C₂×D₆₆₄	—	2	2	no	—
21	C₂×D₆₆₄	—	2	2	no	—
22	C₂×D₆₆₄	—	2	2	no	—
23	C₂×D₆₆₄	D₄	1	4	yes	—
24	C₂×#13050574	—	2	2	no	—
25	D₁₃₂₈	—	2	8	no	—
26	#14581728	—	2	2	no	—
27	#14581728	—	2	2	no	—
28	#14581712	—	4	4	no	—
29	#14581708	—	4	4	no	—
30	C₂×M₁₆×C₈₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
32	#117×C₈₃	D₄	1	1	yes	characteristic
33	M₃₂×C₈₃	—	2	2	no	—
34	C₂×C₁₆×C₈₃	—	2	2	no	—
35	C₂×C₁₆×C₈₃	—	2	1	no	—
36	C₂×C₁₆×C₈₃	D₄	1	2	yes	—
37	#97×C₈₃	—	2	1	no	—
38	C₄×C₈×C₈₃	—	2	1	no	—
39	C₂³×D₈₃	—	4	2	no	—
40	C₂×D₃₃₂	—	2	4	no	—
41	C₂×D₃₃₂	—	4	2	no	—
42	C₂×D₃₃₂	—	4	2	no	—
43	C₂×D₃₃₂	—	4	2	no	—
44	C₂×D₃₃₂	—	4	2	no	—
45	C₂×D₃₃₂	—	4	2	no	—
46	D₆₆₄	—	2	4	no	—
47	D₆₆₄	—	2	4	no	—
48	D₆₆₄	—	4	2	no	—
49	D₆₆₄	—	4	2	no	—
50	#13050574	—	4	2	no	—
51	#13050574	—	4	2	no	—
52	C₂×C₈₃⋊C₈	—	4	2	no	—
53	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
54	M₁₆×C₈₃	—	2	2	no	—
55	C₂×C₈×C₈₃	—	2	1	no	—
56	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
57	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
59	C₂×C₈×C₈₃	#41	1	2	yes	—
60	C₄⋊C₄×C₈₃	D₈	1	2	yes	—
61	C₂²⋊C₄×C₈₃	—	2	1	no	—
62	C₄²×C₈₃	—	2	1	no	—
63	C₁₆×C₈₃	—	2	2	no	—
64	C₁₆×C₈₃	—	4	1	no	—
65	C₂³×C₈₃	#135	1	1	yes	characteristic
66	C₂²×D₈₃	—	4	4	no	—
67	C₂²×D₈₃	—	8	2	no	—
68	C₂²×D₈₃	—	8	2	no	—
69	C₂²×D₈₃	—	16	4	no	—
70	C₂×C₄×C₈₃	—	2	2	no	—
71	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
72	C₂×C₄×C₈₃	#121	1	2	yes	—
73	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
74	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
75	D₃₃₂	—	4	4	no	—
76	D₃₃₂	—	4	4	no	—
77	D₃₃₂	—	8	2	no	—
78	D₃₃₂	—	8	2	no	—
79	C₈×C₈₃	—	4	1	no	—
80	C₈×C₈₃	#120	1	1	yes	characteristic
81	C₈×C₈₃	#123	1	2	yes	—
82	C₈×C₈₃	#124	1	1	yes	characteristic
83	C₈₃⋊C₈	—	8	2	no	—
84	C₂²×C₈₃	—	2	2	no	—
85	C₂²×C₈₃	#463	1	2	yes	—
86	C₂²×C₈₃	#466	1	1	yes	characteristic
87	C₂×D₈₃	—	8	4	no	—
88	C₂×D₈₃	—	16	4	no	—
89	C₂×D₈₃	—	16	4	no	—
90	C₂×D₈₃	—	16	4	no	—
91	C₄×C₈₃	—	4	2	no	—
92	C₄×C₈₃	#394	1	1	yes	characteristic
93	C₄×C₈₃	#459	1	2	yes	—
94	C₄×C₈₃	#469	1	1	yes	characteristic
95	#13439	—	83	1	no	maximal, Sylow
96	C₂×C₈₃	—	2	2	no	—
97	C₂×C₈₃	#2054	1	1	yes	characteristic
98	C₂×C₈₃	#2221	1	2	yes	—
99	D₈₃	—	16	4	no	—
100	C₂²×D₁₆	—	83	1	no	—
101	#2151	D₈₃	1	1	yes	characteristic, complemented
102	C₂×#361	—	83	2	no	—
103	#1978	—	83	2	no	—
104	#1381	—	83	1	no	—
105	C₈₃	#13439	1	1	yes	characteristic, complemented, Sylow
106	C₂²×D₈	—	83	2	no	—
107	C₂×D₁₆	—	166	4	no	—
108	C₂×D₁₆	—	166	2	no	—
109	C₂×M₃₂	C₂×D₈₃	1	1	yes	characteristic
110	C₂²×C₁₆	C₂×D₈₃	1	1	yes	characteristic
111	C₂×#108	—	83	2	no	—
112	#418	—	166	2	no	—
113	C₂×#100	—	166	2	no	—
114	#405	C₂×D₈₃	1	1	yes	characteristic
115	C₄⋊C₁₆	—	2	2	no	—
116	#361	—	166	4	no	—
117	C₂²×D₄	—	166	2	no	—
118	C₂×D₈	—	83	4	no	—
119	C₂×D₈	—	166	2	no	—
120	C₂×D₈	—	166	2	no	—
121	C₂×D₈	—	166	2	no	—
122	C₂×D₈	—	166	2	no	—
123	C₂×D₈	—	166	2	no	—
124	C₂×M₁₆	—	166	2	no	—
125	C₂×M₁₆	D₃₃₂	1	1	yes	characteristic
126	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
127	#117	D₃₃₂	1	1	yes	characteristic
128	D₁₆	—	166	8	no	—
129	M₃₂	—	2	2	no	—
130	C₂×C₁₆	—	2	2	no	—
131	C₂×C₁₆	—	2	1	no	—
132	C₂×C₁₆	C₂²⋊D₈₃	1	2	yes	—
133	#108	—	166	2	no	—
134	#108	—	166	2	no	—
135	#102	—	332	4	no	—
136	#100	—	332	4	no	—
137	#97	—	2	1	no	—
138	C₄×C₈	—	2	1	no	—
139	C₂⁴	—	332	2	no	—
140	C₂×D₄	—	332	2	no	—
141	C₂×D₄	—	332	2	no	—
142	C₂×D₄	—	332	2	no	—
143	C₂×D₄	—	332	2	no	—
144	C₂×D₄	—	332	2	no	—
145	C₂×D₄	—	166	4	no	—
146	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
147	D₈	—	332	2	no	—
148	D₈	—	332	2	no	—
149	D₈	—	166	4	no	—
150	D₈	—	166	4	no	—
151	M₁₆	—	2	2	no	—
152	M₁₆	—	332	2	no	—
153	M₁₆	—	332	2	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	332	2	no	—
156	C₂×C₈	D₄×D₈₃	1	2	yes	—
157	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
158	C₂×C₈	#13050590	1	1	yes	characteristic
159	C₂×C₈	#13050591	1	2	yes	—
160	C₄⋊C₄	D₆₆₄	1	2	yes	—
161	C₂²⋊C₄	—	2	1	no	—
162	C₄²	—	2	1	no	—
163	C₁₆	—	2	2	no	—
164	C₁₆	—	4	1	no	—
165	C₂³	—	1328	4	no	—
166	C₂³	—	332	4	no	—
167	C₂³	—	664	2	no	—
168	C₂³	—	664	2	no	—
169	C₂³	#14581814	1	1	yes	characteristic
170	D₄	—	332	4	no	—
171	D₄	—	332	4	no	—
172	D₄	—	664	2	no	—
173	D₄	—	664	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	#14581766	1	1	yes	characteristic
176	C₂×C₄	#14581767	1	2	yes	—
177	C₂×C₄	C₂×D₆₆₄	1	1	yes	characteristic
178	C₂×C₄	#14581820	1	2	yes	—
179	C₈	—	4	1	no	—
180	C₈	—	664	2	no	—
181	C₈	#14581760	1	1	yes	characteristic
182	C₈	#14581777	1	2	yes	—
183	C₈	#14581781	1	1	yes	characteristic
184	C₂²	—	2	2	no	—
185	C₂²	—	1328	4	no	—
186	C₂²	—	1328	4	no	—
187	C₂²	—	1328	4	no	—
188	C₂²	—	664	4	no	—
189	C₂²	#19127917	1	2	yes	—
190	C₂²	#19127937	1	1	yes	characteristic
191	C₄	—	4	2	no	—
192	C₄	#19127618	1	1	yes	characteristic
193	C₄	#19127897	1	2	yes	—
194	C₄	#19127954	1	1	yes	characteristic
195	C₂	—	2	2	no	—
196	C₂	—	1328	4	no	—
197	C₂	#24908774	1	1	yes	characteristic
198	C₂	#24909583	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₁₃₂₈

C₂

yes

maximal, characteristic

C₂×#19127515

C₂

yes

maximal, complemented

#24908373

C₂

yes

maximal, complemented

#24905557

C₂

yes

maximal, characteristic, complemented

C₈₃×#2151

C₂

yes

maximal, characteristic, complemented

C₂²×D₆₆₄

C₂²

yes

—

C₂×D₁₃₂₈

—

C₂×D₁₃₂₈

—

C₂×#14581728

C₂²

yes

—

#19127705

—

C₂×#14581708

—

#19127515

—

C₂×M₃₂×C₈₃

C₂²

yes

characteristic

C₂²×C₁₆×C₈₃

C₂²

yes

characteristic

#405×C₈₃

C₂²

yes

characteristic

C₄⋊C₁₆×C₈₃

—

C₂²×D₃₃₂

—

C₂×D₆₆₄

—

C₂×D₆₆₄

—

C₂×D₆₆₄

—

C₂×D₆₆₄

—

C₂×D₆₆₄

—

C₂×D₆₆₄

D₄

yes

—

C₂×#13050574

—

D₁₃₂₈

—

—

—

—

—

C₂×M₁₆×C₈₃

D₄

yes

characteristic

C₂²×C₈×C₈₃

C₂³

yes

characteristic

#117×C₈₃

D₄

yes

characteristic

M₃₂×C₈₃

—

C₂×C₁₆×C₈₃

—

C₂×C₁₆×C₈₃

—

C₂×C₁₆×C₈₃

D₄

yes

—

#97×C₈₃

—

C₄×C₈×C₈₃

—

C₂³×D₈₃

—

C₂×D₃₃₂

—

C₂×D₃₃₂

—

C₂×D₃₃₂

—

C₂×D₃₃₂

—

C₂×D₃₃₂

—

C₂×D₃₃₂

—

D₆₆₄

—

D₆₆₄

—

D₆₆₄

—

D₆₆₄

—

#13050574

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

M₁₆×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₂×D₄

yes

—

C₂×C₈×C₈₃

C₂×D₄

yes

C₂×C₈×C₈₃

yes

C₂×C₈×C₈₃

yes

—

C₄⋊C₄×C₈₃

D₈

yes

—

C₂²⋊C₄×C₈₃

—

C₄²×C₈₃

—

C₁₆×C₈₃

—

C₁₆×C₈₃

—

C₂³×C₈₃

#135

yes

characteristic

C₂²×D₈₃

—

C₂²×D₈₃

—

C₂²×D₈₃

—

C₂²×D₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

#121

yes

characteristic

C₂×C₄×C₈₃

#121

yes

—

C₂×C₄×C₈₃

C₂×D₈

yes

characteristic

C₂×C₄×C₈₃

AGL(1,ℤ/8)

yes

—

D₃₃₂

—

D₃₃₂

—

D₃₃₂

—

D₃₃₂

—

C₈×C₈₃

—

C₈×C₈₃

#120

yes

characteristic

C₈×C₈₃

#123

yes

—

C₈×C₈₃

#124

yes

characteristic

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

#463

yes

—

C₂²×C₈₃

#466

yes

characteristic

C₂×D₈₃

—

C₂×D₈₃

—

C₂×D₈₃

—

C₂×D₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#394

yes

characteristic

C₄×C₈₃

#459

yes

—

C₄×C₈₃

yes

—

C₂×C₈₃

—

C₂×C₈₃

#2054

yes

characteristic

C₂×C₈₃

#2221

yes

—

D₈₃

—

100

C₂²×D₁₆

—

101

#2151

D₈₃

yes

characteristic, complemented

102

C₂×#361

—

103

#1978

—

104

#1381

—

105

C₈₃

#13439

yes

characteristic, complemented, Sylow

106

C₂²×D₈

—

107

C₂×D₁₆

—

166

—

108

C₂×D₁₆

—

166

—

109

C₂×M₃₂

C₂×D₈₃

yes

characteristic

110

C₂²×C₁₆

C₂×D₈₃

yes

characteristic

111

C₂×#108

—

112

#418

—

166

—

113

C₂×#100

—

166

—

114

#405

C₂×D₈₃

yes

characteristic

115

C₄⋊C₁₆

—

116

#361

—

166

—

117

C₂²×D₄

—

166

—

118

C₂×D₈

—

119

C₂×D₈

—

166

—

120

C₂×D₈

—

166

—

121

C₂×D₈

—

166

—

122

C₂×D₈

—

166

—

123

C₂×D₈

—

166

—

124

C₂×M₁₆

—

166

—

125

C₂×M₁₆

D₃₃₂

yes

characteristic

126

C₂²×C₈

C₂²×D₈₃

yes

characteristic

127

#117

D₃₃₂

yes

characteristic

128

D₁₆

—

166

—

129

M₃₂

—

130

C₂×C₁₆

—

131

C₂×C₁₆

—

132

C₂×C₁₆

C₂²⋊D₈₃

yes

—

133

#108

—

166

—

134

#108

—

166

—

135

#102

—

332

—

136

#100

—

332

—

137

#97

—

138

C₄×C₈

—

139

C₂⁴

—

332

—

140

C₂×D₄

—

332

—

141

C₂×D₄

—

332

—

142

C₂×D₄

—

332

—

143

C₂×D₄

—

332

—

144

C₂×D₄

—

332

—

145

C₂×D₄

—

166

—

146

C₂²×C₄

C₂×D₃₃₂

yes

characteristic

147

D₈

—

332

—

148

D₈

—

332

—

149

D₈

—

166

—

150

D₈

—

166

—

151

M₁₆

—

152

M₁₆

—

332

—

153

M₁₆

—

332

—

154

C₂×C₈

—

155

C₂×C₈

—

332

—

156

C₂×C₈

D₄×D₈₃

yes

—

157

C₂×C₈

C₂×C₂²⋊D₈₃

yes

characteristic

158

C₂×C₈

#13050590

yes

characteristic

159

C₂×C₈

#13050591

yes

—

160

C₄⋊C₄

D₆₆₄

yes

—

161

C₂²⋊C₄

—

162

C₄²

—

163

C₁₆

—

164

C₁₆

—

165

C₂³

—

1328

—

166

C₂³

—

332

—

167

C₂³

—

664

—

168

C₂³

—

664

—

169

C₂³

#14581814

yes

characteristic

170

D₄

—

332

—

171

D₄

—

332

—

172

D₄

—

664

—

173

D₄

—

664

—

174

C₂×C₄

—

175

C₂×C₄

#14581766

yes

characteristic

176

C₂×C₄

#14581767

yes

—

177

C₂×C₄

C₂×D₆₆₄

yes

characteristic

178

C₂×C₄

#14581820

yes

—

179

C₈

—

180

C₈

—

664

—

181

C₈

#14581760

yes

characteristic

182

C₈

#14581777

yes

—

183

C₈

#14581781

yes

characteristic

184

C₂²

—

185

C₂²

—

1328

—

186

C₂²

—

1328

—

187

C₂²

—

1328

—

188

C₂²

—

664

—

189

C₂²

#19127917

yes

—

190

C₂²

#19127937

yes

characteristic

191

C₄

—

192

C₄

#19127618

yes

characteristic

193

C₄

#19127897

yes

—

194

C₄

#19127954

yes

characteristic

195

C₂

—

196

C₂

—

1328

—

197

C₂

#24908774

yes

characteristic

198

C₂

#24909583

yes

—

	subgroup	G/H
center	C₂²	#19127937
commutator, cosocle	C₂²×C₈×C₈₃	C₂³
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₈₃×#2151	C₂
socle	C₂²×C₈₃	#466
prime radical	C₂²×D₁₃₂₈	C₂
hypercenter	#2151	D₈₃
hypocenter	C₈₃	#13439
LCS	G ▷ C₂²×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ #2151

subgroup

G/H

center

C₂²

#19127937

commutator, cosocle

C₂²×C₈×C₈₃

C₂³

Frattini

C₂²×C₈

C₂²×D₈₃

Fitting

C₈₃×#2151

C₂

socle

C₂²×C₈₃

#466

prime radical

C₂²×D₁₃₂₈

C₂

D₈₃

C₈₃

G ▷ C₂²×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ #2151

N	H ≅ G/N	type
2. C₂×#19127515	C₂	semidirect
3. #24908373	C₂	semidirect
4. #24905557	C₂	semidirect
5. C₈₃×#2151	C₂	semidirect
101. #2151	D₈₃	semidirect
105. C₈₃	#13439	semidirect

H ≅ G/N

type

2. C₂×#19127515

C₂

semidirect

3. #24908373

C₂

semidirect

4. #24905557

C₂

semidirect

5. C₈₃×#2151

C₂

semidirect

101. #2151

D₈₃

semidirect

105. C₈₃

#13439

semidirect

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

subgroup	index
C₂²×D₁₃₂₈	2
C₂×#19127515	2
#24908373	2
#24905557	2
C₈₃×#2151	2
#13439	83

subgroup

index

C₂²×D₁₃₂₈

C₈₃×#2151

Subgroups — #35323944

Conjugacy classes of subgroups

Complements

Maximal subgroups