Group #25015168 — subgroups

Subgroups — #25015168

Order 11008.

398 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 159 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	C₂×#14583304	C₂	1	1	yes	maximal, characteristic
2	C₄×#13063259	C₂	1	1	yes	maximal, characteristic
3	#19132687	C₂	1	4	yes	maximal
4	C₂×C₄₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂×#13063325	—	2	1	no	—
6	#14583304	—	2	2	no	—
7	#14583304	C₄	1	4	yes	complemented
8	C₂×#13063259	—	2	1	no	—
9	C₂×#13063259	C₂²	1	2	yes	—
10	C₂×#13063259	C₂²	1	2	yes	—
11	C₄×C₄⋊Dic₄₃	—	2	1	no	—
12	#14582662	—	2	1	no	—
13	C₂²×C₁₆×C₄₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
15	C₄₃×C₄⋊C₁₆	—	2	2	no	—
16	C₂²×Dic₈₆	—	4	1	no	—
17	#13063325	—	2	4	no	—
18	#13063325	—	4	2	no	—
19	C₂×C₄⋊Dic₄₃	—	2	2	no	—
20	C₂×C₄⋊Dic₄₃	—	2	2	no	—
21	C₂×C₄⋊Dic₄₃	—	4	1	no	—
22	C₂×C₄×Dic₄₃	—	4	1	no	—
23	#13063259	—	2	4	no	—
24	#13063259	—	2	2	no	—
25	#13063259	—	2	2	no	—
26	#13063259	C₂×C₄	1	4	yes	—
27	#13063259	Q₈	1	4	yes	complemented
28	C₄₃⋊#95	—	4	1	no	—
29	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
32	C₂×C₁₆×C₄₃	—	2	2	no	—
33	C₂×C₁₆×C₄₃	—	2	4	no	—
34	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₄₃	D₄	1	2	yes	—
36	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
37	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
38	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
39	C₂²×Dic₄₃	—	4	2	no	—
40	C₂²×Dic₄₃	—	4	2	no	—
41	C₂×Dic₈₆	—	4	4	no	—
42	C₂×Dic₈₆	—	8	2	no	—
43	C₂×Dic₈₆	—	8	2	no	—
44	C₂×C₈×C₄₃	—	2	2	no	—
45	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
46	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
49	C₂×C₈×C₄₃	#41	1	2	yes	—
50	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
51	C₄²×C₄₃	SD₁₆	1	2	yes	—
52	C₄⋊Dic₄₃	—	2	4	no	—
53	C₄⋊Dic₄₃	—	2	4	no	—
54	C₄⋊Dic₄₃	—	4	4	no	—
55	C₄⋊Dic₄₃	—	4	2	no	—
56	C₄⋊Dic₄₃	—	4	2	no	—
57	C₄×Dic₄₃	—	8	2	no	—
58	C₁₆×C₄₃	—	4	2	no	—
59	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₄₃	—	2	2	no	—
61	C₂×C₄×C₄₃	#102	1	2	yes	—
62	C₂×C₄×C₄₃	#102	1	2	yes	—
63	C₂×C₄×C₄₃	D₁₆	1	2	yes	—
64	C₂×C₄×C₄₃	SD₃₂	1	2	yes	—
65	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
67	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
68	C₂×C₄×C₄₃	#137	1	2	yes	—
69	C₂×Dic₄₃	—	4	4	no	—
70	C₂×Dic₄₃	—	4	4	no	—
71	C₂×Dic₄₃	—	8	2	no	—
72	C₂×Dic₄₃	—	8	2	no	—
73	C₂×Dic₄₃	—	8	2	no	—
74	C₂×Dic₄₃	—	8	2	no	—
75	Dic₈₆	—	8	8	no	—
76	C₈×C₄₃	—	2	2	no	—
77	C₈×C₄₃	C₄×Q₈	1	2	yes	—
78	C₈×C₄₃	#124	1	1	yes	characteristic
79	C₈×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
80	#13275	—	43	1	no	maximal, Sylow
81	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
82	C₂²×C₄₃	#474	1	2	yes	—
83	C₂²×C₄₃	C₂×D₁₆	1	1	yes	characteristic
84	C₂²×C₄₃	C₂×SD₃₂	1	1	yes	characteristic
85	C₂²×C₄₃	#510	1	2	yes	—
86	C₄×C₄₃	—	2	2	no	—
87	C₄×C₄₃	#357	1	4	yes	—
88	C₄×C₄₃	#391	1	1	yes	characteristic
89	C₄×C₄₃	#419	1	1	yes	characteristic
90	C₄×C₄₃	#475	1	2	yes	—
91	Dic₄₃	—	8	4	no	—
92	Dic₄₃	—	8	4	no	—
93	C₂×#500	—	43	1	no	—
94	C₂×C₄⋊C₁₆	D₄₃	1	1	yes	characteristic
95	C₄×#107	—	43	1	no	—
96	#1380	—	43	4	no	—
97	C₂×C₄₃	#1988	1	1	yes	characteristic
98	C₂×C₄₃	C₂×#357	1	1	yes	characteristic
99	C₂×C₄₃	#2141	1	1	yes	characteristic
100	C₂×C₄₃	#2224	1	2	yes	—
101	C₂×C₄₃	#2225	1	2	yes	—
102	C₂×C₄⋊Q₈	—	86	1	no	—
103	C₂²×C₁₆	C₂×D₄₃	1	2	yes	—
104	#500	—	43	4	no	—
105	#500	—	86	2	no	—
106	C₂×#107	—	43	2	no	—
107	C₂×#107	—	43	2	no	—
108	C₂×#107	—	86	1	no	—
109	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
110	C₄×C₄⋊C₄	—	86	1	no	—
111	C₄⋊C₁₆	—	2	2	no	—
112	#340	—	86	1	no	—
113	C₄₃	#13275	1	1	yes	characteristic, complemented, Sylow
114	C₂²×Q₈	—	172	1	no	—
115	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
116	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
117	C₄⋊Q₈	—	86	4	no	—
118	C₄⋊Q₈	—	172	2	no	—
119	C₂×C₄⋊C₄	—	86	2	no	—
120	C₂×C₄⋊C₄	—	86	2	no	—
121	C₂×C₄⋊C₄	—	172	1	no	—
122	C₂×C₄²	—	172	1	no	—
123	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
124	C₂×C₁₆	—	2	4	no	—
125	C₂×C₁₆	—	2	2	no	—
126	#107	—	43	4	no	—
127	#107	—	43	4	no	—
128	#107	—	86	4	no	—
129	#107	—	86	2	no	—
130	#107	—	86	2	no	—
131	C₄×C₈	C₄×D₄₃	1	2	yes	—
132	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
133	#95	—	172	1	no	—
134	C₂×Q₈	—	344	2	no	—
135	C₂×Q₈	—	344	2	no	—
136	C₂×Q₈	—	172	4	no	—
137	C₂²×C₄	—	172	2	no	—
138	C₂²×C₄	—	172	2	no	—
139	C₂²×C₄	D₃₄₄	1	1	yes	characteristic
140	C₂²×C₄	D₃₄₄	1	1	yes	characteristic
141	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	(C₂×D₄₃)⋊C₄	1	2	yes	—
144	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
145	C₂×C₈	#10621900	1	2	yes	—
146	C₂×C₈	Q₈×D₄₃	1	2	yes	—
147	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
148	C₄⋊C₄	—	86	4	no	—
149	C₄⋊C₄	—	86	4	no	—
150	C₄⋊C₄	—	172	4	no	—
151	C₄⋊C₄	—	172	2	no	—
152	C₄⋊C₄	—	172	2	no	—
153	C₄²	—	344	2	no	—
154	C₄²	#10621873	1	2	yes	—
155	C₄²	(C₂×D₄₃)⋊C₄	1	2	yes	—
156	C₁₆	—	4	2	no	—
157	C₂³	C₂×D₃₄₄	1	1	yes	characteristic
158	Q₈	—	344	8	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	344	2	no	—
161	C₂×C₄	—	344	2	no	—
162	C₂×C₄	—	344	2	no	—
163	C₂×C₄	—	344	2	no	—
164	C₂×C₄	—	172	4	no	—
165	C₂×C₄	—	172	4	no	—
166	C₂×C₄	#13063247	1	2	yes	—
167	C₂×C₄	#13063247	1	2	yes	—
168	C₂×C₄	D₆₈₈	1	2	yes	—
169	C₂×C₄	#13063270	1	2	yes	—
170	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
171	C₂×C₄	#13063305	1	2	yes	—
172	C₂×C₄	C₂×#10621873	1	1	yes	characteristic
173	C₂×C₄	#13063358	1	2	yes	—
174	C₈	—	2	2	no	—
175	C₈	#13063290	1	2	yes	—
176	C₈	#13063312	1	1	yes	characteristic
177	C₈	#13063323	1	1	yes	characteristic
178	C₂²	C₂×#13063247	1	1	yes	characteristic
179	C₂²	#14583180	1	2	yes	—
180	C₂²	C₂×D₆₈₈	1	1	yes	characteristic
181	C₂²	C₂×#13063270	1	1	yes	characteristic
182	C₂²	#14583342	1	2	yes	—
183	C₄	—	2	2	no	—
184	C₄	—	344	4	no	—
185	C₄	—	344	4	no	—
186	C₄	#14582701	1	4	yes	—
187	C₄	#14582804	1	1	yes	characteristic
188	C₄	#14582909	1	1	yes	characteristic
189	C₄	#14583185	1	2	yes	—
190	C₂	#19135546	1	1	yes	characteristic
191	C₂	C₂×#14582701	1	1	yes	characteristic
192	C₂	#19136316	1	1	yes	characteristic
193	C₂	#19136733	1	2	yes	—
194	C₂	#19136738	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14583304

C₂

yes

maximal, characteristic

C₄×#13063259

C₂

yes

maximal, characteristic

#19132687

C₂

yes

maximal

C₂×C₄₃×C₄⋊C₁₆

C₂

yes

maximal, characteristic

C₂×#13063325

—

#14583304

—

#14583304

C₄

yes

complemented

C₂×#13063259

—

C₂×#13063259

C₂²

yes

—

C₂×#13063259

C₂²

yes

—

C₄×C₄⋊Dic₄₃

—

#14582662

—

C₂²×C₁₆×C₄₃

C₂²

yes

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₄₃×C₄⋊C₁₆

—

C₂²×Dic₈₆

—

#13063325

—

#13063325

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄×Dic₄₃

—

—

—

—

C₂×C₄

yes

—

#13063259

Q₈

yes

complemented

C₄₃⋊#95

—

C₂²×C₈×C₄₃

D₄

yes

characteristic

C₂²×C₈×C₄₃

C₂³

yes

characteristic

C₂×C₄²×C₄₃

D₄

yes

characteristic

C₂×C₁₆×C₄₃

—

C₂×C₁₆×C₄₃

—

C₄×C₈×C₄₃

C₂×C₄

yes

—

C₄×C₈×C₄₃

D₄

yes

—

C₂²×C₄×C₄₃

D₈

yes

characteristic

C₂²×C₄×C₄₃

D₈

yes

characteristic

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂²⋊C₄

yes

—

C₂×C₈×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₃

C₂×D₄

yes

characteristic

C₂×C₈×C₄₃

C₂×Q₈

yes

—

C₂×C₈×C₄₃

#41

yes

—

C₄²×C₄₃

C₂²⋊C₄

yes

—

C₄²×C₄₃

SD₁₆

yes

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄×Dic₄₃

—

C₁₆×C₄₃

—

C₂³×C₄₃

C₂×D₈

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#102

yes

—

C₂×C₄×C₄₃

#102

yes

—

C₂×C₄×C₄₃

D₁₆

yes

—

C₂×C₄×C₄₃

SD₃₂

yes

—

C₂×C₄×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₄₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₄₃

#137

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

C₄×Q₈

yes

—

C₈×C₄₃

#124

yes

characteristic

C₈×C₄₃

C₄⋊Q₈

yes

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

—

C₂²×C₄₃

C₂×D₁₆

yes

characteristic

C₂²×C₄₃

C₂×SD₃₂

yes

characteristic

C₂²×C₄₃

#510

yes

—

C₄×C₄₃

—

C₄×C₄₃

#357

yes

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

—

Dic₄₃

—

Dic₄₃

—

C₂×#500

—

C₂×C₄⋊C₁₆

D₄₃

yes

characteristic

C₄×#107

—

#1380

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

100

C₂×C₄₃

#2224

yes

—

101

C₂×C₄₃

#2225

yes

—

102

C₂×C₄⋊Q₈

—

103

C₂²×C₁₆

C₂×D₄₃

yes

—

104

#500

—

105

#500

—

106

C₂×#107

—

107

C₂×#107

—

108

C₂×#107

—

109

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

110

C₄×C₄⋊C₄

—

111

C₄⋊C₁₆

—

112

#340

—

113

C₄₃

#13275

yes

characteristic, complemented, Sylow

114

C₂²×Q₈

—

172

—

115

C₂²×C₈

D₁₇₂

yes

characteristic

116

C₂²×C₈

C₂²×D₄₃

yes

characteristic

117

C₄⋊Q₈

—

118

C₄⋊Q₈

—

172

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

—

172

—

122

C₂×C₄²

—

172

—

123

C₂×C₄²

D₁₇₂

yes

characteristic

124

C₂×C₁₆

—

125

C₂×C₁₆

—

126

#107

—

127

#107

—

128

#107

—

129

#107

—

130

#107

—

131

C₄×C₈

C₄×D₄₃

yes

—

132

C₄×C₈

C₂²⋊D₄₃

yes

—

133

#95

—

172

—

134

C₂×Q₈

—

344

—

135

C₂×Q₈

—

344

—

136

C₂×Q₈

—

172

—

137

C₂²×C₄

—

172

—

138

C₂²×C₄

—

172

—

139

C₂²×C₄

D₃₄₄

yes

characteristic

140

C₂²×C₄

D₃₄₄

yes

characteristic

141

C₂²×C₄

C₂×D₁₇₂

yes

characteristic

142

C₂×C₈

—

143

C₂×C₈

(C₂×D₄₃)⋊C₄

yes

—

144

C₂×C₈

C₂×C₄×D₄₃

yes

characteristic

145

C₂×C₈

#10621900

yes

—

146

C₂×C₈

Q₈×D₄₃

yes

—

147

C₂×C₈

C₂×C₂²⋊D₄₃

yes

characteristic

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

172

—

151

C₄⋊C₄

—

172

—

152

C₄⋊C₄

—

172

—

153

C₄²

—

344

—

154

C₄²

#10621873

yes

—

155

C₄²

(C₂×D₄₃)⋊C₄

yes

—

156

C₁₆

—

157

C₂³

C₂×D₃₄₄

yes

characteristic

158

Q₈

—

344

—

159

C₂×C₄

—

160

C₂×C₄

—

344

—

161

C₂×C₄

—

344

—

162

C₂×C₄

—

344

—

163

C₂×C₄

—

344

—

164

C₂×C₄

—

172

—

165

C₂×C₄

—

172

—

166

C₂×C₄

#13063247

yes

—

167

C₂×C₄

#13063247

yes

—

168

C₂×C₄

D₆₈₈

yes

—

169

C₂×C₄

#13063270

yes

—

170

C₂×C₄

C₂×(C₂×D₄₃)⋊C₄

yes

characteristic

171

C₂×C₄

#13063305

yes

—

172

C₂×C₄

C₂×#10621873

yes

characteristic

173

C₂×C₄

#13063358

yes

—

174

C₈

—

175

C₈

#13063290

yes

—

176

C₈

#13063312

yes

characteristic

177

C₈

#13063323

yes

characteristic

178

C₂²

C₂×#13063247

yes

characteristic

179

C₂²

#14583180

yes

—

180

C₂²

C₂×D₆₈₈

yes

characteristic

181

C₂²

C₂×#13063270

yes

characteristic

182

C₂²

#14583342

yes

—

183

C₄

—

184

C₄

—

344

—

185

C₄

—

344

—

186

C₄

#14582701

yes

—

187

C₄

#14582804

yes

characteristic

188

C₄

#14582909

yes

characteristic

189

C₄

#14583185

yes

—

190

C₂

#19135546

yes

characteristic

191

C₂

C₂×#14582701

yes

characteristic

192

C₂

#19136316

yes

characteristic

193

C₂

#19136733

yes

—

194

C₂

#19136738

yes

—

	subgroup	G/H
center	C₂³	C₂×D₃₄₄
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₂×C₄₃×C₄⋊C₁₆	C₂
socle, prime radical	C₂³×C₄₃	C₂×D₈
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₄₃
hypocenter	C₄₃	#13275
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

subgroup

G/H

center

C₂³

C₂×D₃₄₄

commutator

C₂×C₈×C₄₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₄₃

Fitting

C₂×C₄₃×C₄⋊C₁₆

C₂

socle, prime radical

C₂³×C₄₃

C₂×D₈

cosocle

C₂²×C₈×C₄₃

C₂³

hypercenter

C₂×C₄⋊C₁₆

D₄₃

hypocenter

C₄₃

#13275

LCS

G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

N	H ≅ G/N	type
7. #14583304	C₄	semidirect
27. #13063259	Q₈	semidirect
113. C₄₃	#13275	semidirect

H ≅ G/N

type

7. #14583304

C₄

semidirect

27. #13063259

Q₈

semidirect

113. C₄₃

#13275

semidirect

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

subgroup	index
C₂×#14583304	2
C₄×#13063259	2
#19132687	2
C₂×C₄₃×C₄⋊C₁₆	2
#13275	43

subgroup

index

C₂×C₄₃×C₄⋊C₁₆

Subgroups — #25015168

Conjugacy classes of subgroups

Complements

Maximal subgroups