Group #35303111 — subgroups

Subgroups — #35303111

Order 21248.

290 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 119 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	#24908236	C₂	1	2	yes	maximal
2	C₂×#19127454	C₂	1	1	yes	maximal, characteristic
3	#24905752	C₂	1	1	yes	maximal, characteristic
4	#24905358	C₂	1	2	yes	maximal
5	C₈₃×#1566	C₂	1	1	yes	maximal, characteristic
6	#19127754	C₂²	1	1	yes	characteristic
7	C₂×#14581727	—	2	2	no	—
8	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
9	C₄×#13050574	C₂²	1	1	yes	characteristic
10	#19127465	—	2	2	no	—
11	#19127454	C₄	1	2	yes	—
12	#19127454	C₄	1	2	yes	—
13	#19127423	—	2	1	no	—
14	#19127422	—	2	1	no	—
15	#423×C₈₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
17	#334×C₈₃	—	2	1	no	—
18	#334×C₈₃	—	2	1	no	—
19	C₂×#13050574	D₄	1	2	yes	—
20	C₂²×C₈₃⋊C₈	—	2	2	no	—
21	#14581727	—	4	4	no	—
22	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
23	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
24	C₄⋊(C₈₃⋊C₈)	C₂×C₄	1	4	yes	—
25	C₂²⋊(C₈₃⋊C₈)	Q₈	1	2	yes	—
26	#14581703	—	2	1	no	—
27	C₄×C₈₃⋊C₈	—	2	1	no	—
28	C₄×C₈₃⋊C₈	—	2	1	no	—
29	C₂×M₁₆×C₈₃	—	2	2	no	—
30	C₂²×C₈×C₈₃	Q₈	1	1	yes	characteristic
31	C₂²×C₈×C₈₃	Q₈	1	1	yes	characteristic
32	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₈₃	—	2	2	no	—
34	C₄⋊C₈×C₈₃	—	2	2	no	—
35	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
38	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
39	#13050574	—	2	2	no	—
40	#13050574	—	2	2	no	—
41	C₂×C₈₃⋊C₈	—	2	4	no	—
42	C₂×C₈₃⋊C₈	—	2	2	no	—
43	C₂×C₈₃⋊C₈	—	2	2	no	—
44	C₂×C₈₃⋊C₈	—	4	2	no	—
45	C₂×C₈₃⋊C₈	—	4	2	no	—
46	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
48	C₂²×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
49	M₁₆×C₈₃	—	4	2	no	—
50	M₁₆×C₈₃	—	4	2	no	—
51	C₂×C₈×C₈₃	—	2	2	no	—
52	C₂×C₈×C₈₃	—	4	1	no	—
53	C₂×C₈×C₈₃	—	4	2	no	—
54	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
58	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
59	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
60	C₄²×C₈₃	#41	1	1	yes	characteristic
61	C₄²×C₈₃	#41	1	1	yes	characteristic
62	C₂³×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
63	C₂×C₄×C₈₃	—	2	1	no	—
64	C₂×C₄×C₈₃	—	2	1	no	—
65	C₂×C₄×C₈₃	—	4	1	no	—
66	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
68	C₂×C₄×C₈₃	C₄×Q₈	1	1	yes	characteristic
69	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
70	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₈₃	#125	1	1	yes	characteristic
72	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
73	C₂×C₄×C₈₃	#137	1	2	yes	—
74	C₈×C₈₃	—	2	1	no	—
75	C₈×C₈₃	—	2	1	no	—
76	C₈×C₈₃	—	2	1	no	—
77	C₈×C₈₃	—	8	2	no	—
78	C₈×C₈₃	#108	1	2	yes	complemented
79	C₈₃⋊C₈	—	4	2	no	—
80	C₈₃⋊C₈	—	4	4	no	—
81	C₈₃⋊C₈	—	4	2	no	—
82	C₂²×C₈₃	—	2	1	no	—
83	C₂²×C₈₃	—	2	1	no	—
84	C₂²×C₈₃	#389	1	1	yes	characteristic
85	C₂²×C₈₃	#501	1	2	yes	—
86	C₄×C₈₃	—	2	1	no	—
87	C₄×C₈₃	—	2	1	no	—
88	C₄×C₈₃	—	2	1	no	—
89	C₄×C₈₃	C₂×#108	1	1	yes	characteristic
90	C₄×C₈₃	#430	1	1	yes	characteristic
91	C₄×C₈₃	#441	1	1	yes	characteristic
92	C₄×C₈₃	#442	1	1	yes	characteristic
93	C₄×C₈₃	#468	1	2	yes	—
94	#9625	—	83	1	no	maximal, Sylow
95	C₂×C₈₃	—	2	2	no	—
96	C₂×C₈₃	#1943	1	1	yes	characteristic
97	C₂×C₈₃	#1950	1	1	yes	characteristic
98	C₂×C₈₃	#1953	1	1	yes	characteristic
99	#1950	—	83	2	no	—
100	#1566	D₈₃	1	1	yes	characteristic
101	C₂×#335	—	83	1	no	—
102	#1451	—	83	1	no	—
103	#1301	—	83	2	no	—
104	C₈₃	#9625	1	1	yes	characteristic, complemented, Sylow
105	#431	—	83	1	no	—
106	C₂×#108	—	166	2	no	—
107	#423	C₂×D₈₃	1	2	yes	—
108	C₂×C₄⋊C₈	—	83	2	no	—
109	C₄×M₁₆	—	83	1	no	—
110	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
111	#341	—	166	2	no	—
112	#335	—	83	2	no	—
113	#335	—	83	2	no	—
114	#334	—	2	1	no	—
115	#334	—	2	1	no	—
116	#322	—	166	1	no	—
117	#322	—	166	1	no	—
118	C₂×M₁₆	—	2	2	no	—
119	C₂×M₁₆	—	83	2	no	—
120	C₂²×C₈	—	166	2	no	—
121	C₂²×C₈	Dic₁₆₆	1	1	yes	characteristic
122	C₂²×C₈	Dic₁₆₆	1	1	yes	characteristic
123	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
124	#108	—	332	4	no	—
125	C₄⋊C₈	—	2	2	no	—
126	C₄⋊C₈	—	2	2	no	—
127	C₄⋊C₈	—	83	4	no	—
128	C₄⋊C₈	—	166	2	no	—
129	C₄⋊C₈	—	166	1	no	—
130	C₂²⋊C₈	—	83	2	no	—
131	#97	—	166	1	no	—
132	C₄×C₈	—	166	1	no	—
133	C₄×C₈	—	166	1	no	—
134	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
135	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
136	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
137	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
138	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
139	C₂²×C₄	Q₈×D₈₃	1	1	yes	characteristic
140	C₂²×C₄	C₂×Dic₁₆₆	1	1	yes	characteristic
141	M₁₆	—	4	2	no	—
142	M₁₆	—	4	2	no	—
143	M₁₆	—	166	2	no	—
144	M₁₆	—	166	2	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	4	1	no	—
147	C₂×C₈	—	4	2	no	—
148	C₂×C₈	—	332	2	no	—
149	C₂×C₈	—	332	2	no	—
150	C₂×C₈	—	166	2	no	—
151	C₂×C₈	—	166	2	no	—
152	C₂×C₈	—	166	4	no	—
153	C₂×C₈	Dic₈₃⋊C₄	1	1	yes	characteristic
154	C₂×C₈	Dic₈₃⋊C₄	1	1	yes	characteristic
155	C₂×C₈	Dic₈₃⋊C₄	1	1	yes	characteristic
156	C₂×C₈	Dic₈₃⋊C₄	1	1	yes	characteristic
157	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
158	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
159	C₄²	#13050589	1	1	yes	characteristic
160	C₄²	#13050591	1	1	yes	characteristic
161	C₂³	#14581791	1	1	yes	characteristic
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	4	1	no	—
165	C₂×C₄	C₂×Dic₈₃⋊C₄	1	1	yes	characteristic
166	C₂×C₄	#14581754	1	1	yes	characteristic
167	C₂×C₄	#14581757	1	1	yes	characteristic
168	C₂×C₄	#14581763	1	1	yes	characteristic
169	C₂×C₄	#14581768	1	1	yes	characteristic
170	C₂×C₄	#14581783	1	1	yes	characteristic
171	C₂×C₄	#14581819	1	2	yes	—
172	C₂×C₄	#14581821	1	2	yes	—
173	C₈	—	2	1	no	—
174	C₈	—	2	1	no	—
175	C₈	—	2	1	no	—
176	C₈	—	8	2	no	—
177	C₈	—	332	2	no	—
178	C₈	—	332	4	no	—
179	C₈	—	332	2	no	—
180	C₈	#14581727	1	2	yes	complemented
181	C₂²	—	2	1	no	—
182	C₂²	—	2	1	no	—
183	C₂²	#19127597	1	1	yes	characteristic
184	C₂²	#19128108	1	2	yes	—
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	2	1	no	—
188	C₄	C₂×#14581727	1	1	yes	characteristic
189	C₄	#19127749	1	1	yes	characteristic
190	C₄	#19127807	1	1	yes	characteristic
191	C₄	#19127812	1	1	yes	characteristic
192	C₄	#19127944	1	2	yes	—
193	C₂	—	2	2	no	—
194	C₂	#24908199	1	1	yes	characteristic
195	C₂	#24908236	1	1	yes	characteristic
196	C₂	#24908251	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24908236

C₂

yes

maximal

C₂×#19127454

C₂

yes

maximal, characteristic

#24905752

C₂

yes

maximal, characteristic

#24905358

C₂

yes

maximal

C₈₃×#1566

C₂

yes

maximal, characteristic

#19127754

C₂²

yes

characteristic

C₂×#14581727

—

C₂×C₄⋊(C₈₃⋊C₈)

C₂²

yes

—

C₄×#13050574

C₂²

yes

characteristic

#19127465

—

#19127454

C₄

yes

—

#19127454

C₄

yes

—

#19127423

—

#19127422

—

#423×C₈₃

C₂²

yes

—

C₂×C₄×C₈×C₈₃

C₂²

yes

characteristic

#334×C₈₃

—

#334×C₈₃

—

C₂×#13050574

D₄

yes

—

C₂²×C₈₃⋊C₈

—

#14581727

—

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

C₂×C₄

yes

—

C₂²⋊(C₈₃⋊C₈)

Q₈

yes

—

#14581703

—

C₄×C₈₃⋊C₈

—

C₄×C₈₃⋊C₈

—

C₂×M₁₆×C₈₃

—

C₂²×C₈×C₈₃

Q₈

yes

characteristic

C₂²×C₈×C₈₃

Q₈

yes

characteristic

C₂×C₄²×C₈₃

C₂³

yes

characteristic

C₄⋊C₈×C₈₃

—

C₄⋊C₈×C₈₃

—

C₄×C₈×C₈₃

C₂×C₄

yes

characteristic

C₄×C₈×C₈₃

C₂×C₄

yes

characteristic

C₄×C₈×C₈₃

D₄

yes

characteristic

C₄×C₈×C₈₃

D₄

yes

characteristic

#13050574

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₃

C₂×Q₈

yes

characteristic

C₂²×C₄×C₈₃

C₂×Q₈

yes

characteristic

M₁₆×C₈₃

—

M₁₆×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₈₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₈₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₈₃

C₄⋊C₄

yes

characteristic

C₄²×C₈₃

C₂²×C₄

yes

characteristic

C₄²×C₈₃

C₂×D₄

yes

C₄²×C₈₃

yes

C₄²×C₈₃

yes

C₂³×C₈₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₄×D₄

yes

characteristic

C₂×C₄×C₈₃

C₄×Q₈

yes

characteristic

C₂×C₄×C₈₃

#121

yes

characteristic

C₂×C₄×C₈₃

C₂²⋊Q₈

yes

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

—

C₂×C₄×C₈₃

#137

yes

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

#108

yes

complemented

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

#389

yes

characteristic

C₂²×C₈₃

#501

yes

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

—

#9625

—

maximal, Sylow

C₂×C₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

100

#1566

D₈₃

yes

characteristic

101

C₂×#335

—

102

#1451

—

103

#1301

—

104

C₈₃

#9625

yes

characteristic, complemented, Sylow

105

#431

—

106

C₂×#108

—

166

—

107

#423

C₂×D₈₃

yes

—

108

C₂×C₄⋊C₈

—

109

C₄×M₁₆

—

110

C₂×C₄×C₈

C₂×D₈₃

yes

characteristic

111

#341

—

166

—

112

#335

—

113

#335

—

114

#334

—

115

#334

—

116

#322

—

166

—

117

#322

—

166

—

118

C₂×M₁₆

—

119

C₂×M₁₆

—

120

C₂²×C₈

—

166

—

121

C₂²×C₈

Dic₁₆₆

yes

characteristic

122

C₂²×C₈

Dic₁₆₆

yes

characteristic

123

C₂×C₄²

C₂²×D₈₃

yes

characteristic

124

#108

—

332

—

125

C₄⋊C₈

—

126

C₄⋊C₈

—

127

C₄⋊C₈

—

128

C₄⋊C₈

—

166

—

129

C₄⋊C₈

—

166

—

130

C₂²⋊C₈

—

131

#97

—

166

—

132

C₄×C₈

—

166

—

133

C₄×C₈

—

166

—

134

C₄×C₈

C₄×D₈₃

yes

characteristic

135

C₄×C₈

C₄×D₈₃

yes

characteristic

136

C₄×C₈

C₂²⋊D₈₃

yes

characteristic

137

C₄×C₈

C₂²⋊D₈₃

yes

characteristic

138

C₂²×C₄

D₄×D₈₃

yes

characteristic

139

C₂²×C₄

Q₈×D₈₃

yes

characteristic

140

C₂²×C₄

C₂×Dic₁₆₆

yes

characteristic

141

M₁₆

—

142

M₁₆

—

143

M₁₆

—

166

—

144

M₁₆

—

166

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

332

—

149

C₂×C₈

—

332

—

150

C₂×C₈

—

166

—

151

C₂×C₈

—

166

—

152

C₂×C₈

—

166

—

153

C₂×C₈

Dic₈₃⋊C₄

yes

characteristic

154

C₂×C₈

Dic₈₃⋊C₄

yes

characteristic

155

C₂×C₈

Dic₈₃⋊C₄

yes

characteristic

156

C₂×C₈

Dic₈₃⋊C₄

yes

characteristic

157

C₄²

C₂×C₄×D₈₃

yes

characteristic

158

C₄²

C₂×C₂²⋊D₈₃

yes

characteristic

159

C₄²

#13050589

yes

characteristic

160

C₄²

#13050591

yes

characteristic

161

C₂³

#14581791

yes

characteristic

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

C₂×Dic₈₃⋊C₄

yes

characteristic

166

C₂×C₄

#14581754

yes

characteristic

167

C₂×C₄

#14581757

yes

characteristic

168

C₂×C₄

#14581763

yes

characteristic

169

C₂×C₄

#14581768

yes

characteristic

170

C₂×C₄

#14581783

yes

characteristic

171

C₂×C₄

#14581819

yes

—

172

C₂×C₄

#14581821

yes

—

173

C₈

—

174

C₈

—

175

C₈

—

176

C₈

—

177

C₈

—

332

—

178

C₈

—

332

—

179

C₈

—

332

—

180

C₈

#14581727

yes

complemented

181

C₂²

—

182

C₂²

—

183

C₂²

#19127597

yes

characteristic

184

C₂²

#19128108

yes

—

185

C₄

—

186

C₄

—

187

C₄

—

188

C₄

C₂×#14581727

yes

characteristic

189

C₄

#19127749

yes

characteristic

190

C₄

#19127807

yes

characteristic

191

C₄

#19127812

yes

characteristic

192

C₄

#19127944

yes

—

193

C₂

—

194

C₂

#24908199

yes

characteristic

195

C₂

#24908236

yes

characteristic

196

C₂

#24908251

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14581763
commutator	C₄²×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1566	C₂
socle	C₂²×C₈₃	#389
prime radical	C₂³×C₈₃	C₄⋊Q₈
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	#1566	D₈₃
hypocenter	C₈₃	#9625
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

subgroup

G/H

center

C₂×C₄

#14581763

commutator

C₄²×C₈₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₃

Fitting

C₈₃×#1566

C₂

socle

C₂²×C₈₃

#389

prime radical

C₂³×C₈₃

C₄⋊Q₈

cosocle

C₂×C₄²×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

N	H ≅ G/N	type
78. C₈×C₈₃	#108	semidirect
104. C₈₃	#9625	semidirect
180. C₈	#14581727	semidirect

H ≅ G/N

type

78. C₈×C₈₃

#108

semidirect

104. C₈₃

#9625

semidirect

180. C₈

#14581727

semidirect

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

subgroup	index
#24908236	2
C₂×#19127454	2
#24905752	2
#24905358	2
C₈₃×#1566	2
#9625	83

subgroup

index

C₈₃×#1566

Subgroups — #35303111

Conjugacy classes of subgroups

Complements

Maximal subgroups