Group #25110442 — subgroups

Subgroups — #25110442

Order 11008.

1050 conjugacy classes of subgroups; only the 259 normal subgroups are shown.

normal subgroups only

This group has 1050 conjugacy classes of subgroups — too many to draw. Only its 259 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	#19137321	C₂	1	2	yes	maximal, complemented
2	#19135420	C₂	1	2	yes	maximal, complemented
3	#19134945	C₂	1	8	yes	maximal, complemented
4	#19134726	C₂	1	1	yes	maximal, characteristic, complemented
5	#19134346	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄₃×#493	C₂	1	1	yes	maximal, characteristic
7	C₂×D₄×Dic₄₃	C₂²	1	4	yes	—
8	C₂×#13063246	C₂²	1	8	yes	—
9	C₂×#13063236	C₂²	1	1	yes	characteristic
10	#14582802	C₂²	1	4	yes	—
11	#14582779	—	2	8	no	—
12	#14582765	C₂²	1	2	yes	—
13	#14582745	C₂²	1	1	yes	characteristic
14	#14582662	C₂²	1	4	yes	—
15	#14582653	C₂²	1	4	yes	—
16	C₂²×D₈×C₄₃	C₂²	1	4	yes	—
17	C₂×#127×C₄₃	C₂²	1	2	yes	—
18	C₄₃×#493	C₄	1	8	yes	complemented
19	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
20	C₂³×Dic₄₃	—	2	4	no	—
21	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
22	D₄×Dic₄₃	—	4	8	no	—
23	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
24	C₂×C₂²⋊Dic₄₃	—	2	4	no	—
25	C₂×C₂²⋊Dic₄₃	—	2	8	no	—
26	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
27	#13063246	—	4	8	no	—
28	#13063236	—	2	2	no	—
29	C₄₃⋊#95	—	2	4	no	—
30	C₄₃⋊#95	D₄	1	4	yes	—
31	C₂²×D₄×C₄₃	—	2	4	no	—
32	C₂²×D₄×C₄₃	C₂³	1	4	yes	—
33	C₂×D₈×C₄₃	—	4	8	no	—
34	C₂×D₈×C₄₃	C₂×C₄	1	16	yes	complemented
35	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
36	#127×C₄₃	—	2	4	no	—
37	#127×C₄₃	C₂×C₄	1	8	yes	complemented
38	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
39	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	complemented
40	C₂⁴×C₄₃	—	2	4	no	—
41	C₂×D₄×C₄₃	—	2	16	no	—
42	C₂×D₄×C₄₃	—	2	4	no	—
43	C₂×D₄×C₄₃	—	2	8	no	—
44	C₂×D₄×C₄₃	—	2	8	no	—
45	C₂×D₄×C₄₃	—	8	8	no	—
46	C₂×D₄×C₄₃	C₂²×C₄	1	8	yes	—
47	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
49	C₂²×Dic₄₃	—	2	4	no	—
50	C₂²×Dic₄₃	—	4	8	no	—
51	C₂²×Dic₄₃	—	8	8	no	—
52	C₂²×Dic₄₃	C₂×D₄	1	4	yes	—
53	D₈×C₄₃	—	4	16	no	—
54	C₂×C₈×C₄₃	—	2	4	no	—
55	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
56	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
57	C₄²×C₄₃	#41	1	2	yes	—
58	C₂²⋊Dic₄₃	—	4	8	no	—
59	C₂²⋊Dic₄₃	—	8	8	no	—
60	C₄⋊Dic₄₃	—	2	8	no	—
61	C₄×Dic₄₃	—	2	4	no	—
62	C₂×C₄₃⋊C₈	—	2	2	no	—
63	C₂×C₄₃⋊C₈	—	2	4	no	—
64	C₂³×C₄₃	—	2	8	no	—
65	C₂³×C₄₃	—	4	4	no	—
66	C₂³×C₄₃	—	4	4	no	—
67	C₂³×C₄₃	—	4	8	no	—
68	C₂³×C₄₃	—	4	8	no	—
69	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
70	D₄×C₄₃	—	2	16	no	—
71	D₄×C₄₃	—	4	8	no	—
72	D₄×C₄₃	—	8	16	no	—
73	C₂×C₄×C₄₃	—	2	4	no	—
74	C₂×C₄×C₄₃	C₄×D₄	1	4	yes	—
75	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
76	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
77	C₂×C₄×C₄₃	H(5,𝔽₂)	1	4	yes	—
78	C₂×Dic₄₃	—	2	4	no	—
79	C₂×Dic₄₃	—	2	8	no	—
80	C₂×Dic₄₃	—	4	4	no	—
81	C₂×Dic₄₃	—	4	8	no	—
82	C₂×Dic₄₃	—	8	16	no	—
83	C₈×C₄₃	—	2	4	no	—
84	C₄₃⋊C₈	—	4	2	no	—
85	#22276	—	43	1	no	maximal, Sylow
86	C₂²×C₄₃	—	4	8	no	—
87	C₂²×C₄₃	—	4	16	no	—
88	C₂²×C₄₃	—	8	4	no	—
89	C₂²×C₄₃	—	8	8	no	—
90	C₂²×C₄₃	—	8	4	no	—
91	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
92	C₂²×C₄₃	#535	1	2	yes	—
93	C₂²×C₄₃	#576	1	4	yes	—
94	C₄×C₄₃	—	2	4	no	—
95	C₄×C₄₃	#518	1	2	yes	—
96	C₄×C₄₃	#540	1	2	yes	—
97	Dic₄₃	—	4	4	no	—
98	Dic₄₃	—	8	4	no	—
99	C₂×#493	D₄₃	1	1	yes	characteristic
100	#2328	—	43	2	no	—
101	#1961	—	43	2	no	—
102	#1879	—	43	8	no	—
103	#1835	—	43	1	no	—
104	#1746	—	43	1	no	—
105	C₂×C₄₃	—	8	8	no	—
106	C₂×C₄₃	#2328	1	1	yes	characteristic
107	C₂×C₄₃	#2948	1	2	yes	—
108	C₂×C₄₃	#3220	1	4	yes	—
109	C₂²×D₈	C₂×D₄₃	1	4	yes	—
110	C₂×#127	C₂×D₄₃	1	2	yes	—
111	C₂×C₄×D₄	—	43	4	no	—
112	#493	Dic₄₃	1	8	yes	complemented
113	C₂×#102	—	43	8	no	—
114	C₂×#97	—	43	1	no	—
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	—	43	1	no	—
120	#340	—	43	4	no	—
121	#336	—	43	4	no	—
122	C₄₃	#22276	1	1	yes	characteristic, complemented, Sylow
123	C₂²×D₄	—	2	4	no	—
124	C₂²×D₄	C₂²×D₄₃	1	4	yes	—
125	C₂³×C₄	—	86	4	no	—
126	C₂×D₈	—	4	8	no	—
127	C₂×D₈	C₂×Dic₄₃	1	16	yes	complemented
128	C₂²×C₈	—	43	2	no	—
129	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
130	#127	—	2	4	no	—
131	#127	C₂×Dic₄₃	1	8	yes	complemented
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	—	43	2	no	—
137	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
138	#102	—	172	8	no	—
139	#97	—	86	2	no	—
140	C₄×C₈	C₂×Dic₄₃	1	4	yes	complemented
141	#95	—	43	4	no	—
142	#95	—	86	4	no	—
143	C₂⁴	—	2	4	no	—
144	C₂×D₄	—	2	16	no	—
145	C₂×D₄	—	2	4	no	—
146	C₂×D₄	—	2	8	no	—
147	C₂×D₄	—	2	8	no	—
148	C₂×D₄	—	8	8	no	—
149	C₂×D₄	C₂²×Dic₄₃	1	8	yes	—
150	C₂²×C₄	—	43	4	no	—
151	C₂²×C₄	—	86	4	no	—
152	C₂²×C₄	—	344	8	no	—
153	C₂²×C₄	—	172	8	no	—
154	C₂²×C₄	D₄×D₄₃	1	2	yes	—
155	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
156	D₈	—	4	16	no	—
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	—	86	2	no	—
159	C₂×C₈	—	86	4	no	—
160	C₂×C₈	C₂²×Dic₄₃	1	4	yes	—
161	C₄⋊C₄	—	86	8	no	—
162	C₂²⋊C₄	—	344	8	no	—
163	C₂²⋊C₄	—	172	8	no	—
164	C₄²	—	86	4	no	—
165	C₄²	#10621900	1	2	yes	—
166	C₄²	C₂²×Dic₄₃	1	2	yes	—
167	C₂³	—	2	8	no	—
168	C₂³	—	4	8	no	—
169	C₂³	—	4	8	no	—
170	C₂³	—	4	4	no	—
171	C₂³	—	4	4	no	—
172	C₂³	C₂×D₄×D₄₃	1	1	yes	characteristic
173	D₄	—	2	16	no	—
174	D₄	—	4	8	no	—
175	D₄	—	8	16	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	86	8	no	—
178	C₂×C₄	—	86	4	no	—
179	C₂×C₄	—	344	16	no	—
180	C₂×C₄	—	172	8	no	—
181	C₂×C₄	—	172	4	no	—
182	C₂×C₄	D₄×Dic₄₃	1	4	yes	—
183	C₂×C₄	C₂³×Dic₄₃	1	1	yes	characteristic
184	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
185	C₂×C₄	#13063370	1	4	yes	—
186	C₈	—	2	4	no	—
187	C₈	—	172	2	no	—
188	C₂²	—	4	16	no	—
189	C₂²	—	4	8	no	—
190	C₂²	—	8	4	no	—
191	C₂²	—	8	4	no	—
192	C₂²	—	8	8	no	—
193	C₂²	C₂×D₄×Dic₄₃	1	1	yes	characteristic
194	C₂²	#14583462	1	2	yes	—
195	C₂²	#14583700	1	4	yes	—
196	C₄	—	2	4	no	—
197	C₄	—	344	4	no	—
198	C₄	—	172	4	no	—
199	C₄	#14583374	1	2	yes	—
200	C₄	#14583503	1	2	yes	—
201	C₂	—	8	8	no	—
202	C₂	#19137321	1	1	yes	characteristic
203	C₂	#19143447	1	2	yes	—
204	C₂	#19145638	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19137321

C₂

yes

maximal, complemented

#19135420

C₂

yes

maximal, complemented

#19134945

C₂

yes

maximal, complemented

#19134726

C₂

yes

maximal, characteristic, complemented

#19134346

C₂

yes

maximal, characteristic, complemented

C₂×C₄₃×#493

C₂

yes

maximal, characteristic

C₂×D₄×Dic₄₃

C₂²

yes

—

C₂×#13063246

C₂²

yes

—

C₂×#13063236

C₂²

yes

characteristic

#14582802

C₂²

yes

—

#14582779

—

#14582765

C₂²

yes

—

#14582745

C₂²

yes

characteristic

#14582662

C₂²

yes

—

#14582653

C₂²

yes

—

C₂²×D₈×C₄₃

C₂²

yes

—

C₂×#127×C₄₃

C₂²

yes

—

C₄₃×#493

C₄

yes

complemented

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₂³×Dic₄₃

—

C₂²×C₄₃⋊C₈

C₂³

yes

—

D₄×Dic₄₃

—

C₂×C₄⋊Dic₄₃

C₂³

yes

—

C₂×C₂²⋊Dic₄₃

—

C₂×C₂²⋊Dic₄₃

—

C₂×C₄×Dic₄₃

C₂³

yes

—

#13063246

—

#13063236

—

C₄₃⋊#95

—

C₄₃⋊#95

D₄

yes

—

C₂²×D₄×C₄₃

—

C₂²×D₄×C₄₃

C₂³

yes

—

C₂×D₈×C₄₃

—

C₂×D₈×C₄₃

C₂×C₄

yes

complemented

C₂²×C₈×C₄₃

C₂³

yes

—

#127×C₄₃

—

#127×C₄₃

C₂×C₄

yes

complemented

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄×C₈×C₄₃

C₂×C₄

yes

complemented

C₂⁴×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

C₂²×C₄

yes

—

C₂²×C₄×C₄₃

C₂×D₄

yes

—

C₂²×C₄×C₄₃

C₂⁴

yes

characteristic

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂²×Dic₄₃

C₂×D₄

yes

—

D₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂²×C₄

yes

—

C₄²×C₄₃

C₂²×C₄

yes

—

C₄²×C₄₃

#41

yes

—

C₂²⋊Dic₄₃

—

C₂²⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

C₂²×D₄

yes

characteristic

D₄×C₄₃

—

D₄×C₄₃

—

D₄×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

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

#22276

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂×C₄×D₄

yes

characteristic

C₂²×C₄₃

#535

yes

—

C₂²×C₄₃

#576

yes

—

C₄×C₄₃

—

C₄×C₄₃

#518

yes

—

C₄×C₄₃

#540

yes

—

Dic₄₃

—

Dic₄₃

—

C₂×#493

D₄₃

yes

characteristic

100

#2328

—

101

#1961

—

102

#1879

—

103

#1835

—

104

#1746

—

105

C₂×C₄₃

—

106

C₂×C₄₃

#2328

yes

characteristic

107

C₂×C₄₃

#2948

yes

—

108

C₂×C₄₃

#3220

yes

—

109

C₂²×D₈

C₂×D₄₃

yes

—

110

C₂×#127

C₂×D₄₃

yes

—

111

C₂×C₄×D₄

—

112

#493

Dic₄₃

yes

complemented

113

C₂×#102

—

114

C₂×#97

—

115

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

116

#390

—

117

#386

—

118

#382

—

119

#376

—

120

#340

—

121

#336

—

122

C₄₃

#22276

yes

characteristic, complemented, Sylow

123

C₂²×D₄

—

124

C₂²×D₄

C₂²×D₄₃

yes

—

125

C₂³×C₄

—

126

C₂×D₈

—

127

C₂×D₈

C₂×Dic₄₃

yes

complemented

128

C₂²×C₈

—

129

C₂²×C₈

C₂²×D₄₃

yes

—

130

#127

—

131

#127

C₂×Dic₄₃

yes

complemented

132

C₄×D₄

—

172

—

133

C₂×C₄⋊C₄

—

134

C₂×C₂²⋊C₄

—

135

C₂×C₂²⋊C₄

—

136

C₂×C₄²

—

137

C₂×C₄²

C₂²×D₄₃

yes

characteristic

138

#102

—

172

—

139

#97

—

140

C₄×C₈

C₂×Dic₄₃

yes

complemented

141

#95

—

142

#95

—

143

C₂⁴

—

144

C₂×D₄

—

145

C₂×D₄

—

146

C₂×D₄

—

147

C₂×D₄

—

148

C₂×D₄

—

149

C₂×D₄

C₂²×Dic₄₃

yes

—

150

C₂²×C₄

—

151

C₂²×C₄

—

152

C₂²×C₄

—

344

—

153

C₂²×C₄

—

172

—

154

C₂²×C₄

D₄×D₄₃

yes

—

155

C₂²×C₄

C₂³×D₄₃

yes

characteristic

156

D₈

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

C₂²×Dic₄₃

yes

—

161

C₄⋊C₄

—

162

C₂²⋊C₄

—

344

—

163

C₂²⋊C₄

—

172

—

164

C₄²

—

165

C₄²

#10621900

yes

—

166

C₄²

C₂²×Dic₄₃

yes

—

167

C₂³

—

168

C₂³

—

169

C₂³

—

170

C₂³

—

171

C₂³

—

172

C₂³

C₂×D₄×D₄₃

yes

characteristic

173

D₄

—

174

D₄

—

175

D₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

344

—

180

C₂×C₄

—

172

—

181

C₂×C₄

—

172

—

182

C₂×C₄

D₄×Dic₄₃

yes

—

183

C₂×C₄

C₂³×Dic₄₃

yes

characteristic

184

C₂×C₄

C₂×#10621900

yes

characteristic

185

C₂×C₄

#13063370

yes

—

186

C₈

—

187

C₈

—

172

—

188

C₂²

—

189

C₂²

—

190

C₂²

—

191

C₂²

—

192

C₂²

—

193

C₂²

C₂×D₄×Dic₄₃

yes

characteristic

194

C₂²

#14583462

yes

—

195

C₂²

#14583700

yes

—

196

C₄

—

197

C₄

—

344

—

198

C₄

—

172

—

199

C₄

#14583374

yes

—

200

C₄

#14583503

yes

—

201

C₂

—

202

C₂

#19137321

yes

characteristic

203

C₂

#19143447

yes

—

204

C₂

#19145638

yes

—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting, prime radical	C₂×C₄₃×#493	C₂
socle	C₂³×C₄₃	C₂²×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₂×#493	D₄₃
hypocenter	C₄₃	#22276
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#493

subgroup

G/H

center

C₂³

C₂×D₄×D₄₃

commutator

C₂×C₄×C₄₃

C₂³×C₄

Frattini

C₂²×C₄

C₂³×D₄₃

Fitting, prime radical

C₂×C₄₃×#493

C₂

socle

C₂³×C₄₃

C₂²×D₄

cosocle

C₂²×C₄×C₄₃

C₂⁴

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#493

N	H ≅ G/N	type
1. #19137321	C₂	semidirect
2. #19135420	C₂	semidirect
3. #19134945	C₂	semidirect
4. #19134726	C₂	semidirect
5. #19134346	C₂	semidirect
18. C₄₃×#493	C₄	semidirect
34. C₂×D₈×C₄₃	C₂×C₄	semidirect
37. #127×C₄₃	C₂×C₄	semidirect
39. C₄×C₈×C₄₃	C₂×C₄	semidirect
112. #493	Dic₄₃	semidirect
122. C₄₃	#22276	semidirect
127. C₂×D₈	C₂×Dic₄₃	semidirect
131. #127	C₂×Dic₄₃	semidirect
140. C₄×C₈	C₂×Dic₄₃	semidirect

H ≅ G/N

type

1. #19137321

C₂

semidirect

2. #19135420

C₂

semidirect

3. #19134945

C₂

semidirect

4. #19134726

C₂

semidirect

5. #19134346

C₂

semidirect

18. C₄₃×#493

C₄

semidirect

34. C₂×D₈×C₄₃

C₂×C₄

semidirect

37. #127×C₄₃

C₂×C₄

semidirect

39. C₄×C₈×C₄₃

C₂×C₄

semidirect

112. #493

Dic₄₃

semidirect

122. C₄₃

#22276

semidirect

127. C₂×D₈

C₂×Dic₄₃

semidirect

131. #127

C₂×Dic₄₃

semidirect

140. C₄×C₈

C₂×Dic₄₃

semidirect

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

subgroup	index
#19137321	2
#19135420	2
#19134945	2
#19134726	2
#19134346	2
C₂×C₄₃×#493	2
#22276	43

subgroup

index

C₂×C₄₃×#493

Subgroups — #25110442

Conjugacy classes of subgroups

Complements

Maximal subgroups