Group #35304533 — subgroups

Subgroups — #35304533

Order 21248.

430 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 163 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	#24915826	C₂	1	1	yes	maximal, characteristic
2	#24907580	C₂	1	1	yes	maximal, characteristic
3	#24905347	C₂	1	4	yes	maximal
4	C₄×C₄⋊C₈×C₈₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14581784	—	2	1	no	—
6	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
7	#19127599	—	2	2	no	—
8	#19127599	C₄	1	4	yes	—
9	#19127571	C₂²	1	2	yes	—
10	#19127567	—	2	1	no	—
11	#19127456	—	2	2	no	—
12	C₂×C₄⋊C₈×C₈₃	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₈₃	—	2	1	no	—
14	C₄³×C₈₃	C₂²	1	1	yes	characteristic
15	#336×C₈₃	—	2	1	no	—
16	C₂²×C₈₃⋊C₈	—	2	2	no	—
17	#14581784	—	4	2	no	—
18	#14581784	—	4	2	no	—
19	C₂×C₄⋊Dic₈₃	—	2	2	no	—
20	C₂×C₄⋊Dic₈₃	—	2	2	no	—
21	C₂×C₄⋊Dic₈₃	C₂×C₄	1	4	yes	—
22	C₂×Dic₈₃⋊C₄	—	2	4	no	—
23	C₄⋊(C₈₃⋊C₈)	—	2	4	no	—
24	C₈₃⋊#95	—	2	4	no	—
25	C₂²×C₈×C₈₃	—	2	2	no	—
26	C₂×C₄²×C₈₃	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₈₃	D₄	1	2	yes	—
28	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₈₃	—	2	2	no	—
32	C₄⋊C₈×C₈₃	Q₈	1	4	yes	—
33	C₄×C₈×C₈₃	—	4	2	no	—
34	C₂²×Dic₈₃	—	2	4	no	—
35	C₂×C₈₃⋊C₈	—	4	2	no	—
36	C₂×C₈₃⋊C₈	—	4	4	no	—
37	C₄⋊Dic₈₃	—	4	2	no	—
38	C₄⋊Dic₈₃	—	4	2	no	—
39	C₄⋊Dic₈₃	—	4	4	no	—
40	C₄⋊Dic₈₃	—	4	4	no	—
41	Dic₈₃⋊C₄	—	4	8	no	—
42	C₂²×C₄×C₈₃	C₂²⋊C₄	1	2	yes	—
43	C₂²×C₄×C₈₃	C₂²⋊C₄	1	2	yes	—
44	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₈₃	—	2	4	no	—
48	C₂×C₈×C₈₃	—	4	2	no	—
49	C₂×C₈×C₈₃	—	4	2	no	—
50	C₄²×C₈₃	—	2	4	no	—
51	C₄²×C₈₃	—	2	2	no	—
52	C₄²×C₈₃	—	4	2	no	—
53	C₄²×C₈₃	D₈	1	2	yes	—
54	C₄²×C₈₃	SD₁₆	1	2	yes	—
55	C₄²×C₈₃	C₂×Q₈	1	2	yes	—
56	C₄²×C₈₃	#41	1	2	yes	—
57	C₂³×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₈₃	—	2	2	no	—
59	C₂×C₄×C₈₃	—	2	2	no	—
60	C₂×C₄×C₈₃	—	2	2	no	—
61	C₂×C₄×C₈₃	—	2	2	no	—
62	C₂×C₄×C₈₃	—	2	2	no	—
63	C₂×C₄×C₈₃	—	2	2	no	—
64	C₂×C₄×C₈₃	#102	1	4	yes	—
65	C₂×C₄×C₈₃	C₄×Q₈	1	2	yes	—
66	C₂×C₄×C₈₃	C₂²⋊Q₈	1	2	yes	—
67	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
68	C₂×C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
69	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
70	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₈₃	#135	1	4	yes	—
72	C₂×C₄×C₈₃	#137	1	2	yes	—
73	C₂×Dic₈₃	—	4	4	no	—
74	C₂×Dic₈₃	—	4	4	no	—
75	C₂×Dic₈₃	—	4	4	no	—
76	C₈×C₈₃	—	4	4	no	—
77	C₈₃⋊C₈	—	8	2	no	—
78	C₂²×C₈₃	#391	1	1	yes	characteristic
79	C₂²×C₈₃	#413	1	2	yes	—
80	C₂²×C₈₃	C₂×#102	1	1	yes	characteristic
81	C₂²×C₈₃	#416	1	2	yes	—
82	C₂²×C₈₃	#419	1	1	yes	characteristic
83	C₄×C₈₃	—	2	4	no	—
84	C₄×C₈₃	—	2	2	no	—
85	C₄×C₈₃	—	4	2	no	—
86	C₄×C₈₃	#474	1	2	yes	—
87	C₄×C₈₃	#475	1	2	yes	—
88	C₄×C₈₃	#488	1	2	yes	—
89	C₄×C₈₃	#499	1	2	yes	—
90	Dic₈₃	—	8	4	no	—
91	#9899	—	83	1	no	maximal, Sylow
92	C₂×C₈₃	#1540	1	2	yes	—
93	C₂×C₈₃	#1554	1	2	yes	—
94	C₂×C₈₃	#1988	1	1	yes	characteristic
95	C₂×C₈₃	#1991	1	1	yes	characteristic
96	C₂×C₈₃	#1998	1	1	yes	characteristic
97	#2859	—	83	1	no	—
98	#1831	—	83	1	no	—
99	C₄×C₄⋊C₈	D₈₃	1	1	yes	characteristic
100	#1296	—	83	4	no	—
101	C₈₃	#9899	1	1	yes	characteristic, complemented, Sylow
102	C₂×#125	—	166	1	no	—
103	C₂×C₄⋊C₈	—	83	2	no	—
104	C₂×C₄⋊C₈	C₂×D₈₃	1	2	yes	—
105	C₂×C₄×C₈	—	2	1	no	—
106	#389	—	83	4	no	—
107	#389	—	166	2	no	—
108	#384	—	83	2	no	—
109	#382	—	166	1	no	—
110	C₄³	C₂×D₈₃	1	1	yes	characteristic
111	#336	—	2	1	no	—
112	#336	—	166	2	no	—
113	C₂²×C₈	—	2	2	no	—
114	C₂²×C₈	—	166	2	no	—
115	#125	—	332	2	no	—
116	#125	—	332	2	no	—
117	C₂×C₄⋊C₄	—	83	4	no	—
118	C₂×C₄⋊C₄	—	166	4	no	—
119	C₂×C₄⋊C₄	—	166	2	no	—
120	C₂×C₄⋊C₄	—	166	2	no	—
121	C₂×C₄²	C₄×D₈₃	1	2	yes	—
122	C₂×C₄²	D₃₃₂	1	2	yes	—
123	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
124	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
125	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
126	C₄⋊C₈	—	2	2	no	—
127	C₄⋊C₈	—	166	4	no	—
128	C₄⋊C₈	Dic₁₆₆	1	4	yes	—
129	C₄×C₈	—	4	2	no	—
130	#95	—	166	4	no	—
131	C₂²×C₄	—	166	4	no	—
132	C₂²×C₄	(C₂×D₈₃)⋊C₄	1	2	yes	—
133	C₂²×C₄	(C₂×D₈₃)⋊C₄	1	2	yes	—
134	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
135	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
136	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
137	C₂×C₈	—	2	4	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	4	2	no	—
140	C₂×C₈	—	332	2	no	—
141	C₂×C₈	—	332	4	no	—
142	C₄⋊C₄	—	332	4	no	—
143	C₄⋊C₄	—	332	2	no	—
144	C₄⋊C₄	—	332	2	no	—
145	C₄⋊C₄	—	332	4	no	—
146	C₄⋊C₄	—	332	8	no	—
147	C₄²	—	2	4	no	—
148	C₄²	—	2	2	no	—
149	C₄²	—	4	2	no	—
150	C₄²	#13050575	1	2	yes	—
151	C₄²	#13050577	1	2	yes	—
152	C₄²	C₂×Dic₁₆₆	1	2	yes	—
153	C₄²	#13050590	1	2	yes	—
154	C₂³	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	332	4	no	—
162	C₂×C₄	—	332	4	no	—
163	C₂×C₄	—	332	4	no	—
164	C₂×C₄	#14581712	1	4	yes	—
165	C₂×C₄	C₄×Dic₁₆₆	1	2	yes	—
166	C₂×C₄	#14581771	1	2	yes	—
167	C₂×C₄	#14581780	1	1	yes	characteristic
168	C₂×C₄	#14581792	1	1	yes	characteristic
169	C₂×C₄	C₂×#13050575	1	1	yes	characteristic
170	C₂×C₄	C₂×#13050577	1	1	yes	characteristic
171	C₂×C₄	#14581814	1	4	yes	—
172	C₂×C₄	#14581822	1	2	yes	—
173	C₈	—	4	4	no	—
174	C₈	—	664	2	no	—
175	C₂²	#19127607	1	1	yes	characteristic
176	C₂²	#19127683	1	2	yes	—
177	C₂²	C₂×#14581712	1	1	yes	characteristic
178	C₂²	#19127698	1	2	yes	—
179	C₂²	#19127709	1	1	yes	characteristic
180	C₄	—	2	4	no	—
181	C₄	—	2	2	no	—
182	C₄	—	4	2	no	—
183	C₄	—	664	4	no	—
184	C₄	#19127981	1	2	yes	—
185	C₄	#19127986	1	2	yes	—
186	C₄	#19128056	1	2	yes	—
187	C₄	#19128103	1	2	yes	—
188	C₂	#24906113	1	2	yes	—
189	C₂	#24906171	1	2	yes	—
190	C₂	#24908415	1	1	yes	characteristic
191	C₂	#24908428	1	1	yes	characteristic
192	C₂	#24908456	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24915826

C₂

yes

maximal, characteristic

#24907580

C₂

yes

maximal, characteristic

#24905347

C₂

yes

maximal

C₄×C₄⋊C₈×C₈₃

C₂

yes

maximal, characteristic

C₂×#14581784

—

C₂×C₄⋊(C₈₃⋊C₈)

C₂²

yes

—

#19127599

—

#19127599

C₄

yes

—

#19127571

C₂²

yes

—

#19127567

—

#19127456

—

C₂×C₄⋊C₈×C₈₃

C₂²

yes

—

C₂×C₄×C₈×C₈₃

—

C₄³×C₈₃

C₂²

yes

characteristic

#336×C₈₃

—

C₂²×C₈₃⋊C₈

—

#14581784

—

#14581784

—

C₂×C₄⋊Dic₈₃

—

C₂×C₄⋊Dic₈₃

—

C₂×C₄⋊Dic₈₃

C₂×C₄

yes

—

C₂×Dic₈₃⋊C₄

—

C₄⋊(C₈₃⋊C₈)

—

C₈₃⋊#95

—

C₂²×C₈×C₈₃

—

C₂×C₄²×C₈₃

C₂×C₄

yes

—

C₂×C₄²×C₈₃

D₄

yes

—

C₂×C₄²×C₈₃

D₄

yes

characteristic

C₂×C₄²×C₈₃

D₄

yes

characteristic

C₂×C₄²×C₈₃

C₂³

yes

characteristic

C₄⋊C₈×C₈₃

—

C₄⋊C₈×C₈₃

Q₈

yes

—

C₄×C₈×C₈₃

—

C₂²×Dic₈₃

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

Dic₈₃⋊C₄

—

C₂²×C₄×C₈₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₈₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

D₈

yes

—

C₄²×C₈₃

SD₁₆

yes

—

C₄²×C₈₃

C₂×Q₈

yes

—

C₄²×C₈₃

#41

yes

—

C₂³×C₈₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

#102

yes

—

C₂×C₄×C₈₃

C₄×Q₈

yes

—

C₂×C₄×C₈₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₈₃

#124

yes

characteristic

C₂×C₄×C₈₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₈₃

C₂×D₈

yes

characteristic

C₂×C₄×C₈₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₈₃

#135

yes

—

C₂×C₄×C₈₃

#137

yes

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₂²×C₈₃

#391

yes

characteristic

C₂²×C₈₃

#413

yes

—

C₂²×C₈₃

C₂×#102

yes

characteristic

C₂²×C₈₃

#416

yes

—

C₂²×C₈₃

#419

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#474

yes

—

C₄×C₈₃

#475

yes

—

C₄×C₈₃

#488

yes

—

C₄×C₈₃

#499

yes

—

Dic₈₃

—

#9899

—

maximal, Sylow

C₂×C₈₃

#1540

yes

—

C₂×C₈₃

#1554

yes

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

—

C₄×C₄⋊C₈

D₈₃

yes

characteristic

100

#1296

—

101

C₈₃

#9899

yes

characteristic, complemented, Sylow

102

C₂×#125

—

166

—

103

C₂×C₄⋊C₈

—

104

C₂×C₄⋊C₈

C₂×D₈₃

yes

—

105

C₂×C₄×C₈

—

106

#389

—

107

#389

—

166

—

108

#384

—

109

#382

—

166

—

110

C₄³

C₂×D₈₃

yes

characteristic

111

#336

—

112

#336

—

166

—

113

C₂²×C₈

—

114

C₂²×C₈

—

166

—

115

#125

—

332

—

116

#125

—

332

—

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

—

166

—

119

C₂×C₄⋊C₄

—

166

—

120

C₂×C₄⋊C₄

—

166

—

121

C₂×C₄²

C₄×D₈₃

yes

—

122

C₂×C₄²

D₃₃₂

yes

—

123

C₂×C₄²

C₂²⋊D₈₃

yes

characteristic

124

C₂×C₄²

C₂²⋊D₈₃

yes

characteristic

125

C₂×C₄²

C₂²×D₈₃

yes

characteristic

126

C₄⋊C₈

—

127

C₄⋊C₈

—

166

—

128

C₄⋊C₈

Dic₁₆₆

yes

—

129

C₄×C₈

—

130

#95

—

166

—

131

C₂²×C₄

—

166

—

132

C₂²×C₄

(C₂×D₈₃)⋊C₄

yes

—

133

C₂²×C₄

(C₂×D₈₃)⋊C₄

yes

—

134

C₂²×C₄

C₂×C₄×D₈₃

yes

characteristic

135

C₂²×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

136

C₂²×C₄

C₂×D₃₃₂

yes

characteristic

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

332

—

141

C₂×C₈

—

332

—

142

C₄⋊C₄

—

332

—

143

C₄⋊C₄

—

332

—

144

C₄⋊C₄

—

332

—

145

C₄⋊C₄

—

332

—

146

C₄⋊C₄

—

332

—

147

C₄²

—

148

C₄²

—

149

C₄²

—

150

C₄²

#13050575

yes

—

151

C₄²

#13050577

yes

—

152

C₄²

C₂×Dic₁₆₆

yes

—

153

C₄²

#13050590

yes

—

154

C₂³

C₂×(C₂×D₈₃)⋊C₄

yes

characteristic

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

332

—

162

C₂×C₄

—

332

—

163

C₂×C₄

—

332

—

164

C₂×C₄

#14581712

yes

—

165

C₂×C₄

C₄×Dic₁₆₆

yes

—

166

C₂×C₄

#14581771

yes

—

167

C₂×C₄

#14581780

yes

characteristic

168

C₂×C₄

#14581792

yes

characteristic

169

C₂×C₄

C₂×#13050575

yes

characteristic

170

C₂×C₄

C₂×#13050577

yes

characteristic

171

C₂×C₄

#14581814

yes

—

172

C₂×C₄

#14581822

yes

—

173

C₈

—

174

C₈

—

664

—

175

C₂²

#19127607

yes

characteristic

176

C₂²

#19127683

yes

—

177

C₂²

C₂×#14581712

yes

characteristic

178

C₂²

#19127698

yes

—

179

C₂²

#19127709

yes

characteristic

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

664

—

184

C₄

#19127981

yes

—

185

C₄

#19127986

yes

—

186

C₄

#19128056

yes

—

187

C₄

#19128103

yes

—

188

C₂

#24906113

yes

—

189

C₂

#24906171

yes

—

190

C₂

#24908415

yes

characteristic

191

C₂

#24908428

yes

characteristic

192

C₂

#24908456

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₈₃)⋊C₄
commutator	C₂²×C₄×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₄×C₄⋊C₈×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	C₄×C₄⋊C₈	D₈₃
hypocenter	C₈₃	#9899
LCS	G ▷ C₂²×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×(C₂×D₈₃)⋊C₄

commutator

C₂²×C₄×C₈₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₃

Fitting

C₄×C₄⋊C₈×C₈₃

C₂

socle, prime radical

C₂³×C₈₃

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₈₃

C₂³

hypercenter

C₄×C₄⋊C₈

D₈₃

hypocenter

C₈₃

#9899

LCS

G ▷ C₂²×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊C₈

N	H ≅ G/N	type
101. C₈₃	#9899	semidirect

H ≅ G/N

type

101. C₈₃

#9899

semidirect

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

subgroup	index
#24915826	2
#24907580	2
#24905347	2
C₄×C₄⋊C₈×C₈₃	2
#9899	83

subgroup

index

C₄×C₄⋊C₈×C₈₃

Subgroups — #35304533

Conjugacy classes of subgroups

Complements

Maximal subgroups