Group #24906137 — subgroups

Subgroups — #24906137

Order 10624.

260 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 73 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	#19128249	C₂	1	1	yes	maximal, characteristic
2	#19127723	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊C₈×D₈₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#19127448	C₂	1	1	yes	maximal, characteristic, complemented
5	#19127446	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127445	C₂	1	1	yes	maximal, characteristic, complemented
7	#331×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13050589	—	2	1	no	—
9	C₂×#13050573	—	2	1	no	—
10	C₂×C₈×D₈₃	—	2	1	no	—
11	#14581792	C₂²	1	1	yes	characteristic
12	#14581779	—	2	1	no	—
13	#14581763	—	2	1	no	—
14	#14581763	C₄	1	2	yes	—
15	D₄×Dic₈₃	—	2	1	no	—
16	D₄×Dic₈₃	C₄	1	2	yes	—
17	C₄²×D₈₃	C₂²	1	1	yes	characteristic
18	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
19	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
20	Dic₈₃⋊C₈	—	2	1	no	—
21	Dic₈₃⋊C₈	—	2	1	no	—
22	#127×C₈₃	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
24	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
25	#13050589	—	4	2	no	—
26	#13050589	—	4	2	no	—
27	C₂×Dic₁₆₆	—	2	1	no	—
28	C₂×C₂²⋊D₈₃	—	2	2	no	—
29	C₂²×Dic₈₃	—	2	2	no	—
30	C₂×C₄×D₈₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×D₈₃	D₄	1	1	yes	characteristic
32	C₂×C₄×D₈₃	D₄	1	1	yes	characteristic
33	#13050573	—	4	2	no	—
34	C₂×C₈₃⋊C₈	—	2	1	no	—
35	C₂×C₈₃⋊C₈	—	2	1	no	—
36	C₈×D₈₃	—	4	2	no	—
37	C₄⋊Dic₈₃	—	2	1	no	—
38	C₄⋊Dic₈₃	C₂×C₄	1	2	yes	—
39	C₂²⋊Dic₈₃	—	2	2	no	—
40	C₂²⋊Dic₈₃	—	2	2	no	—
41	C₄×Dic₈₃	C₂×C₄	1	1	yes	characteristic
42	C₄×Dic₈₃	D₄	1	1	yes	characteristic
43	C₄×Dic₈₃	D₄	1	1	yes	characteristic
44	C₂×D₄×C₈₃	—	2	1	no	—
45	C₂×D₄×C₈₃	—	2	1	no	—
46	C₂×D₄×C₈₃	C₂×C₄	1	2	yes	—
47	C₂×C₈×C₈₃	—	2	1	no	—
48	C₂×C₈×C₈₃	—	2	1	no	—
49	C₄²×C₈₃	C₂³	1	1	yes	characteristic
50	C₂³×C₈₃	—	2	2	no	—
51	C₂²×D₈₃	C₂²⋊C₄	1	1	yes	characteristic
52	D₄×C₈₃	—	4	2	no	—
53	D₄×C₈₃	—	4	1	no	—
54	D₄×C₈₃	—	4	1	no	—
55	D₄×C₈₃	—	4	1	no	—
56	D₄×C₈₃	—	4	1	no	—
57	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
60	C₂²⋊D₈₃	—	4	4	no	—
61	C₂×Dic₈₃	—	2	2	no	—
62	C₂×Dic₈₃	—	4	4	no	—
63	C₂×Dic₈₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×Dic₈₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×Dic₈₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₄×D₈₃	—	2	1	no	—
67	C₄×D₈₃	—	2	1	no	—
68	C₄×D₈₃	—	4	1	no	—
69	C₄×D₈₃	D₈	1	2	yes	—
70	C₄×D₈₃	SD₁₆	1	2	yes	—
71	Dic₁₆₆	—	4	1	no	—
72	Dic₁₆₆	—	4	1	no	—
73	C₈×C₈₃	—	4	1	no	—
74	C₈×C₈₃	—	4	1	no	—
75	C₈₃⋊C₈	—	4	1	no	—
76	C₈₃⋊C₈	—	4	1	no	—
77	C₂²×C₈₃	—	4	2	no	—
78	C₂²×C₈₃	—	4	2	no	—
79	C₂²×C₈₃	—	4	2	no	—
80	C₂²×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×D₈₃	—	2	1	no	—
82	C₂×D₈₃	—	2	1	no	—
83	C₂×D₈₃	#102	1	2	yes	—
84	C₄×C₈₃	—	2	1	no	—
85	C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
86	C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
87	C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
88	C₄×C₈₃	#137	1	1	yes	characteristic
89	Dic₈₃	—	2	1	no	—
90	Dic₈₃	—	2	1	no	—
91	Dic₈₃	—	4	2	no	—
92	Dic₈₃	#102	1	2	yes	—
93	C₂×C₈₃	—	8	2	no	—
94	C₂×C₈₃	#413	1	1	yes	characteristic
95	C₂×C₈₃	C₂×#102	1	1	yes	characteristic
96	C₂×C₈₃	#417	1	1	yes	characteristic
97	D₈₃	—	2	2	no	—
98	#1547	—	83	1	no	maximal, Sylow
99	C₈₃	#1547	1	1	yes	characteristic, complemented, Sylow
100	#532	—	83	1	no	—
101	#423	—	83	1	no	—
102	C₂×C₄⋊C₈	—	83	1	no	—
103	#332	—	83	1	no	—
104	#332	—	83	1	no	—
105	#331	—	83	1	no	—
106	#331	D₈₃	1	1	yes	characteristic, complemented
107	C₂×#41	—	166	1	no	—
108	C₂×M₁₆	—	166	1	no	—
109	C₂²×C₈	—	166	1	no	—
110	C₄⋊Q₈	—	83	1	no	—
111	#127	C₂×D₈₃	1	1	yes	characteristic
112	#124	—	166	1	no	—
113	#121	—	83	2	no	—
114	#121	—	166	1	no	—
115	C₄×D₄	—	83	2	no	—
116	C₄×D₄	—	166	1	no	—
117	C₂×C₄²	—	83	1	no	—
118	C₄⋊C₈	—	83	1	no	—
119	C₄⋊C₈	—	83	1	no	—
120	C₄⋊C₈	—	166	1	no	—
121	C₄⋊C₈	—	166	1	no	—
122	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
123	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
124	#41	—	332	2	no	—
125	#41	—	332	2	no	—
126	C₂×Q₈	—	166	1	no	—
127	C₂×D₄	—	2	1	no	—
128	C₂×D₄	—	2	1	no	—
129	C₂×D₄	—	166	2	no	—
130	C₂×D₄	C₄×D₈₃	1	2	yes	—
131	C₂²×C₄	—	83	1	no	—
132	C₂²×C₄	—	83	1	no	—
133	C₂²×C₄	—	83	1	no	—
134	C₂²×C₄	—	166	2	no	—
135	M₁₆	—	332	2	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	332	2	no	—
139	C₂×C₈	—	166	1	no	—
140	C₂×C₈	—	166	1	no	—
141	C₄⋊C₄	—	83	2	no	—
142	C₄⋊C₄	—	166	1	no	—
143	C₂²⋊C₄	—	166	2	no	—
144	C₂²⋊C₄	—	166	2	no	—
145	C₄²	—	83	1	no	—
146	C₄²	—	83	1	no	—
147	C₄²	—	83	1	no	—
148	C₄²	C₂²×D₈₃	1	1	yes	characteristic
149	C₂³	—	2	2	no	—
150	C₂³	—	83	1	no	—
151	Q₈	—	332	1	no	—
152	Q₈	—	332	1	no	—
153	D₄	—	4	2	no	—
154	D₄	—	4	1	no	—
155	D₄	—	4	1	no	—
156	D₄	—	4	1	no	—
157	D₄	—	4	1	no	—
158	D₄	—	332	4	no	—
159	C₂×C₄	—	332	1	no	—
160	C₂×C₄	—	332	4	no	—
161	C₂×C₄	—	83	1	no	—
162	C₂×C₄	—	83	2	no	—
163	C₂×C₄	—	83	2	no	—
164	C₂×C₄	—	83	1	no	—
165	C₂×C₄	—	83	1	no	—
166	C₂×C₄	—	166	2	no	—
167	C₂×C₄	—	166	1	no	—
168	C₂×C₄	—	166	1	no	—
169	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
170	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
171	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
172	C₈	—	4	1	no	—
173	C₈	—	4	1	no	—
174	C₈	—	332	1	no	—
175	C₈	—	332	1	no	—
176	C₂²	—	4	2	no	—
177	C₂²	—	4	2	no	—
178	C₂²	—	4	2	no	—
179	C₂²	—	83	2	no	—
180	C₂²	—	166	1	no	—
181	C₂²	—	166	1	no	—
182	C₂²	C₂²⋊C₄×D₈₃	1	1	yes	characteristic
183	C₄	—	2	1	no	—
184	C₄	—	332	2	no	—
185	C₄	—	83	2	no	—
186	C₄	—	166	1	no	—
187	C₄	—	166	1	no	—
188	C₄	D₈×D₈₃	1	1	yes	characteristic
189	C₄	SD₁₆×D₈₃	1	1	yes	characteristic
190	C₄	#14581816	1	1	yes	characteristic
191	C₄	#14581821	1	1	yes	characteristic
192	C₂	—	8	2	no	—
193	C₂	—	166	2	no	—
194	C₂	#19127682	1	1	yes	characteristic
195	C₂	#102×D₈₃	1	1	yes	characteristic
196	C₂	#19127699	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19128249

C₂

yes

maximal, characteristic

#19127723

C₂

yes

maximal, characteristic, complemented

C₄⋊C₈×D₈₃

C₂

yes

maximal, characteristic, complemented

#19127448

C₂

yes

maximal, characteristic, complemented

#19127446

C₂

yes

maximal, characteristic, complemented

#19127445

C₂

yes

maximal, characteristic, complemented

#331×C₈₃

C₂

yes

maximal, characteristic, complemented

C₂×#13050589

—

C₂×#13050573

—

C₂×C₈×D₈₃

—

#14581792

C₂²

yes

—

—

C₄

yes

—

D₄×Dic₈₃

—

D₄×Dic₈₃

C₄

yes

—

C₄²×D₈₃

C₂²

yes

characteristic

C₄⋊(C₈₃⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₈₃⋊C₈)

C₂²

yes

characteristic

Dic₈₃⋊C₈

—

Dic₈₃⋊C₈

—

#127×C₈₃

C₂²

yes

characteristic

C₄⋊C₈×C₈₃

C₂²

yes

characteristic

C₄⋊C₈×C₈₃

C₂²

yes

characteristic

#13050589

—

#13050589

—

C₂×Dic₁₆₆

—

C₂×C₂²⋊D₈₃

—

C₂²×Dic₈₃

—

C₂×C₄×D₈₃

C₂×C₄

yes

characteristic

C₂×C₄×D₈₃

D₄

yes

characteristic

C₂×C₄×D₈₃

D₄

yes

characteristic

#13050573

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₈×D₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

C₂×C₄

yes

—

C₂²⋊Dic₈₃

—

C₂²⋊Dic₈₃

—

C₄×Dic₈₃

C₂×C₄

yes

characteristic

C₄×Dic₈₃

D₄

yes

characteristic

C₄×Dic₈₃

D₄

yes

characteristic

C₂×D₄×C₈₃

—

C₂×D₄×C₈₃

—

C₂×D₄×C₈₃

C₂×C₄

yes

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₄²×C₈₃

C₂³

yes

characteristic

C₂³×C₈₃

—

C₂²×D₈₃

C₂²⋊C₄

yes

characteristic

D₄×C₈₃

—

D₄×C₈₃

—

D₄×C₈₃

—

D₄×C₈₃

—

D₄×C₈₃

—

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂²⋊D₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₈₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₈₃

C₂²⋊C₄

yes

characteristic

C₄×D₈₃

—

C₄×D₈₃

—

C₄×D₈₃

—

C₄×D₈₃

D₈

yes

—

C₄×D₈₃

SD₁₆

yes

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₈₃

—

C₂×D₈₃

—

C₂×D₈₃

#102

yes

—

C₄×C₈₃

—

C₄×C₈₃

C₂×D₈

yes

characteristic

C₄×C₈₃

C₂×SD₁₆

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

Dic₈₃

—

Dic₈₃

—

Dic₈₃

—

Dic₈₃

#102

yes

—

C₂×C₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

D₈₃

—

#1547

—

maximal, Sylow

C₈₃

#1547

yes

characteristic, complemented, Sylow

100

#532

—

101

#423

—

102

C₂×C₄⋊C₈

—

103

#332

—

104

#332

—

105

#331

—

106

#331

D₈₃

yes

characteristic, complemented

107

C₂×#41

—

166

—

108

C₂×M₁₆

—

166

—

109

C₂²×C₈

—

166

—

110

C₄⋊Q₈

—

111

#127

C₂×D₈₃

yes

characteristic

112

#124

—

166

—

113

#121

—

114

#121

—

166

—

115

C₄×D₄

—

116

C₄×D₄

—

166

—

117

C₂×C₄²

—

118

C₄⋊C₈

—

119

C₄⋊C₈

—

120

C₄⋊C₈

—

166

—

121

C₄⋊C₈

—

166

—

122

C₄⋊C₈

C₂×D₈₃

yes

characteristic

123

C₄⋊C₈

C₂×D₈₃

yes

characteristic

124

#41

—

332

—

125

#41

—

332

—

126

C₂×Q₈

—

166

—

127

C₂×D₄

—

128

C₂×D₄

—

129

C₂×D₄

—

166

—

130

C₂×D₄

C₄×D₈₃

yes

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

—

134

C₂²×C₄

—

166

—

135

M₁₆

—

332

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

332

—

139

C₂×C₈

—

166

—

140

C₂×C₈

—

166

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

166

—

143

C₂²⋊C₄

—

166

—

144

C₂²⋊C₄

—

166

—

145

C₄²

—

146

C₄²

—

147

C₄²

—

148

C₄²

C₂²×D₈₃

yes

characteristic

149

C₂³

—

150

C₂³

—

151

Q₈

—

332

—

152

Q₈

—

332

—

153

D₄

—

154

D₄

—

155

D₄

—

156

D₄

—

157

D₄

—

158

D₄

—

332

—

159

C₂×C₄

—

332

—

160

C₂×C₄

—

332

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

166

—

167

C₂×C₄

—

166

—

168

C₂×C₄

—

166

—

169

C₂×C₄

C₂×C₄×D₈₃

yes

characteristic

170

C₂×C₄

D₄×D₈₃

yes

characteristic

171

C₂×C₄

D₄×D₈₃

yes

characteristic

172

C₈

—

173

C₈

—

174

C₈

—

332

—

175

C₈

—

332

—

176

C₂²

—

177

C₂²

—

178

C₂²

—

179

C₂²

—

180

C₂²

—

166

—

181

C₂²

—

166

—

182

C₂²

C₂²⋊C₄×D₈₃

yes

characteristic

183

C₄

—

184

C₄

—

332

—

185

C₄

—

186

C₄

—

166

—

187

C₄

—

166

—

188

C₄

D₈×D₈₃

yes

characteristic

189

C₄

SD₁₆×D₈₃

yes

characteristic

190

C₄

#14581816

yes

characteristic

191

C₄

#14581821

yes

characteristic

192

C₂

—

193

C₂

—

166

—

194

C₂

#19127682

yes

characteristic

195

C₂

#102×D₈₃

yes

characteristic

196

C₂

#19127699

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₈₃
commutator	C₂×C₄×C₈₃	C₂²×C₄
Frattini	C₄²	C₂²×D₈₃
Fitting	#331×C₈₃	C₂
socle	C₂²×C₈₃	C₂×C₂²⋊C₄
prime radical	#19128249	C₂
cosocle	C₄²×C₈₃	C₂³
hypercenter	#331	D₈₃
hypocenter	C₈₃	#1547
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #331

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₈₃

commutator

C₂×C₄×C₈₃

C₂²×C₄

Frattini

C₄²

C₂²×D₈₃

Fitting

#331×C₈₃

C₂

socle

C₂²×C₈₃

C₂×C₂²⋊C₄

prime radical

#19128249

C₂

cosocle

C₄²×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #331

N	H ≅ G/N	type
2. #19127723	C₂	semidirect
3. C₄⋊C₈×D₈₃	C₂	semidirect
4. #19127448	C₂	semidirect
5. #19127446	C₂	semidirect
6. #19127445	C₂	semidirect
7. #331×C₈₃	C₂	semidirect
99. C₈₃	#1547	semidirect
106. #331	D₈₃	semidirect

H ≅ G/N

type

2. #19127723

C₂

semidirect

3. C₄⋊C₈×D₈₃

C₂

semidirect

4. #19127448

C₂

semidirect

5. #19127446

C₂

semidirect

6. #19127445

C₂

semidirect

7. #331×C₈₃

C₂

semidirect

99. C₈₃

#1547

semidirect

106. #331

D₈₃

semidirect

Subgroups — #24906137

Conjugacy classes of subgroups

Complements