Group #25015358 — subgroups

Subgroups — #25015358

Order 11008.

516 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 99 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×#14583171	C₂	1	2	yes	maximal
2	C₂×#14582715	C₂	1	2	yes	maximal, complemented
3	#19136208	C₂	1	1	yes	maximal, characteristic
4	#19132689	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×Q₃₂×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13063356	—	2	2	no	—
7	C₂×#13063331	C₂²	1	1	yes	characteristic
8	C₂×#13063265	C₂²	1	1	yes	characteristic
9	#14583171	—	2	4	no	—
10	#14583171	—	2	4	no	—
11	C₂×#13063261	C₂²	1	2	yes	—
12	C₂×#13063244	—	2	2	no	—
13	#14582715	—	2	4	no	—
14	(C₂×D₄₃)⋊C₁₆	—	2	2	no	—
15	C₂²×Q₁₆×C₄₃	C₂²	1	2	yes	—
16	C₂×Q₃₂×C₄₃	—	2	4	no	—
17	C₂×Q₃₂×C₄₃	—	2	2	no	—
18	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
19	C₂×#10621898	D₄	1	1	yes	characteristic
20	#13063356	—	2	8	no	—
21	#13063356	—	4	4	no	—
22	C₂×#10621885	—	4	2	no	—
23	C₂×#10621884	—	4	2	no	—
24	#13063331	—	2	2	no	—
25	#13063331	—	2	2	no	—
26	C₂×#10621877	—	2	2	no	—
27	C₂×#10621877	D₄	1	1	yes	characteristic
28	C₂×#10621872	—	2	1	no	—
29	C₂×C₈×D₄₃	—	2	1	no	—
30	#13063265	—	2	2	no	—
31	C₂×C₄₃⋊C₁₆	—	2	1	no	—
32	#13063261	—	2	2	no	—
33	#13063261	D₄	1	4	yes	—
34	#13063244	—	2	8	no	—
35	C₂²×Q₈×C₄₃	—	2	2	no	—
36	C₂×Q₁₆×C₄₃	—	2	2	no	—
37	C₂×Q₁₆×C₄₃	—	2	2	no	—
38	C₂×Q₁₆×C₄₃	—	2	2	no	—
39	C₂×Q₁₆×C₄₃	—	2	2	no	—
40	C₂×Q₁₆×C₄₃	—	2	2	no	—
41	C₂×Q₁₆×C₄₃	D₄	1	4	yes	—
42	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
43	Q₃₂×C₄₃	—	2	8	no	—
44	C₂×C₁₆×C₄₃	—	2	2	no	—
45	C₂×C₁₆×C₄₃	D₄	1	2	yes	—
46	C₂×Q₈×C₄₃	—	2	4	no	—
47	C₂×Q₈×C₄₃	—	4	2	no	—
48	C₂×Q₈×C₄₃	—	4	2	no	—
49	C₂×Q₈×C₄₃	—	4	2	no	—
50	C₂×Q₈×C₄₃	—	4	2	no	—
51	C₂×Q₈×C₄₃	—	4	2	no	—
52	C₂²×C₄×C₄₃	—	4	2	no	—
53	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₂²⋊D₄₃	—	2	1	no	—
55	#10621898	—	2	2	no	—
56	#10621898	—	2	2	no	—
57	C₂×D₁₇₂	D₈	1	1	yes	characteristic
58	C₂×C₄×D₄₃	—	2	1	no	—
59	C₂×Dic₈₆	D₈	1	1	yes	characteristic
60	Q₁₆×C₄₃	—	2	4	no	—
61	Q₁₆×C₄₃	—	2	4	no	—
62	Q₁₆×C₄₃	—	4	2	no	—
63	Q₁₆×C₄₃	—	4	2	no	—
64	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
65	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
68	#10621885	—	4	8	no	—
69	#10621884	—	4	8	no	—
70	#10621877	—	2	2	no	—
71	#10621877	—	2	4	no	—
72	#10621877	—	4	2	no	—
73	C₂×C₄₃⋊C₈	—	2	1	no	—
74	C₂×C₄₃⋊C₈	—	4	2	no	—
75	#10621872	—	4	2	no	—
76	C₈×D₄₃	—	4	1	no	—
77	C₈×D₄₃	—	4	1	no	—
78	C₁₆×C₄₃	—	2	2	no	—
79	C₄₃⋊C₁₆	—	4	1	no	—
80	C₂³×C₄₃	#135	1	1	yes	characteristic
81	C₂²×D₄₃	—	2	1	no	—
82	Q₈×C₄₃	—	4	4	no	—
83	Q₈×C₄₃	—	4	4	no	—
84	Q₈×C₄₃	—	8	2	no	—
85	Q₈×C₄₃	—	8	2	no	—
86	C₂×C₄×C₄₃	—	4	4	no	—
87	C₂×C₄×C₄₃	—	8	2	no	—
88	C₂×C₄×C₄₃	—	8	2	no	—
89	C₂×C₄×C₄₃	#120	1	2	yes	—
90	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
91	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
92	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
93	C₂²⋊D₄₃	—	4	2	no	—
94	C₂×Dic₄₃	—	2	1	no	—
95	D₁₇₂	—	2	2	no	—
96	C₄×D₄₃	—	4	1	no	—
97	C₄×D₄₃	—	4	1	no	—
98	Dic₈₆	—	2	2	no	—
99	C₈×C₄₃	#120	1	1	yes	characteristic
100	C₈×C₄₃	#121	1	2	yes	—
101	C₈×C₄₃	#127	1	1	yes	characteristic
102	C₄₃⋊C₈	—	4	1	no	—
103	C₄₃⋊C₈	—	4	4	no	—
104	#13315	—	43	1	no	maximal, Sylow
105	C₂²×C₄₃	—	2	2	no	—
106	C₂²×C₄₃	#449	1	2	yes	—
107	C₂²×C₄₃	#466	1	1	yes	characteristic
108	C₂×D₄₃	—	4	2	no	—
109	C₂×D₄₃	—	4	1	no	—
110	C₄×C₄₃	—	8	4	no	—
111	C₄×C₄₃	#392	1	1	yes	characteristic
112	C₄×C₄₃	#447	1	2	yes	—
113	C₄×C₄₃	#469	1	1	yes	characteristic
114	Dic₄₃	—	4	1	no	—
115	C₂²×Q₃₂	D₄₃	1	1	yes	characteristic, complemented
116	C₂×#473	—	43	2	no	—
117	C₂×#362	—	43	2	no	—
118	#2113	—	43	1	no	—
119	#1381	—	43	1	no	—
120	C₂×C₄₃	—	2	2	no	—
121	C₂×C₄₃	#1999	1	1	yes	characteristic
122	C₂×C₄₃	#2222	1	2	yes	—
123	D₄₃	—	8	1	no	—
124	C₂×#137	—	86	2	no	—
125	C₂²×Q₁₆	C₂×D₄₃	1	2	yes	—
126	C₂×#131	—	43	1	no	—
127	C₂×Q₃₂	—	2	4	no	—
128	C₂×Q₃₂	—	2	2	no	—
129	C₂×M₃₂	—	43	1	no	—
130	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
131	#473	—	86	4	no	—
132	#473	—	86	4	no	—
133	C₂×#108	—	43	2	no	—
134	C₂×#101	—	86	2	no	—
135	#362	—	86	4	no	—
136	C₂²⋊C₁₆	—	86	2	no	—
137	C₄₃	#13315	1	1	yes	characteristic, complemented, Sylow
138	C₂×#41	—	43	1	no	—
139	C₂²×Q₈	—	2	2	no	—
140	#137	—	86	8	no	—
141	#137	—	172	4	no	—
142	C₂×Q₁₆	—	2	2	no	—
143	C₂×Q₁₆	—	2	2	no	—
144	C₂×Q₁₆	—	2	2	no	—
145	C₂×Q₁₆	—	2	2	no	—
146	C₂×Q₁₆	—	2	2	no	—
147	C₂×Q₁₆	—	172	2	no	—
148	C₂×Q₁₆	C₂²⋊D₄₃	1	4	yes	—
149	C₂×SD₁₆	—	172	2	no	—
150	#131	—	86	2	no	—
151	#131	—	86	2	no	—
152	C₂×M₁₆	—	43	1	no	—
153	C₂×M₁₆	—	86	1	no	—
154	C₂×M₁₆	—	86	2	no	—
155	C₂²×C₈	—	86	1	no	—
156	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
157	Q₃₂	—	2	8	no	—
158	M₃₂	—	86	2	no	—
159	C₂×C₁₆	—	2	2	no	—
160	C₂×C₁₆	—	86	1	no	—
161	C₂×C₁₆	C₂²⋊D₄₃	1	2	yes	—
162	#108	—	43	4	no	—
163	#108	—	86	2	no	—
164	#101	—	86	8	no	—
165	#41	—	86	2	no	—
166	#41	—	86	2	no	—
167	C₂×Q₈	—	2	4	no	—
168	C₂×Q₈	—	4	2	no	—
169	C₂×Q₈	—	4	2	no	—
170	C₂×Q₈	—	4	2	no	—
171	C₂×Q₈	—	4	2	no	—
172	C₂×Q₈	—	4	2	no	—
173	C₂×Q₈	—	43	1	no	—
174	C₂×D₄	—	43	1	no	—
175	C₂×D₄	—	86	1	no	—
176	C₂²×C₄	—	4	2	no	—
177	C₂²×C₄	—	86	1	no	—
178	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
179	Q₁₆	—	2	4	no	—
180	Q₁₆	—	2	4	no	—
181	Q₁₆	—	4	2	no	—
182	Q₁₆	—	4	2	no	—
183	Q₁₆	—	172	8	no	—
184	SD₁₆	—	172	8	no	—
185	M₁₆	—	86	2	no	—
186	M₁₆	—	86	4	no	—
187	M₁₆	—	172	2	no	—
188	M₁₆	—	172	2	no	—
189	C₂×C₈	—	86	1	no	—
190	C₂×C₈	—	172	2	no	—
191	C₂×C₈	—	172	1	no	—
192	C₂×C₈	—	172	1	no	—
193	C₂×C₈	D₄×D₄₃	1	2	yes	—
194	C₂×C₈	#10621900	1	1	yes	characteristic
195	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
196	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
197	C₁₆	—	2	2	no	—
198	C₁₆	—	172	1	no	—
199	C₂³	—	86	1	no	—
200	C₂³	#13063344	1	1	yes	characteristic
201	Q₈	—	4	4	no	—
202	Q₈	—	4	4	no	—
203	Q₈	—	8	2	no	—
204	Q₈	—	8	2	no	—
205	Q₈	—	86	2	no	—
206	D₄	—	86	2	no	—
207	D₄	—	172	2	no	—
208	C₂×C₄	—	4	4	no	—
209	C₂×C₄	—	8	2	no	—
210	C₂×C₄	—	8	2	no	—
211	C₂×C₄	—	86	1	no	—
212	C₂×C₄	—	172	1	no	—
213	C₂×C₄	—	172	1	no	—
214	C₂×C₄	#13063293	1	2	yes	—
215	C₂×C₄	#13063296	1	1	yes	characteristic
216	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
217	C₂×C₄	#13063349	1	2	yes	—
218	C₈	—	172	1	no	—
219	C₈	—	172	4	no	—
220	C₈	#13063294	1	1	yes	characteristic
221	C₈	#13063297	1	2	yes	—
222	C₈	#13063321	1	1	yes	characteristic
223	C₂²	—	2	2	no	—
224	C₂²	—	172	1	no	—
225	C₂²	—	172	2	no	—
226	C₂²	#14583041	1	2	yes	—
227	C₂²	#14583132	1	1	yes	characteristic
228	C₄	—	8	4	no	—
229	C₄	—	172	1	no	—
230	C₄	#14582807	1	1	yes	characteristic
231	C₄	#14583031	1	2	yes	—
232	C₄	#14583149	1	1	yes	characteristic
233	C₂	—	2	2	no	—
234	C₂	—	344	1	no	—
235	C₂	#19135590	1	1	yes	characteristic
236	C₂	#19136717	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14583171

C₂

yes

maximal

C₂×#14582715

C₂

yes

maximal, complemented

#19136208

C₂

yes

maximal, characteristic

#19132689

C₂

yes

maximal, characteristic, complemented

C₂²×Q₃₂×C₄₃

C₂

yes

maximal, characteristic, complemented

C₂×#13063356

—

C₂×#13063331

C₂²

yes

characteristic

C₂×#13063265

C₂²

yes

—

—

C₂²

yes

—

C₂×#13063244

—

#14582715

—

(C₂×D₄₃)⋊C₁₆

—

C₂²×Q₁₆×C₄₃

C₂²

yes

—

C₂×Q₃₂×C₄₃

—

C₂×Q₃₂×C₄₃

—

C₂²×C₁₆×C₄₃

C₂²

yes

characteristic

C₂×#10621898

D₄

yes

—

—

—

—

—

—

—

D₄

yes

characteristic

C₂×#10621872

—

C₂×C₈×D₄₃

—

#13063265

—

C₂×C₄₃⋊C₁₆

—

#13063261

—

#13063261

D₄

yes

—

#13063244

—

C₂²×Q₈×C₄₃

—

C₂×Q₁₆×C₄₃

—

C₂×Q₁₆×C₄₃

—

C₂×Q₁₆×C₄₃

—

C₂×Q₁₆×C₄₃

—

C₂×Q₁₆×C₄₃

—

C₂×Q₁₆×C₄₃

D₄

yes

—

C₂²×C₈×C₄₃

C₂³

yes

characteristic

Q₃₂×C₄₃

—

C₂×C₁₆×C₄₃

—

C₂×C₁₆×C₄₃

D₄

yes

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₂²⋊D₄₃

—

#10621898

—

#10621898

—

C₂×D₁₇₂

D₈

yes

characteristic

C₂×C₄×D₄₃

—

C₂×Dic₈₆

D₈

yes

characteristic

Q₁₆×C₄₃

—

Q₁₆×C₄₃

—

Q₁₆×C₄₃

—

Q₁₆×C₄₃

—

C₂×C₈×C₄₃

C₂×D₄

yes

—

C₂×C₈×C₄₃

C₂×D₄

yes

—

C₂×C₈×C₄₃

C₂×D₄

yes

C₂×C₈×C₄₃

yes

—

—

—

—

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

#10621872

—

C₈×D₄₃

—

C₈×D₄₃

—

C₁₆×C₄₃

—

C₄₃⋊C₁₆

—

C₂³×C₄₃

#135

yes

characteristic

C₂²×D₄₃

—

Q₈×C₄₃

—

Q₈×C₄₃

—

Q₈×C₄₃

—

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#120

yes

—

C₂×C₄×C₄₃

#121

yes

characteristic

C₂×C₄×C₄₃

C₂×D₈

yes

characteristic

C₂×C₄×C₄₃

AGL(1,ℤ/8)

yes

—

C₂²⋊D₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

C₄×D₄₃

—

C₄×D₄₃

—

Dic₈₆

—

C₈×C₄₃

#120

yes

characteristic

100

C₈×C₄₃

#121

yes

—

101

C₈×C₄₃

#127

yes

characteristic

102

C₄₃⋊C₈

—

103

C₄₃⋊C₈

—

104

#13315

—

maximal, Sylow

105

C₂²×C₄₃

—

106

C₂²×C₄₃

#449

yes

—

107

C₂²×C₄₃

#466

yes

characteristic

108

C₂×D₄₃

—

109

C₂×D₄₃

—

110

C₄×C₄₃

—

111

C₄×C₄₃

#392

yes

characteristic

112

C₄×C₄₃

#447

yes

—

113

C₄×C₄₃

#469

yes

characteristic

114

Dic₄₃

—

115

C₂²×Q₃₂

D₄₃

yes

characteristic, complemented

116

C₂×#473

—

117

C₂×#362

—

118

#2113

—

119

#1381

—

120

C₂×C₄₃

—

121

C₂×C₄₃

#1999

yes

characteristic

122

C₂×C₄₃

#2222

yes

—

123

D₄₃

—

124

C₂×#137

—

125

C₂²×Q₁₆

C₂×D₄₃

yes

—

126

C₂×#131

—

127

C₂×Q₃₂

—

128

C₂×Q₃₂

—

129

C₂×M₃₂

—

130

C₂²×C₁₆

C₂×D₄₃

yes

characteristic

131

#473

—

132

#473

—

133

C₂×#108

—

134

C₂×#101

—

135

#362

—

136

C₂²⋊C₁₆

—

137

C₄₃

#13315

yes

characteristic, complemented, Sylow

138

C₂×#41

—

139

C₂²×Q₈

—

140

#137

—

141

#137

—

172

—

142

C₂×Q₁₆

—

143

C₂×Q₁₆

—

144

C₂×Q₁₆

—

145

C₂×Q₁₆

—

146

C₂×Q₁₆

—

147

C₂×Q₁₆

—

172

—

148

C₂×Q₁₆

C₂²⋊D₄₃

yes

—

149

C₂×SD₁₆

—

172

—

150

#131

—

151

#131

—

152

C₂×M₁₆

—

153

C₂×M₁₆

—

154

C₂×M₁₆

—

155

C₂²×C₈

—

156

C₂²×C₈

C₂²×D₄₃

yes

characteristic

157

Q₃₂

—

158

M₃₂

—

159

C₂×C₁₆

—

160

C₂×C₁₆

—

161

C₂×C₁₆

C₂²⋊D₄₃

yes

—

162

#108

—

163

#108

—

164

#101

—

165

#41

—

166

#41

—

167

C₂×Q₈

—

168

C₂×Q₈

—

169

C₂×Q₈

—

170

C₂×Q₈

—

171

C₂×Q₈

—

172

C₂×Q₈

—

173

C₂×Q₈

—

174

C₂×D₄

—

175

C₂×D₄

—

176

C₂²×C₄

—

177

C₂²×C₄

—

178

C₂²×C₄

D₄×D₄₃

yes

characteristic

179

Q₁₆

—

180

Q₁₆

—

181

Q₁₆

—

182

Q₁₆

—

183

Q₁₆

—

172

—

184

SD₁₆

—

172

—

185

M₁₆

—

186

M₁₆

—

187

M₁₆

—

172

—

188

M₁₆

—

172

—

189

C₂×C₈

—

190

C₂×C₈

—

172

—

191

C₂×C₈

—

172

—

192

C₂×C₈

—

172

—

193

C₂×C₈

D₄×D₄₃

yes

—

194

C₂×C₈

#10621900

yes

characteristic

195

C₂×C₈

C₂×C₂²⋊D₄₃

yes

—

196

C₂×C₈

C₂×C₂²⋊D₄₃

yes

characteristic

197

C₁₆

—

198

C₁₆

—

172

—

199

C₂³

—

200

C₂³

#13063344

yes

characteristic

201

Q₈

—

202

Q₈

—

203

Q₈

—

204

Q₈

—

205

Q₈

—

206

D₄

—

207

D₄

—

172

—

208

C₂×C₄

—

209

C₂×C₄

—

210

C₂×C₄

—

211

C₂×C₄

—

212

C₂×C₄

—

172

—

213

C₂×C₄

—

172

—

214

C₂×C₄

#13063293

yes

—

215

C₂×C₄

#13063296

yes

characteristic

216

C₂×C₄

D₈×D₄₃

yes

characteristic

217

C₂×C₄

#13063349

yes

—

218

C₈

—

172

—

219

C₈

—

172

—

220

C₈

#13063294

yes

characteristic

221

C₈

#13063297

yes

—

222

C₈

#13063321

yes

characteristic

223

C₂²

—

224

C₂²

—

172

—

225

C₂²

—

172

—

226

C₂²

#14583041

yes

—

227

C₂²

#14583132

yes

characteristic

228

C₄

—

229

C₄

—

172

—

230

C₄

#14582807

yes

characteristic

231

C₄

#14583031

yes

—

232

C₄

#14583149

yes

characteristic

233

C₂

—

234

C₂

—

344

—

235

C₂

#19135590

yes

characteristic

236

C₂

#19136717

yes

—

	subgroup	G/H
center	C₂²	#14583132
commutator, cosocle	C₂²×C₈×C₄₃	C₂³
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₂²×Q₃₂×C₄₃	C₂
socle	C₂²×C₄₃	#466
prime radical	C₂×#10621898	D₄
hypercenter	C₂²×Q₃₂	D₄₃
hypocenter	C₄₃	#13315
LCS	G ▷ C₂²×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₃₂

subgroup

G/H

center

C₂²

#14583132

commutator, cosocle

C₂²×C₈×C₄₃

C₂³

Frattini

C₂²×C₈

C₂²×D₄₃

Fitting

C₂²×Q₃₂×C₄₃

C₂

C₂²×C₄₃

D₄

C₂²×Q₃₂

D₄₃

hypocenter

C₄₃

#13315

LCS

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₃₂

N	H ≅ G/N	type
2. C₂×#14582715	C₂	semidirect
4. #19132689	C₂	semidirect
5. C₂²×Q₃₂×C₄₃	C₂	semidirect
115. C₂²×Q₃₂	D₄₃	semidirect
137. C₄₃	#13315	semidirect

H ≅ G/N

type

2. C₂×#14582715

C₂

semidirect

4. #19132689

C₂

semidirect

5. C₂²×Q₃₂×C₄₃

C₂

semidirect

115. C₂²×Q₃₂

D₄₃

semidirect

137. C₄₃

#13315

semidirect

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

subgroup	index
C₂×#14583171	2
C₂×#14582715	2
#19136208	2
#19132689	2
C₂²×Q₃₂×C₄₃	2
#13315	43

subgroup

index

C₂²×Q₃₂×C₄₃

Subgroups — #25015358

Conjugacy classes of subgroups

Complements

Maximal subgroups