Group #35320763 — subgroups

Subgroups — #35320763

Order 21248.

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

normal subgroups only

This group has 342 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	#24909218	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₈₃	C₂	1	1	yes	maximal, characteristic
3	#24905545	C₂	1	1	yes	maximal, characteristic
4	#24905544	C₂	1	2	yes	maximal
5	C₈₃×#1376	C₂	1	1	yes	maximal, characteristic
6	C₂×#14581732	C₂²	1	2	yes	—
7	C₂×C₄×C₈₃⋊C₈	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₈₃	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₈₃	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₈₃	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₈₃	C₂²	1	1	yes	characteristic
12	C₄²×Dic₈₃	C₄	1	1	yes	characteristic
13	Dic₈₃⋊C₁₆	—	2	4	no	—
14	C₂×M₃₂×C₈₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
16	C₂²×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×Dic₈₃	—	2	2	no	—
19	C₂×C₄×Dic₈₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₈₃	C₂×C₄	1	1	yes	characteristic
21	#14581732	—	2	2	no	—
22	#14581732	—	2	2	no	—
23	C₂×C₈₃⋊C₁₆	—	2	2	no	—
24	C₄×C₈₃⋊C₈	—	2	2	no	—
25	C₈×Dic₈₃	—	2	2	no	—
26	C₈×Dic₈₃	—	2	2	no	—
27	C₈×Dic₈₃	C₈	1	2	yes	—
28	C₈×Dic₈₃	C₈	1	2	yes	—
29	C₈×Dic₈₃	D₄	1	2	yes	—
30	C₈×Dic₈₃	Q₈	1	2	yes	—
31	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
34	M₃₂×C₈₃	—	2	2	no	—
35	M₃₂×C₈₃	—	2	2	no	—
36	C₂×C₁₆×C₈₃	—	2	2	no	—
37	C₄×C₈×C₈₃	—	2	2	no	—
38	C₂²×Dic₈₃	—	2	1	no	—
39	C₂²×Dic₈₃	M₁₆	1	2	yes	—
40	C₂×C₈₃⋊C₈	—	2	1	no	—
41	C₂×C₈₃⋊C₈	—	2	1	no	—
42	C₂×C₈₃⋊C₈	—	2	1	no	—
43	C₂×C₈₃⋊C₈	—	2	1	no	—
44	C₂×C₈₃⋊C₈	C₄⋊C₄	1	2	yes	—
45	C₂×C₈₃⋊C₈	C₂×C₈	1	2	yes	—
46	C₄×Dic₈₃	—	2	8	no	—
47	C₄×Dic₈₃	—	2	1	no	—
48	C₄×Dic₈₃	C₄⋊C₄	1	2	yes	—
49	C₄×Dic₈₃	C₂×C₈	1	1	yes	characteristic
50	C₄×Dic₈₃	C₂×C₈	1	1	yes	characteristic
51	C₄×Dic₈₃	M₁₆	1	2	yes	—
52	C₈₃⋊C₁₆	—	4	2	no	—
53	C₂²×C₄×C₈₃	—	2	1	no	—
54	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₈₃	—	2	1	no	—
56	C₂×C₈×C₈₃	—	2	1	no	—
57	C₂×C₈×C₈₃	C₂×C₈	1	2	yes	—
58	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₈×C₈₃	#41	1	2	yes	—
61	C₂×C₈×C₈₃	#41	1	2	yes	—
62	C₄²×C₈₃	—	2	1	no	—
63	C₄²×C₈₃	M₁₆	1	2	yes	—
64	C₁₆×C₈₃	—	4	2	no	—
65	C₂³×C₈₃	C₂×M₁₆	1	1	yes	characteristic
66	C₂×C₄×C₈₃	—	2	4	no	—
67	C₂×C₄×C₈₃	—	2	2	no	—
68	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₈₃	#117	1	1	yes	characteristic
70	C₂×C₄×C₈₃	#117	1	1	yes	characteristic
71	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
73	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
74	C₂×Dic₈₃	—	2	4	no	—
75	C₂×Dic₈₃	—	2	2	no	—
76	C₂×Dic₈₃	—	2	4	no	—
77	C₂×Dic₈₃	C₄⋊C₈	1	2	yes	—
78	C₂×Dic₈₃	C₄⋊C₈	1	2	yes	—
79	C₈×C₈₃	—	2	2	no	—
80	C₈×C₈₃	#124	1	2	yes	—
81	C₈×C₈₃	#125	1	2	yes	—
82	C₈₃⋊C₈	—	2	2	no	—
83	C₈₃⋊C₈	—	2	2	no	—
84	C₂²×C₈₃	—	2	2	no	—
85	C₂²×C₈₃	C₂×C₄⋊C₈	1	1	yes	characteristic
86	C₂²×C₈₃	#432	1	1	yes	characteristic
87	C₂²×C₈₃	#432	1	1	yes	characteristic
88	C₄×C₈₃	—	2	4	no	—
89	C₄×C₈₃	#382	1	1	yes	characteristic
90	C₄×C₈₃	#423	1	1	yes	characteristic
91	C₄×C₈₃	#431	1	1	yes	characteristic
92	C₄×C₈₃	#431	1	1	yes	characteristic
93	Dic₈₃	—	2	4	no	—
94	Dic₈₃	—	2	2	no	—
95	Dic₈₃	#364	1	4	yes	—
96	#12883	—	83	1	no	maximal, Sylow
97	C₂×C₈₃	—	2	1	no	—
98	C₂×C₈₃	—	2	1	no	—
99	C₂×C₈₃	#1831	1	1	yes	characteristic
100	C₂×C₈₃	C₂×#364	1	2	yes	—
101	#2150	—	83	2	no	—
102	C₄²×C₈	—	83	1	no	—
103	#1376	—	83	2	no	—
104	#1376	—	83	1	no	—
105	#1376	D₈₃	1	1	yes	characteristic
106	C₈₃	#12883	1	1	yes	characteristic, complemented, Sylow
107	C₂×M₃₂	—	83	2	no	—
108	C₂×M₃₂	C₂×D₈₃	1	2	yes	—
109	C₂×C₄×C₈	—	83	1	no	—
110	C₂×C₄×C₈	—	83	1	no	—
111	C₂×C₄×C₈	—	83	1	no	—
112	C₂×C₄×C₈	—	83	1	no	—
113	C₂×C₄×C₈	—	83	1	no	—
114	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
115	C₄³	—	83	1	no	—
116	C₄⋊C₁₆	—	166	4	no	—
117	C₂²×C₈	—	83	1	no	—
118	C₂²×C₈	—	83	1	no	—
119	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
120	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
121	C₂×C₄²	—	83	1	no	—
122	C₂×C₄²	—	83	1	no	—
123	C₂×C₄²	—	166	2	no	—
124	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
125	M₃₂	—	2	2	no	—
126	M₃₂	—	2	2	no	—
127	M₃₂	—	166	2	no	—
128	M₃₂	—	166	2	no	—
129	C₂×C₁₆	—	2	2	no	—
130	C₂×C₁₆	—	166	2	no	—
131	C₄×C₈	—	2	2	no	—
132	C₄×C₈	—	83	2	no	—
133	C₄×C₈	—	83	2	no	—
134	C₄×C₈	—	83	2	no	—
135	C₄×C₈	—	83	2	no	—
136	C₄×C₈	—	166	2	no	—
137	C₄×C₈	—	166	2	no	—
138	C₄×C₈	—	166	2	no	—
139	C₂²×C₄	—	2	1	no	—
140	C₂²×C₄	—	83	2	no	—
141	C₂²×C₄	—	166	1	no	—
142	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	83	2	no	—
146	C₂×C₈	—	83	2	no	—
147	C₂×C₈	—	166	1	no	—
148	C₂×C₈	—	166	1	no	—
149	C₂×C₈	—	166	1	no	—
150	C₂×C₈	—	166	1	no	—
151	C₂×C₈	C₈×D₈₃	1	2	yes	—
152	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
153	C₂×C₈	Q₈×D₈₃	1	1	yes	characteristic
154	C₂×C₈	#13050589	1	2	yes	—
155	C₂×C₈	#13050590	1	2	yes	—
156	C₄²	—	2	1	no	—
157	C₄²	—	83	1	no	—
158	C₄²	—	83	1	no	—
159	C₄²	—	83	2	no	—
160	C₄²	—	83	2	no	—
161	C₄²	—	166	8	no	—
162	C₄²	—	166	1	no	—
163	C₄²	#13050573	1	2	yes	—
164	C₁₆	—	4	2	no	—
165	C₁₆	—	332	2	no	—
166	C₂³	M₁₆×D₈₃	1	1	yes	characteristic
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	83	2	no	—
170	C₂×C₄	—	83	2	no	—
171	C₂×C₄	—	166	4	no	—
172	C₂×C₄	—	166	4	no	—
173	C₂×C₄	—	166	2	no	—
174	C₂×C₄	C₄⋊C₄×D₈₃	1	1	yes	characteristic
175	C₂×C₄	#14581748	1	1	yes	characteristic
176	C₂×C₄	#14581749	1	1	yes	characteristic
177	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
178	C₂×C₄	M₁₆×D₈₃	1	1	yes	characteristic
179	C₂×C₄	C₂×#13050573	1	1	yes	characteristic
180	C₈	—	2	2	no	—
181	C₈	—	166	2	no	—
182	C₈	—	166	2	no	—
183	C₈	#14581778	1	2	yes	—
184	C₈	#14581782	1	2	yes	—
185	C₂²	—	2	2	no	—
186	C₂²	C₄⋊C₈×D₈₃	1	1	yes	characteristic
187	C₂²	#19127755	1	1	yes	characteristic
188	C₂²	#19127756	1	1	yes	characteristic
189	C₄	—	2	4	no	—
190	C₄	—	83	4	no	—
191	C₄	—	166	4	no	—
192	C₄	—	166	2	no	—
193	C₄	#19127565	1	1	yes	characteristic
194	C₄	#19127723	1	1	yes	characteristic
195	C₄	#19127751	1	1	yes	characteristic
196	C₄	#19127752	1	1	yes	characteristic
197	C₂	—	2	1	no	—
198	C₂	—	2	1	no	—
199	C₂	#24907578	1	1	yes	characteristic
200	C₂	#364×D₈₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#24909218

C₂

yes

maximal

C₄×C₈×Dic₈₃

C₂

yes

maximal, characteristic

#24905545

C₂

yes

maximal, characteristic

#24905544

C₂

yes

maximal

C₈₃×#1376

C₂

yes

maximal, characteristic

C₂×#14581732

C₂²

yes

—

C₂×C₄×C₈₃⋊C₈

C₄

yes

characteristic

C₂×C₈×Dic₈₃

C₄

yes

characteristic

C₂×C₈×Dic₈₃

C₄

yes

characteristic

C₂×C₈×Dic₈₃

C₂²

yes

characteristic

C₂×C₈×Dic₈₃

C₂²

yes

characteristic

C₄²×Dic₈₃

C₄

yes

characteristic

Dic₈₃⋊C₁₆

—

C₂×M₃₂×C₈₃

C₂²

yes

—

C₂×C₄×C₈×C₈₃

C₂²

yes

characteristic

C₂²×C₈₃⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₈₃⋊C₈

C₂×C₄

yes

characteristic

C₂×C₄×Dic₈₃

—

C₂×C₄×Dic₈₃

C₂×C₄

yes

characteristic

C₂×C₄×Dic₈₃

C₂×C₄

yes

characteristic

#14581732

—

#14581732

—

C₂×C₈₃⋊C₁₆

—

C₄×C₈₃⋊C₈

—

C₈×Dic₈₃

—

C₈×Dic₈₃

—

C₈×Dic₈₃

C₈

yes

—

C₈×Dic₈₃

C₈

yes

—

C₈×Dic₈₃

D₄

yes

—

C₈×Dic₈₃

Q₈

yes

—

C₂²×C₈×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₈₃

C₂³

yes

characteristic

C₂×C₄²×C₈₃

C₂×C₄

yes

characteristic

M₃₂×C₈₃

—

M₃₂×C₈₃

—

C₂×C₁₆×C₈₃

—

C₄×C₈×C₈₃

—

C₂²×Dic₈₃

—

C₂²×Dic₈₃

M₁₆

yes

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

C₄⋊C₄

yes

—

C₂×C₈₃⋊C₈

C₂×C₈

yes

—

C₄×Dic₈₃

—

C₄×Dic₈₃

—

C₄×Dic₈₃

C₄⋊C₄

yes

—

C₄×Dic₈₃

C₂×C₈

yes

characteristic

C₄×Dic₈₃

C₂×C₈

yes

characteristic

C₄×Dic₈₃

M₁₆

yes

—

C₈₃⋊C₁₆

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₂×C₈

yes

—

C₂×C₈×C₈₃

C₂×D₄

yes

characteristic

C₂×C₈×C₈₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₈₃

#41

yes

—

C₂×C₈×C₈₃

#41

yes

—

C₄²×C₈₃

—

C₄²×C₈₃

M₁₆

yes

—

C₁₆×C₈₃

—

C₂³×C₈₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×C₄⋊C₄

yes

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₈₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₈₃

C₂×M₁₆

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

C₄⋊C₈

yes

—

C₂×Dic₈₃

C₄⋊C₈

yes

—

C₈×C₈₃

—

C₈×C₈₃

#124

yes

—

C₈×C₈₃

#125

yes

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

C₂×C₄⋊C₈

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₄×C₈₃

—

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

Dic₈₃

—

Dic₈₃

—

Dic₈₃

#364

yes

—

#12883

—

maximal, Sylow

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

#1831

yes

characteristic

100

C₂×C₈₃

C₂×#364

yes

—

101

#2150

—

102

C₄²×C₈

—

103

#1376

—

104

#1376

—

105

#1376

D₈₃

yes

characteristic

106

C₈₃

#12883

yes

characteristic, complemented, Sylow

107

C₂×M₃₂

—

108

C₂×M₃₂

C₂×D₈₃

yes

—

109

C₂×C₄×C₈

—

110

C₂×C₄×C₈

—

111

C₂×C₄×C₈

—

112

C₂×C₄×C₈

—

113

C₂×C₄×C₈

—

114

C₂×C₄×C₈

C₂×D₈₃

yes

characteristic

115

C₄³

—

116

C₄⋊C₁₆

—

166

—

117

C₂²×C₈

—

118

C₂²×C₈

—

119

C₂²×C₈

C₄×D₈₃

yes

characteristic

120

C₂²×C₈

C₂²×D₈₃

yes

characteristic

121

C₂×C₄²

—

122

C₂×C₄²

—

123

C₂×C₄²

—

166

—

124

C₂×C₄²

C₄×D₈₃

yes

characteristic

125

M₃₂

—

126

M₃₂

—

127

M₃₂

—

166

—

128

M₃₂

—

166

—

129

C₂×C₁₆

—

130

C₂×C₁₆

—

166

—

131

C₄×C₈

—

132

C₄×C₈

—

133

C₄×C₈

—

134

C₄×C₈

—

135

C₄×C₈

—

136

C₄×C₈

—

166

—

137

C₄×C₈

—

166

—

138

C₄×C₈

—

166

—

139

C₂²×C₄

—

140

C₂²×C₄

—

141

C₂²×C₄

—

166

—

142

C₂²×C₄

C₂×C₄×D₈₃

yes

characteristic

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

166

—

148

C₂×C₈

—

166

—

149

C₂×C₈

—

166

—

150

C₂×C₈

—

166

—

151

C₂×C₈

C₈×D₈₃

yes

—

152

C₂×C₈

D₄×D₈₃

yes

characteristic

153

C₂×C₈

Q₈×D₈₃

yes

characteristic

154

C₂×C₈

#13050589

yes

—

155

C₂×C₈

#13050590

yes

—

156

C₄²

—

157

C₄²

—

158

C₄²

—

159

C₄²

—

160

C₄²

—

161

C₄²

—

166

—

162

C₄²

—

166

—

163

C₄²

#13050573

yes

—

164

C₁₆

—

165

C₁₆

—

332

—

166

C₂³

M₁₆×D₈₃

yes

characteristic

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

166

—

172

C₂×C₄

—

166

—

173

C₂×C₄

—

166

—

174

C₂×C₄

C₄⋊C₄×D₈₃

yes

characteristic

175

C₂×C₄

#14581748

yes

characteristic

176

C₂×C₄

#14581749

yes

characteristic

177

C₂×C₄

C₂×C₈×D₈₃

yes

characteristic

178

C₂×C₄

M₁₆×D₈₃

yes

characteristic

179

C₂×C₄

C₂×#13050573

yes

characteristic

180

C₈

—

181

C₈

—

166

—

182

C₈

—

166

—

183

C₈

#14581778

yes

—

184

C₈

#14581782

yes

—

185

C₂²

—

186

C₂²

C₄⋊C₈×D₈₃

yes

characteristic

187

C₂²

#19127755

yes

characteristic

188

C₂²

#19127756

yes

characteristic

189

C₄

—

190

C₄

—

191

C₄

—

166

—

192

C₄

—

166

—

193

C₄

#19127565

yes

characteristic

194

C₄

#19127723

yes

characteristic

195

C₄

#19127751

yes

characteristic

196

C₄

#19127752

yes

characteristic

197

C₂

—

198

C₂

—

199

C₂

#24907578

yes

characteristic

200

C₂

#364×D₈₃

yes

—

	subgroup	G/H
center	C₂×C₈	D₄×D₈₃
commutator	C₂×C₄×C₈₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₈₃×#1376	C₂
socle	C₂²×C₈₃	C₂×C₄⋊C₈
prime radical	C₂³×C₈₃	C₂×M₁₆
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	#1376	D₈₃
hypocenter	C₈₃	#12883
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1376

subgroup

G/H

center

C₂×C₈

D₄×D₈₃

commutator

C₂×C₄×C₈₃

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₈₃

Fitting

C₈₃×#1376

C₂

socle

C₂²×C₈₃

C₂×C₄⋊C₈

prime radical

C₂³×C₈₃

C₂×M₁₆

cosocle

C₂²×C₈×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1376

N	H ≅ G/N	type
106. C₈₃	#12883	semidirect

H ≅ G/N

type

106. C₈₃

#12883

semidirect

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

subgroup	index
#24909218	2
C₄×C₈×Dic₈₃	2
#24905545	2
#24905544	2
C₈₃×#1376	2
#12883	83

subgroup

index

C₄×C₈×Dic₈₃

C₈₃×#1376

Subgroups — #35320763

Conjugacy classes of subgroups

Complements

Maximal subgroups