Group #25016205 — subgroups

Subgroups — #25016205

Order 11008.

410 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 109 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	C₂×Dic₄₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582702	C₂	1	2	yes	maximal
4	#19135454	C₂	1	2	yes	maximal
5	C₄₃×#1380	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₇₂	C₂²	1	2	yes	—
7	C₂×Dic₃₄₄	—	2	4	no	—
8	C₂×Dic₃₄₄	—	2	2	no	—
9	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂×#13063248	—	2	2	no	—
11	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
12	#14582804	—	2	2	no	—
13	Dic₄₃⋊C₁₆	—	2	2	no	—
14	#14582702	—	2	4	no	—
15	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
16	C₂×#107×C₄₃	C₂²	1	2	yes	—
17	C₂²×Dic₈₆	—	2	2	no	—
18	C₂×Dic₁₇₂	—	2	2	no	—
19	C₂×Dic₁₇₂	—	2	2	no	—
20	C₂×Dic₁₇₂	—	2	2	no	—
21	C₂×Dic₁₇₂	—	2	2	no	—
22	C₂×Dic₁₇₂	—	2	2	no	—
23	C₂×Dic₁₇₂	D₄	1	4	yes	—
24	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
25	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
26	Dic₃₄₄	—	2	8	no	—
27	C₂×C₄₃⋊C₁₆	—	2	2	no	—
28	C₂×C₄₃⋊C₁₆	—	2	1	no	—
29	#13063248	—	4	4	no	—
30	C₈×Dic₄₃	—	2	1	no	—
31	C₈×Dic₄₃	—	2	1	no	—
32	C₄₃⋊#95	—	4	2	no	—
33	C₄₃⋊#95	—	4	2	no	—
34	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
36	C₂×C₁₆×C₄₃	—	2	2	no	—
37	C₂×C₁₆×C₄₃	D₄	1	2	yes	—
38	#107×C₄₃	—	2	2	no	—
39	#107×C₄₃	—	2	2	no	—
40	C₂²×C₄×C₄₃	—	4	2	no	—
41	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×Dic₄₃	—	4	2	no	—
43	C₂²×Dic₄₃	D₈	1	1	yes	characteristic
44	C₂²×Dic₄₃	D₈	1	1	yes	characteristic
45	C₂×Dic₈₆	—	2	4	no	—
46	C₂×Dic₈₆	—	4	2	no	—
47	C₂×Dic₈₆	—	4	2	no	—
48	C₂×Dic₈₆	—	4	2	no	—
49	C₂×Dic₈₆	—	4	2	no	—
50	C₂×Dic₈₆	—	4	2	no	—
51	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
53	C₂×C₈×C₄₃	#41	1	2	yes	—
54	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
55	C₄⋊C₄×C₄₃	—	4	2	no	—
56	C₄⋊C₄×C₄₃	—	4	2	no	—
57	C₄×Dic₄₃	—	2	1	no	—
58	C₄×Dic₄₃	—	2	1	no	—
59	C₂×C₄₃⋊C₈	—	2	1	no	—
60	C₂×C₄₃⋊C₈	—	2	2	no	—
61	Dic₁₇₂	—	2	4	no	—
62	Dic₁₇₂	—	2	4	no	—
63	Dic₁₇₂	—	4	2	no	—
64	Dic₁₇₂	—	4	2	no	—
65	C₁₆×C₄₃	—	2	2	no	—
66	C₄₃⋊C₁₆	—	4	1	no	—
67	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
68	C₂×C₄×C₄₃	—	8	2	no	—
69	C₂×C₄×C₄₃	—	8	2	no	—
70	C₂×C₄×C₄₃	—	8	2	no	—
71	C₂×C₄×C₄₃	#121	1	2	yes	—
72	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
73	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
74	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
75	C₂×Dic₄₃	—	2	1	no	—
76	C₂×Dic₄₃	—	2	1	no	—
77	C₂×Dic₄₃	—	2	1	no	—
78	C₂×Dic₄₃	—	2	2	no	—
79	C₂×Dic₄₃	—	4	4	no	—
80	C₂×Dic₄₃	—	8	2	no	—
81	C₂×Dic₄₃	—	8	2	no	—
82	C₂×Dic₄₃	Q₃₂	1	2	yes	complemented
83	Dic₈₆	—	4	4	no	—
84	Dic₈₆	—	4	4	no	—
85	Dic₈₆	—	8	2	no	—
86	Dic₈₆	—	8	2	no	—
87	C₈×C₄₃	#120	1	1	yes	characteristic
88	C₈×C₄₃	#123	1	2	yes	—
89	C₈×C₄₃	#124	1	1	yes	characteristic
90	C₄₃⋊C₈	—	4	1	no	—
91	#13450	—	43	1	no	maximal, Sylow
92	C₂²×C₄₃	#459	1	2	yes	—
93	C₂²×C₄₃	#466	1	1	yes	characteristic
94	C₂²×C₄₃	C₂×Q₃₂	1	1	yes	characteristic
95	C₂²×C₄₃	#508	1	1	yes	characteristic
96	C₂²×C₄₃	#510	1	2	yes	—
97	C₄×C₄₃	—	16	2	no	—
98	C₄×C₄₃	#394	1	1	yes	characteristic
99	C₄×C₄₃	#463	1	2	yes	—
100	C₄×C₄₃	#469	1	1	yes	characteristic
101	Dic₄₃	—	2	2	no	—
102	Dic₄₃	—	4	1	no	—
103	Dic₄₃	—	8	4	no	—
104	C₂²×Q₃₂	—	43	1	no	—
105	C₂×C₄⋊C₁₆	—	43	1	no	—
106	C₂×#358	—	43	2	no	—
107	#1969	—	43	2	no	—
108	#1380	D₄₃	1	1	yes	characteristic
109	C₂×C₄₃	#2054	1	1	yes	characteristic
110	C₂×C₄₃	#2209	1	2	yes	—
111	C₂×C₄₃	#2211	1	2	yes	—
112	C₂×C₄₃	#2216	1	1	yes	characteristic
113	C₂×C₄₃	#2219	1	1	yes	characteristic
114	C₂²×Q₁₆	—	43	2	no	—
115	C₂×Q₃₂	—	86	4	no	—
116	C₂×Q₃₂	—	86	2	no	—
117	C₂²×C₁₆	—	43	1	no	—
118	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
119	C₂×#107	C₂×D₄₃	1	2	yes	—
120	C₂×#103	—	86	2	no	—
121	C₂×C₄×C₈	—	43	1	no	—
122	#391	—	86	2	no	—
123	C₄⋊C₁₆	—	86	2	no	—
124	#358	—	86	4	no	—
125	C₄₃	#13450	1	1	yes	characteristic, complemented, Sylow
126	C₂²×Q₈	—	86	2	no	—
127	C₂×Q₁₆	—	43	4	no	—
128	C₂×Q₁₆	—	86	2	no	—
129	C₂×Q₁₆	—	86	2	no	—
130	C₂×Q₁₆	—	86	2	no	—
131	C₂×Q₁₆	—	86	2	no	—
132	C₂×Q₁₆	—	86	2	no	—
133	C₂²×C₈	—	43	1	no	—
134	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄²	—	43	1	no	—
137	Q₃₂	—	86	8	no	—
138	C₂×C₁₆	—	2	2	no	—
139	C₂×C₁₆	—	86	1	no	—
140	C₂×C₁₆	—	86	2	no	—
141	C₂×C₁₆	D₁₇₂	1	2	yes	—
142	#107	—	2	2	no	—
143	#107	—	2	2	no	—
144	#103	—	172	4	no	—
145	C₄×C₈	—	86	1	no	—
146	C₄×C₈	—	86	1	no	—
147	#95	—	172	2	no	—
148	#95	—	172	2	no	—
149	C₂×Q₈	—	86	4	no	—
150	C₂×Q₈	—	172	2	no	—
151	C₂×Q₈	—	172	2	no	—
152	C₂×Q₈	—	172	2	no	—
153	C₂×Q₈	—	172	2	no	—
154	C₂×Q₈	—	172	2	no	—
155	C₂²×C₄	—	4	2	no	—
156	C₂²×C₄	—	43	1	no	—
157	C₂²×C₄	—	43	1	no	—
158	C₂²×C₄	—	172	2	no	—
159	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
160	Q₁₆	—	86	4	no	—
161	Q₁₆	—	86	4	no	—
162	Q₁₆	—	172	2	no	—
163	Q₁₆	—	172	2	no	—
164	C₂×C₈	—	86	1	no	—
165	C₂×C₈	—	86	2	no	—
166	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
167	C₂×C₈	#10621898	1	2	yes	—
168	C₂×C₈	D₄×D₄₃	1	2	yes	—
169	C₂×C₈	#10621902	1	1	yes	characteristic
170	C₄⋊C₄	—	4	2	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄²	—	86	1	no	—
173	C₄²	—	86	1	no	—
174	C₁₆	—	2	2	no	—
175	C₁₆	—	172	1	no	—
176	C₂³	D₈×D₄₃	1	1	yes	characteristic
177	Q₈	—	344	2	no	—
178	Q₈	—	344	2	no	—
179	Q₈	—	172	4	no	—
180	Q₈	—	172	4	no	—
181	C₂×C₄	—	8	2	no	—
182	C₂×C₄	—	8	2	no	—
183	C₂×C₄	—	8	2	no	—
184	C₂×C₄	—	43	2	no	—
185	C₂×C₄	—	86	1	no	—
186	C₂×C₄	—	86	1	no	—
187	C₂×C₄	—	86	2	no	—
188	C₂×C₄	—	86	1	no	—
189	C₂×C₄	—	344	2	no	—
190	C₂×C₄	—	344	2	no	—
191	C₂×C₄	—	172	4	no	—
192	C₂×C₄	#13063298	1	2	yes	—
193	C₂×C₄	#13063300	1	1	yes	characteristic
194	C₂×C₄	#13063348	1	1	yes	characteristic
195	C₂×C₄	#13063352	1	2	yes	—
196	C₈	—	172	1	no	—
197	C₈	#13063295	1	1	yes	characteristic
198	C₈	#13063310	1	2	yes	—
199	C₈	#13063313	1	1	yes	characteristic
200	C₂²	#14583093	1	2	yes	—
201	C₂²	#14583136	1	1	yes	characteristic
202	C₂²	Q₃₂×D₄₃	1	1	yes	characteristic
203	C₂²	#14583328	1	1	yes	characteristic
204	C₂²	#14583338	1	2	yes	—
205	C₄	—	16	2	no	—
206	C₄	—	86	2	no	—
207	C₄	—	344	4	no	—
208	C₄	—	172	1	no	—
209	C₄	#14582817	1	1	yes	characteristic
210	C₄	#14583117	1	2	yes	—
211	C₄	#14583153	1	1	yes	characteristic
212	C₂	#19135907	1	1	yes	characteristic
213	C₂	#19136644	1	2	yes	—
214	C₂	#19136656	1	2	yes	—
215	C₂	#19136687	1	1	yes	characteristic
216	C₂	#19136704	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₃₄₄

C₂

yes

maximal, characteristic

C₂×Dic₄₃⋊C₁₆

C₂

yes

maximal, characteristic

C₂×#14582702

C₂

yes

maximal

#19135454

C₂

yes

maximal

C₄₃×#1380

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₂

C₂²

yes

—

C₂×Dic₃₄₄

—

C₂×Dic₃₄₄

—

C₂²×C₄₃⋊C₁₆

C₂²

yes

characteristic

C₂×#13063248

—

C₂×C₈×Dic₄₃

C₂²

yes

characteristic

#14582804

—

Dic₄₃⋊C₁₆

—

#14582702

—

C₂²×C₁₆×C₄₃

C₂²

yes

characteristic

C₂×#107×C₄₃

C₂²

yes

—

C₂²×Dic₈₆

—

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

D₄

yes

—

C₂²×C₄₃⋊C₈

D₄

yes

characteristic

C₂×C₄×Dic₄₃

D₄

yes

characteristic

Dic₃₄₄

—

C₂×C₄₃⋊C₁₆

—

C₂×C₄₃⋊C₁₆

—

#13063248

—

C₈×Dic₄₃

—

C₈×Dic₄₃

—

C₄₃⋊#95

—

C₄₃⋊#95

—

C₂²×C₈×C₄₃

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₄₃

—

C₂×C₁₆×C₄₃

—

C₂×C₁₆×C₄₃

D₄

yes

—

#107×C₄₃

—

#107×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×Dic₄₃

—

C₂²×Dic₄₃

D₈

yes

characteristic

C₂²×Dic₄₃

D₈

yes

characteristic

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

C₂×D₄

yes

characteristic

C₂×C₈×C₄₃

C₂×D₄

yes

—

C₂×C₈×C₄₃

#41

yes

—

C₂×C₈×C₄₃

#41

yes

characteristic

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄×Dic₄₃

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

Dic₁₇₂

—

Dic₁₇₂

—

Dic₁₇₂

—

Dic₁₇₂

—

C₁₆×C₄₃

—

C₄₃⋊C₁₆

—

C₂³×C₄₃

C₂×D₈

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#121

yes

—

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

Q₃₂

yes

complemented

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

#120

yes

characteristic

C₈×C₄₃

#123

yes

—

C₈×C₄₃

#124

yes

characteristic

C₄₃⋊C₈

—

#13450

—

maximal, Sylow

C₂²×C₄₃

#459

yes

—

C₂²×C₄₃

#466

yes

characteristic

C₂²×C₄₃

C₂×Q₃₂

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

—

C₄×C₄₃

—

C₄×C₄₃

#394

yes

characteristic

C₄×C₄₃

#463

yes

—

100

C₄×C₄₃

#469

yes

characteristic

101

Dic₄₃

—

102

Dic₄₃

—

103

Dic₄₃

—

104

C₂²×Q₃₂

—

105

C₂×C₄⋊C₁₆

—

106

C₂×#358

—

107

#1969

—

108

#1380

D₄₃

yes

characteristic

109

C₂×C₄₃

#2054

yes

characteristic

110

C₂×C₄₃

#2209

yes

—

111

C₂×C₄₃

#2211

yes

—

112

C₂×C₄₃

#2216

yes

characteristic

113

C₂×C₄₃

#2219

yes

characteristic

114

C₂²×Q₁₆

—

115

C₂×Q₃₂

—

116

C₂×Q₃₂

—

117

C₂²×C₁₆

—

118

C₂²×C₁₆

C₂×D₄₃

yes

characteristic

119

C₂×#107

C₂×D₄₃

yes

—

120

C₂×#103

—

121

C₂×C₄×C₈

—

122

#391

—

123

C₄⋊C₁₆

—

124

#358

—

125

C₄₃

#13450

yes

characteristic, complemented, Sylow

126

C₂²×Q₈

—

127

C₂×Q₁₆

—

128

C₂×Q₁₆

—

129

C₂×Q₁₆

—

130

C₂×Q₁₆

—

131

C₂×Q₁₆

—

132

C₂×Q₁₆

—

133

C₂²×C₈

—

134

C₂²×C₈

C₂²×D₄₃

yes

characteristic

135

C₂×C₄⋊C₄

—

136

C₂×C₄²

—

137

Q₃₂

—

138

C₂×C₁₆

—

139

C₂×C₁₆

—

140

C₂×C₁₆

—

141

C₂×C₁₆

D₁₇₂

yes

—

142

#107

—

143

#107

—

144

#103

—

172

—

145

C₄×C₈

—

146

C₄×C₈

—

147

#95

—

172

—

148

#95

—

172

—

149

C₂×Q₈

—

150

C₂×Q₈

—

172

—

151

C₂×Q₈

—

172

—

152

C₂×Q₈

—

172

—

153

C₂×Q₈

—

172

—

154

C₂×Q₈

—

172

—

155

C₂²×C₄

—

156

C₂²×C₄

—

157

C₂²×C₄

—

158

C₂²×C₄

—

172

—

159

C₂²×C₄

D₄×D₄₃

yes

characteristic

160

Q₁₆

—

161

Q₁₆

—

162

Q₁₆

—

172

—

163

Q₁₆

—

172

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₂×C₈

C₂×D₁₇₂

yes

characteristic

167

C₂×C₈

#10621898

yes

—

168

C₂×C₈

D₄×D₄₃

yes

—

169

C₂×C₈

#10621902

yes

characteristic

170

C₄⋊C₄

—

171

C₄⋊C₄

—

172

C₄²

—

173

C₄²

—

174

C₁₆

—

175

C₁₆

—

172

—

176

C₂³

D₈×D₄₃

yes

characteristic

177

Q₈

—

344

—

178

Q₈

—

344

—

179

Q₈

—

172

—

180

Q₈

—

172

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

344

—

190

C₂×C₄

—

344

—

191

C₂×C₄

—

172

—

192

C₂×C₄

#13063298

yes

—

193

C₂×C₄

#13063300

yes

characteristic

194

C₂×C₄

#13063348

yes

characteristic

195

C₂×C₄

#13063352

yes

—

196

C₈

—

172

—

197

C₈

#13063295

yes

characteristic

198

C₈

#13063310

yes

—

199

C₈

#13063313

yes

characteristic

200

C₂²

#14583093

yes

—

201

C₂²

#14583136

yes

characteristic

202

C₂²

Q₃₂×D₄₃

yes

characteristic

203

C₂²

#14583328

yes

characteristic

204

C₂²

#14583338

yes

—

205

C₄

—

206

C₄

—

207

C₄

—

344

—

208

C₄

—

172

—

209

C₄

#14582817

yes

characteristic

210

C₄

#14583117

yes

—

211

C₄

#14583153

yes

characteristic

212

C₂

#19135907

yes

characteristic

213

C₂

#19136644

yes

—

214

C₂

#19136656

yes

—

215

C₂

#19136687

yes

characteristic

216

C₂

#19136704

yes

characteristic

	subgroup	G/H
center	C₂³	D₈×D₄₃
commutator, cosocle	C₂²×C₈×C₄₃	C₂³
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#1380	C₂
socle, prime radical	C₂³×C₄₃	C₂×D₈
hypercenter	#1380	D₄₃
hypocenter	C₄₃	#13450
LCS	G ▷ C₂²×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1380

subgroup

G/H

center

C₂³

D₈×D₄₃

commutator, cosocle

C₂²×C₈×C₄₃

C₂³

Frattini

C₂²×C₈

C₂²×D₄₃

Fitting

C₄₃×#1380

C₂

socle, prime radical

C₂³×C₄₃

C₂×D₈

D₄₃

C₄₃

G ▷ C₂²×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1380

N	H ≅ G/N	type
82. C₂×Dic₄₃	Q₃₂	semidirect
125. C₄₃	#13450	semidirect

H ≅ G/N

type

82. C₂×Dic₄₃

Q₃₂

semidirect

125. C₄₃

#13450

semidirect

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

subgroup	index
C₂²×Dic₃₄₄	2
C₂×Dic₄₃⋊C₁₆	2
C₂×#14582702	2
#19135454	2
C₄₃×#1380	2
#13450	43

subgroup

index

C₂²×Dic₃₄₄

C₂×Dic₄₃⋊C₁₆

C₂×#14582702

#19135454

C₄₃×#1380

#13450

Subgroups — #25016205

Conjugacy classes of subgroups

Complements

Maximal subgroups