Group #25110493 — subgroups

Subgroups — #25110493

Order 11008.

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

normal subgroups only

This group has 826 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	#19145041	C₂	1	1	yes	maximal, characteristic
2	#19143314	C₂	1	2	yes	maximal, complemented
3	#19135423	C₂	1	2	yes	maximal, complemented
4	#19134921	C₂	1	8	yes	maximal, complemented
5	#19134736	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₃×#1745	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#13063347	C₂²	1	4	yes	—
8	C₂×#13063331	C₂²	1	4	yes	—
9	#14583449	C₂²	1	2	yes	—
10	#14583293	C₄	1	4	yes	complemented
11	#14583288	C₄	1	4	yes	complemented
12	#14582972	—	2	8	no	—
13	#14582923	C₂²	1	2	yes	—
14	C₂×#13063253	C₂²	1	8	yes	—
15	#14582865	C₂²	1	4	yes	—
16	#14582664	C₂²	1	4	yes	—
17	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
18	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
19	C₄³×C₄₃	C₂²	1	1	yes	characteristic
20	C₄₃×#336	C₂²	1	4	yes	—
21	C₂×#10621898	C₂³	1	4	yes	—
22	#13063347	—	4	8	no	—
23	C₂×Dic₁₇₂	C₂×C₄	1	4	yes	—
24	C₂×#10621873	C₂×C₄	1	8	yes	—
25	C₂×D₃₄₄	C₂×C₄	1	4	yes	—
26	#13063331	—	4	8	no	—
27	C₂×#10621877	C₂³	1	4	yes	—
28	C₂×#10621872	—	2	4	no	—
29	C₂×C₈×D₄₃	—	2	4	no	—
30	#13063325	C₂×C₄	1	2	yes	—
31	#13063322	C₂×C₄	1	2	yes	—
32	#13063313	C₂×C₄	1	4	yes	—
33	#13063299	—	2	4	no	—
34	C₄×D₁₇₂	—	2	4	no	—
35	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
36	#13063253	—	4	8	no	—
37	(C₂×D₄₃)⋊C₈	—	2	8	no	—
38	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
39	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
40	C₂×C₄²×C₄₃	—	2	2	no	—
41	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
42	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
43	#97×C₄₃	C₂×C₄	1	4	yes	—
44	#97×C₄₃	D₄	1	4	yes	—
45	C₂²×C₄×C₄₃	—	2	2	no	—
46	C₂²×C₄×C₄₃	#41	1	2	yes	—
47	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
48	C₂×C₂²⋊D₄₃	—	2	4	no	—
49	#10621898	—	2	8	no	—
50	#10621898	—	2	8	no	—
51	C₂×D₁₇₂	—	2	4	no	—
52	C₂×D₁₇₂	C₂²×C₄	1	4	yes	—
53	C₂×C₄×D₄₃	—	2	4	no	—
54	C₂×Dic₈₆	C₂²×C₄	1	4	yes	—
55	C₂×C₈×C₄₃	—	2	4	no	—
56	C₂×C₈×C₄₃	—	2	4	no	—
57	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
58	C₂×C₈×C₄₃	C₂×D₄	1	4	yes	—
59	C₄²×C₄₃	—	2	4	no	—
60	C₄²×C₄₃	—	4	4	no	—
61	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
62	C₄²×C₄₃	C₂×D₄	1	2	yes	—
63	(C₂×D₄₃)⋊C₄	—	2	8	no	—
64	C₄⋊Dic₄₃	—	2	4	no	—
65	#10621877	—	2	8	no	—
66	C₂×C₄₃⋊C₈	—	2	4	no	—
67	Dic₁₇₂	—	4	4	no	—
68	D₃₄₄	—	4	4	no	—
69	#10621873	—	4	8	no	—
70	#10621872	—	4	8	no	—
71	C₈×D₄₃	—	4	8	no	—
72	C₂³×C₄₃	C₂×#41	1	1	yes	characteristic
73	C₂²×D₄₃	—	2	4	no	—
74	C₂×C₄×C₄₃	—	2	4	no	—
75	C₂×C₄×C₄₃	—	4	2	no	—
76	C₂×C₄×C₄₃	—	4	2	no	—
77	C₂×C₄×C₄₃	—	4	2	no	—
78	C₂×C₄×C₄₃	C₄×D₄	1	4	yes	—
79	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
80	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
81	C₂×C₄×C₄₃	H(5,𝔽₂)	1	4	yes	—
82	C₂²⋊D₄₃	—	4	8	no	—
83	C₂×Dic₄₃	—	2	4	no	—
84	D₁₇₂	—	2	8	no	—
85	D₁₇₂	—	8	4	no	—
86	C₄×D₄₃	—	4	4	no	—
87	C₄×D₄₃	—	4	4	no	—
88	Dic₈₆	—	2	8	no	—
89	C₈×C₄₃	—	2	4	no	—
90	C₈×C₄₃	—	2	4	no	—
91	C₄₃⋊C₈	—	4	4	no	—
92	#22283	—	43	1	no	maximal, Sylow
93	C₂²×C₄₃	—	2	2	no	—
94	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
95	C₂²×C₄₃	#540	1	2	yes	—
96	C₂×D₄₃	—	4	4	no	—
97	C₂×D₄₃	—	4	8	no	—
98	C₄×C₄₃	—	2	4	no	—
99	C₄×C₄₃	—	4	4	no	—
100	C₄×C₄₃	#518	1	2	yes	—
101	C₄×C₄₃	#535	1	2	yes	—
102	Dic₄₃	—	4	4	no	—
103	#3150	—	43	1	no	—
104	#2932	—	43	2	no	—
105	#1962	—	43	2	no	—
106	#1876	—	43	8	no	—
107	#1838	—	43	1	no	—
108	#1745	D₄₃	1	1	yes	characteristic, complemented
109	C₂×C₄₃	—	2	2	no	—
110	C₂×C₄₃	#2328	1	1	yes	characteristic
111	C₂×C₄₃	#2956	1	2	yes	—
112	D₄₃	—	8	4	no	—
113	C₂×#135	—	43	4	no	—
114	C₂×#131	—	43	4	no	—
115	#532	—	43	2	no	—
116	#497	—	43	4	no	—
117	#496	—	43	4	no	—
118	#435	—	86	8	no	—
119	#423	—	43	2	no	—
120	C₂×#104	—	43	8	no	—
121	#408	—	43	4	no	—
122	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
123	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
124	C₄³	C₂×D₄₃	1	1	yes	characteristic
125	#341	—	43	4	no	—
126	#336	C₂×D₄₃	1	4	yes	—
127	C₄₃	#22283	1	1	yes	characteristic, complemented, Sylow
128	C₂×#41	—	43	4	no	—
129	#135	—	172	8	no	—
130	C₂×Q₁₆	—	43	4	no	—
131	C₂×SD₁₆	—	43	8	no	—
132	C₂×D₈	—	43	4	no	—
133	#131	—	172	8	no	—
134	C₂×M₁₆	—	43	4	no	—
135	C₂×M₁₆	—	86	4	no	—
136	C₂²×C₈	—	86	4	no	—
137	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
138	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
139	C₄⋊Q₈	—	43	2	no	—
140	#127	—	43	2	no	—
141	#124	—	43	4	no	—
142	#121	—	86	4	no	—
143	C₄×D₄	—	86	4	no	—
144	C₂×C₄²	—	2	2	no	—
145	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
146	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
147	C₄⋊C₈	—	86	4	no	—
148	#104	—	172	8	no	—
149	C₂²⋊C₈	—	86	8	no	—
150	#97	C₄×D₄₃	1	4	yes	—
151	#97	D₁₇₂	1	4	yes	—
152	#41	—	86	8	no	—
153	#41	—	86	8	no	—
154	C₂×Q₈	—	43	4	no	—
155	C₂×D₄	—	43	4	no	—
156	C₂×D₄	—	86	4	no	—
157	C₂×D₄	—	86	4	no	—
158	C₂²×C₄	—	2	2	no	—
159	C₂²×C₄	—	86	4	no	—
160	C₂²×C₄	#10621898	1	2	yes	—
161	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
162	Q₁₆	—	172	4	no	—
163	SD₁₆	—	172	8	no	—
164	D₈	—	172	4	no	—
165	M₁₆	—	86	8	no	—
166	M₁₆	—	172	8	no	—
167	C₂×C₈	—	2	4	no	—
168	C₂×C₈	—	2	4	no	—
169	C₂×C₈	—	86	4	no	—
170	C₂×C₈	—	172	8	no	—
171	C₂×C₈	C₂×C₄×D₄₃	1	4	yes	—
172	C₂×C₈	C₂×D₁₇₂	1	4	yes	—
173	C₄⋊C₄	—	86	4	no	—
174	C₂²⋊C₄	—	86	8	no	—
175	C₄²	—	2	4	no	—
176	C₄²	—	4	4	no	—
177	C₄²	C₂×C₄×D₄₃	1	2	yes	—
178	C₄²	C₂×D₁₇₂	1	2	yes	—
179	C₂³	—	86	4	no	—
180	C₂³	C₂×#10621898	1	1	yes	characteristic
181	Q₈	—	86	8	no	—
182	D₄	—	86	8	no	—
183	D₄	—	344	4	no	—
184	D₄	—	172	8	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	86	4	no	—
190	C₂×C₄	—	172	4	no	—
191	C₂×C₄	—	172	4	no	—
192	C₂×C₄	C₄×D₁₇₂	1	4	yes	—
193	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
194	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
195	C₂×C₄	#13063371	1	4	yes	—
196	C₈	—	2	4	no	—
197	C₈	—	2	4	no	—
198	C₈	—	172	4	no	—
199	C₂²	—	2	2	no	—
200	C₂²	—	172	4	no	—
201	C₂²	—	172	8	no	—
202	C₂²	C₂×C₄×D₁₇₂	1	1	yes	characteristic
203	C₂²	#14583504	1	2	yes	—
204	C₄	—	2	4	no	—
205	C₄	—	4	4	no	—
206	C₄	—	172	4	no	—
207	C₄	#14583373	1	2	yes	—
208	C₄	#14583468	1	2	yes	—
209	C₂	—	2	2	no	—
210	C₂	—	344	4	no	—
211	C₂	#19137320	1	1	yes	characteristic
212	C₂	#19143506	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19145041

C₂

yes

maximal, characteristic

#19143314

C₂

yes

maximal, complemented

#19135423

C₂

yes

maximal, complemented

#19134921

C₂

yes

maximal, complemented

#19134736

C₂

yes

maximal, characteristic, complemented

C₄₃×#1745

C₂

yes

maximal, characteristic, complemented

C₂×#13063347

C₂²

yes

—

C₂×#13063331

C₂²

yes

—

#14583449

C₂²

yes

—

#14583293

C₄

yes

complemented

#14583288

C₄

yes

complemented

#14582972

—

#14582923

C₂²

yes

—

C₂×#13063253

C₂²

yes

—

#14582865

C₂²

yes

—

#14582664

C₂²

yes

—

C₂×#97×C₄₃

C₂²

yes

characteristic

C₂×#97×C₄₃

C₂²

yes

characteristic

C₄³×C₄₃

C₂²

yes

characteristic

C₄₃×#336

C₂²

yes

—

C₂×#10621898

C₂³

yes

—

#13063347

—

C₂×Dic₁₇₂

C₂×C₄

yes

—

C₂×#10621873

C₂×C₄

yes

—

C₂×D₃₄₄

C₂×C₄

yes

—

#13063331

—

C₂×#10621877

C₂³

yes

—

C₂×#10621872

—

C₂×C₈×D₄₃

—

#13063325

C₂×C₄

yes

—

#13063322

C₂×C₄

yes

—

#13063313

C₂×C₄

yes

—

#13063299

—

C₄×D₁₇₂

—

C₄⋊(C₄₃⋊C₈)

—

#13063253

—

(C₂×D₄₃)⋊C₈

—

C₂²×C₈×C₄₃

C₂³

yes

—

C₂²×C₈×C₄₃

C₂³

yes

—

C₂×C₄²×C₄₃

—

C₂×C₄²×C₄₃

C₂³

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

#97×C₄₃

C₂×C₄

yes

—

#97×C₄₃

D₄

yes

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

#41

yes

—

C₂²×C₄×C₄₃

C₂⁴

yes

characteristic

C₂×C₂²⋊D₄₃

—

#10621898

—

#10621898

—

C₂×D₁₇₂

—

C₂×D₁₇₂

C₂²×C₄

yes

—

C₂×C₄×D₄₃

—

C₂×Dic₈₆

C₂²×C₄

yes

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂²×C₄

yes

—

C₂×C₈×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

C₂²×C₄

yes

—

C₄²×C₄₃

C₂×D₄

yes

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

#10621877

—

C₂×C₄₃⋊C₈

—

Dic₁₇₂

—

D₃₄₄

—

#10621873

—

#10621872

—

C₈×D₄₃

—

C₂³×C₄₃

C₂×#41

yes

characteristic

C₂²×D₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₄×D₄

yes

—

C₂×C₄×C₄₃

C₂³×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂²×D₄

yes

characteristic

C₂×C₄×C₄₃

H(5,𝔽₂)

yes

—

C₂²⋊D₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

C₄×D₄₃

—

C₄×D₄₃

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

#22283

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

C₂×C₄×D₄

yes

characteristic

C₂²×C₄₃

#540

yes

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

100

C₄×C₄₃

#518

yes

—

101

C₄×C₄₃

#535

yes

—

102

Dic₄₃

—

103

#3150

—

104

#2932

—

105

#1962

—

106

#1876

—

107

#1838

—

108

#1745

D₄₃

yes

characteristic, complemented

109

C₂×C₄₃

—

110

C₂×C₄₃

#2328

yes

characteristic

111

C₂×C₄₃

#2956

yes

—

112

D₄₃

—

113

C₂×#135

—

114

C₂×#131

—

115

#532

—

116

#497

—

117

#496

—

118

#435

—

119

#423

—

120

C₂×#104

—

121

#408

—

122

C₂×#97

C₂×D₄₃

yes

characteristic

123

C₂×#97

C₂×D₄₃

yes

characteristic

124

C₄³

C₂×D₄₃

yes

characteristic

125

#341

—

126

#336

C₂×D₄₃

yes

—

127

C₄₃

#22283

yes

characteristic, complemented, Sylow

128

C₂×#41

—

129

#135

—

172

—

130

C₂×Q₁₆

—

131

C₂×SD₁₆

—

132

C₂×D₈

—

133

#131

—

172

—

134

C₂×M₁₆

—

135

C₂×M₁₆

—

136

C₂²×C₈

—

137

C₂²×C₈

C₂²×D₄₃

yes

—

138

C₂²×C₈

C₂²×D₄₃

yes

—

139

C₄⋊Q₈

—

140

#127

—

141

#124

—

142

#121

—

143

C₄×D₄

—

144

C₂×C₄²

—

145

C₂×C₄²

C₂²×D₄₃

yes

—

146

C₂×C₄²

C₂²×D₄₃

yes

characteristic

147

C₄⋊C₈

—

148

#104

—

172

—

149

C₂²⋊C₈

—

150

#97

C₄×D₄₃

yes

—

151

#97

D₁₇₂

yes

—

152

#41

—

153

#41

—

154

C₂×Q₈

—

155

C₂×D₄

—

156

C₂×D₄

—

157

C₂×D₄

—

158

C₂²×C₄

—

159

C₂²×C₄

—

160

C₂²×C₄

#10621898

yes

—

161

C₂²×C₄

C₂³×D₄₃

yes

characteristic

162

Q₁₆

—

172

—

163

SD₁₆

—

172

—

164

D₈

—

172

—

165

M₁₆

—

166

M₁₆

—

172

—

167

C₂×C₈

—

168

C₂×C₈

—

169

C₂×C₈

—

170

C₂×C₈

—

172

—

171

C₂×C₈

C₂×C₄×D₄₃

yes

—

172

C₂×C₈

C₂×D₁₇₂

yes

—

173

C₄⋊C₄

—

174

C₂²⋊C₄

—

175

C₄²

—

176

C₄²

—

177

C₄²

C₂×C₄×D₄₃

yes

—

178

C₄²

C₂×D₁₇₂

yes

—

179

C₂³

—

180

C₂³

C₂×#10621898

yes

characteristic

181

Q₈

—

182

D₄

—

183

D₄

—

344

—

184

D₄

—

172

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

172

—

191

C₂×C₄

—

172

—

192

C₂×C₄

C₄×D₁₇₂

yes

—

193

C₂×C₄

C₂²×C₄×D₄₃

yes

characteristic

194

C₂×C₄

C₂²×D₁₇₂

yes

characteristic

195

C₂×C₄

#13063371

yes

—

196

C₈

—

197

C₈

—

198

C₈

—

172

—

199

C₂²

—

200

C₂²

—

172

—

201

C₂²

—

172

—

202

C₂²

C₂×C₄×D₁₇₂

yes

characteristic

203

C₂²

#14583504

yes

—

204

C₄

—

205

C₄

—

206

C₄

—

172

—

207

C₄

#14583373

yes

—

208

C₄

#14583468

yes

—

209

C₂

—

210

C₂

—

344

—

211

C₂

#19137320

yes

characteristic

212

C₂

#19143506

yes

—

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1745	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
prime radical	#19145041	C₂
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1745	D₄₃
hypocenter	C₄₃	#22283
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ #1745

subgroup

G/H

center

C₂×C₄

C₂²×D₁₇₂

commutator

C₂×C₄×C₄₃

C₂³×C₄

Frattini

C₂²×C₄

C₂³×D₄₃

Fitting

C₄₃×#1745

C₂

socle

C₂²×C₄₃

C₂×C₄×D₄

prime radical

#19145041

C₂

cosocle

C₂²×C₄×C₄₃

C₂⁴

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ #1745

N	H ≅ G/N	type
2. #19143314	C₂	semidirect
3. #19135423	C₂	semidirect
4. #19134921	C₂	semidirect
5. #19134736	C₂	semidirect
6. C₄₃×#1745	C₂	semidirect
10. #14583293	C₄	semidirect
11. #14583288	C₄	semidirect
108. #1745	D₄₃	semidirect
127. C₄₃	#22283	semidirect

H ≅ G/N

type

2. #19143314

C₂

semidirect

3. #19135423

C₂

semidirect

4. #19134921

C₂

semidirect

5. #19134736

C₂

semidirect

6. C₄₃×#1745

C₂

semidirect

10. #14583293

C₄

semidirect

11. #14583288

C₄

semidirect

108. #1745

D₄₃

semidirect

127. C₄₃

#22283

semidirect

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

subgroup	index
#19145041	2
#19143314	2
#19135423	2
#19134921	2
#19134736	2
C₄₃×#1745	2
#22283	43

subgroup

index

C₄₃×#1745

Subgroups — #25110493

Conjugacy classes of subgroups

Complements

Maximal subgroups