Group #35320766 — subgroups

Subgroups — #35320766

Order 21248.

302 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 123 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	#24907149	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×#14581703	C₄	1	1	yes	characteristic
8	C₂×#14581702	C₄	1	1	yes	characteristic
9	C₂×#14581702	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₈₃	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₈₃	C₂²	1	1	yes	characteristic
12	#19127547	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	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₈₃	C₂×C₄	1	1	yes	characteristic
20	#14581732	—	2	2	no	—
21	#14581732	—	2	2	no	—
22	C₂×C₈₃⋊C₁₆	—	2	2	no	—
23	#14581703	—	2	2	no	—
24	#14581702	—	2	2	no	—
25	#14581702	C₈	1	4	yes	—
26	C₈×Dic₈₃	—	2	2	no	—
27	C₈×Dic₈₃	D₄	1	2	yes	—
28	C₈×Dic₈₃	Q₈	1	2	yes	—
29	C₈₃⋊#95	—	2	2	no	—
30	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
33	M₃₂×C₈₃	—	2	2	no	—
34	M₃₂×C₈₃	—	2	2	no	—
35	C₂×C₁₆×C₈₃	—	2	2	no	—
36	C₄×C₈×C₈₃	—	2	2	no	—
37	C₂²×Dic₈₃	—	2	1	no	—
38	C₂²×Dic₈₃	M₁₆	1	2	yes	—
39	C₂×C₈₃⋊C₈	—	2	1	no	—
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	C₄⋊C₄	1	2	yes	—
44	C₂×C₈₃⋊C₈	C₂×C₈	1	2	yes	—
45	C₄×Dic₈₃	—	2	1	no	—
46	C₄×Dic₈₃	C₄⋊C₄	1	2	yes	—
47	C₄×Dic₈₃	C₂×C₈	1	2	yes	—
48	C₄×Dic₈₃	M₁₆	1	2	yes	—
49	C₈₃⋊C₁₆	—	4	2	no	—
50	C₂²×C₄×C₈₃	—	2	1	no	—
51	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₈₃	—	2	1	no	—
53	C₂×C₈×C₈₃	—	2	1	no	—
54	C₂×C₈×C₈₃	C₂×C₈	1	2	yes	—
55	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₈×C₈₃	#41	1	2	yes	—
58	C₂×C₈×C₈₃	#41	1	2	yes	—
59	C₄²×C₈₃	—	2	1	no	—
60	C₄²×C₈₃	M₁₆	1	2	yes	—
61	C₁₆×C₈₃	—	4	2	no	—
62	C₂³×C₈₃	C₂×M₁₆	1	1	yes	characteristic
63	C₂×C₄×C₈₃	—	2	2	no	—
64	C₂×C₄×C₈₃	—	4	2	no	—
65	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₈₃	#117	1	1	yes	characteristic
67	C₂×C₄×C₈₃	#117	1	1	yes	characteristic
68	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
70	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
71	C₂×Dic₈₃	—	2	2	no	—
72	C₂×Dic₈₃	—	4	2	no	—
73	C₂×Dic₈₃	—	4	2	no	—
74	C₂×Dic₈₃	C₄⋊C₈	1	4	yes	—
75	C₈×C₈₃	—	2	2	no	—
76	C₈×C₈₃	#124	1	2	yes	—
77	C₈×C₈₃	#125	1	2	yes	—
78	C₈₃⋊C₈	—	2	2	no	—
79	C₈₃⋊C₈	—	2	2	no	—
80	C₂²×C₈₃	—	2	2	no	—
81	C₂²×C₈₃	C₂×C₄⋊C₈	1	1	yes	characteristic
82	C₂²×C₈₃	#432	1	1	yes	characteristic
83	C₂²×C₈₃	#432	1	1	yes	characteristic
84	C₄×C₈₃	—	4	2	no	—
85	C₄×C₈₃	#382	1	1	yes	characteristic
86	C₄×C₈₃	#423	1	1	yes	characteristic
87	C₄×C₈₃	#431	1	1	yes	characteristic
88	C₄×C₈₃	#431	1	1	yes	characteristic
89	Dic₈₃	—	2	4	no	—
90	Dic₈₃	—	4	2	no	—
91	#12884	—	83	1	no	maximal, Sylow
92	C₂×C₈₃	—	2	1	no	—
93	C₂×C₈₃	—	2	1	no	—
94	C₂×C₈₃	#1831	1	1	yes	characteristic
95	C₂×C₈₃	#2153	1	2	yes	—
96	#2150	—	83	2	no	—
97	#1726	—	83	1	no	—
98	#1376	—	83	2	no	—
99	#1376	—	83	1	no	—
100	#1376	D₈₃	1	1	yes	characteristic
101	C₈₃	#12884	1	1	yes	characteristic, complemented, Sylow
102	C₂×M₃₂	—	83	2	no	—
103	C₂×M₃₂	C₂×D₈₃	1	2	yes	—
104	C₂×#97	—	83	1	no	—
105	C₂×#97	—	83	1	no	—
106	C₂×#97	—	83	1	no	—
107	C₂×C₄×C₈	—	83	1	no	—
108	C₂×C₄×C₈	—	83	1	no	—
109	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
110	#376	—	83	1	no	—
111	C₄⋊C₁₆	—	166	4	no	—
112	C₂²×C₈	—	83	1	no	—
113	C₂²×C₈	—	83	1	no	—
114	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
115	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
116	C₂×C₄²	—	83	1	no	—
117	C₂×C₄²	—	83	1	no	—
118	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
119	M₃₂	—	2	2	no	—
120	M₃₂	—	2	2	no	—
121	M₃₂	—	166	2	no	—
122	M₃₂	—	166	2	no	—
123	C₂×C₁₆	—	2	2	no	—
124	C₂×C₁₆	—	166	2	no	—
125	#97	—	83	4	no	—
126	#97	—	166	2	no	—
127	#97	—	166	2	no	—
128	C₄×C₈	—	2	2	no	—
129	C₄×C₈	—	83	2	no	—
130	C₄×C₈	—	83	2	no	—
131	C₄×C₈	—	166	2	no	—
132	#95	—	166	2	no	—
133	C₂²×C₄	—	2	1	no	—
134	C₂²×C₄	—	83	2	no	—
135	C₂²×C₄	—	166	1	no	—
136	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	83	2	no	—
140	C₂×C₈	—	83	2	no	—
141	C₂×C₈	—	166	1	no	—
142	C₂×C₈	—	166	1	no	—
143	C₂×C₈	—	166	1	no	—
144	C₂×C₈	—	166	1	no	—
145	C₂×C₈	C₈×D₈₃	1	2	yes	—
146	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
147	C₂×C₈	Q₈×D₈₃	1	1	yes	characteristic
148	C₂×C₈	#13050589	1	2	yes	—
149	C₂×C₈	#13050590	1	2	yes	—
150	C₄²	—	2	1	no	—
151	C₄²	—	83	2	no	—
152	C₄²	—	83	2	no	—
153	C₄²	—	83	2	no	—
154	C₄²	—	166	1	no	—
155	C₄²	#13050573	1	2	yes	—
156	C₁₆	—	4	2	no	—
157	C₁₆	—	332	2	no	—
158	C₂³	M₁₆×D₈₃	1	1	yes	characteristic
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	332	2	no	—
162	C₂×C₄	—	332	2	no	—
163	C₂×C₄	—	83	4	no	—
164	C₂×C₄	—	166	2	no	—
165	C₂×C₄	C₄⋊C₄×D₈₃	1	1	yes	characteristic
166	C₂×C₄	#14581748	1	1	yes	characteristic
167	C₂×C₄	#14581749	1	1	yes	characteristic
168	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
169	C₂×C₄	M₁₆×D₈₃	1	1	yes	characteristic
170	C₂×C₄	C₂×#13050573	1	1	yes	characteristic
171	C₈	—	2	2	no	—
172	C₈	—	166	2	no	—
173	C₈	—	166	2	no	—
174	C₈	#14581778	1	2	yes	—
175	C₈	#14581782	1	2	yes	—
176	C₂²	—	2	2	no	—
177	C₂²	C₄⋊C₈×D₈₃	1	1	yes	characteristic
178	C₂²	#19127755	1	1	yes	characteristic
179	C₂²	#19127756	1	1	yes	characteristic
180	C₄	—	4	2	no	—
181	C₄	—	332	2	no	—
182	C₄	—	166	4	no	—
183	C₄	#19127565	1	1	yes	characteristic
184	C₄	#19127723	1	1	yes	characteristic
185	C₄	#19127751	1	1	yes	characteristic
186	C₄	#19127752	1	1	yes	characteristic
187	C₂	—	2	1	no	—
188	C₂	—	2	1	no	—
189	C₂	#24907578	1	1	yes	characteristic
190	C₂	#24909227	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#24909218

C₂

yes

maximal

#24907149

C₂

yes

maximal, characteristic

#24905545

C₂

yes

maximal, characteristic

#24905544

C₂

yes

maximal

C₈₃×#1376

C₂

yes

maximal, characteristic

C₂×#14581732

C₂²

yes

—

C₂×#14581703

C₄

yes

characteristic

C₂×#14581702

C₄

yes

characteristic

C₂×#14581702

C₄

yes

characteristic

C₂×C₈×Dic₈₃

C₂²

yes

characteristic

C₂×C₈×Dic₈₃

C₂²

yes

characteristic

#19127547

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

yes

characteristic

C₂×C₄×Dic₈₃

C₂×C₄

yes

characteristic

#14581732

—

#14581732

—

C₂×C₈₃⋊C₁₆

—

#14581703

—

#14581702

—

#14581702

C₈

yes

—

C₈×Dic₈₃

—

C₈×Dic₈₃

D₄

yes

—

C₈×Dic₈₃

Q₈

yes

—

C₈₃⋊#95

—

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⋊C₄

yes

—

C₄×Dic₈₃

C₂×C₈

yes

—

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

—

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

—

#12884

—

maximal, Sylow

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

#1831

yes

characteristic

C₂×C₈₃

#2153

yes

—

—

—

—

—

100

#1376

D₈₃

yes

characteristic

101

C₈₃

#12884

yes

characteristic, complemented, Sylow

102

C₂×M₃₂

—

103

C₂×M₃₂

C₂×D₈₃

yes

—

104

C₂×#97

—

105

C₂×#97

—

106

C₂×#97

—

107

C₂×C₄×C₈

—

108

C₂×C₄×C₈

—

109

C₂×C₄×C₈

C₂×D₈₃

yes

characteristic

110

#376

—

111

C₄⋊C₁₆

—

166

—

112

C₂²×C₈

—

113

C₂²×C₈

—

114

C₂²×C₈

C₄×D₈₃

yes

characteristic

115

C₂²×C₈

C₂²×D₈₃

yes

characteristic

116

C₂×C₄²

—

117

C₂×C₄²

—

118

C₂×C₄²

C₄×D₈₃

yes

characteristic

119

M₃₂

—

120

M₃₂

—

121

M₃₂

—

166

—

122

M₃₂

—

166

—

123

C₂×C₁₆

—

124

C₂×C₁₆

—

166

—

125

#97

—

126

#97

—

166

—

127

#97

—

166

—

128

C₄×C₈

—

129

C₄×C₈

—

130

C₄×C₈

—

131

C₄×C₈

—

166

—

132

#95

—

166

—

133

C₂²×C₄

—

134

C₂²×C₄

—

135

C₂²×C₄

—

166

—

136

C₂²×C₄

C₂×C₄×D₈₃

yes

characteristic

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

166

—

142

C₂×C₈

—

166

—

143

C₂×C₈

—

166

—

144

C₂×C₈

—

166

—

145

C₂×C₈

C₈×D₈₃

yes

—

146

C₂×C₈

D₄×D₈₃

yes

characteristic

147

C₂×C₈

Q₈×D₈₃

yes

characteristic

148

C₂×C₈

#13050589

yes

—

149

C₂×C₈

#13050590

yes

—

150

C₄²

—

151

C₄²

—

152

C₄²

—

153

C₄²

—

154

C₄²

—

166

—

155

C₄²

#13050573

yes

—

156

C₁₆

—

157

C₁₆

—

332

—

158

C₂³

M₁₆×D₈₃

yes

characteristic

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

332

—

162

C₂×C₄

—

332

—

163

C₂×C₄

—

164

C₂×C₄

—

166

—

165

C₂×C₄

C₄⋊C₄×D₈₃

yes

characteristic

166

C₂×C₄

#14581748

yes

characteristic

167

C₂×C₄

#14581749

yes

characteristic

168

C₂×C₄

C₂×C₈×D₈₃

yes

characteristic

169

C₂×C₄

M₁₆×D₈₃

yes

characteristic

170

C₂×C₄

C₂×#13050573

yes

characteristic

171

C₈

—

172

C₈

—

166

—

173

C₈

—

166

—

174

C₈

#14581778

yes

—

175

C₈

#14581782

yes

—

176

C₂²

—

177

C₂²

C₄⋊C₈×D₈₃

yes

characteristic

178

C₂²

#19127755

yes

characteristic

179

C₂²

#19127756

yes

characteristic

180

C₄

—

181

C₄

—

332

—

182

C₄

—

166

—

183

C₄

#19127565

yes

characteristic

184

C₄

#19127723

yes

characteristic

185

C₄

#19127751

yes

characteristic

186

C₄

#19127752

yes

characteristic

187

C₂

—

188

C₂

—

189

C₂

#24907578

yes

characteristic

190

C₂

#24909227

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	#12884
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
101. C₈₃	#12884	semidirect

H ≅ G/N

type

101. C₈₃

#12884

semidirect

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

subgroup	index
#24909218	2
#24907149	2
#24905545	2
#24905544	2
C₈₃×#1376	2
#12884	83

subgroup

index

C₈₃×#1376

Subgroups — #35320766

Conjugacy classes of subgroups

Complements

Maximal subgroups