Group #25014635 — subgroups

Subgroups — #25014635

Order 11008.

318 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 131 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×#14582727	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582674	C₂	1	1	yes	maximal, characteristic
3	#19135326	C₂	1	2	yes	maximal
4	#19132686	C₂	1	2	yes	maximal
5	C₂×C₄₃×#366	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#13063259	C₂²	1	2	yes	—
8	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
9	#14582795	—	2	2	no	—
10	#14582727	C₄	1	4	yes	complemented
11	#14582674	—	2	2	no	—
12	#14582661	—	2	2	no	—
13	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
14	C₂×#107×C₄₃	C₂²	1	2	yes	—
15	C₄₃×#366	—	2	2	no	—
16	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
17	C₂×C₄⋊Dic₄₃	—	2	2	no	—
18	C₂×Dic₄₃⋊C₄	—	4	2	no	—
19	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
20	C₂×C₄₃⋊C₁₆	—	2	1	no	—
21	C₂×C₄₃⋊C₁₆	—	2	1	no	—
22	C₂×C₄₃⋊C₁₆	Q₈	1	2	yes	—
23	#13063259	—	2	2	no	—
24	#13063259	C₂×C₄	1	4	yes	—
25	C₈×Dic₄₃	—	2	1	no	—
26	C₈×Dic₄₃	—	2	1	no	—
27	C₄₃⋊#95	—	4	2	no	—
28	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
30	C₂×C₁₆×C₄₃	C₂×C₄	1	2	yes	—
31	C₂×C₁₆×C₄₃	D₄	1	2	yes	—
32	C₂×C₁₆×C₄₃	Q₈	1	2	yes	—
33	#107×C₄₃	—	2	2	no	—
34	#107×C₄₃	—	2	2	no	—
35	C₂²×C₄×C₄₃	—	4	2	no	—
36	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
37	C₂²×Dic₄₃	—	4	2	no	—
38	C₂²×Dic₄₃	D₈	1	1	yes	characteristic
39	C₂²×Dic₄₃	D₈	1	1	yes	characteristic
40	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
45	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
46	C₄⋊C₄×C₄₃	—	4	2	no	—
47	C₄⋊C₄×C₄₃	—	4	2	no	—
48	C₄⋊Dic₄₃	—	2	4	no	—
49	C₄⋊Dic₄₃	—	4	2	no	—
50	Dic₄₃⋊C₄	—	8	4	no	—
51	C₄×Dic₄₃	—	2	1	no	—
52	C₄×Dic₄₃	—	2	1	no	—
53	C₂×C₄₃⋊C₈	—	2	1	no	—
54	C₂×C₄₃⋊C₈	—	2	1	no	—
55	C₂×C₄₃⋊C₈	Q₁₆	1	2	yes	—
56	C₁₆×C₄₃	C₄⋊C₄	1	4	yes	complemented
57	C₄₃⋊C₁₆	—	2	2	no	—
58	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₄₃	—	8	2	no	—
60	C₂×C₄×C₄₃	—	8	2	no	—
61	C₂×C₄×C₄₃	—	8	2	no	—
62	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
63	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
64	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
66	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
67	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
68	C₂×Dic₄₃	—	2	1	no	—
69	C₂×Dic₄₃	—	2	1	no	—
70	C₂×Dic₄₃	—	2	2	no	—
71	C₂×Dic₄₃	—	2	2	no	—
72	C₂×Dic₄₃	—	4	4	no	—
73	C₂×Dic₄₃	—	8	2	no	—
74	C₂×Dic₄₃	—	8	2	no	—
75	C₈×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₈×C₄₃	C₄×Q₈	1	1	yes	characteristic
77	C₈×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
78	C₈×C₄₃	#125	1	1	yes	characteristic
79	C₄₃⋊C₈	—	2	2	no	—
80	#13167	—	43	1	no	maximal, Sylow
81	C₂²×C₄₃	C₄×D₈	1	1	yes	characteristic
82	C₂²×C₄₃	#466	1	1	yes	characteristic
83	C₂²×C₄₃	#500	1	1	yes	characteristic
84	C₂²×C₄₃	#509	1	2	yes	—
85	C₂²×C₄₃	#510	1	2	yes	—
86	C₄×C₄₃	—	16	2	no	—
87	C₄×C₄₃	#389	1	1	yes	characteristic
88	C₄×C₄₃	C₄×Q₁₆	1	1	yes	characteristic
89	C₄×C₄₃	#467	1	1	yes	characteristic
90	C₄×C₄₃	#499	1	1	yes	characteristic
91	Dic₄₃	—	4	1	no	—
92	Dic₄₃	—	4	1	no	—
93	Dic₄₃	—	8	4	no	—
94	C₂×#367	—	43	1	no	—
95	C₂×#366	D₄₃	1	1	yes	characteristic
96	C₂×#346	—	43	1	no	—
97	#1942	—	43	2	no	—
98	#1380	—	43	2	no	—
99	C₂×C₄₃	#1942	1	1	yes	characteristic
100	C₂×C₄₃	#2175	1	2	yes	—
101	C₂×C₄₃	#2218	1	1	yes	characteristic
102	C₂×C₄₃	#2219	1	1	yes	characteristic
103	C₂×C₄₃	#2255	1	2	yes	—
104	C₂²×C₁₆	—	43	1	no	—
105	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
106	C₂×#107	—	43	2	no	—
107	C₂×#107	C₂×D₄₃	1	2	yes	—
108	C₂×C₄×C₈	—	43	1	no	—
109	#389	—	86	2	no	—
110	#367	—	43	4	no	—
111	#366	—	2	2	no	—
112	#346	—	86	2	no	—
113	#340	—	86	2	no	—
114	C₄₃	#13167	1	1	yes	characteristic, complemented, Sylow
115	C₂²×C₈	—	43	1	no	—
116	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
117	C₂×C₄⋊C₄	—	2	2	no	—
118	C₂×C₄⋊C₄	—	86	2	no	—
119	C₂×C₄⋊C₄	—	172	2	no	—
120	C₂×C₄²	—	43	1	no	—
121	C₂×C₁₆	—	43	2	no	—
122	C₂×C₁₆	—	86	1	no	—
123	C₂×C₁₆	—	86	1	no	—
124	C₂×C₁₆	Dic₈₆	1	2	yes	—
125	C₂×C₁₆	C₄×D₄₃	1	2	yes	—
126	C₂×C₁₆	C₂²⋊D₄₃	1	2	yes	—
127	#107	—	2	2	no	—
128	#107	—	2	2	no	—
129	#107	—	43	4	no	—
130	#107	—	86	2	no	—
131	C₄×C₈	—	86	1	no	—
132	C₄×C₈	—	86	1	no	—
133	#95	—	172	2	no	—
134	C₂²×C₄	—	4	2	no	—
135	C₂²×C₄	—	43	1	no	—
136	C₂²×C₄	—	43	1	no	—
137	C₂²×C₄	—	172	2	no	—
138	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
139	C₂×C₈	—	43	2	no	—
140	C₂×C₈	—	86	1	no	—
141	C₂×C₈	—	86	1	no	—
142	C₂×C₈	C₂×Dic₈₆	1	1	yes	characteristic
143	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
144	C₂×C₈	#10621900	1	1	yes	characteristic
145	C₂×C₈	Q₈×D₄₃	1	1	yes	characteristic
146	C₂×C₈	#10621902	1	1	yes	characteristic
147	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
148	C₄⋊C₄	—	4	2	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄⋊C₄	—	86	4	no	—
151	C₄⋊C₄	—	344	4	no	—
152	C₄⋊C₄	—	172	2	no	—
153	C₄²	—	86	1	no	—
154	C₄²	—	86	1	no	—
155	C₁₆	—	86	2	no	—
156	C₁₆	Dic₄₃⋊C₄	1	4	yes	complemented
157	C₂³	D₈×D₄₃	1	1	yes	characteristic
158	C₂×C₄	—	8	2	no	—
159	C₂×C₄	—	8	2	no	—
160	C₂×C₄	—	8	2	no	—
161	C₂×C₄	—	86	2	no	—
162	C₂×C₄	—	86	2	no	—
163	C₂×C₄	—	86	1	no	—
164	C₂×C₄	—	86	1	no	—
165	C₂×C₄	—	344	2	no	—
166	C₂×C₄	—	344	2	no	—
167	C₂×C₄	—	172	4	no	—
168	C₂×C₄	#13063287	1	1	yes	characteristic
169	C₂×C₄	#13063296	1	1	yes	characteristic
170	C₂×C₄	#13063324	1	1	yes	characteristic
171	C₂×C₄	Q₁₆×D₄₃	1	1	yes	characteristic
172	C₂×C₄	#13063344	1	1	yes	characteristic
173	C₂×C₄	#13063348	1	1	yes	characteristic
174	C₈	—	86	2	no	—
175	C₈	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
176	C₈	#13063290	1	1	yes	characteristic
177	C₈	#13063301	1	1	yes	characteristic
178	C₈	#13063316	1	1	yes	characteristic
179	C₂²	#14582981	1	1	yes	characteristic
180	C₂²	#14583132	1	1	yes	characteristic
181	C₂²	#14583303	1	1	yes	characteristic
182	C₂²	#14583333	1	2	yes	—
183	C₂²	#14583341	1	2	yes	—
184	C₄	—	16	2	no	—
185	C₄	—	344	4	no	—
186	C₄	—	172	1	no	—
187	C₄	—	172	1	no	—
188	C₄	#14582795	1	1	yes	characteristic
189	C₄	#14582993	1	1	yes	characteristic
190	C₄	#14583137	1	1	yes	characteristic
191	C₄	#14583300	1	1	yes	characteristic
192	C₂	#19135326	1	1	yes	characteristic
193	C₂	#19136458	1	2	yes	—
194	C₂	#19136695	1	1	yes	characteristic
195	C₂	#19136700	1	1	yes	characteristic
196	C₂	#19136879	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14582727

C₂

yes

maximal, characteristic

C₂×#14582674

C₂

yes

maximal, characteristic

#19135326

C₂

yes

maximal

#19132686

C₂

yes

maximal

C₂×C₄₃×#366

C₂

yes

maximal, characteristic

C₂²×C₄₃⋊C₁₆

C₂²

yes

characteristic

C₂×#13063259

C₂²

yes

—

C₂×C₈×Dic₄₃

C₂²

yes

characteristic

#14582795

—

#14582727

C₄

yes

complemented

#14582674

—

#14582661

—

C₂²×C₁₆×C₄₃

C₂²

yes

characteristic

C₂×#107×C₄₃

C₂²

yes

—

C₄₃×#366

—

C₂²×C₄₃⋊C₈

D₄

yes

characteristic

C₂×C₄⋊Dic₄₃

—

C₂×Dic₄₃⋊C₄

—

C₂×C₄×Dic₄₃

D₄

yes

characteristic

C₂×C₄₃⋊C₁₆

—

C₂×C₄₃⋊C₁₆

—

C₂×C₄₃⋊C₁₆

Q₈

yes

—

#13063259

—

#13063259

C₂×C₄

yes

—

C₈×Dic₄₃

—

C₈×Dic₄₃

—

C₄₃⋊#95

—

C₂²×C₈×C₄₃

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₄₃

—

C₂×C₁₆×C₄₃

C₂×C₄

yes

—

C₂×C₁₆×C₄₃

D₄

yes

—

C₂×C₁₆×C₄₃

Q₈

yes

—

#107×C₄₃

—

#107×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×Dic₄₃

—

C₂²×Dic₄₃

D₈

yes

characteristic

C₂²×Dic₄₃

D₈

yes

characteristic

C₂×C₈×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₃

C₂×D₄

yes

characteristic

C₂×C₈×C₄₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₄₃

C₂×Q₈

yes

C₂×C₈×C₄₃

yes

C₂×C₈×C₄₃

yes

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

Q₁₆

yes

—

C₁₆×C₄₃

C₄⋊C₄

yes

complemented

C₄₃⋊C₁₆

—

C₂³×C₄₃

C₂×D₈

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₄×D₄

yes

characteristic

C₂×C₄×C₄₃

#121

yes

characteristic

C₂×C₄×C₄₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₄₃

C₂×Q₁₆

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₈×C₄₃

C₂×C₄⋊C₄

yes

characteristic

C₈×C₄₃

C₄×Q₈

yes

characteristic

C₈×C₄₃

C₂²⋊Q₈

yes

characteristic

C₈×C₄₃

#125

yes

characteristic

C₄₃⋊C₈

—

#13167

—

maximal, Sylow

C₂²×C₄₃

C₄×D₈

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

—

C₂²×C₄₃

#510

yes

—

C₄×C₄₃

—

C₄×C₄₃

#389

yes

characteristic

C₄×C₄₃

C₄×Q₁₆

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

C₂×#367

—

C₂×#366

D₄₃

yes

—

—

—

C₂×C₄₃

#1942

yes

characteristic

100

C₂×C₄₃

#2175

yes

—

101

C₂×C₄₃

#2218

yes

characteristic

102

C₂×C₄₃

#2219

yes

characteristic

103

C₂×C₄₃

#2255

yes

—

104

C₂²×C₁₆

—

105

C₂²×C₁₆

C₂×D₄₃

yes

characteristic

106

C₂×#107

—

107

C₂×#107

C₂×D₄₃

yes

—

108

C₂×C₄×C₈

—

109

#389

—

110

#367

—

111

#366

—

112

#346

—

113

#340

—

114

C₄₃

#13167

yes

characteristic, complemented, Sylow

115

C₂²×C₈

—

116

C₂²×C₈

C₂²×D₄₃

yes

characteristic

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

172

—

120

C₂×C₄²

—

121

C₂×C₁₆

—

122

C₂×C₁₆

—

123

C₂×C₁₆

—

124

C₂×C₁₆

Dic₈₆

yes

—

125

C₂×C₁₆

C₄×D₄₃

yes

—

126

C₂×C₁₆

C₂²⋊D₄₃

yes

—

127

#107

—

128

#107

—

129

#107

—

130

#107

—

131

C₄×C₈

—

132

C₄×C₈

—

133

#95

—

172

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

172

—

138

C₂²×C₄

D₄×D₄₃

yes

characteristic

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

C₂×Dic₈₆

yes

characteristic

143

C₂×C₈

C₂×C₄×D₄₃

yes

characteristic

144

C₂×C₈

#10621900

yes

characteristic

145

C₂×C₈

Q₈×D₄₃

yes

characteristic

146

C₂×C₈

#10621902

yes

characteristic

147

C₂×C₈

C₂×C₂²⋊D₄₃

yes

characteristic

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

344

—

152

C₄⋊C₄

—

172

—

153

C₄²

—

154

C₄²

—

155

C₁₆

—

156

C₁₆

Dic₄₃⋊C₄

yes

complemented

157

C₂³

D₈×D₄₃

yes

characteristic

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

344

—

166

C₂×C₄

—

344

—

167

C₂×C₄

—

172

—

168

C₂×C₄

#13063287

yes

characteristic

169

C₂×C₄

#13063296

yes

characteristic

170

C₂×C₄

#13063324

yes

characteristic

171

C₂×C₄

Q₁₆×D₄₃

yes

characteristic

172

C₂×C₄

#13063344

yes

characteristic

173

C₂×C₄

#13063348

yes

characteristic

174

C₈

—

175

C₈

C₂×Dic₄₃⋊C₄

yes

characteristic

176

C₈

#13063290

yes

characteristic

177

C₈

#13063301

yes

characteristic

178

C₈

#13063316

yes

characteristic

179

C₂²

#14582981

yes

characteristic

180

C₂²

#14583132

yes

characteristic

181

C₂²

#14583303

yes

characteristic

182

C₂²

#14583333

yes

—

183

C₂²

#14583341

yes

—

184

C₄

—

185

C₄

—

344

—

186

C₄

—

172

—

187

C₄

—

172

—

188

C₄

#14582795

yes

characteristic

189

C₄

#14582993

yes

characteristic

190

C₄

#14583137

yes

characteristic

191

C₄

#14583300

yes

characteristic

192

C₂

#19135326

yes

characteristic

193

C₂

#19136458

yes

—

194

C₂

#19136695

yes

characteristic

195

C₂

#19136700

yes

characteristic

196

C₂

#19136879

yes

—

	subgroup	G/H
center	C₂³	D₈×D₄₃
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₂×C₄₃×#366	C₂
socle, prime radical	C₂³×C₄₃	C₂×D₈
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	C₂×#366	D₄₃
hypocenter	C₄₃	#13167
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

subgroup

G/H

center

C₂³

D₈×D₄₃

commutator

C₂×C₈×C₄₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₄₃

Fitting

C₂×C₄₃×#366

C₂

socle, prime radical

C₂³×C₄₃

C₂×D₈

cosocle

C₂²×C₈×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

N	H ≅ G/N	type
10. #14582727	C₄	semidirect
56. C₁₆×C₄₃	C₄⋊C₄	semidirect
114. C₄₃	#13167	semidirect
156. C₁₆	Dic₄₃⋊C₄	semidirect

H ≅ G/N

type

10. #14582727

C₄

semidirect

56. C₁₆×C₄₃

C₄⋊C₄

semidirect

114. C₄₃

#13167

semidirect

156. C₁₆

Dic₄₃⋊C₄

semidirect

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

subgroup	index
C₂×#14582727	2
C₂×#14582674	2
#19135326	2
#19132686	2
C₂×C₄₃×#366	2
#13167	43

subgroup

index

C₂×C₄₃×#366

Subgroups — #25014635

Conjugacy classes of subgroups

Complements

Maximal subgroups