Group #35304077 — subgroups

Subgroups — #35304077

Order 21248.

434 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 135 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	#24908308	C₂	1	2	yes	maximal
2	#24906219	C₂	1	1	yes	maximal, characteristic
3	#24906135	C₂	1	2	yes	maximal
4	#24905752	C₂	1	1	yes	maximal, characteristic
5	C₂×#500×C₈₃	C₂	1	1	yes	maximal, characteristic
6	#19127754	C₂²	1	1	yes	characteristic
7	#19127725	C₂²	1	2	yes	—
8	C₂×#14581711	—	2	4	no	—
9	C₄×#13050574	C₂²	1	1	yes	characteristic
10	#19127452	—	2	2	no	—
11	#19127447	—	2	4	no	—
12	#19127423	—	2	1	no	—
13	#19127422	—	2	1	no	—
14	C₂×C₄⋊Q₈×C₈₃	C₂²	1	2	yes	—
15	#500×C₈₃	—	2	1	no	—
16	#500×C₈₃	—	2	1	no	—
17	#500×C₈₃	—	2	2	no	—
18	C₂×#107×C₈₃	C₄	1	4	yes	—
19	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
20	C₂×#13050574	—	2	2	no	—
21	C₂×#13050574	D₄	1	2	yes	—
22	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
23	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
24	C₄⋊(C₈₃⋊C₈)	D₄	1	2	yes	—
25	#14581711	—	4	8	no	—
26	C₂²⋊(C₈₃⋊C₈)	—	2	1	no	—
27	#14581703	—	2	1	no	—
28	C₄×C₈₃⋊C₈	—	2	1	no	—
29	C₄×C₈₃⋊C₈	—	2	1	no	—
30	C₂²×Q₈×C₈₃	—	2	2	no	—
31	C₂²×C₈×C₈₃	C₂×C₄	1	2	yes	—
32	C₄⋊Q₈×C₈₃	—	2	2	no	—
33	C₄⋊Q₈×C₈₃	—	2	2	no	—
34	C₄⋊Q₈×C₈₃	—	2	2	no	—
35	C₄⋊Q₈×C₈₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
37	C₂×C₄⋊C₄×C₈₃	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
39	#107×C₈₃	—	2	4	no	—
40	#107×C₈₃	—	2	4	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	#13050574	—	2	4	no	—
45	#13050574	—	4	4	no	—
46	C₂×C₈₃⋊C₈	—	2	2	no	—
47	C₂×C₈₃⋊C₈	—	2	2	no	—
48	C₂×C₈₃⋊C₈	—	4	2	no	—
49	C₂×Q₈×C₈₃	—	4	2	no	—
50	C₂×Q₈×C₈₃	—	4	2	no	—
51	C₂×Q₈×C₈₃	—	4	4	no	—
52	C₂×Q₈×C₈₃	—	4	4	no	—
53	C₂²×C₄×C₈₃	—	2	4	no	—
54	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
57	C₂×C₈×C₈₃	—	2	4	no	—
58	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
59	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
60	C₄⋊C₄×C₈₃	—	2	4	no	—
61	C₄⋊C₄×C₈₃	—	2	4	no	—
62	C₄⋊C₄×C₈₃	—	4	2	no	—
63	C₄⋊C₄×C₈₃	—	4	2	no	—
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	Q₈×C₈₃	—	8	4	no	—
70	Q₈×C₈₃	—	8	2	no	—
71	Q₈×C₈₃	—	8	2	no	—
72	C₂×C₄×C₈₃	—	2	2	no	—
73	C₂×C₄×C₈₃	—	2	2	no	—
74	C₂×C₄×C₈₃	—	4	4	no	—
75	C₂×C₄×C₈₃	—	4	8	no	—
76	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
78	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
79	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
80	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
81	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
82	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	4	yes	—
83	C₈×C₈₃	—	2	2	no	—
84	C₈×C₈₃	#101	1	4	yes	complemented
85	C₈₃⋊C₈	—	4	2	no	—
86	C₈₃⋊C₈	—	4	2	no	—
87	C₈₃⋊C₈	—	8	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	—	8	4	no	—
95	C₄×C₈₃	C₂×#101	1	2	yes	—
96	C₄×C₈₃	#469	1	2	yes	—
97	C₄×C₈₃	#473	1	2	yes	—
98	C₄×C₈₃	#496	1	2	yes	—
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	D₈₃	1	1	yes	characteristic
105	#1966	—	83	2	no	—
106	#1566	—	83	1	no	—
107	#1546	—	83	2	no	—
108	#1451	—	83	1	no	—
109	C₈₃	#9808	1	1	yes	characteristic, complemented, Sylow
110	C₂×C₄⋊Q₈	C₂×D₈₃	1	2	yes	—
111	#500	—	2	1	no	—
112	#500	—	2	1	no	—
113	#500	—	2	2	no	—
114	#431	—	83	1	no	—
115	C₂×#107	Dic₈₃	1	4	yes	—
116	#423	—	83	2	no	—
117	C₂×#101	—	166	4	no	—
118	C₄×M₁₆	—	83	1	no	—
119	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
120	#334	—	166	2	no	—
121	#332	—	166	4	no	—
122	#322	—	166	1	no	—
123	#322	—	166	1	no	—
124	C₂²×Q₈	—	2	2	no	—
125	C₂×M₁₆	—	83	2	no	—
126	C₂×M₁₆	—	166	2	no	—
127	C₂²×C₈	C₂×Dic₈₃	1	2	yes	—
128	C₄⋊Q₈	—	2	2	no	—
129	C₄⋊Q₈	—	2	2	no	—
130	C₄⋊Q₈	—	2	2	no	—
131	C₄⋊Q₈	—	2	2	no	—
132	C₂×C₄⋊C₄	—	2	2	no	—
133	C₂×C₄⋊C₄	C₂×Dic₈₃	1	4	yes	—
134	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
135	#107	—	2	4	no	—
136	#107	—	2	4	no	—
137	C₄⋊C₈	—	83	2	no	—
138	C₄⋊C₈	—	166	2	no	—
139	C₄⋊C₈	—	166	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	C₂²⋊D₈₃	1	1	yes	characteristic
146	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
147	C₄×C₈	C₂²⋊D₈₃	1	2	yes	—
148	C₂×Q₈	—	4	2	no	—
149	C₂×Q₈	—	4	2	no	—
150	C₂×Q₈	—	4	4	no	—
151	C₂×Q₈	—	4	4	no	—
152	C₂²×C₄	—	2	4	no	—
153	C₂²×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
154	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
155	C₂²×C₄	#13050589	1	1	yes	characteristic
156	M₁₆	—	332	4	no	—
157	M₁₆	—	166	4	no	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	—	332	2	no	—
160	C₂×C₈	—	166	2	no	—
161	C₂×C₈	—	166	2	no	—
162	C₂×C₈	C₂²⋊Dic₈₃	1	2	yes	—
163	C₂×C₈	C₂²⋊Dic₈₃	1	2	yes	—
164	C₄⋊C₄	—	2	4	no	—
165	C₄⋊C₄	—	2	4	no	—
166	C₄⋊C₄	—	4	2	no	—
167	C₄⋊C₄	—	4	2	no	—
168	C₄²	D₄×D₈₃	1	1	yes	characteristic
169	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
170	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
171	C₄²	#13050589	1	1	yes	characteristic
172	C₂³	D₄×Dic₈₃	1	1	yes	characteristic
173	Q₈	—	8	2	no	—
174	Q₈	—	8	2	no	—
175	Q₈	—	8	4	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	4	8	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	C₂×C₂²⋊Dic₈₃	1	1	yes	characteristic
181	C₂×C₄	D₄×Dic₈₃	1	1	yes	characteristic
182	C₂×C₄	#14581763	1	1	yes	characteristic
183	C₂×C₄	#14581763	1	1	yes	characteristic
184	C₂×C₄	#14581779	1	1	yes	characteristic
185	C₂×C₄	#14581788	1	1	yes	characteristic
186	C₂×C₄	#14581816	1	4	yes	—
187	C₈	—	2	2	no	—
188	C₈	—	332	2	no	—
189	C₈	—	332	2	no	—
190	C₈	—	664	2	no	—
191	C₈	#14581711	1	4	yes	complemented
192	C₂²	—	2	1	no	—
193	C₂²	—	2	1	no	—
194	C₂²	#19127604	1	1	yes	characteristic
195	C₂²	#19127813	1	2	yes	—
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	—	8	4	no	—
199	C₄	C₂×#14581711	1	2	yes	—
200	C₄	#19127951	1	2	yes	—
201	C₄	#19127973	1	2	yes	—
202	C₄	#19128089	1	2	yes	—
203	C₂	—	2	2	no	—
204	C₂	#24908308	1	1	yes	characteristic
205	C₂	#24908340	1	1	yes	characteristic
206	C₂	#24908379	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24908308

C₂

yes

maximal

#24906219

C₂

yes

maximal, characteristic

#24906135

C₂

yes

maximal

#24905752

C₂

yes

maximal, characteristic

C₂×#500×C₈₃

C₂

yes

maximal, characteristic

#19127754

C₂²

yes

characteristic

#19127725

C₂²

yes

—

C₂×#14581711

—

C₄×#13050574

C₂²

yes

—

—

—

—

C₂×C₄⋊Q₈×C₈₃

C₂²

yes

—

#500×C₈₃

—

#500×C₈₃

—

#500×C₈₃

—

C₂×#107×C₈₃

C₄

yes

—

C₂×C₄×C₈×C₈₃

C₂²

yes

characteristic

C₂×#13050574

—

C₂×#13050574

D₄

yes

—

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

D₄

yes

—

#14581711

—

C₂²⋊(C₈₃⋊C₈)

—

#14581703

—

C₄×C₈₃⋊C₈

—

C₄×C₈₃⋊C₈

—

C₂²×Q₈×C₈₃

—

C₂²×C₈×C₈₃

C₂×C₄

yes

—

C₄⋊Q₈×C₈₃

—

C₄⋊Q₈×C₈₃

—

C₄⋊Q₈×C₈₃

—

C₄⋊Q₈×C₈₃

—

C₂×C₄⋊C₄×C₈₃

—

C₂×C₄⋊C₄×C₈₃

C₂×C₄

yes

—

C₂×C₄²×C₈₃

C₂³

yes

characteristic

#107×C₈₃

—

#107×C₈₃

—

C₄×C₈×C₈₃

D₄

yes

characteristic

C₄×C₈×C₈₃

D₄

yes

characteristic

C₄×C₈×C₈₃

D₄

yes

—

#13050574

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₃

#41

yes

characteristic

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₂²⋊C₄

yes

—

C₂×C₈×C₈₃

C₂²⋊C₄

yes

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

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

Q₈×C₈₃

—

Q₈×C₈₃

—

Q₈×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

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

—

C₈×C₈₃

—

C₈×C₈₃

#101

yes

complemented

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

#390

yes

characteristic

C₂²×C₈₃

#442

yes

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₂×#101

yes

—

C₄×C₈₃

#469

yes

—

C₄×C₈₃

#473

yes

—

C₄×C₈₃

#496

yes

—

#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

D₈₃

yes

characteristic

105

#1966

—

106

#1566

—

107

#1546

—

108

#1451

—

109

C₈₃

#9808

yes

characteristic, complemented, Sylow

110

C₂×C₄⋊Q₈

C₂×D₈₃

yes

—

111

#500

—

112

#500

—

113

#500

—

114

#431

—

115

C₂×#107

Dic₈₃

yes

—

116

#423

—

117

C₂×#101

—

166

—

118

C₄×M₁₆

—

119

C₂×C₄×C₈

C₂×D₈₃

yes

characteristic

120

#334

—

166

—

121

#332

—

166

—

122

#322

—

166

—

123

#322

—

166

—

124

C₂²×Q₈

—

125

C₂×M₁₆

—

126

C₂×M₁₆

—

166

—

127

C₂²×C₈

C₂×Dic₈₃

yes

—

128

C₄⋊Q₈

—

129

C₄⋊Q₈

—

130

C₄⋊Q₈

—

131

C₄⋊Q₈

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄⋊C₄

C₂×Dic₈₃

yes

—

134

C₂×C₄²

C₂²×D₈₃

yes

characteristic

135

#107

—

136

#107

—

137

C₄⋊C₈

—

138

C₄⋊C₈

—

166

—

139

C₄⋊C₈

—

166

—

140

#101

—

332

—

141

C₂²⋊C₈

—

166

—

142

#97

—

166

—

143

C₄×C₈

—

166

—

144

C₄×C₈

—

166

—

145

C₄×C₈

C₂²⋊D₈₃

yes

characteristic

146

C₄×C₈

C₂²⋊D₈₃

yes

characteristic

147

C₄×C₈

C₂²⋊D₈₃

yes

—

148

C₂×Q₈

—

149

C₂×Q₈

—

150

C₂×Q₈

—

151

C₂×Q₈

—

152

C₂²×C₄

—

153

C₂²×C₄

C₂²×Dic₈₃

yes

characteristic

154

C₂²×C₄

D₄×D₈₃

yes

characteristic

155

C₂²×C₄

#13050589

yes

characteristic

156

M₁₆

—

332

—

157

M₁₆

—

166

—

158

C₂×C₈

—

159

C₂×C₈

—

332

—

160

C₂×C₈

—

166

—

161

C₂×C₈

—

166

—

162

C₂×C₈

C₂²⋊Dic₈₃

yes

—

163

C₂×C₈

C₂²⋊Dic₈₃

yes

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

168

C₄²

D₄×D₈₃

yes

characteristic

169

C₄²

C₂×C₂²⋊D₈₃

yes

characteristic

170

C₄²

C₂×C₂²⋊D₈₃

yes

characteristic

171

C₄²

#13050589

yes

characteristic

172

C₂³

D₄×Dic₈₃

yes

characteristic

173

Q₈

—

174

Q₈

—

175

Q₈

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

C₂×C₂²⋊Dic₈₃

yes

characteristic

181

C₂×C₄

D₄×Dic₈₃

yes

characteristic

182

C₂×C₄

#14581763

yes

characteristic

183

C₂×C₄

#14581763

yes

characteristic

184

C₂×C₄

#14581779

yes

characteristic

185

C₂×C₄

#14581788

yes

characteristic

186

C₂×C₄

#14581816

yes

—

187

C₈

—

188

C₈

—

332

—

189

C₈

—

332

—

190

C₈

—

664

—

191

C₈

#14581711

yes

complemented

192

C₂²

—

193

C₂²

—

194

C₂²

#19127604

yes

characteristic

195

C₂²

#19127813

yes

—

196

C₄

—

197

C₄

—

198

C₄

—

199

C₄

C₂×#14581711

yes

—

200

C₄

#19127951

yes

—

201

C₄

#19127973

yes

—

202

C₄

#19128089

yes

—

203

C₂

—

204

C₂

#24908308

yes

characteristic

205

C₂

#24908340

yes

characteristic

206

C₂

#24908379

yes

characteristic

	subgroup	G/H
center	C₂²	#19127604
commutator	C₂²×C₄×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₂×#500×C₈₃	C₂
socle	C₂²×C₈₃	#390
prime radical	C₂³×C₈₃	C₄×D₄
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	C₂×#500	D₈₃
hypocenter	C₈₃	#9808
LCS	G ▷ C₂²×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#500

subgroup

G/H

center

C₂²

#19127604

commutator

C₂²×C₄×C₈₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₃

Fitting

C₂×#500×C₈₃

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 ◁ C₂×#500

N	H ≅ G/N	type
84. C₈×C₈₃	#101	semidirect
109. C₈₃	#9808	semidirect
191. C₈	#14581711	semidirect

H ≅ G/N

type

84. C₈×C₈₃

#101

semidirect

109. C₈₃

#9808

semidirect

191. C₈

#14581711

semidirect

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

subgroup	index
#24908308	2
#24906219	2
#24906135	2
#24905752	2
C₂×#500×C₈₃	2
#9808	83

subgroup

index

C₂×#500×C₈₃

Subgroups — #35304077

Conjugacy classes of subgroups

Complements

Maximal subgroups