Group #25015146 — subgroups

Subgroups — #25015146

Order 11008.

316 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 115 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	#19143527	C₂	1	1	yes	maximal, characteristic
2	#19136361	C₂	1	1	yes	maximal, characteristic
3	#19132682	C₂	1	2	yes	maximal
4	#19132676	C₂	1	2	yes	maximal
5	C₄₃×#2110	C₂	1	1	yes	maximal, characteristic
6	#14583452	C₄	1	1	yes	characteristic
7	C₂×#13063265	C₂²	1	2	yes	—
8	#14583027	—	2	2	no	—
9	C₈×Dic₈₆	—	2	2	no	—
10	#14582927	C₄	1	1	yes	characteristic
11	#14582926	C₄	1	1	yes	characteristic
12	#14582922	C₄	1	1	yes	characteristic
13	#14582854	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
15	#14582720	—	2	2	no	—
16	C₂×M₃₂×C₄₃	C₂²	1	2	yes	—
17	C₄₃×#405	C₂²	1	1	yes	characteristic
18	C₄₃×#347	—	2	2	no	—
19	C₂×#10621877	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
21	#13063324	—	2	1	no	—
22	#13063316	—	2	1	no	—
23	#13063302	—	2	2	no	—
24	C₄×Dic₈₆	—	2	2	no	—
25	#13063284	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×Dic₄₃	C₂×C₄	1	1	yes	characteristic
27	#13063265	—	2	2	no	—
28	#13063265	—	2	2	no	—
29	C₂×C₄₃⋊C₁₆	—	2	2	no	—
30	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
31	Dic₄₃⋊C₈	—	2	2	no	—
32	Dic₄₃⋊C₈	—	2	2	no	—
33	#13063235	—	2	1	no	—
34	C₈×Dic₄₃	—	2	1	no	—
35	C₈×Dic₄₃	—	2	1	no	—
36	C₈×Dic₄₃	—	2	1	no	—
37	C₂×M₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
39	#117×C₄₃	C₂×C₄	1	1	yes	characteristic
40	M₃₂×C₄₃	—	2	2	no	—
41	M₃₂×C₄₃	Q₈	1	4	yes	complemented
42	C₂×C₁₆×C₄₃	—	2	2	no	—
43	#97×C₄₃	D₄	1	2	yes	—
44	C₄×C₈×C₄₃	D₄	1	2	yes	—
45	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×Dic₄₃	—	2	1	no	—
47	C₂×Dic₈₆	—	2	2	no	—
48	M₁₆×C₄₃	—	2	2	no	—
49	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
50	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
53	C₂×C₈×C₄₃	#41	1	2	yes	—
54	C₄⋊C₄×C₄₃	—	2	1	no	—
55	C₂²⋊C₄×C₄₃	—	2	1	no	—
56	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
57	C₂²⋊Dic₄₃	M₁₆	1	2	yes	—
58	C₄⋊Dic₄₃	—	2	2	no	—
59	C₄⋊Dic₄₃	—	2	1	no	—
60	Dic₄₃⋊C₄	—	2	2	no	—
61	Dic₄₃⋊C₄	—	2	2	no	—
62	C₄×Dic₄₃	—	2	1	no	—
63	C₄×Dic₄₃	—	2	1	no	—
64	C₄×Dic₄₃	M₁₆	1	1	yes	characteristic
65	C₄×Dic₄₃	M₁₆	1	1	yes	characteristic
66	#10621877	—	2	2	no	—
67	C₂×C₄₃⋊C₈	—	2	1	no	—
68	C₂×C₄₃⋊C₈	—	2	1	no	—
69	C₂×C₄₃⋊C₈	—	2	2	no	—
70	C₁₆×C₄₃	—	2	4	no	—
71	C₄₃⋊C₁₆	—	4	2	no	—
72	C₂³×C₄₃	C₂×M₁₆	1	1	yes	characteristic
73	C₂×C₄×C₄₃	—	2	2	no	—
74	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₄₃	C₄×Q₈	1	2	yes	—
76	C₂×C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₄×C₄₃	#131	1	2	yes	—
78	C₂×Dic₄₃	—	2	1	no	—
79	C₂×Dic₄₃	—	2	1	no	—
80	C₂×Dic₄₃	—	2	2	no	—
81	C₂×Dic₄₃	—	4	2	no	—
82	Dic₈₆	—	4	4	no	—
83	C₈×C₄₃	—	2	2	no	—
84	C₈×C₄₃	C₂²⋊Q₈	1	2	yes	—
85	C₈×C₄₃	#124	1	1	yes	characteristic
86	C₈×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
87	C₄₃⋊C₈	—	4	1	no	—
88	C₄₃⋊C₈	—	4	1	no	—
89	#13270	—	43	1	no	maximal, Sylow
90	C₂²×C₄₃	—	4	1	no	—
91	C₂²×C₄₃	#407	1	1	yes	characteristic
92	C₂²×C₄₃	#446	1	2	yes	—
93	C₄×C₄₃	—	4	2	no	—
94	C₄×C₄₃	#391	1	1	yes	characteristic
95	C₄×C₄₃	#408	1	1	yes	characteristic
96	C₄×C₄₃	#446	1	2	yes	—
97	Dic₄₃	—	2	2	no	—
98	Dic₄₃	—	4	2	no	—
99	Dic₄₃	—	4	1	no	—
100	#2960	—	43	1	no	—
101	#2153	—	43	1	no	—
102	#2110	D₄₃	1	1	yes	characteristic
103	#1378	—	43	2	no	—
104	#1376	—	43	2	no	—
105	C₂×C₄₃	—	2	1	no	—
106	C₂×C₄₃	—	2	2	no	—
107	C₂×C₄₃	#1985	1	1	yes	characteristic
108	#533	—	43	1	no	—
109	C₂×M₃₂	—	43	2	no	—
110	C₂×M₃₂	C₂×D₄₃	1	2	yes	—
111	#446	—	86	2	no	—
112	C₈×Q₈	—	86	2	no	—
113	#424	—	43	1	no	—
114	#424	—	43	1	no	—
115	#423	—	43	1	no	—
116	#405	—	43	1	no	—
117	#405	C₂×D₄₃	1	1	yes	characteristic
118	C₂×C₄×C₈	—	43	1	no	—
119	#364	—	86	2	no	—
120	#347	—	2	2	no	—
121	C₄₃	#13270	1	1	yes	characteristic, complemented, Sylow
122	C₂×M₁₆	—	43	1	no	—
123	C₂×M₁₆	C₄×D₄₃	1	1	yes	characteristic
124	C₂²×C₈	—	43	1	no	—
125	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
126	C₄⋊Q₈	—	86	1	no	—
127	#125	—	86	1	no	—
128	C₂²⋊Q₈	—	86	2	no	—
129	C₄×Q₈	—	86	2	no	—
130	#117	—	43	1	no	—
131	#117	C₄×D₄₃	1	1	yes	characteristic
132	C₂×C₄²	—	43	1	no	—
133	M₃₂	—	2	2	no	—
134	M₃₂	—	86	2	no	—
135	M₃₂	—	86	2	no	—
136	M₃₂	Dic₈₆	1	4	yes	complemented
137	C₂×C₁₆	—	2	2	no	—
138	C₂×C₁₆	—	86	2	no	—
139	C₄⋊C₈	—	86	2	no	—
140	C₄⋊C₈	—	86	2	no	—
141	C₄⋊C₈	—	86	2	no	—
142	#97	—	86	1	no	—
143	#97	C₂²⋊D₄₃	1	2	yes	—
144	C₄×C₈	—	86	1	no	—
145	C₄×C₈	—	86	1	no	—
146	C₄×C₈	—	86	1	no	—
147	C₄×C₈	D₁₇₂	1	2	yes	—
148	C₂×Q₈	—	86	2	no	—
149	C₂²×C₄	—	86	1	no	—
150	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
151	M₁₆	—	2	2	no	—
152	M₁₆	—	86	2	no	—
153	C₂×C₈	—	86	1	no	—
154	C₂×C₈	—	86	1	no	—
155	C₂×C₈	—	86	2	no	—
156	C₂×C₈	(C₂×D₄₃)⋊C₄	1	2	yes	—
157	C₂×C₈	C₂×Dic₈₆	1	2	yes	—
158	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
159	C₂×C₈	#10621898	1	2	yes	—
160	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
161	C₄⋊C₄	—	2	1	no	—
162	C₄⋊C₄	—	86	2	no	—
163	C₄⋊C₄	—	86	2	no	—
164	C₄⋊C₄	—	86	2	no	—
165	C₄⋊C₄	—	86	1	no	—
166	C₂²⋊C₄	—	2	1	no	—
167	C₂²⋊C₄	—	43	2	no	—
168	C₄²	—	43	1	no	—
169	C₄²	—	43	1	no	—
170	C₄²	—	86	1	no	—
171	C₄²	—	86	1	no	—
172	C₄²	(C₂×D₄₃)⋊C₄	1	2	yes	—
173	C₁₆	—	2	4	no	—
174	C₁₆	—	172	2	no	—
175	C₂³	M₁₆×D₄₃	1	1	yes	characteristic
176	Q₈	—	172	4	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	86	1	no	—
179	C₂×C₄	—	86	1	no	—
180	C₂×C₄	—	86	2	no	—
181	C₂×C₄	—	172	2	no	—
182	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
183	C₂×C₄	C₄×Dic₈₆	1	2	yes	—
184	C₂×C₄	M₁₆×D₄₃	1	1	yes	characteristic
185	C₂×C₄	#13063333	1	2	yes	—
186	C₈	—	2	2	no	—
187	C₈	—	172	1	no	—
188	C₈	—	172	1	no	—
189	C₈	#13063304	1	2	yes	—
190	C₈	#13063313	1	1	yes	characteristic
191	C₈	#13063325	1	1	yes	characteristic
192	C₂²	—	4	1	no	—
193	C₂²	#14582862	1	1	yes	characteristic
194	C₂²	#14583027	1	2	yes	—
195	C₄	—	4	2	no	—
196	C₄	—	86	2	no	—
197	C₄	—	172	1	no	—
198	C₄	—	172	2	no	—
199	C₄	#14582805	1	1	yes	characteristic
200	C₄	#14582867	1	1	yes	characteristic
201	C₄	#14583027	1	2	yes	—
202	C₂	—	2	1	no	—
203	C₂	—	2	2	no	—
204	C₂	#19135535	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19143527

C₂

yes

maximal, characteristic

#19136361

C₂

yes

maximal, characteristic

#19132682

C₂

yes

maximal

#19132676

C₂

yes

maximal

C₄₃×#2110

C₂

yes

maximal, characteristic

#14583452

C₄

yes

characteristic

C₂×#13063265

C₂²

yes

—

#14583027

—

C₈×Dic₈₆

—

#14582927

C₄

yes

characteristic

#14582926

C₄

yes

characteristic

#14582922

C₄

yes

characteristic

#14582854

C₂²

yes

characteristic

C₂×C₈×Dic₄₃

C₂²

yes

characteristic

#14582720

—

C₂×M₃₂×C₄₃

C₂²

yes

—

C₄₃×#405

C₂²

yes

characteristic

C₄₃×#347

—

C₂×#10621877

C₂×C₄

yes

characteristic

C₂²×C₄₃⋊C₈

C₂×C₄

yes

—

—

—

C₄×Dic₈₆

—

#13063284

C₂×C₄

yes

characteristic

C₂×C₄×Dic₄₃

C₂×C₄

yes

characteristic

#13063265

—

#13063265

—

C₂×C₄₃⋊C₁₆

—

C₄⋊(C₄₃⋊C₈)

—

Dic₄₃⋊C₈

—

Dic₄₃⋊C₈

—

#13063235

—

C₈×Dic₄₃

—

C₈×Dic₄₃

—

C₈×Dic₄₃

—

C₂×M₁₆×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₃

C₂³

yes

characteristic

#117×C₄₃

C₂×C₄

yes

characteristic

M₃₂×C₄₃

—

M₃₂×C₄₃

Q₈

yes

complemented

C₂×C₁₆×C₄₃

—

#97×C₄₃

D₄

yes

—

C₄×C₈×C₄₃

D₄

yes

—

C₂²×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂²×Dic₄₃

—

C₂×Dic₈₆

—

M₁₆×C₄₃

—

C₂×C₈×C₄₃

C₂²⋊C₄

yes

—

C₂×C₈×C₄₃

C₂×D₄

yes

characteristic

C₂×C₈×C₄₃

C₂×D₄

yes

characteristic

C₂×C₈×C₄₃

C₂×Q₈

yes

—

C₂×C₈×C₄₃

#41

yes

—

C₄⋊C₄×C₄₃

—

C₂²⋊C₄×C₄₃

—

C₄²×C₄₃

C₂²⋊C₄

yes

—

C₂²⋊Dic₄₃

M₁₆

yes

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₄×Dic₄₃

—

C₄×Dic₄₃

M₁₆

yes

characteristic

C₄×Dic₄₃

M₁₆

yes

characteristic

#10621877

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₁₆×C₄₃

—

C₄₃⋊C₁₆

—

C₂³×C₄₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₄×Q₈

yes

—

C₂×C₄×C₄₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₄₃

#131

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

C₂²⋊Q₈

yes

—

C₈×C₄₃

#124

yes

characteristic

C₈×C₄₃

C₄⋊Q₈

yes

characteristic

C₄₃⋊C₈

—

C₄₃⋊C₈

—

#13270

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

#407

yes

characteristic

C₂²×C₄₃

#446

yes

—

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

100

#2960

—

101

#2153

—

102

#2110

D₄₃

yes

characteristic

103

#1378

—

104

#1376

—

105

C₂×C₄₃

—

106

C₂×C₄₃

—

107

C₂×C₄₃

#1985

yes

characteristic

108

#533

—

109

C₂×M₃₂

—

110

C₂×M₃₂

C₂×D₄₃

yes

—

111

#446

—

112

C₈×Q₈

—

113

#424

—

114

#424

—

115

#423

—

116

#405

—

117

#405

C₂×D₄₃

yes

characteristic

118

C₂×C₄×C₈

—

119

#364

—

120

#347

—

121

C₄₃

#13270

yes

characteristic, complemented, Sylow

122

C₂×M₁₆

—

123

C₂×M₁₆

C₄×D₄₃

yes

characteristic

124

C₂²×C₈

—

125

C₂²×C₈

C₂²×D₄₃

yes

characteristic

126

C₄⋊Q₈

—

127

#125

—

128

C₂²⋊Q₈

—

129

C₄×Q₈

—

130

#117

—

131

#117

C₄×D₄₃

yes

characteristic

132

C₂×C₄²

—

133

M₃₂

—

134

M₃₂

—

135

M₃₂

—

136

M₃₂

Dic₈₆

yes

complemented

137

C₂×C₁₆

—

138

C₂×C₁₆

—

139

C₄⋊C₈

—

140

C₄⋊C₈

—

141

C₄⋊C₈

—

142

#97

—

143

#97

C₂²⋊D₄₃

yes

—

144

C₄×C₈

—

145

C₄×C₈

—

146

C₄×C₈

—

147

C₄×C₈

D₁₇₂

yes

—

148

C₂×Q₈

—

149

C₂²×C₄

—

150

C₂²×C₄

C₂×C₄×D₄₃

yes

characteristic

151

M₁₆

—

152

M₁₆

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

(C₂×D₄₃)⋊C₄

yes

—

157

C₂×C₈

C₂×Dic₈₆

yes

—

158

C₂×C₈

C₂×D₁₇₂

yes

characteristic

159

C₂×C₈

#10621898

yes

—

160

C₂×C₈

C₂×C₂²⋊D₄₃

yes

characteristic

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₂²⋊C₄

—

167

C₂²⋊C₄

—

168

C₄²

—

169

C₄²

—

170

C₄²

—

171

C₄²

—

172

C₄²

(C₂×D₄₃)⋊C₄

yes

—

173

C₁₆

—

174

C₁₆

—

172

—

175

C₂³

M₁₆×D₄₃

yes

characteristic

176

Q₈

—

172

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

172

—

182

C₂×C₄

C₂×(C₂×D₄₃)⋊C₄

yes

characteristic

183

C₂×C₄

C₄×Dic₈₆

yes

—

184

C₂×C₄

M₁₆×D₄₃

yes

characteristic

185

C₂×C₄

#13063333

yes

—

186

C₈

—

187

C₈

—

172

—

188

C₈

—

172

—

189

C₈

#13063304

yes

—

190

C₈

#13063313

yes

characteristic

191

C₈

#13063325

yes

characteristic

192

C₂²

—

193

C₂²

#14582862

yes

characteristic

194

C₂²

#14583027

yes

—

195

C₄

—

196

C₄

—

197

C₄

—

172

—

198

C₄

—

172

—

199

C₄

#14582805

yes

characteristic

200

C₄

#14582867

yes

characteristic

201

C₄

#14583027

yes

—

202

C₂

—

203

C₂

—

204

C₂

#19135535

yes

characteristic

	subgroup	G/H
center	C₈	#13063313
commutator	C₂²×C₄×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#2110	C₂
socle	C₂×C₄₃	#1985
prime radical	C₂³×C₄₃	C₂×M₁₆
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	#2110	D₄₃
hypocenter	C₄₃	#13270
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2110

subgroup

G/H

center

C₈

#13063313

commutator

C₂²×C₄×C₄₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₄₃

Fitting

C₄₃×#2110

C₂

socle

C₂×C₄₃

#1985

prime radical

C₂³×C₄₃

C₂×M₁₆

cosocle

C₂²×C₈×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2110

N	H ≅ G/N	type
41. M₃₂×C₄₃	Q₈	semidirect
121. C₄₃	#13270	semidirect
136. M₃₂	Dic₈₆	semidirect

H ≅ G/N

type

41. M₃₂×C₄₃

Q₈

semidirect

121. C₄₃

#13270

semidirect

136. M₃₂

Dic₈₆

semidirect

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

subgroup	index
#19143527	2
#19136361	2
#19132682	2
#19132676	2
C₄₃×#2110	2
#13270	43

subgroup

index

C₄₃×#2110

Subgroups — #25015146

Conjugacy classes of subgroups

Complements

Maximal subgroups