Group #35298051 — subgroups

Subgroups — #35298051

Order 21248.

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

normal subgroups only

This group has 310 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	#24907151	C₂	1	1	yes	maximal, characteristic
2	#24906209	C₂	1	2	yes	maximal
3	#24905308	C₂	1	1	yes	maximal, characteristic
4	#24905291	C₂	1	2	yes	maximal
5	C₈₃×#1281	C₂	1	1	yes	maximal, characteristic
6	#19127754	C₂²	1	2	yes	—
7	C₂×#14581702	C₄	1	1	yes	characteristic
8	C₂×#14581702	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₈₃	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₈₃	C₄	1	1	yes	characteristic
12	#19127547	C₂²	1	1	yes	characteristic
13	#19127453	—	2	2	no	—
14	#19127451	—	2	2	no	—
15	#431×C₈₃	C₂²	1	2	yes	—
16	C₂×#97×C₈₃	C₂²	1	1	yes	characteristic
17	C₂²×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₈₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₈₃	C₂×C₄	1	1	yes	characteristic
21	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
22	C₂²⋊(C₈₃⋊C₈)	—	2	2	no	—
23	#14581702	—	2	2	no	—
24	#14581702	C₈	1	4	yes	complemented
25	C₄×C₈₃⋊C₈	—	2	2	no	—
26	C₄×C₈₃⋊C₈	D₄	1	2	yes	—
27	C₄×C₈₃⋊C₈	Q₈	1	2	yes	—
28	C₈×Dic₈₃	—	2	2	no	—
29	C₈×Dic₈₃	—	2	2	no	—
30	C₈₃⋊#95	—	2	2	no	—
31	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₈₃	—	2	2	no	—
35	C₂²⋊C₈×C₈₃	—	2	2	no	—
36	#97×C₈₃	—	2	2	no	—
37	C₄×C₈×C₈₃	—	2	2	no	—
38	C₂²×Dic₈₃	—	2	1	no	—
39	C₂²×Dic₈₃	—	2	1	no	—
40	C₂×C₈₃⋊C₈	—	2	1	no	—
41	C₂×C₈₃⋊C₈	—	2	1	no	—
42	C₂×C₈₃⋊C₈	—	4	2	no	—
43	C₂×C₈₃⋊C₈	C₄⋊C₄	1	2	yes	—
44	C₂×C₈₃⋊C₈	C₄⋊C₄	1	2	yes	—
45	C₂×C₈₃⋊C₈	C₂×C₈	1	2	yes	—
46	C₂×C₈₃⋊C₈	M₁₆	1	2	yes	—
47	C₄×Dic₈₃	—	2	1	no	—
48	C₄×Dic₈₃	—	2	1	no	—
49	C₄×Dic₈₃	C₂×C₈	1	2	yes	—
50	C₄×Dic₈₃	M₁₆	1	2	yes	—
51	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₈₃	#41	1	2	yes	—
53	C₂×C₈×C₈₃	—	2	1	no	—
54	C₂×C₈×C₈₃	—	2	1	no	—
55	C₂×C₈×C₈₃	—	2	1	no	—
56	C₂×C₈×C₈₃	—	2	1	no	—
57	C₂×C₈×C₈₃	—	4	2	no	—
58	C₂×C₈×C₈₃	C₂×C₈	1	2	yes	—
59	C₂×C₈×C₈₃	M₁₆	1	2	yes	—
60	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
62	C₄²×C₈₃	#41	1	2	yes	—
63	C₂³×C₈₃	#117	1	1	yes	characteristic
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	#124	1	2	yes	—
68	C₂×C₄×C₈₃	#125	1	2	yes	—
69	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
71	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
72	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
73	C₂×Dic₈₃	—	2	2	no	—
74	C₂×Dic₈₃	—	2	2	no	—
75	C₂×Dic₈₃	—	4	2	no	—
76	C₂×Dic₈₃	—	4	2	no	—
77	C₈×C₈₃	—	2	2	no	—
78	C₈×C₈₃	—	2	2	no	—
79	C₈×C₈₃	—	8	2	no	—
80	C₈₃⋊C₈	—	2	2	no	—
81	C₈₃⋊C₈	—	8	2	no	—
82	C₈₃⋊C₈	C₄⋊C₈	1	4	yes	—
83	C₂²×C₈₃	—	2	2	no	—
84	C₂²×C₈₃	#382	1	1	yes	characteristic
85	C₂²×C₈₃	#431	1	1	yes	characteristic
86	C₂²×C₈₃	#432	1	1	yes	characteristic
87	C₄×C₈₃	—	2	2	no	—
88	C₄×C₈₃	—	2	2	no	—
89	C₄×C₈₃	C₂×C₄⋊C₈	1	1	yes	characteristic
90	C₄×C₈₃	#423	1	1	yes	characteristic
91	C₄×C₈₃	#431	1	1	yes	characteristic
92	C₄×C₈₃	#432	1	1	yes	characteristic
93	Dic₈₃	—	4	2	no	—
94	Dic₈₃	—	4	2	no	—
95	#8781	—	83	1	no	maximal, Sylow
96	C₂×C₈₃	—	2	1	no	—
97	C₂×C₈₃	—	2	1	no	—
98	C₂×C₈₃	#1831	1	1	yes	characteristic
99	C₂×C₈₃	#1833	1	2	yes	—
100	#1726	—	83	1	no	—
101	#1564	—	83	2	no	—
102	#1281	—	83	1	no	—
103	#1281	D₈₃	1	1	yes	characteristic
104	#1274	—	83	2	no	—
105	C₈₃	#8781	1	1	yes	characteristic, complemented, Sylow
106	#431	—	83	2	no	—
107	#431	C₂×D₈₃	1	2	yes	—
108	C₂×#97	—	83	1	no	—
109	C₂×#97	—	83	1	no	—
110	C₂×#97	C₂×D₈₃	1	1	yes	characteristic
111	C₂×C₄×C₈	—	83	1	no	—
112	C₂×C₄×C₈	—	83	1	no	—
113	C₂×C₄×C₈	—	83	1	no	—
114	#376	—	83	1	no	—
115	#335	—	166	2	no	—
116	#334	—	166	2	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	C₄×D₈₃	1	1	yes	characteristic
121	C₂×C₄²	—	83	1	no	—
122	C₂×C₄²	—	83	1	no	—
123	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	166	2	no	—
126	C₂²⋊C₈	—	2	2	no	—
127	C₂²⋊C₈	—	166	2	no	—
128	#97	—	2	2	no	—
129	#97	—	83	4	no	—
130	#97	—	166	2	no	—
131	C₄×C₈	—	2	2	no	—
132	C₄×C₈	—	83	2	no	—
133	C₄×C₈	—	83	2	no	—
134	C₄×C₈	—	166	2	no	—
135	C₄×C₈	—	166	2	no	—
136	C₄×C₈	—	166	2	no	—
137	#95	—	166	2	no	—
138	C₂²×C₄	—	166	1	no	—
139	C₂²×C₄	—	166	1	no	—
140	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
141	C₂²×C₄	#13050590	1	2	yes	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	4	2	no	—
147	C₂×C₈	—	332	2	no	—
148	C₂×C₈	—	83	2	no	—
149	C₂×C₈	—	83	2	no	—
150	C₂×C₈	—	83	2	no	—
151	C₂×C₈	—	83	2	no	—
152	C₂×C₈	—	166	1	no	—
153	C₂×C₈	—	166	1	no	—
154	C₂×C₈	C₈×D₈₃	1	2	yes	—
155	C₂×C₈	#13050573	1	2	yes	—
156	C₄²	—	83	2	no	—
157	C₄²	—	83	2	no	—
158	C₄²	—	166	1	no	—
159	C₄²	—	166	1	no	—
160	C₄²	D₄×D₈₃	1	1	yes	characteristic
161	C₄²	Q₈×D₈₃	1	1	yes	characteristic
162	C₄²	#13050589	1	2	yes	—
163	C₂³	#14581749	1	1	yes	characteristic
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	332	2	no	—
166	C₂×C₄	—	332	2	no	—
167	C₂×C₄	—	166	2	no	—
168	C₂×C₄	—	166	2	no	—
169	C₂×C₄	C₄⋊C₄×D₈₃	1	1	yes	characteristic
170	C₂×C₄	#14581748	1	1	yes	characteristic
171	C₂×C₄	#14581778	1	2	yes	—
172	C₂×C₄	#14581782	1	2	yes	—
173	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
174	C₂×C₄	M₁₆×D₈₃	1	1	yes	characteristic
175	C₂×C₄	M₁₆×D₈₃	1	1	yes	characteristic
176	C₂×C₄	C₂×#13050573	1	1	yes	characteristic
177	C₈	—	2	2	no	—
178	C₈	—	2	2	no	—
179	C₈	—	8	2	no	—
180	C₈	—	83	4	no	—
181	C₈	—	664	2	no	—
182	C₈	—	166	2	no	—
183	C₂²	—	2	2	no	—
184	C₂²	#19127565	1	1	yes	characteristic
185	C₂²	#19127752	1	1	yes	characteristic
186	C₂²	#19127756	1	1	yes	characteristic
187	C₄	—	2	2	no	—
188	C₄	—	2	2	no	—
189	C₄	—	332	2	no	—
190	C₄	—	332	2	no	—
191	C₄	C₄⋊C₈×D₈₃	1	1	yes	characteristic
192	C₄	#19127723	1	1	yes	characteristic
193	C₄	#19127751	1	1	yes	characteristic
194	C₄	#19127755	1	1	yes	characteristic
195	C₂	—	2	1	no	—
196	C₂	—	2	1	no	—
197	C₂	#24907578	1	1	yes	characteristic
198	C₂	#24907584	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#24907151

C₂

yes

maximal, characteristic

#24906209

C₂

yes

maximal

#24905308

C₂

yes

maximal, characteristic

#24905291

C₂

yes

maximal

C₈₃×#1281

C₂

yes

maximal, characteristic

#19127754

C₂²

yes

—

C₂×#14581702

C₄

yes

characteristic

C₂×#14581702

C₄

yes

characteristic

C₂×C₄×C₈₃⋊C₈

C₂²

yes

characteristic

C₂×C₈×Dic₈₃

C₄

yes

characteristic

C₂×C₈×Dic₈₃

C₄

yes

characteristic

#19127547

C₂²

yes

characteristic

#19127453

—

#19127451

—

#431×C₈₃

C₂²

yes

—

C₂×#97×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

C₄⋊(C₈₃⋊C₈)

—

C₂²⋊(C₈₃⋊C₈)

—

#14581702

—

#14581702

C₈

yes

complemented

C₄×C₈₃⋊C₈

—

C₄×C₈₃⋊C₈

D₄

yes

—

C₄×C₈₃⋊C₈

Q₈

yes

—

C₈×Dic₈₃

—

C₈×Dic₈₃

—

C₈₃⋊#95

—

C₂²×C₈×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₈₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₈₃

C₂³

yes

characteristic

C₄⋊C₈×C₈₃

—

C₂²⋊C₈×C₈₃

—

#97×C₈₃

—

C₄×C₈×C₈₃

—

C₂²×Dic₈₃

—

C₂²×Dic₈₃

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

C₄⋊C₄

yes

—

C₂×C₈₃⋊C₈

C₄⋊C₄

yes

—

C₂×C₈₃⋊C₈

C₂×C₈

yes

—

C₂×C₈₃⋊C₈

M₁₆

yes

—

C₄×Dic₈₃

—

C₄×Dic₈₃

—

C₄×Dic₈₃

C₂×C₈

yes

—

C₄×Dic₈₃

M₁₆

yes

—

C₂²×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₈₃

#41

yes

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₂×C₈

yes

—

C₂×C₈×C₈₃

M₁₆

yes

—

C₄²×C₈₃

C₂×D₄

yes

characteristic

C₄²×C₈₃

C₂×Q₈

yes

characteristic

C₄²×C₈₃

#41

yes

—

C₂³×C₈₃

#117

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×C₄⋊C₄

yes

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

—

C₂×C₄×C₈₃

#125

yes

—

C₂×C₄×C₈₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₈₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₈₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₈₃

C₂×M₁₆

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₈₃⋊C₈

C₄⋊C₈

yes

—

C₂²×C₈₃

—

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₂×C₄⋊C₈

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

Dic₈₃

—

Dic₈₃

—

#8781

—

maximal, Sylow

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

#1831

yes

characteristic

C₂×C₈₃

#1833

yes

—

100

#1726

—

101

#1564

—

102

#1281

—

103

#1281

D₈₃

yes

characteristic

104

#1274

—

105

C₈₃

#8781

yes

characteristic, complemented, Sylow

106

#431

—

107

#431

C₂×D₈₃

yes

—

108

C₂×#97

—

109

C₂×#97

—

110

C₂×#97

C₂×D₈₃

yes

characteristic

111

C₂×C₄×C₈

—

112

C₂×C₄×C₈

—

113

C₂×C₄×C₈

—

114

#376

—

115

#335

—

166

—

116

#334

—

166

—

117

C₂²×C₈

—

118

C₂²×C₈

—

119

C₂²×C₈

C₄×D₈₃

yes

characteristic

120

C₂²×C₈

C₄×D₈₃

yes

characteristic

121

C₂×C₄²

—

122

C₂×C₄²

—

123

C₂×C₄²

C₂²×D₈₃

yes

characteristic

124

C₄⋊C₈

—

125

C₄⋊C₈

—

166

—

126

C₂²⋊C₈

—

127

C₂²⋊C₈

—

166

—

128

#97

—

129

#97

—

130

#97

—

166

—

131

C₄×C₈

—

132

C₄×C₈

—

133

C₄×C₈

—

134

C₄×C₈

—

166

—

135

C₄×C₈

—

166

—

136

C₄×C₈

—

166

—

137

#95

—

166

—

138

C₂²×C₄

—

166

—

139

C₂²×C₄

—

166

—

140

C₂²×C₄

C₂×C₄×D₈₃

yes

characteristic

141

C₂²×C₄

#13050590

yes

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

332

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

166

—

153

C₂×C₈

—

166

—

154

C₂×C₈

C₈×D₈₃

yes

—

155

C₂×C₈

#13050573

yes

—

156

C₄²

—

157

C₄²

—

158

C₄²

—

166

—

159

C₄²

—

166

—

160

C₄²

D₄×D₈₃

yes

characteristic

161

C₄²

Q₈×D₈₃

yes

characteristic

162

C₄²

#13050589

yes

—

163

C₂³

#14581749

yes

characteristic

164

C₂×C₄

—

165

C₂×C₄

—

332

—

166

C₂×C₄

—

332

—

167

C₂×C₄

—

166

—

168

C₂×C₄

—

166

—

169

C₂×C₄

C₄⋊C₄×D₈₃

yes

characteristic

170

C₂×C₄

#14581748

yes

characteristic

171

C₂×C₄

#14581778

yes

—

172

C₂×C₄

#14581782

yes

—

173

C₂×C₄

C₂×C₈×D₈₃

yes

characteristic

174

C₂×C₄

M₁₆×D₈₃

yes

characteristic

175

C₂×C₄

M₁₆×D₈₃

yes

characteristic

176

C₂×C₄

C₂×#13050573

yes

characteristic

177

C₈

—

178

C₈

—

179

C₈

—

180

C₈

—

181

C₈

—

664

—

182

C₈

—

166

—

183

C₂²

—

184

C₂²

#19127565

yes

characteristic

185

C₂²

#19127752

yes

characteristic

186

C₂²

#19127756

yes

characteristic

187

C₄

—

188

C₄

—

189

C₄

—

332

—

190

C₄

—

332

—

191

C₄

C₄⋊C₈×D₈₃

yes

characteristic

192

C₄

#19127723

yes

characteristic

193

C₄

#19127751

yes

characteristic

194

C₄

#19127755

yes

characteristic

195

C₂

—

196

C₂

—

197

C₂

#24907578

yes

characteristic

198

C₂

#24907584

yes

—

	subgroup	G/H
center	C₂×C₄	#14581748
commutator	C₂×C₄×C₈₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1281	C₂
socle	C₂²×C₈₃	#382
prime radical	C₂³×C₈₃	#117
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	#1281	D₈₃
hypocenter	C₈₃	#8781
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1281

subgroup

G/H

center

C₂×C₄

#14581748

commutator

C₂×C₄×C₈₃

C₂²×C₈

Frattini

C₂×C₄²

C₂²×D₈₃

Fitting

C₈₃×#1281

C₂

C₂²×C₈₃

C₂³×C₈₃

C₂×C₄²×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1281

N	H ≅ G/N	type
24. #14581702	C₈	semidirect
105. C₈₃	#8781	semidirect

H ≅ G/N

type

24. #14581702

C₈

semidirect

105. C₈₃

#8781

semidirect

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

subgroup	index
#24907151	2
#24906209	2
#24905308	2
#24905291	2
C₈₃×#1281	2
#8781	83

subgroup

index

C₈₃×#1281

Subgroups — #35298051

Conjugacy classes of subgroups

Complements

Maximal subgroups