Group #25016713 — subgroups

Subgroups — #25016713

Order 11008.

318 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 97 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×#14582723	C₂	1	2	yes	maximal
2	C₂×#14582714	C₂	1	2	yes	maximal
3	#19135466	C₂	1	1	yes	maximal, characteristic
4	#19132679	C₂	1	1	yes	maximal, characteristic
5	C₄₃×#1849	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₇₂	C₂²	1	1	yes	characteristic
7	C₂×#13063265	C₂²	1	2	yes	—
8	C₂×#13063257	C₂²	1	1	yes	characteristic
9	C₂×#13063250	—	2	1	no	—
10	#14582805	—	2	1	no	—
11	#14582723	—	2	4	no	—
12	#14582714	—	2	4	no	—
13	C₂×#108×C₄₃	—	2	1	no	—
14	C₂×#108×C₄₃	C₂²	1	2	yes	—
15	C₄₃×#423	—	2	1	no	—
16	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
17	C₂²×Dic₈₆	—	2	1	no	—
18	C₂×Dic₁₇₂	—	2	2	no	—
19	C₂×Dic₁₇₂	—	2	2	no	—
20	C₂×Dic₁₇₂	—	2	1	no	—
21	C₂×Dic₁₇₂	Q₈	1	2	yes	—
22	C₂×C₄⋊Dic₄₃	—	2	1	no	—
23	#13063265	—	2	2	no	—
24	#13063265	D₄	1	4	yes	—
25	C₂×C₄₃⋊C₁₆	—	2	2	no	—
26	#13063257	—	2	2	no	—
27	#13063250	—	4	2	no	—
28	C₄₃⋊#95	—	4	2	no	—
29	C₂×M₁₆×C₄₃	—	2	2	no	—
30	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
33	#108×C₄₃	—	2	2	no	—
34	#108×C₄₃	—	2	2	no	—
35	#108×C₄₃	—	4	2	no	—
36	C₄⋊C₈×C₄₃	—	4	2	no	—
37	#97×C₄₃	—	2	2	no	—
38	C₂²×C₄×C₄₃	SD₁₆	1	2	yes	—
39	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
40	C₂²×Dic₄₃	—	4	1	no	—
41	C₂²×Dic₄₃	—	4	1	no	—
42	C₂×Dic₈₆	—	2	2	no	—
43	C₂×Dic₈₆	—	4	1	no	—
44	C₂×Dic₈₆	—	4	1	no	—
45	C₂×Dic₈₆	—	4	1	no	—
46	C₂×Dic₈₆	—	4	1	no	—
47	C₂×Dic₈₆	—	4	1	no	—
48	M₁₆×C₄₃	—	4	2	no	—
49	M₁₆×C₄₃	—	4	2	no	—
50	C₂×C₈×C₄₃	—	2	1	no	—
51	C₂×C₈×C₄₃	—	2	1	no	—
52	C₂×C₈×C₄₃	—	4	2	no	—
53	C₂×C₈×C₄₃	SD₁₆	1	2	yes	—
54	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
55	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₈×C₄₃	#41	1	2	yes	—
57	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
58	C₄²×C₄₃	—	2	1	no	—
59	C₄²×C₄₃	—	2	1	no	—
60	C₄⋊Dic₄₃	—	4	1	no	—
61	C₄⋊Dic₄₃	—	4	1	no	—
62	Dic₁₇₂	—	2	4	no	—
63	Dic₁₇₂	—	4	2	no	—
64	C₄₃⋊C₁₆	—	2	4	no	—
65	C₂³×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
66	C₂×C₄×C₄₃	—	2	2	no	—
67	C₂×C₄×C₄₃	—	4	2	no	—
68	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
69	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
70	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
71	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
72	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
73	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
74	C₂×Dic₄₃	—	4	2	no	—
75	C₂×Dic₄₃	—	8	1	no	—
76	C₂×Dic₄₃	—	8	1	no	—
77	C₂×Dic₄₃	—	8	2	no	—
78	C₂×Dic₄₃	—	8	1	no	—
79	Dic₈₆	—	4	2	no	—
80	Dic₈₆	—	4	2	no	—
81	Dic₈₆	—	8	1	no	—
82	Dic₈₆	—	8	1	no	—
83	C₈×C₄₃	—	2	2	no	—
84	C₈×C₄₃	—	8	2	no	—
85	C₈×C₄₃	C₂²⋊Q₈	1	2	yes	—
86	C₈×C₄₃	#123	1	2	yes	—
87	#13536	—	43	1	no	maximal, Sylow
88	C₂²×C₄₃	—	2	2	no	—
89	C₂²×C₄₃	#476	1	1	yes	characteristic
90	C₂²×C₄₃	#481	1	1	yes	characteristic
91	C₂²×C₄₃	#492	1	1	yes	characteristic
92	C₄×C₄₃	—	4	2	no	—
93	C₄×C₄₃	#399	1	1	yes	characteristic
94	C₄×C₄₃	#476	1	1	yes	characteristic
95	C₄×C₄₃	#481	1	1	yes	characteristic
96	C₄×C₄₃	#496	1	1	yes	characteristic
97	Dic₄₃	—	8	2	no	—
98	Dic₄₃	—	16	1	no	—
99	C₂×#365	—	43	2	no	—
100	C₂×#362	—	43	2	no	—
101	#1971	—	43	1	no	—
102	#1849	D₄₃	1	1	yes	characteristic
103	#1377	—	43	1	no	—
104	C₂×C₄₃	—	2	1	no	—
105	C₂×C₄₃	—	2	1	no	—
106	C₂×C₄₃	#2094	1	1	yes	characteristic
107	C₂×C₄₃	#2239	1	2	yes	—
108	C₂²×Q₁₆	—	43	1	no	—
109	C₂×M₃₂	—	43	2	no	—
110	C₂×#108	—	2	1	no	—
111	C₂×#108	C₂×D₄₃	1	2	yes	—
112	C₂×#106	—	43	1	no	—
113	#423	—	2	1	no	—
114	C₂×#103	—	86	1	no	—
115	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
116	#391	—	86	1	no	—
117	#365	—	86	4	no	—
118	#362	—	86	4	no	—
119	C₄₃	#13536	1	1	yes	characteristic, complemented, Sylow
120	C₂²×Q₈	—	86	1	no	—
121	C₂×Q₁₆	—	43	2	no	—
122	C₂×Q₁₆	—	86	2	no	—
123	C₂×Q₁₆	—	86	2	no	—
124	C₂×Q₁₆	—	86	1	no	—
125	C₂×M₁₆	—	2	2	no	—
126	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
127	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
128	C₂×C₄⋊C₄	—	86	1	no	—
129	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
130	M₃₂	—	43	4	no	—
131	M₃₂	—	86	2	no	—
132	C₂×C₁₆	—	86	2	no	—
133	#108	—	2	2	no	—
134	#108	—	2	2	no	—
135	#108	—	4	2	no	—
136	#106	—	86	2	no	—
137	C₄⋊C₈	—	4	2	no	—
138	#103	—	172	2	no	—
139	#97	—	2	2	no	—
140	#95	—	172	2	no	—
141	C₂×Q₈	—	86	2	no	—
142	C₂×Q₈	—	172	1	no	—
143	C₂×Q₈	—	172	1	no	—
144	C₂×Q₈	—	172	1	no	—
145	C₂×Q₈	—	172	1	no	—
146	C₂×Q₈	—	172	1	no	—
147	C₂²×C₄	—	172	1	no	—
148	C₂²×C₄	—	172	1	no	—
149	C₂²×C₄	#10621884	1	2	yes	—
150	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
151	Q₁₆	—	86	4	no	—
152	Q₁₆	—	172	2	no	—
153	M₁₆	—	4	2	no	—
154	M₁₆	—	4	2	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	4	2	no	—
158	C₂×C₈	#10621884	1	2	yes	—
159	C₂×C₈	#10621898	1	2	yes	—
160	C₂×C₈	D₄×D₄₃	1	2	yes	—
161	C₂×C₈	Q₈×D₄₃	1	1	yes	characteristic
162	C₂×C₈	#10621902	1	1	yes	characteristic
163	C₄⋊C₄	—	172	1	no	—
164	C₄⋊C₄	—	172	1	no	—
165	C₄²	—	2	1	no	—
166	C₄²	—	2	1	no	—
167	C₁₆	—	86	4	no	—
168	C₂³	C₂×#10621884	1	1	yes	characteristic
169	Q₈	—	344	1	no	—
170	Q₈	—	344	1	no	—
171	Q₈	—	172	2	no	—
172	Q₈	—	172	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	344	1	no	—
176	C₂×C₄	—	344	1	no	—
177	C₂×C₄	—	344	2	no	—
178	C₂×C₄	—	344	1	no	—
179	C₂×C₄	—	172	2	no	—
180	C₂×C₄	#13063301	1	1	yes	characteristic
181	C₂×C₄	#13063308	1	1	yes	characteristic
182	C₂×C₄	#13063321	1	1	yes	characteristic
183	C₂×C₄	C₂×#10621884	1	1	yes	characteristic
184	C₂×C₄	#13063351	1	1	yes	characteristic
185	C₂×C₄	#13063351	1	1	yes	characteristic
186	C₈	—	2	2	no	—
187	C₈	—	8	2	no	—
188	C₈	#13063303	1	2	yes	—
189	C₈	#13063310	1	2	yes	—
190	C₂²	—	2	2	no	—
191	C₂²	#14583186	1	1	yes	characteristic
192	C₂²	#14583217	1	1	yes	characteristic
193	C₂²	#14583273	1	1	yes	characteristic
194	C₄	—	4	2	no	—
195	C₄	—	344	2	no	—
196	C₄	—	688	1	no	—
197	C₄	#14582834	1	1	yes	characteristic
198	C₄	#14583186	1	1	yes	characteristic
199	C₄	#14583217	1	1	yes	characteristic
200	C₄	#14583286	1	1	yes	characteristic
201	C₂	—	2	1	no	—
202	C₂	—	2	1	no	—
203	C₂	#19136129	1	1	yes	characteristic
204	C₂	#19136814	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14582723

C₂

yes

maximal

C₂×#14582714

C₂

yes

maximal

#19135466

C₂

yes

maximal, characteristic

#19132679

C₂

yes

maximal, characteristic

C₄₃×#1849

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₂

C₂²

yes

characteristic

C₂×#13063265

C₂²

yes

—

C₂×#13063257

C₂²

yes

—

—

—

—

C₂×#108×C₄₃

—

C₂×#108×C₄₃

C₂²

yes

—

C₄₃×#423

—

C₂×#97×C₄₃

C₂²

yes

characteristic

C₂²×Dic₈₆

—

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

Q₈

yes

—

C₂×C₄⋊Dic₄₃

—

#13063265

—

#13063265

D₄

yes

—

C₂×C₄₃⋊C₁₆

—

#13063257

—

#13063250

—

C₄₃⋊#95

—

C₂×M₁₆×C₄₃

—

C₂²×C₈×C₄₃

D₄

yes

characteristic

C₂²×C₈×C₄₃

C₂³

yes

characteristic

C₂×C₄²×C₄₃

D₄

yes

characteristic

#108×C₄₃

—

#108×C₄₃

—

#108×C₄₃

—

C₄⋊C₈×C₄₃

—

#97×C₄₃

—

C₂²×C₄×C₄₃

SD₁₆

yes

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

M₁₆×C₄₃

—

M₁₆×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

SD₁₆

yes

—

C₂×C₈×C₄₃

C₂×D₄

yes

—

C₂×C₈×C₄₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₄₃

#41

yes

—

C₂×C₈×C₄₃

#41

yes

characteristic

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₁₇₂

—

Dic₁₇₂

—

C₄₃⋊C₁₆

—

C₂³×C₄₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²⋊Q₈

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

C₂×SD₁₆

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

C₂²⋊Q₈

yes

—

C₈×C₄₃

#123

yes

—

#13536

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

Dic₄₃

—

C₂×#365

—

100

C₂×#362

—

101

#1971

—

102

#1849

D₄₃

yes

characteristic

103

#1377

—

104

C₂×C₄₃

—

105

C₂×C₄₃

—

106

C₂×C₄₃

#2094

yes

characteristic

107

C₂×C₄₃

#2239

yes

—

108

C₂²×Q₁₆

—

109

C₂×M₃₂

—

110

C₂×#108

—

111

C₂×#108

C₂×D₄₃

yes

—

112

C₂×#106

—

113

#423

—

114

C₂×#103

—

115

C₂×#97

C₂×D₄₃

yes

characteristic

116

#391

—

117

#365

—

118

#362

—

119

C₄₃

#13536

yes

characteristic, complemented, Sylow

120

C₂²×Q₈

—

121

C₂×Q₁₆

—

122

C₂×Q₁₆

—

123

C₂×Q₁₆

—

124

C₂×Q₁₆

—

125

C₂×M₁₆

—

126

C₂²×C₈

C₂²⋊D₄₃

yes

characteristic

127

C₂²×C₈

C₂²×D₄₃

yes

characteristic

128

C₂×C₄⋊C₄

—

129

C₂×C₄²

C₂²⋊D₄₃

yes

characteristic

130

M₃₂

—

131

M₃₂

—

132

C₂×C₁₆

—

133

#108

—

134

#108

—

135

#108

—

136

#106

—

137

C₄⋊C₈

—

138

#103

—

172

—

139

#97

—

140

#95

—

172

—

141

C₂×Q₈

—

142

C₂×Q₈

—

172

—

143

C₂×Q₈

—

172

—

144

C₂×Q₈

—

172

—

145

C₂×Q₈

—

172

—

146

C₂×Q₈

—

172

—

147

C₂²×C₄

—

172

—

148

C₂²×C₄

—

172

—

149

C₂²×C₄

#10621884

yes

—

150

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

151

Q₁₆

—

152

Q₁₆

—

172

—

153

M₁₆

—

154

M₁₆

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

#10621884

yes

—

159

C₂×C₈

#10621898

yes

—

160

C₂×C₈

D₄×D₄₃

yes

—

161

C₂×C₈

Q₈×D₄₃

yes

characteristic

162

C₂×C₈

#10621902

yes

characteristic

163

C₄⋊C₄

—

172

—

164

C₄⋊C₄

—

172

—

165

C₄²

—

166

C₄²

—

167

C₁₆

—

168

C₂³

C₂×#10621884

yes

characteristic

169

Q₈

—

344

—

170

Q₈

—

344

—

171

Q₈

—

172

—

172

Q₈

—

172

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

344

—

176

C₂×C₄

—

344

—

177

C₂×C₄

—

344

—

178

C₂×C₄

—

344

—

179

C₂×C₄

—

172

—

180

C₂×C₄

#13063301

yes

characteristic

181

C₂×C₄

#13063308

yes

characteristic

182

C₂×C₄

#13063321

yes

characteristic

183

C₂×C₄

C₂×#10621884

yes

characteristic

184

C₂×C₄

#13063351

yes

characteristic

185

C₂×C₄

#13063351

yes

characteristic

186

C₈

—

187

C₈

—

188

C₈

#13063303

yes

—

189

C₈

#13063310

yes

—

190

C₂²

—

191

C₂²

#14583186

yes

characteristic

192

C₂²

#14583217

yes

characteristic

193

C₂²

#14583273

yes

characteristic

194

C₄

—

195

C₄

—

344

—

196

C₄

—

688

—

197

C₄

#14582834

yes

characteristic

198

C₄

#14583186

yes

characteristic

199

C₄

#14583217

yes

characteristic

200

C₄

#14583286

yes

characteristic

201

C₂

—

202

C₂

—

203

C₂

#19136129

yes

characteristic

204

C₂

#19136814

yes

—

	subgroup	G/H
center	C₂²	#14583217
commutator, cosocle	C₂²×C₈×C₄₃	C₂³
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#1849	C₂
socle	C₂²×C₄₃	#481
prime radical	C₂³×C₄₃	C₂×SD₁₆
hypercenter	#1849	D₄₃
hypocenter	C₄₃	#13536
LCS	G ▷ C₂²×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1849

subgroup

G/H

center

C₂²

#14583217

commutator, cosocle

C₂²×C₈×C₄₃

C₂³

Frattini

C₂²×C₈

C₂²×D₄₃

Fitting

C₄₃×#1849

C₂

socle

C₂²×C₄₃

#481

prime radical

C₂³×C₄₃

C₂×SD₁₆

D₄₃

C₄₃

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1849

N	H ≅ G/N	type
119. C₄₃	#13536	semidirect

H ≅ G/N

type

119. C₄₃

#13536

semidirect

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

subgroup	index
C₂×#14582723	2
C₂×#14582714	2
#19135466	2
#19132679	2
C₄₃×#1849	2
#13536	43

subgroup

index

C₄₃×#1849

Subgroups — #25016713

Conjugacy classes of subgroups

Complements

Maximal subgroups