Group #25252021 — subgroups

Subgroups — #25252021

Order 11008.

826 conjugacy classes of subgroups; only the 219 normal subgroups are shown.

normal subgroups only

This group has 826 conjugacy classes of subgroups — too many to draw. Only its 219 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×#14583331	C₂	1	4	yes	maximal, complemented
2	#19145023	C₂	1	2	yes	maximal, complemented
3	#19136866	C₂	1	4	yes	maximal, complemented
4	#19136861	C₂	1	4	yes	maximal, complemented
5	C₂×C₄₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13063347	—	2	2	no	—
7	C₂×#13063347	C₂²	1	4	yes	—
8	#14583449	—	2	2	no	—
9	#14583331	—	2	16	no	—
10	C₂×Dic₃₄₄	C₂²	1	4	yes	—
11	C₂×#13063270	C₂²	1	8	yes	—
12	C₂×D₆₈₈	C₂²	1	4	yes	—
13	#14583283	—	2	2	no	—
14	#14583283	C₂²	1	4	yes	—
15	#14583279	C₂²	1	2	yes	—
16	#14583276	C₂²	1	2	yes	—
17	#14583272	—	2	2	no	—
18	C₂²×C₁₆×C₄₃	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
20	C₄₃×#346	C₂²	1	4	yes	—
21	C₂×#10621898	—	2	4	no	—
22	#13063347	—	2	16	no	—
23	#13063347	—	4	8	no	—
24	C₂×Dic₁₇₂	—	2	2	no	—
25	C₂×Dic₁₇₂	C₂³	1	4	yes	—
26	C₂×#10621873	—	2	4	no	—
27	C₂×#10621873	—	2	4	no	—
28	C₂×D₃₄₄	—	2	2	no	—
29	C₂×D₃₄₄	C₂³	1	4	yes	—
30	#13063325	—	2	2	no	—
31	#13063322	—	2	2	no	—
32	#13063313	—	2	4	no	—
33	#13063299	—	4	4	no	—
34	C₄×D₁₇₂	—	4	4	no	—
35	Dic₃₄₄	—	2	8	no	—
36	#13063270	—	2	16	no	—
37	D₆₈₈	—	2	8	no	—
38	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
41	C₂×C₁₆×C₄₃	D₄	1	8	yes	complemented
42	C₂×C₁₆×C₄₃	C₂³	1	4	yes	—
43	C₄×C₈×C₄₃	D₄	1	2	yes	—
44	C₄×C₈×C₄₃	C₂³	1	2	yes	—
45	C₂²×C₄×C₄₃	D₈	1	2	yes	—
46	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₂²⋊D₄₃	—	4	4	no	—
48	#10621898	—	4	8	no	—
49	#10621898	—	4	8	no	—
50	C₂×D₁₇₂	—	2	4	no	—
51	C₂×D₁₇₂	—	4	4	no	—
52	C₂×C₄×D₄₃	—	4	4	no	—
53	C₂×Dic₈₆	—	2	4	no	—
54	C₂×C₈×C₄₃	D₈	1	4	yes	—
55	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₄₃	C₂×D₄	1	4	yes	—
57	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
58	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
59	C₄²×C₄₃	D₈	1	2	yes	—
60	C₄²×C₄₃	C₂×D₄	1	2	yes	—
61	(C₂×D₄₃)⋊C₄	—	4	8	no	—
62	C₄⋊Dic₄₃	—	4	4	no	—
63	Dic₁₇₂	—	2	8	no	—
64	Dic₁₇₂	—	4	4	no	—
65	D₃₄₄	—	2	8	no	—
66	D₃₄₄	—	4	4	no	—
67	#10621873	—	4	8	no	—
68	#10621873	—	4	8	no	—
69	C₁₆×C₄₃	C₂×D₄	1	8	yes	—
70	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
71	C₂²×D₄₃	—	4	4	no	—
72	C₂×C₄×C₄₃	—	4	2	no	—
73	C₂×C₄×C₄₃	#127	1	2	yes	—
74	C₂×C₄×C₄₃	C₂×D₈	1	4	yes	—
75	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
76	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
77	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
78	C₂²⋊D₄₃	—	8	8	no	—
79	C₂×Dic₄₃	—	4	4	no	—
80	D₁₇₂	—	4	4	no	—
81	D₁₇₂	—	4	4	no	—
82	D₁₇₂	—	8	8	no	—
83	C₄×D₄₃	—	8	8	no	—
84	Dic₈₆	—	4	4	no	—
85	Dic₈₆	—	4	4	no	—
86	C₈×C₄₃	#127	1	2	yes	—
87	C₈×C₄₃	C₂×D₈	1	4	yes	—
88	C₈×C₄₃	C₂²×D₄	1	2	yes	—
89	#33716	—	43	1	no	maximal, Sylow
90	C₂²×C₄₃	—	2	2	no	—
91	C₂²×C₄₃	#493	1	2	yes	—
92	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
93	C₂×D₄₃	—	8	4	no	—
94	C₂×D₄₃	—	8	8	no	—
95	C₄×C₄₃	—	2	4	no	—
96	C₄×C₄₃	#493	1	2	yes	—
97	C₄×C₄₃	C₂×#127	1	1	yes	characteristic
98	C₄×C₄₃	C₂²×D₈	1	1	yes	characteristic
99	Dic₄₃	—	8	4	no	—
100	C₂×#508	—	43	4	no	—
101	#3147	—	43	2	no	—
102	#2251	—	43	4	no	—
103	#2250	—	43	4	no	—
104	C₂×#346	D₄₃	1	1	yes	characteristic, complemented
105	C₂×C₄₃	—	2	2	no	—
106	C₂×C₄₃	C₂×#493	1	1	yes	characteristic
107	C₂×C₄₃	#3414	1	2	yes	—
108	D₄₃	—	16	4	no	—
109	C₂×#135	—	43	4	no	—
110	C₂×#135	—	86	2	no	—
111	#532	—	86	2	no	—
112	#508	—	86	16	no	—
113	C₂×Q₃₂	—	43	4	no	—
114	C₂×SD₃₂	—	43	8	no	—
115	C₂×D₁₆	—	43	4	no	—
116	C₂²×C₁₆	C₂×D₄₃	1	2	yes	—
117	#495	—	43	4	no	—
118	#495	—	86	2	no	—
119	#494	—	43	2	no	—
120	#493	—	43	2	no	—
121	#492	—	86	2	no	—
122	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
123	#346	C₂×D₄₃	1	4	yes	—
124	C₄₃	#33716	1	1	yes	characteristic, complemented, Sylow
125	C₂×#41	—	86	4	no	—
126	#135	—	86	16	no	—
127	#135	—	172	8	no	—
128	C₂×Q₁₆	—	43	4	no	—
129	C₂×Q₁₆	—	86	2	no	—
130	C₂×SD₁₆	—	86	4	no	—
131	C₂×SD₁₆	—	86	4	no	—
132	C₂×D₈	—	43	4	no	—
133	C₂×D₈	—	86	2	no	—
134	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
136	C₄⋊Q₈	—	86	2	no	—
137	#127	—	86	2	no	—
138	#124	—	86	4	no	—
139	#121	—	172	4	no	—
140	C₄×D₄	—	172	4	no	—
141	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
142	Q₃₂	—	86	8	no	—
143	SD₃₂	—	86	16	no	—
144	D₁₆	—	86	8	no	—
145	C₂×C₁₆	D₁₇₂	1	8	yes	complemented
146	C₂×C₁₆	C₂²×D₄₃	1	4	yes	—
147	C₄×C₈	D₁₇₂	1	2	yes	—
148	C₄×C₈	C₂²×D₄₃	1	2	yes	—
149	#41	—	172	8	no	—
150	#41	—	172	8	no	—
151	C₂×Q₈	—	86	4	no	—
152	C₂×D₄	—	86	4	no	—
153	C₂×D₄	—	172	4	no	—
154	C₂×D₄	—	172	4	no	—
155	C₂²×C₄	—	172	4	no	—
156	C₂²×C₄	D₃₄₄	1	2	yes	—
157	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
158	Q₁₆	—	86	8	no	—
159	Q₁₆	—	172	4	no	—
160	SD₁₆	—	172	8	no	—
161	SD₁₆	—	172	8	no	—
162	D₈	—	86	8	no	—
163	D₈	—	172	4	no	—
164	C₂×C₈	D₃₄₄	1	4	yes	—
165	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
166	C₂×C₈	C₂×D₁₇₂	1	4	yes	—
167	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
168	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
169	C₄⋊C₄	—	172	4	no	—
170	C₂²⋊C₄	—	172	8	no	—
171	C₄²	D₃₄₄	1	2	yes	—
172	C₄²	C₂×D₁₇₂	1	2	yes	—
173	C₁₆	C₂×D₁₇₂	1	8	yes	—
174	C₂³	—	172	4	no	—
175	C₂³	C₂×D₃₄₄	1	1	yes	characteristic
176	Q₈	—	172	4	no	—
177	Q₈	—	172	4	no	—
178	D₄	—	344	8	no	—
179	D₄	—	344	8	no	—
180	D₄	—	172	4	no	—
181	D₄	—	172	4	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	344	8	no	—
184	C₂×C₄	—	172	4	no	—
185	C₂×C₄	#13063322	1	2	yes	—
186	C₂×C₄	C₂×D₃₄₄	1	4	yes	—
187	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
188	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
189	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
190	C₈	#13063322	1	2	yes	—
191	C₈	C₂×D₃₄₄	1	4	yes	—
192	C₈	C₂²×D₁₇₂	1	2	yes	—
193	C₂²	—	2	2	no	—
194	C₂²	—	344	4	no	—
195	C₂²	—	344	8	no	—
196	C₂²	#14583276	1	2	yes	—
197	C₂²	C₂²×D₃₄₄	1	1	yes	characteristic
198	C₄	—	2	4	no	—
199	C₄	—	344	4	no	—
200	C₄	#14583276	1	2	yes	—
201	C₄	C₂×#13063322	1	1	yes	characteristic
202	C₄	C₂²×D₃₄₄	1	1	yes	characteristic
203	C₂	—	2	2	no	—
204	C₂	—	688	4	no	—
205	C₂	C₂×#14583276	1	1	yes	characteristic
206	C₂	#19147731	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14583331

C₂

yes

maximal, complemented

#19145023

C₂

yes

maximal, complemented

#19136866

C₂

yes

maximal, complemented

#19136861

C₂

yes

maximal, complemented

C₂×C₄₃×#346

C₂

yes

maximal, characteristic, complemented

C₂×#13063347

—

C₂×#13063347

C₂²

yes

—

#14583449

—

#14583331

—

C₂×Dic₃₄₄

C₂²

yes

—

C₂×#13063270

C₂²

yes

—

C₂×D₆₈₈

C₂²

yes

—

#14583283

—

#14583283

C₂²

yes

—

#14583279

C₂²

yes

—

#14583276

C₂²

yes

—

#14583272

—

C₂²×C₁₆×C₄₃

C₂²

yes

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₄₃×#346

C₂²

yes

—

C₂×#10621898

—

#13063347

—

#13063347

—

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

C₂³

yes

—

C₂×#10621873

—

C₂×#10621873

—

C₂×D₃₄₄

—

C₂×D₃₄₄

C₂³

yes

—

—

—

—

—

C₄×D₁₇₂

—

Dic₃₄₄

—

#13063270

—

D₆₈₈

—

C₂²×C₈×C₄₃

D₄

yes

characteristic

C₂²×C₈×C₄₃

C₂³

yes

characteristic

C₂×C₄²×C₄₃

D₄

yes

characteristic

C₂×C₁₆×C₄₃

D₄

yes

complemented

C₂×C₁₆×C₄₃

C₂³

yes

—

C₄×C₈×C₄₃

D₄

yes

—

C₄×C₈×C₄₃

C₂³

yes

—

C₂²×C₄×C₄₃

D₈

yes

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₂²⋊D₄₃

—

#10621898

—

#10621898

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×C₄×D₄₃

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

D₈

yes

—

C₂×C₈×C₄₃

C₂×D₄

yes

characteristic

C₂×C₈×C₄₃

C₂×D₄

yes

—

C₂×C₈×C₄₃

C₂×D₄

yes

—

C₂×C₈×C₄₃

C₂⁴

yes

characteristic

C₄²×C₄₃

D₈

yes

—

C₄²×C₄₃

C₂×D₄

yes

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

Dic₁₇₂

—

Dic₁₇₂

—

D₃₄₄

—

D₃₄₄

—

#10621873

—

#10621873

—

C₁₆×C₄₃

C₂×D₄

yes

—

C₂³×C₄₃

C₂×D₈

yes

characteristic

C₂²×D₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#127

yes

—

C₂×C₄×C₄₃

C₂×D₈

yes

—

C₂×C₄×C₄₃

C₂×D₈

yes

—

C₂×C₄×C₄₃

C₂×D₈

yes

characteristic

C₂×C₄×C₄₃

C₂²×D₄

yes

characteristic

C₂²⋊D₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

C₄×D₄₃

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

#127

yes

—

C₈×C₄₃

C₂×D₈

yes

—

C₈×C₄₃

C₂²×D₄

yes

—

#33716

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

#493

yes

—

C₂²×C₄₃

C₂²×D₈

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#493

yes

—

C₄×C₄₃

C₂×#127

yes

characteristic

C₄×C₄₃

C₂²×D₈

yes

characteristic

Dic₄₃

—

100

C₂×#508

—

101

#3147

—

102

#2251

—

103

#2250

—

104

C₂×#346

D₄₃

yes

characteristic, complemented

105

C₂×C₄₃

—

106

C₂×C₄₃

C₂×#493

yes

characteristic

107

C₂×C₄₃

#3414

yes

—

108

D₄₃

—

109

C₂×#135

—

110

C₂×#135

—

111

#532

—

112

#508

—

113

C₂×Q₃₂

—

114

C₂×SD₃₂

—

115

C₂×D₁₆

—

116

C₂²×C₁₆

C₂×D₄₃

yes

—

117

#495

—

118

#495

—

119

#494

—

120

#493

—

121

#492

—

122

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

123

#346

C₂×D₄₃

yes

—

124

C₄₃

#33716

yes

characteristic, complemented, Sylow

125

C₂×#41

—

126

#135

—

127

#135

—

172

—

128

C₂×Q₁₆

—

129

C₂×Q₁₆

—

130

C₂×SD₁₆

—

131

C₂×SD₁₆

—

132

C₂×D₈

—

133

C₂×D₈

—

134

C₂²×C₈

D₁₇₂

yes

characteristic

135

C₂²×C₈

C₂²×D₄₃

yes

characteristic

136

C₄⋊Q₈

—

137

#127

—

138

#124

—

139

#121

—

172

—

140

C₄×D₄

—

172

—

141

C₂×C₄²

D₁₇₂

yes

characteristic

142

Q₃₂

—

143

SD₃₂

—

144

D₁₆

—

145

C₂×C₁₆

D₁₇₂

yes

complemented

146

C₂×C₁₆

C₂²×D₄₃

yes

—

147

C₄×C₈

D₁₇₂

yes

—

148

C₄×C₈

C₂²×D₄₃

yes

—

149

#41

—

172

—

150

#41

—

172

—

151

C₂×Q₈

—

152

C₂×D₄

—

153

C₂×D₄

—

172

—

154

C₂×D₄

—

172

—

155

C₂²×C₄

—

172

—

156

C₂²×C₄

D₃₄₄

yes

—

157

C₂²×C₄

C₂×D₁₇₂

yes

characteristic

158

Q₁₆

—

159

Q₁₆

—

172

—

160

SD₁₆

—

172

—

161

SD₁₆

—

172

—

162

D₈

—

163

D₈

—

172

—

164

C₂×C₈

D₃₄₄

yes

—

165

C₂×C₈

C₂×D₁₇₂

yes

characteristic

166

C₂×C₈

C₂×D₁₇₂

yes

—

167

C₂×C₈

C₂×D₁₇₂

yes

—

168

C₂×C₈

C₂³×D₄₃

yes

characteristic

169

C₄⋊C₄

—

172

—

170

C₂²⋊C₄

—

172

—

171

C₄²

D₃₄₄

yes

—

172

C₄²

C₂×D₁₇₂

yes

—

173

C₁₆

C₂×D₁₇₂

yes

—

174

C₂³

—

172

—

175

C₂³

C₂×D₃₄₄

yes

characteristic

176

Q₈

—

172

—

177

Q₈

—

172

—

178

D₄

—

344

—

179

D₄

—

344

—

180

D₄

—

172

—

181

D₄

—

172

—

182

C₂×C₄

—

183

C₂×C₄

—

344

—

184

C₂×C₄

—

172

—

185

C₂×C₄

#13063322

yes

—

186

C₂×C₄

C₂×D₃₄₄

yes

—

187

C₂×C₄

C₂×D₃₄₄

yes

—

188

C₂×C₄

C₂×D₃₄₄

yes

characteristic

189

C₂×C₄

C₂²×D₁₇₂

yes

characteristic

190

C₈

#13063322

yes

—

191

C₈

C₂×D₃₄₄

yes

—

192

C₈

C₂²×D₁₇₂

yes

—

193

C₂²

—

194

C₂²

—

344

—

195

C₂²

—

344

—

196

C₂²

#14583276

yes

—

197

C₂²

C₂²×D₃₄₄

yes

characteristic

198

C₄

—

199

C₄

—

344

—

200

C₄

#14583276

yes

—

201

C₄

C₂×#13063322

yes

characteristic

202

C₄

C₂²×D₃₄₄

yes

characteristic

203

C₂

—

204

C₂

—

688

—

205

C₂

C₂×#14583276

yes

characteristic

206

C₂

#19147731

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₃₄₄
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₂×C₄₃×#346	C₂
socle	C₂²×C₄₃	C₂²×D₈
hypercenter	C₂×#346	D₄₃
hypocenter	C₄₃	#33716
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂×C₄

C₂×D₃₄₄

commutator, cosocle

C₂×C₈×C₄₃

C₂⁴

Frattini

C₂×C₈

C₂³×D₄₃

Fitting

C₂×C₄₃×#346

C₂

socle

C₂²×C₄₃

C₂²×D₈

D₄₃

C₄₃

G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
1. C₂×#14583331	C₂	semidirect
2. #19145023	C₂	semidirect
3. #19136866	C₂	semidirect
4. #19136861	C₂	semidirect
5. C₂×C₄₃×#346	C₂	semidirect
41. C₂×C₁₆×C₄₃	D₄	semidirect
104. C₂×#346	D₄₃	semidirect
124. C₄₃	#33716	semidirect
145. C₂×C₁₆	D₁₇₂	semidirect

H ≅ G/N

type

1. C₂×#14583331

C₂

semidirect

2. #19145023

C₂

semidirect

3. #19136866

C₂

semidirect

4. #19136861

C₂

semidirect

5. C₂×C₄₃×#346

C₂

semidirect

41. C₂×C₁₆×C₄₃

D₄

semidirect

104. C₂×#346

D₄₃

semidirect

124. C₄₃

#33716

semidirect

145. C₂×C₁₆

D₁₇₂

semidirect

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

subgroup	index
C₂×#14583331	2
#19145023	2
#19136866	2
#19136861	2
C₂×C₄₃×#346	2
#33716	43

subgroup

index

C₂×C₄₃×#346

Subgroups — #25252021

Conjugacy classes of subgroups

Complements

Maximal subgroups