Group #25249974 — subgroups

Subgroups — #25249974

Order 11008.

646 conjugacy classes of subgroups; only the 211 normal subgroups are shown.

normal subgroups only

This group has 646 conjugacy classes of subgroups — too many to draw. Only its 211 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	C₂²×Dic₃₄₄	C₂	1	1	yes	maximal, characteristic
2	#19143462	C₂	1	2	yes	maximal
3	#19136465	C₂	1	4	yes	maximal, complemented
4	#19136453	C₂	1	4	yes	maximal, complemented
5	#19136314	C₂	1	2	yes	maximal
6	#19136191	C₂	1	1	yes	maximal, characteristic
7	C₄₃×#2158	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₇₂	C₂²	1	2	yes	—
9	#14583376	—	2	2	no	—
10	C₂×Dic₃₄₄	C₄	1	8	yes	complemented
11	C₂×Dic₃₄₄	C₂²	1	2	yes	—
12	C₂×Dic₃₄₄	C₂²	1	2	yes	—
13	C₂×#13063265	C₂²	1	1	yes	characteristic
14	#14583005	—	2	4	no	—
15	#14582993	C₂²	1	8	yes	—
16	#14582907	—	2	2	no	—
17	#14582854	C₂²	1	1	yes	characteristic
18	#14582702	C₂²	1	8	yes	—
19	#14582674	C₂²	1	2	yes	—
20	C₁₆×Dic₄₃	C₂²	1	2	yes	—
21	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
22	C₄₃×#427	C₂²	1	2	yes	—
23	C₄₃×#367	C₂²	1	2	yes	—
24	C₄₃×#366	C₂²	1	2	yes	—
25	C₂²×Dic₈₆	—	2	2	no	—
26	C₂×Dic₁₇₂	—	2	2	no	—
27	C₂×Dic₁₇₂	—	2	4	no	—
28	C₂×Dic₁₇₂	C₂×C₄	1	8	yes	—
29	C₂×Dic₁₇₂	C₂³	1	4	yes	—
30	C₂×#10621877	D₄	1	1	yes	characteristic
31	C₄×Dic₈₆	—	4	4	no	—
32	#13063290	—	2	8	no	—
33	#13063284	D₄	1	1	yes	characteristic
34	#13063281	—	4	4	no	—
35	Dic₃₄₄	C₂×C₄	1	8	yes	—
36	Dic₃₄₄	C₂×C₄	1	8	yes	—
37	#13063265	—	2	2	no	—
38	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
39	#13063248	—	2	8	no	—
40	#13063235	D₄	1	2	yes	—
41	C₈×Dic₄₃	C₂³	1	2	yes	—
42	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
43	#117×C₄₃	—	2	2	no	—
44	C₂×C₁₆×C₄₃	C₂×C₄	1	4	yes	—
45	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
46	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
47	#107×C₄₃	C₂³	1	4	yes	—
48	#106×C₄₃	—	2	2	no	—
49	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×Dic₄₃	—	4	2	no	—
51	C₂×Dic₈₆	—	2	8	no	—
52	C₂×Dic₈₆	—	2	4	no	—
53	C₂×Dic₈₆	—	4	2	no	—
54	C₂×Dic₈₆	—	4	4	no	—
55	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
56	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₄₃	#41	1	2	yes	—
58	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
59	C₄⋊C₄×C₄₃	—	2	4	no	—
60	C₂²⋊C₄×C₄₃	—	4	2	no	—
61	C₄²×C₄₃	—	4	2	no	—
62	C₂²⋊Dic₄₃	D₈	1	2	yes	—
63	C₄⋊Dic₄₃	D₈	1	2	yes	—
64	Dic₄₃⋊C₄	—	4	8	no	—
65	C₄×Dic₄₃	—	4	8	no	—
66	C₄×Dic₄₃	C₂×D₄	1	2	yes	—
67	#10621877	—	2	2	no	—
68	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
69	Dic₁₇₂	—	2	8	no	—
70	Dic₁₇₂	—	2	4	no	—
71	Dic₁₇₂	C₂²×C₄	1	8	yes	—
72	C₁₆×C₄₃	C₂²×C₄	1	2	yes	—
73	C₁₆×C₄₃	C₂²×C₄	1	2	yes	—
74	C₄₃⋊C₁₆	—	2	2	no	—
75	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
76	C₂×C₄×C₄₃	—	4	4	no	—
77	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
78	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₄₃	#135	1	2	yes	—
80	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
81	C₂×Dic₄₃	—	4	4	no	—
82	C₂×Dic₄₃	—	4	8	no	—
83	C₂×Dic₄₃	C₂×D₈	1	4	yes	—
84	Dic₈₆	—	2	8	no	—
85	Dic₈₆	—	4	4	no	—
86	Dic₈₆	—	4	8	no	—
87	C₈×C₄₃	C₄×D₄	1	2	yes	—
88	C₈×C₄₃	C₂³×C₄	1	1	yes	characteristic
89	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
90	C₄₃⋊C₈	—	2	2	no	—
91	#33458	—	43	1	no	maximal, Sylow
92	C₂²×C₄₃	—	2	2	no	—
93	C₂²×C₄₃	C₄×D₈	1	2	yes	—
94	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
95	C₄×C₄₃	—	8	4	no	—
96	C₄×C₄₃	C₄×D₈	1	2	yes	—
97	C₄×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
98	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
99	Dic₄₃	—	2	4	no	—
100	Dic₄₃	—	4	8	no	—
101	C₂²×Q₃₂	—	43	1	no	—
102	#2950	—	43	2	no	—
103	#2176	—	43	4	no	—
104	C₄×Q₃₂	—	43	4	no	—
105	#2158	D₄₃	1	1	yes	characteristic
106	#2141	—	43	2	no	—
107	#2108	—	43	1	no	—
108	C₂×C₄₃	—	2	2	no	—
109	C₂×C₄₃	C₂×C₄×D₈	1	1	yes	characteristic
110	C₂×C₄₃	#3417	1	2	yes	—
111	C₂²×Q₁₆	—	43	2	no	—
112	#519	—	86	2	no	—
113	C₂×Q₃₂	—	43	2	no	—
114	C₂×Q₃₂	—	43	2	no	—
115	C₂×Q₃₂	—	43	8	no	—
116	C₂×M₃₂	—	43	1	no	—
117	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
118	#441	—	86	4	no	—
119	C₄×Q₁₆	—	43	8	no	—
120	#427	C₂×D₄₃	1	2	yes	—
121	#419	—	86	2	no	—
122	#405	—	43	1	no	—
123	#367	C₂×D₄₃	1	2	yes	—
124	#366	C₂×D₄₃	1	2	yes	—
125	#358	—	43	8	no	—
126	#346	—	43	2	no	—
127	C₄×C₁₆	—	43	2	no	—
128	C₄₃	#33458	1	1	yes	characteristic, complemented, Sylow
129	C₂²×Q₈	—	86	2	no	—
130	C₂×Q₁₆	—	43	8	no	—
131	C₂×Q₁₆	—	43	4	no	—
132	C₂×Q₁₆	—	86	2	no	—
133	C₂×Q₁₆	—	86	4	no	—
134	C₂×M₁₆	—	43	1	no	—
135	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
136	C₄×Q₈	—	86	8	no	—
137	C₄×Q₈	—	172	4	no	—
138	#117	—	2	2	no	—
139	#117	—	43	1	no	—
140	#117	—	172	4	no	—
141	Q₃₂	—	43	8	no	—
142	Q₃₂	—	43	8	no	—
143	M₃₂	—	86	2	no	—
144	C₂×C₁₆	—	43	2	no	—
145	C₂×C₁₆	C₄×D₄₃	1	4	yes	—
146	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
147	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
148	#107	C₂²×D₄₃	1	4	yes	—
149	#106	—	2	2	no	—
150	#103	—	86	8	no	—
151	#97	—	43	2	no	—
152	C₄×C₈	—	43	2	no	—
153	C₂×Q₈	—	86	8	no	—
154	C₂×Q₈	—	86	4	no	—
155	C₂×Q₈	—	172	2	no	—
156	C₂×Q₈	—	172	4	no	—
157	C₂²×C₄	—	172	2	no	—
158	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
159	Q₁₆	—	43	8	no	—
160	Q₁₆	—	86	8	no	—
161	Q₁₆	—	86	4	no	—
162	M₁₆	—	86	2	no	—
163	C₂×C₈	—	43	2	no	—
164	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
165	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
166	C₂×C₈	#10621902	1	2	yes	—
167	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
168	C₄⋊C₄	—	2	4	no	—
169	C₄⋊C₄	—	43	2	no	—
170	C₄⋊C₄	—	172	8	no	—
171	C₂²⋊C₄	—	4	2	no	—
172	C₂²⋊C₄	—	43	2	no	—
173	C₄²	—	4	2	no	—
174	C₄²	—	43	2	no	—
175	C₄²	—	172	8	no	—
176	C₁₆	—	86	2	no	—
177	C₁₆	C₂×C₄×D₄₃	1	2	yes	—
178	C₁₆	C₂×C₄×D₄₃	1	2	yes	—
179	C₂³	D₈×D₄₃	1	1	yes	characteristic
180	Q₈	—	86	8	no	—
181	Q₈	—	172	4	no	—
182	Q₈	—	172	8	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	—	43	4	no	—
185	C₂×C₄	—	172	4	no	—
186	C₂×C₄	—	172	8	no	—
187	C₂×C₄	#13063287	1	2	yes	—
188	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
189	C₂×C₄	#13063348	1	2	yes	—
190	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
191	C₈	—	86	2	no	—
192	C₈	#13063287	1	2	yes	—
193	C₈	C₂²×C₄×D₄₃	1	1	yes	characteristic
194	C₈	C₂×#10621902	1	1	yes	characteristic
195	C₂²	—	2	2	no	—
196	C₂²	#14582981	1	2	yes	—
197	C₂²	C₂×D₈×D₄₃	1	1	yes	characteristic
198	C₄	—	8	4	no	—
199	C₄	—	86	4	no	—
200	C₄	—	172	8	no	—
201	C₄	#14582981	1	2	yes	—
202	C₄	C₂×#13063287	1	1	yes	characteristic
203	C₄	C₂×#13063348	1	1	yes	characteristic
204	C₂	—	2	2	no	—
205	C₂	C₂×#14582981	1	1	yes	characteristic
206	C₂	#19147745	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₃₄₄

C₂

yes

maximal, characteristic

#19143462

C₂

yes

maximal

#19136465

C₂

yes

maximal, complemented

#19136453

C₂

yes

maximal, complemented

#19136314

C₂

yes

maximal

#19136191

C₂

yes

maximal, characteristic

C₄₃×#2158

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₂

C₂²

yes

—

#14583376

—

C₂×Dic₃₄₄

C₄

yes

complemented

C₂×Dic₃₄₄

C₂²

yes

—

C₂×Dic₃₄₄

C₂²

yes

—

C₂×#13063265

C₂²

yes

characteristic

#14583005

—

#14582993

C₂²

yes

—

#14582907

—

#14582854

C₂²

yes

characteristic

#14582702

C₂²

yes

—

#14582674

C₂²

yes

—

C₁₆×Dic₄₃

C₂²

yes

—

C₂²×C₁₆×C₄₃

C₂²

yes

characteristic

C₄₃×#427

C₂²

yes

—

C₄₃×#367

C₂²

yes

—

C₄₃×#366

C₂²

yes

—

C₂²×Dic₈₆

—

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

C₂×C₄

yes

—

C₂×Dic₁₇₂

C₂³

yes

—

C₂×#10621877

D₄

yes

characteristic

C₄×Dic₈₆

—

#13063290

—

#13063284

D₄

yes

characteristic

#13063281

—

Dic₃₄₄

C₂×C₄

yes

—

Dic₃₄₄

C₂×C₄

yes

—

#13063265

—

C₂×C₄₃⋊C₁₆

C₂³

yes

—

#13063248

—

#13063235

D₄

yes

—

C₈×Dic₄₃

C₂³

yes

—

C₂²×C₈×C₄₃

C₂³

yes

characteristic

#117×C₄₃

—

C₂×C₁₆×C₄₃

C₂×C₄

yes

—

C₂×C₁₆×C₄₃

C₂³

yes

characteristic

C₂×C₁₆×C₄₃

C₂³

yes

characteristic

#107×C₄₃

C₂³

yes

—

#106×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×Dic₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

C₂²×C₄

yes

—

C₂×C₈×C₄₃

C₂×D₄

yes

characteristic

C₂×C₈×C₄₃

#41

yes

—

C₂×C₈×C₄₃

C₂⁴

yes

characteristic

C₄⋊C₄×C₄₃

—

C₂²⋊C₄×C₄₃

—

C₄²×C₄₃

—

C₂²⋊Dic₄₃

D₈

yes

—

C₄⋊Dic₄₃

D₈

yes

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₄×Dic₄₃

C₂×D₄

yes

—

#10621877

—

C₂×C₄₃⋊C₈

C₂×D₄

yes

—

Dic₁₇₂

—

Dic₁₇₂

—

Dic₁₇₂

C₂²×C₄

yes

—

C₁₆×C₄₃

C₂²×C₄

yes

—

C₁₆×C₄₃

C₂²×C₄

yes

—

C₄₃⋊C₁₆

—

C₂³×C₄₃

C₂×D₈

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₄×D₄

yes

—

C₂×C₄×C₄₃

C₂×D₈

yes

characteristic

C₂×C₄×C₄₃

#135

yes

—

C₂×C₄×C₄₃

C₂²×D₄

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

C₂×D₈

yes

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

C₄×D₄

yes

—

C₈×C₄₃

C₂³×C₄

yes

characteristic

C₈×C₄₃

C₂×#41

yes

characteristic

C₄₃⋊C₈

—

#33458

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

C₄×D₈

yes

—

C₂²×C₄₃

C₂²×D₈

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

C₄×D₈

yes

—

C₄×C₄₃

C₂×C₄×D₄

yes

characteristic

C₄×C₄₃

C₂×#135

yes

characteristic

Dic₄₃

—

100

Dic₄₃

—

101

C₂²×Q₃₂

—

102

#2950

—

103

#2176

—

104

C₄×Q₃₂

—

105

#2158

D₄₃

yes

characteristic

106

#2141

—

107

#2108

—

108

C₂×C₄₃

—

109

C₂×C₄₃

C₂×C₄×D₈

yes

characteristic

110

C₂×C₄₃

#3417

yes

—

111

C₂²×Q₁₆

—

112

#519

—

113

C₂×Q₃₂

—

114

C₂×Q₃₂

—

115

C₂×Q₃₂

—

116

C₂×M₃₂

—

117

C₂²×C₁₆

C₂×D₄₃

yes

characteristic

118

#441

—

119

C₄×Q₁₆

—

120

#427

C₂×D₄₃

yes

—

121

#419

—

122

#405

—

123

#367

C₂×D₄₃

yes

—

124

#366

C₂×D₄₃

yes

—

125

#358

—

126

#346

—

127

C₄×C₁₆

—

128

C₄₃

#33458

yes

characteristic, complemented, Sylow

129

C₂²×Q₈

—

130

C₂×Q₁₆

—

131

C₂×Q₁₆

—

132

C₂×Q₁₆

—

133

C₂×Q₁₆

—

134

C₂×M₁₆

—

135

C₂²×C₈

C₂²×D₄₃

yes

characteristic

136

C₄×Q₈

—

137

C₄×Q₈

—

172

—

138

#117

—

139

#117

—

140

#117

—

172

—

141

Q₃₂

—

142

Q₃₂

—

143

M₃₂

—

144

C₂×C₁₆

—

145

C₂×C₁₆

C₄×D₄₃

yes

—

146

C₂×C₁₆

C₂²×D₄₃

yes

characteristic

147

C₂×C₁₆

C₂²×D₄₃

yes

characteristic

148

#107

C₂²×D₄₃

yes

—

149

#106

—

150

#103

—

151

#97

—

152

C₄×C₈

—

153

C₂×Q₈

—

154

C₂×Q₈

—

155

C₂×Q₈

—

172

—

156

C₂×Q₈

—

172

—

157

C₂²×C₄

—

172

—

158

C₂²×C₄

D₄×D₄₃

yes

characteristic

159

Q₁₆

—

160

Q₁₆

—

161

Q₁₆

—

162

M₁₆

—

163

C₂×C₈

—

164

C₂×C₈

C₂×C₄×D₄₃

yes

—

165

C₂×C₈

D₄×D₄₃

yes

characteristic

166

C₂×C₈

#10621902

yes

—

167

C₂×C₈

C₂³×D₄₃

yes

characteristic

168

C₄⋊C₄

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

172

—

171

C₂²⋊C₄

—

172

C₂²⋊C₄

—

173

C₄²

—

174

C₄²

—

175

C₄²

—

172

—

176

C₁₆

—

177

C₁₆

C₂×C₄×D₄₃

yes

—

178

C₁₆

C₂×C₄×D₄₃

yes

—

179

C₂³

D₈×D₄₃

yes

characteristic

180

Q₈

—

181

Q₈

—

172

—

182

Q₈

—

172

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

172

—

186

C₂×C₄

—

172

—

187

C₂×C₄

#13063287

yes

—

188

C₂×C₄

D₈×D₄₃

yes

characteristic

189

C₂×C₄

#13063348

yes

—

190

C₂×C₄

C₂×D₄×D₄₃

yes

characteristic

191

C₈

—

192

C₈

#13063287

yes

—

193

C₈

C₂²×C₄×D₄₃

yes

characteristic

194

C₈

C₂×#10621902

yes

characteristic

195

C₂²

—

196

C₂²

#14582981

yes

—

197

C₂²

C₂×D₈×D₄₃

yes

characteristic

198

C₄

—

199

C₄

—

200

C₄

—

172

—

201

C₄

#14582981

yes

—

202

C₄

C₂×#13063287

yes

characteristic

203

C₄

C₂×#13063348

yes

characteristic

204

C₂

—

205

C₂

C₂×#14582981

yes

characteristic

206

C₂

#19147745

yes

—

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₃
commutator	C₈×C₄₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄₃×#2158	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	C₂³×C₄₃	C₂×D₈
cosocle	C₂×C₈×C₄₃	C₂⁴
hypercenter	#2158	D₄₃
hypocenter	C₄₃	#33458
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

subgroup

G/H

center

C₂²

C₂×D₈×D₄₃

commutator

C₈×C₄₃

C₂³×C₄

Frattini

C₂×C₈

C₂³×D₄₃

Fitting

C₄₃×#2158

C₂

socle

C₂²×C₄₃

C₂²×D₈

prime radical

C₂³×C₄₃

C₂×D₈

cosocle

C₂×C₈×C₄₃

C₂⁴

D₄₃

C₄₃

G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

N	H ≅ G/N	type
3. #19136465	C₂	semidirect
4. #19136453	C₂	semidirect
10. C₂×Dic₃₄₄	C₄	semidirect
128. C₄₃	#33458	semidirect

H ≅ G/N

type

3. #19136465

C₂

semidirect

4. #19136453

C₂

semidirect

10. C₂×Dic₃₄₄

C₄

semidirect

128. C₄₃

#33458

semidirect

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

subgroup	index
C₂²×Dic₃₄₄	2
#19143462	2
#19136465	2
#19136453	2
#19136314	2
#19136191	2
C₄₃×#2158	2
#33458	43

subgroup

index

C₂²×Dic₃₄₄

C₄₃×#2158

Subgroups — #25249974

Conjugacy classes of subgroups

Complements

Maximal subgroups