Group #25110441 — subgroups

Subgroups — #25110441

Order 11008.

1050 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 1050 conjugacy classes of subgroups — too many to draw. Only its 227 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×#14583276	C₂	1	1	yes	maximal, characteristic
2	#19137320	C₂	1	2	yes	maximal, complemented
3	#19135419	C₂	1	2	yes	maximal, complemented
4	#19134942	C₂	1	8	yes	maximal, complemented
5	#19134726	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₃×#1746	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₃₄₄	C₂²	1	4	yes	—
8	C₂×#13063322	C₂²	1	2	yes	—
9	C₂×C₄×D₁₇₂	C₂²	1	4	yes	—
10	#14583276	C₄	1	8	yes	complemented
11	C₂×#13063247	C₂²	1	8	yes	—
12	#14582801	C₂²	1	4	yes	—
13	#14582777	—	2	8	no	—
14	#14582765	C₂²	1	2	yes	—
15	#14582662	C₂²	1	4	yes	—
16	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
18	C₄₃×#376	C₂²	1	1	yes	characteristic
19	C₄₃×#336	C₂²	1	4	yes	—
20	C₂²×D₁₇₂	—	2	4	no	—
21	C₂²×D₁₇₂	C₂³	1	4	yes	—
22	C₂²×C₄×D₄₃	—	2	4	no	—
23	C₂×D₃₄₄	—	4	8	no	—
24	C₂×D₃₄₄	C₂×C₄	1	16	yes	complemented
25	#13063322	—	2	4	no	—
26	#13063322	C₂×C₄	1	8	yes	complemented
27	C₄×D₁₇₂	—	4	8	no	—
28	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
29	C₂×(C₂×D₄₃)⋊C₄	—	2	4	no	—
30	C₂×(C₂×D₄₃)⋊C₄	—	2	8	no	—
31	#13063247	—	4	8	no	—
32	C₄₃⋊#95	—	2	4	no	—
33	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
34	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
35	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
36	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
37	#97×C₄₃	—	2	2	no	—
38	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	complemented
39	#95×C₄₃	D₄	1	4	yes	—
40	C₂³×D₄₃	—	2	4	no	—
41	C₂²×C₄×C₄₃	C₂×D₄	1	4	yes	—
42	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
43	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
44	C₂²×Dic₄₃	—	2	4	no	—
45	C₂×D₁₇₂	—	2	4	no	—
46	C₂×D₁₇₂	—	2	16	no	—
47	C₂×D₁₇₂	—	2	8	no	—
48	C₂×D₁₇₂	—	2	8	no	—
49	C₂×D₁₇₂	—	8	8	no	—
50	C₂×D₁₇₂	C₂²×C₄	1	8	yes	—
51	C₂×C₄×D₄₃	—	4	8	no	—
52	C₂×C₄×D₄₃	—	8	8	no	—
53	C₂×C₈×C₄₃	—	2	4	no	—
54	C₂×C₈×C₄₃	—	2	2	no	—
55	C₂×C₈×C₄₃	—	2	4	no	—
56	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
57	C₄²×C₄₃	—	2	4	no	—
58	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
59	C₄²×C₄₃	#41	1	2	yes	—
60	(C₂×D₄₃)⋊C₄	—	4	8	no	—
61	(C₂×D₄₃)⋊C₄	—	8	8	no	—
62	C₄⋊Dic₄₃	—	2	8	no	—
63	D₃₄₄	—	4	16	no	—
64	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
65	C₂²×D₄₃	—	2	8	no	—
66	C₂²×D₄₃	—	4	4	no	—
67	C₂²×D₄₃	—	4	4	no	—
68	C₂²×D₄₃	—	4	8	no	—
69	C₂²×D₄₃	—	4	8	no	—
70	C₂×C₄×C₄₃	—	2	4	no	—
71	C₂×C₄×C₄₃	—	2	4	no	—
72	C₂×C₄×C₄₃	—	2	8	no	—
73	C₂×C₄×C₄₃	C₄×D₄	1	4	yes	—
74	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
75	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
76	C₂×C₄×C₄₃	H(5,𝔽₂)	1	4	yes	—
77	C₂×Dic₄₃	—	4	4	no	—
78	C₂×Dic₄₃	—	4	8	no	—
79	D₁₇₂	—	2	16	no	—
80	D₁₇₂	—	4	8	no	—
81	D₁₇₂	—	8	16	no	—
82	C₄×D₄₃	—	8	16	no	—
83	C₈×C₄₃	—	2	4	no	—
84	C₈×C₄₃	—	4	2	no	—
85	#22276	—	43	1	no	maximal, Sylow
86	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
87	C₂²×C₄₃	#535	1	2	yes	—
88	C₂²×C₄₃	#576	1	4	yes	—
89	C₂×D₄₃	—	4	8	no	—
90	C₂×D₄₃	—	4	16	no	—
91	C₂×D₄₃	—	8	4	no	—
92	C₂×D₄₃	—	8	8	no	—
93	C₂×D₄₃	—	8	4	no	—
94	C₄×C₄₃	—	2	4	no	—
95	C₄×C₄₃	—	4	4	no	—
96	C₄×C₄₃	#518	1	2	yes	—
97	C₄×C₄₃	#540	1	2	yes	—
98	Dic₄₃	—	8	4	no	—
99	C₂×#493	—	43	1	no	—
100	#2328	—	43	2	no	—
101	#1961	—	43	2	no	—
102	#1879	—	43	8	no	—
103	#1835	—	43	1	no	—
104	#1746	D₄₃	1	1	yes	characteristic, complemented
105	C₂×C₄₃	#2328	1	1	yes	characteristic
106	C₂×C₄₃	#2948	1	2	yes	—
107	C₂×C₄₃	#3220	1	4	yes	—
108	D₄₃	—	8	8	no	—
109	C₂²×D₈	—	43	4	no	—
110	C₂×#127	—	43	2	no	—
111	C₂×C₄×D₄	—	43	4	no	—
112	#493	—	43	8	no	—
113	C₂×#102	—	43	8	no	—
114	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
115	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
116	#390	—	43	4	no	—
117	#386	—	86	8	no	—
118	#382	—	43	2	no	—
119	#376	C₂×D₄₃	1	1	yes	characteristic
120	#340	—	43	4	no	—
121	#336	C₂×D₄₃	1	4	yes	—
122	C₄₃	#22276	1	1	yes	characteristic, complemented, Sylow
123	C₂²×D₄	—	43	4	no	—
124	C₂²×D₄	—	86	4	no	—
125	C₂³×C₄	—	86	4	no	—
126	C₂×D₈	—	43	16	no	—
127	C₂×D₈	—	172	8	no	—
128	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
129	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
130	#127	—	43	8	no	—
131	#127	—	86	4	no	—
132	C₄×D₄	—	172	8	no	—
133	C₂×C₄⋊C₄	—	43	4	no	—
134	C₂×C₂²⋊C₄	—	86	8	no	—
135	C₂×C₂²⋊C₄	—	86	4	no	—
136	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
137	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
138	#102	—	172	8	no	—
139	#97	—	2	2	no	—
140	C₄×C₈	C₄×D₄₃	1	4	yes	complemented
141	#95	—	86	4	no	—
142	#95	D₁₇₂	1	4	yes	—
143	C₂⁴	—	86	4	no	—
144	C₂×D₄	—	43	8	no	—
145	C₂×D₄	—	86	16	no	—
146	C₂×D₄	—	86	4	no	—
147	C₂×D₄	—	86	8	no	—
148	C₂×D₄	—	86	8	no	—
149	C₂×D₄	—	344	8	no	—
150	C₂²×C₄	—	86	4	no	—
151	C₂²×C₄	—	344	8	no	—
152	C₂²×C₄	—	172	8	no	—
153	C₂²×C₄	C₂×D₁₇₂	1	4	yes	—
154	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
155	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
156	D₈	—	172	16	no	—
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	2	4	no	—
160	C₂×C₈	C₂×C₄×D₄₃	1	4	yes	—
161	C₄⋊C₄	—	86	8	no	—
162	C₂²⋊C₄	—	344	8	no	—
163	C₂²⋊C₄	—	172	8	no	—
164	C₄²	—	2	4	no	—
165	C₄²	C₂×C₄×D₄₃	1	2	yes	—
166	C₄²	#10621898	1	2	yes	—
167	C₂³	—	86	8	no	—
168	C₂³	—	172	8	no	—
169	C₂³	—	172	8	no	—
170	C₂³	—	172	4	no	—
171	C₂³	—	172	4	no	—
172	C₂³	C₂²×D₁₇₂	1	1	yes	characteristic
173	D₄	—	86	16	no	—
174	D₄	—	344	16	no	—
175	D₄	—	172	8	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	8	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	344	16	no	—
180	C₂×C₄	—	172	8	no	—
181	C₂×C₄	—	172	4	no	—
182	C₂×C₄	C₄×D₁₇₂	1	4	yes	—
183	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
184	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
185	C₂×C₄	#13063371	1	4	yes	—
186	C₈	—	2	4	no	—
187	C₈	—	4	2	no	—
188	C₂²	—	344	4	no	—
189	C₂²	—	344	4	no	—
190	C₂²	—	344	8	no	—
191	C₂²	—	172	16	no	—
192	C₂²	—	172	8	no	—
193	C₂²	C₂×C₄×D₁₇₂	1	1	yes	characteristic
194	C₂²	#14583468	1	2	yes	—
195	C₂²	#14583704	1	4	yes	—
196	C₄	—	2	4	no	—
197	C₄	—	4	4	no	—
198	C₄	—	344	4	no	—
199	C₄	#14583373	1	2	yes	—
200	C₄	#14583504	1	2	yes	—
201	C₂	—	344	8	no	—
202	C₂	#19137320	1	1	yes	characteristic
203	C₂	#19143448	1	2	yes	—
204	C₂	#19145647	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14583276

C₂

yes

maximal, characteristic

#19137320

C₂

yes

maximal, complemented

#19135419

C₂

yes

maximal, complemented

#19134942

C₂

yes

maximal, complemented

#19134726

C₂

yes

maximal, characteristic, complemented

C₄₃×#1746

C₂

yes

maximal, characteristic, complemented

C₂²×D₃₄₄

C₂²

yes

—

C₂×#13063322

C₂²

yes

—

C₂×C₄×D₁₇₂

C₂²

yes

—

#14583276

C₄

yes

complemented

C₂×#13063247

C₂²

yes

—

#14582801

C₂²

yes

—

#14582777

—

#14582765

C₂²

yes

—

#14582662

C₂²

yes

—

C₂×#97×C₄₃

C₂²

yes

characteristic

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₄₃×#376

C₂²

yes

characteristic

C₄₃×#336

C₂²

yes

—

C₂²×D₁₇₂

—

C₂²×D₁₇₂

C₂³

yes

—

C₂²×C₄×D₄₃

—

C₂×D₃₄₄

—

C₂×D₃₄₄

C₂×C₄

yes

complemented

#13063322

—

#13063322

C₂×C₄

yes

complemented

C₄×D₁₇₂

—

C₂×C₄⋊Dic₄₃

C₂³

yes

—

C₂×(C₂×D₄₃)⋊C₄

—

C₂×(C₂×D₄₃)⋊C₄

—

#13063247

—

C₄₃⋊#95

—

C₂²×C₈×C₄₃

C₂³

yes

—

C₂²×C₈×C₄₃

C₂³

yes

—

C₂×C₄²×C₄₃

C₂³

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

#97×C₄₃

—

C₄×C₈×C₄₃

C₂×C₄

yes

complemented

#95×C₄₃

D₄

yes

—

C₂³×D₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

—

C₂²×C₄×C₄₃

C₂×D₄

yes

—

C₂²×C₄×C₄₃

C₂⁴

yes

characteristic

C₂²×Dic₄₃

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

C₂²×C₄

yes

—

C₂×C₄×D₄₃

—

C₂×C₄×D₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂²×C₄

yes

—

C₄²×C₄₃

—

C₄²×C₄₃

C₂²×C₄

yes

—

C₄²×C₄₃

#41

yes

—

(C₂×D₄₃)⋊C₄

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

D₃₄₄

—

C₂³×C₄₃

C₂²×D₄

yes

characteristic

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

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

C₂×#41

yes

characteristic

C₂×C₄×C₄₃

H(5,𝔽₂)

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

C₄×D₄₃

—

C₈×C₄₃

—

C₈×C₄₃

—

#22276

—

maximal, Sylow

C₂²×C₄₃

C₂×C₄×D₄

yes

characteristic

C₂²×C₄₃

#535

yes

—

C₂²×C₄₃

#576

yes

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#518

yes

—

C₄×C₄₃

#540

yes

—

Dic₄₃

—

C₂×#493

—

100

#2328

—

101

#1961

—

102

#1879

—

103

#1835

—

104

#1746

D₄₃

yes

characteristic, complemented

105

C₂×C₄₃

#2328

yes

characteristic

106

C₂×C₄₃

#2948

yes

—

107

C₂×C₄₃

#3220

yes

—

108

D₄₃

—

109

C₂²×D₈

—

110

C₂×#127

—

111

C₂×C₄×D₄

—

112

#493

—

113

C₂×#102

—

114

C₂×#97

C₂×D₄₃

yes

characteristic

115

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

116

#390

—

117

#386

—

118

#382

—

119

#376

C₂×D₄₃

yes

characteristic

120

#340

—

121

#336

C₂×D₄₃

yes

—

122

C₄₃

#22276

yes

characteristic, complemented, Sylow

123

C₂²×D₄

—

124

C₂²×D₄

—

125

C₂³×C₄

—

126

C₂×D₈

—

127

C₂×D₈

—

172

—

128

C₂²×C₈

C₂²×D₄₃

yes

—

129

C₂²×C₈

C₂²×D₄₃

yes

—

130

#127

—

131

#127

—

132

C₄×D₄

—

172

—

133

C₂×C₄⋊C₄

—

134

C₂×C₂²⋊C₄

—

135

C₂×C₂²⋊C₄

—

136

C₂×C₄²

C₂²×D₄₃

yes

—

137

C₂×C₄²

C₂²×D₄₃

yes

characteristic

138

#102

—

172

—

139

#97

—

140

C₄×C₈

C₄×D₄₃

yes

complemented

141

#95

—

142

#95

D₁₇₂

yes

—

143

C₂⁴

—

144

C₂×D₄

—

145

C₂×D₄

—

146

C₂×D₄

—

147

C₂×D₄

—

148

C₂×D₄

—

149

C₂×D₄

—

344

—

150

C₂²×C₄

—

151

C₂²×C₄

—

344

—

152

C₂²×C₄

—

172

—

153

C₂²×C₄

C₂×D₁₇₂

yes

—

154

C₂²×C₄

C₂×D₁₇₂

yes

—

155

C₂²×C₄

C₂³×D₄₃

yes

characteristic

156

D₈

—

172

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

C₂×C₄×D₄₃

yes

—

161

C₄⋊C₄

—

162

C₂²⋊C₄

—

344

—

163

C₂²⋊C₄

—

172

—

164

C₄²

—

165

C₄²

C₂×C₄×D₄₃

yes

—

166

C₄²

#10621898

yes

—

167

C₂³

—

168

C₂³

—

172

—

169

C₂³

—

172

—

170

C₂³

—

172

—

171

C₂³

—

172

—

172

C₂³

C₂²×D₁₇₂

yes

characteristic

173

D₄

—

174

D₄

—

344

—

175

D₄

—

172

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

344

—

180

C₂×C₄

—

172

—

181

C₂×C₄

—

172

—

182

C₂×C₄

C₄×D₁₇₂

yes

—

183

C₂×C₄

C₂²×C₄×D₄₃

yes

characteristic

184

C₂×C₄

C₂×#10621898

yes

characteristic

185

C₂×C₄

#13063371

yes

—

186

C₈

—

187

C₈

—

188

C₂²

—

344

—

189

C₂²

—

344

—

190

C₂²

—

344

—

191

C₂²

—

172

—

192

C₂²

—

172

—

193

C₂²

C₂×C₄×D₁₇₂

yes

characteristic

194

C₂²

#14583468

yes

—

195

C₂²

#14583704

yes

—

196

C₄

—

197

C₄

—

198

C₄

—

344

—

199

C₄

#14583373

yes

—

200

C₄

#14583504

yes

—

201

C₂

—

344

—

202

C₂

#19137320

yes

characteristic

203

C₂

#19143448

yes

—

204

C₂

#19145647

yes

—

	subgroup	G/H
center	C₂³	C₂²×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1746	C₂
socle	C₂³×C₄₃	C₂²×D₄
prime radical	C₂×#14583276	C₂
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1746	D₄₃
hypocenter	C₄₃	#22276
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1746

subgroup

G/H

center

C₂³

C₂²×D₁₇₂

commutator

C₂×C₄×C₄₃

C₂³×C₄

Frattini

C₂²×C₄

C₂³×D₄₃

Fitting

C₄₃×#1746

C₂

socle

C₂³×C₄₃

C₂²×D₄

prime radical

C₂×#14583276

C₂

cosocle

C₂²×C₄×C₄₃

C₂⁴

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ #376 ◁ #1746

N	H ≅ G/N	type
2. #19137320	C₂	semidirect
3. #19135419	C₂	semidirect
4. #19134942	C₂	semidirect
5. #19134726	C₂	semidirect
6. C₄₃×#1746	C₂	semidirect
10. #14583276	C₄	semidirect
24. C₂×D₃₄₄	C₂×C₄	semidirect
26. #13063322	C₂×C₄	semidirect
38. C₄×C₈×C₄₃	C₂×C₄	semidirect
104. #1746	D₄₃	semidirect
122. C₄₃	#22276	semidirect
140. C₄×C₈	C₄×D₄₃	semidirect

H ≅ G/N

type

2. #19137320

C₂

semidirect

3. #19135419

C₂

semidirect

4. #19134942

C₂

semidirect

5. #19134726

C₂

semidirect

6. C₄₃×#1746

C₂

semidirect

10. #14583276

C₄

semidirect

24. C₂×D₃₄₄

C₂×C₄

semidirect

26. #13063322

C₂×C₄

semidirect

38. C₄×C₈×C₄₃

C₂×C₄

semidirect

104. #1746

D₄₃

semidirect

122. C₄₃

#22276

semidirect

140. C₄×C₈

C₄×D₄₃

semidirect

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

subgroup	index
C₂×#14583276	2
#19137320	2
#19135419	2
#19134942	2
#19134726	2
C₄₃×#1746	2
#22276	43

subgroup

index

C₄₃×#1746

Subgroups — #25110441

Conjugacy classes of subgroups

Complements

Maximal subgroups