Group #25016712 — subgroups

Subgroups — #25016712

Order 11008.

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

normal subgroups only

This group has 318 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×#14582721	C₂	1	2	yes	maximal
2	C₂×#14582715	C₂	1	2	yes	maximal
3	#19134770	C₂	1	1	yes	maximal, characteristic
4	#19132679	C₂	1	1	yes	maximal, characteristic
5	C₄₃×#1971	C₂	1	1	yes	maximal, characteristic
6	C₂×#13063265	C₂²	1	2	yes	—
7	C₂×#13063261	—	2	1	no	—
8	C₂×#13063261	C₂²	1	2	yes	—
9	#14582923	—	2	1	no	—
10	#14582721	—	2	4	no	—
11	#14582715	—	2	4	no	—
12	C₂²×Q₁₆×C₄₃	C₂²	1	1	yes	characteristic
13	C₂×#106×C₄₃	C₂²	1	1	yes	characteristic
14	C₂×#103×C₄₃	—	2	1	no	—
15	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
16	C₄₃×#391	—	2	1	no	—
17	C₂×#10621877	—	2	2	no	—
18	#13063265	—	2	2	no	—
19	#13063265	D₄	1	4	yes	—
20	C₂×C₄₃⋊C₁₆	—	2	2	no	—
21	#13063261	—	2	2	no	—
22	#13063261	—	2	2	no	—
23	#13063261	—	4	2	no	—
24	C₄⋊(C₄₃⋊C₈)	—	4	2	no	—
25	C₂²×Q₈×C₄₃	—	2	1	no	—
26	C₂×Q₁₆×C₄₃	—	2	2	no	—
27	C₂×Q₁₆×C₄₃	—	2	2	no	—
28	C₂×Q₁₆×C₄₃	—	2	1	no	—
29	C₂×Q₁₆×C₄₃	Q₈	1	2	yes	—
30	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
33	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
34	#106×C₄₃	—	2	2	no	—
35	#103×C₄₃	—	4	2	no	—
36	#97×C₄₃	—	2	2	no	—
37	#95×C₄₃	—	4	2	no	—
38	C₂×Q₈×C₄₃	—	2	2	no	—
39	C₂×Q₈×C₄₃	—	4	1	no	—
40	C₂×Q₈×C₄₃	—	4	1	no	—
41	C₂×Q₈×C₄₃	—	4	1	no	—
42	C₂×Q₈×C₄₃	—	4	1	no	—
43	C₂×Q₈×C₄₃	—	4	1	no	—
44	C₂²×C₄×C₄₃	—	4	1	no	—
45	C₂²×C₄×C₄₃	—	4	1	no	—
46	C₂²×C₄×C₄₃	SD₁₆	1	2	yes	—
47	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
48	Q₁₆×C₄₃	—	2	4	no	—
49	Q₁₆×C₄₃	—	4	2	no	—
50	C₂×C₈×C₄₃	—	2	1	no	—
51	C₂×C₈×C₄₃	—	2	1	no	—
52	C₂×C₈×C₄₃	SD₁₆	1	2	yes	—
53	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
54	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
56	C₂×C₈×C₄₃	#41	1	2	yes	—
57	C₄⋊C₄×C₄₃	—	4	1	no	—
58	C₄⋊C₄×C₄₃	—	4	1	no	—
59	C₄²×C₄₃	—	2	1	no	—
60	C₄²×C₄₃	—	2	1	no	—
61	#10621877	—	4	2	no	—
62	#10621877	—	4	2	no	—
63	C₂×C₄₃⋊C₈	—	4	2	no	—
64	C₄₃⋊C₁₆	—	2	4	no	—
65	C₂³×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
66	Q₈×C₄₃	—	4	2	no	—
67	Q₈×C₄₃	—	4	2	no	—
68	Q₈×C₄₃	—	8	1	no	—
69	Q₈×C₄₃	—	8	1	no	—
70	C₂×C₄×C₄₃	—	2	2	no	—
71	C₂×C₄×C₄₃	—	4	2	no	—
72	C₂×C₄×C₄₃	—	4	2	no	—
73	C₂×C₄×C₄₃	—	8	1	no	—
74	C₂×C₄×C₄₃	—	8	1	no	—
75	C₂×C₄×C₄₃	—	8	2	no	—
76	C₂×C₄×C₄₃	—	8	1	no	—
77	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
78	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
79	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
80	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
81	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
82	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
83	C₈×C₄₃	—	2	2	no	—
84	C₈×C₄₃	C₂²⋊Q₈	1	2	yes	—
85	C₈×C₄₃	#123	1	2	yes	—
86	C₄₃⋊C₈	—	8	2	no	—
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	—	8	2	no	—
94	C₄×C₄₃	—	16	1	no	—
95	C₄×C₄₃	#399	1	1	yes	characteristic
96	C₄×C₄₃	#476	1	1	yes	characteristic
97	C₄×C₄₃	#481	1	1	yes	characteristic
98	C₄×C₄₃	#496	1	1	yes	characteristic
99	C₂×#365	—	43	2	no	—
100	C₂×#362	—	43	2	no	—
101	#1971	D₄₃	1	1	yes	characteristic
102	#1849	—	43	1	no	—
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	C₂×D₄₃	1	1	yes	characteristic
109	C₂×M₃₂	—	43	2	no	—
110	C₂×#108	—	43	2	no	—
111	C₂×#108	—	86	1	no	—
112	C₂×#106	C₂×D₄₃	1	1	yes	characteristic
113	#423	—	86	1	no	—
114	C₂×#103	—	2	1	no	—
115	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
116	#391	—	2	1	no	—
117	#365	—	86	4	no	—
118	#362	—	86	4	no	—
119	C₄₃	#13536	1	1	yes	characteristic, complemented, Sylow
120	C₂²×Q₈	—	2	1	no	—
121	C₂×Q₁₆	—	2	1	no	—
122	C₂×Q₁₆	—	2	2	no	—
123	C₂×Q₁₆	—	2	2	no	—
124	C₂×Q₁₆	Dic₈₆	1	2	yes	—
125	C₂×M₁₆	—	86	2	no	—
126	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
127	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
128	C₂×C₄⋊C₄	—	2	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	—	86	2	no	—
134	#108	—	86	2	no	—
135	#108	—	172	2	no	—
136	#106	—	2	2	no	—
137	C₄⋊C₈	—	172	2	no	—
138	#103	—	4	2	no	—
139	#97	—	2	2	no	—
140	#95	—	4	2	no	—
141	C₂×Q₈	—	2	2	no	—
142	C₂×Q₈	—	4	1	no	—
143	C₂×Q₈	—	4	1	no	—
144	C₂×Q₈	—	4	1	no	—
145	C₂×Q₈	—	4	1	no	—
146	C₂×Q₈	—	4	1	no	—
147	C₂²×C₄	—	4	1	no	—
148	C₂²×C₄	—	4	1	no	—
149	C₂²×C₄	#10621883	1	2	yes	—
150	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
151	Q₁₆	—	2	4	no	—
152	Q₁₆	—	4	2	no	—
153	M₁₆	—	172	2	no	—
154	M₁₆	—	172	2	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	172	2	no	—
158	C₂×C₈	#10621883	1	2	yes	—
159	C₂×C₈	C₂×Dic₈₆	1	1	yes	characteristic
160	C₂×C₈	#10621898	1	1	yes	characteristic
161	C₂×C₈	D₄×D₄₃	1	2	yes	—
162	C₂×C₈	#10621900	1	2	yes	—
163	C₄⋊C₄	—	4	1	no	—
164	C₄⋊C₄	—	4	1	no	—
165	C₄²	—	2	1	no	—
166	C₄²	—	2	1	no	—
167	C₁₆	—	86	4	no	—
168	C₂³	C₂×#10621883	1	1	yes	characteristic
169	Q₈	—	4	2	no	—
170	Q₈	—	4	2	no	—
171	Q₈	—	8	1	no	—
172	Q₈	—	8	1	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	—	8	1	no	—
177	C₂×C₄	—	8	1	no	—
178	C₂×C₄	—	8	2	no	—
179	C₂×C₄	—	8	1	no	—
180	C₂×C₄	#13063304	1	1	yes	characteristic
181	C₂×C₄	#13063306	1	1	yes	characteristic
182	C₂×C₄	#13063321	1	1	yes	characteristic
183	C₂×C₄	C₂×#10621883	1	1	yes	characteristic
184	C₂×C₄	#13063350	1	1	yes	characteristic
185	C₂×C₄	#13063350	1	1	yes	characteristic
186	C₈	—	2	2	no	—
187	C₈	—	344	2	no	—
188	C₈	#13063302	1	2	yes	—
189	C₈	#13063307	1	2	yes	—
190	C₂²	—	2	2	no	—
191	C₂²	#14583189	1	1	yes	characteristic
192	C₂²	#14583215	1	1	yes	characteristic
193	C₂²	#14583271	1	1	yes	characteristic
194	C₄	—	4	2	no	—
195	C₄	—	8	2	no	—
196	C₄	—	16	1	no	—
197	C₄	#14582835	1	1	yes	characteristic
198	C₄	#14583189	1	1	yes	characteristic
199	C₄	#14583215	1	1	yes	characteristic
200	C₄	#14583289	1	1	yes	characteristic
201	C₂	—	2	1	no	—
202	C₂	—	2	1	no	—
203	C₂	#19136128	1	1	yes	characteristic
204	C₂	#19136812	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14582721

C₂

yes

maximal

C₂×#14582715

C₂

yes

maximal

#19134770

C₂

yes

maximal, characteristic

#19132679

C₂

yes

maximal, characteristic

C₄₃×#1971

C₂

yes

maximal, characteristic

C₂×#13063265

C₂²

yes

—

C₂×#13063261

—

C₂×#13063261

C₂²

yes

—

#14582923

—

#14582721

—

#14582715

—

C₂²×Q₁₆×C₄₃

C₂²

yes

characteristic

C₂×#106×C₄₃

C₂²

yes

characteristic

C₂×#103×C₄₃

—

C₂×#97×C₄₃

C₂²

yes

characteristic

C₄₃×#391

—

C₂×#10621877

—

#13063265

—

#13063265

D₄

yes

—

C₂×C₄₃⋊C₁₆

—

#13063261

—

#13063261

—

#13063261

—

C₄⋊(C₄₃⋊C₈)

—

C₂²×Q₈×C₄₃

—

C₂×Q₁₆×C₄₃

—

C₂×Q₁₆×C₄₃

—

C₂×Q₁₆×C₄₃

—

C₂×Q₁₆×C₄₃

Q₈

yes

—

C₂²×C₈×C₄₃

D₄

yes

characteristic

C₂²×C₈×C₄₃

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄²×C₄₃

D₄

yes

characteristic

#106×C₄₃

—

#103×C₄₃

—

#97×C₄₃

—

#95×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

SD₁₆

yes

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

Q₁₆×C₄₃

—

Q₁₆×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

SD₁₆

yes

—

C₂×C₈×C₄₃

C₂×D₄

yes

—

C₂×C₈×C₄₃

C₂×Q₈

yes

C₂×C₈×C₄₃

yes

C₂×C₈×C₄₃

yes

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

#10621877

—

#10621877

—

C₂×C₄₃⋊C₈

—

C₄₃⋊C₁₆

—

C₂³×C₄₃

C₂×SD₁₆

yes

characteristic

Q₈×C₄₃

—

Q₈×C₄₃

—

Q₈×C₄₃

—

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

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

C₈×C₄₃

—

C₈×C₄₃

C₂²⋊Q₈

yes

—

C₈×C₄₃

#123

yes

—

C₄₃⋊C₈

—

#13536

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

—

100

C₂×#362

—

101

#1971

D₄₃

yes

characteristic

102

#1849

—

103

#1377

—

104

C₂×C₄₃

—

105

C₂×C₄₃

—

106

C₂×C₄₃

#2094

yes

characteristic

107

C₂×C₄₃

#2239

yes

—

108

C₂²×Q₁₆

C₂×D₄₃

yes

characteristic

109

C₂×M₃₂

—

110

C₂×#108

—

111

C₂×#108

—

112

C₂×#106

C₂×D₄₃

yes

characteristic

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

Dic₈₆

yes

—

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

—

172

—

136

#106

—

137

C₄⋊C₈

—

172

—

138

#103

—

139

#97

—

140

#95

—

141

C₂×Q₈

—

142

C₂×Q₈

—

143

C₂×Q₈

—

144

C₂×Q₈

—

145

C₂×Q₈

—

146

C₂×Q₈

—

147

C₂²×C₄

—

148

C₂²×C₄

—

149

C₂²×C₄

#10621883

yes

—

150

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

151

Q₁₆

—

152

Q₁₆

—

153

M₁₆

—

172

—

154

M₁₆

—

172

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

172

—

158

C₂×C₈

#10621883

yes

—

159

C₂×C₈

C₂×Dic₈₆

yes

characteristic

160

C₂×C₈

#10621898

yes

characteristic

161

C₂×C₈

D₄×D₄₃

yes

—

162

C₂×C₈

#10621900

yes

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄²

—

166

C₄²

—

167

C₁₆

—

168

C₂³

C₂×#10621883

yes

characteristic

169

Q₈

—

170

Q₈

—

171

Q₈

—

172

Q₈

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

#13063304

yes

characteristic

181

C₂×C₄

#13063306

yes

characteristic

182

C₂×C₄

#13063321

yes

characteristic

183

C₂×C₄

C₂×#10621883

yes

characteristic

184

C₂×C₄

#13063350

yes

characteristic

185

C₂×C₄

#13063350

yes

characteristic

186

C₈

—

187

C₈

—

344

—

188

C₈

#13063302

yes

—

189

C₈

#13063307

yes

—

190

C₂²

—

191

C₂²

#14583189

yes

characteristic

192

C₂²

#14583215

yes

characteristic

193

C₂²

#14583271

yes

characteristic

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

#14582835

yes

characteristic

198

C₄

#14583189

yes

characteristic

199

C₄

#14583215

yes

characteristic

200

C₄

#14583289

yes

characteristic

201

C₂

—

202

C₂

—

203

C₂

#19136128

yes

characteristic

204

C₂

#19136812

yes

—

	subgroup	G/H
center	C₂²	#14583215
commutator, cosocle	C₂²×C₈×C₄₃	C₂³
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#1971	C₂
socle	C₂²×C₄₃	#481
prime radical	C₂³×C₄₃	C₂×SD₁₆
hypercenter	#1971	D₄₃
hypocenter	C₄₃	#13536
LCS	G ▷ C₂²×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1971

subgroup

G/H

center

C₂²

#14583215

commutator, cosocle

C₂²×C₈×C₄₃

C₂³

Frattini

C₂²×C₈

C₂²×D₄₃

Fitting

C₄₃×#1971

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 ◁ #1971

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×#14582721	2
C₂×#14582715	2
#19134770	2
#19132679	2
C₄₃×#1971	2
#13536	43

subgroup

index

C₄₃×#1971

Subgroups — #25016712

Conjugacy classes of subgroups

Complements

Maximal subgroups