Group #19135985 — subgroups

Subgroups — #19135985

Order 5504.

248 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 87 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×Dic₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#13063250	C₂	1	1	yes	maximal, characteristic
3	C₂×#13063248	C₂	1	1	yes	maximal, characteristic
4	#14582804	C₂	1	1	yes	maximal, characteristic
5	#14582798	C₂	1	1	yes	maximal, characteristic
6	#14582661	C₂	1	1	yes	maximal, characteristic
7	C₂×#107×C₄₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₈₆	C₂²	1	1	yes	characteristic
9	C₂²×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₄₃	—	2	1	no	—
11	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
13	Dic₄₃⋊C₈	—	2	2	no	—
14	#13063250	—	2	2	no	—
15	#13063248	—	2	2	no	—
16	C₄₃⋊#95	—	2	1	no	—
17	C₄₃⋊#95	—	2	1	no	—
18	C₄₃⋊#95	—	2	1	no	—
19	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
22	#107×C₄₃	—	2	2	no	—
23	C₂²×C₄×C₄₃	—	2	1	no	—
24	C₂²×C₄×C₄₃	—	2	1	no	—
25	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
26	C₂²×Dic₄₃	—	2	1	no	—
27	C₂²×Dic₄₃	—	2	1	no	—
28	C₂²×Dic₄₃	D₄	1	1	yes	characteristic
29	C₂²×Dic₄₃	D₄	1	1	yes	characteristic
30	C₂×Dic₈₆	—	2	1	no	—
31	C₂×Dic₈₆	—	2	1	no	—
32	C₂×Dic₈₆	—	2	1	no	—
33	C₂×Dic₈₆	—	2	1	no	—
34	C₂×Dic₈₆	—	2	1	no	—
35	C₂×Dic₈₆	Q₈	1	2	yes	—
36	C₂×C₈×C₄₃	—	2	1	no	—
37	C₂×C₈×C₄₃	—	2	1	no	—
38	C₂×C₈×C₄₃	D₄	1	2	yes	—
39	C₄⋊C₄×C₄₃	—	2	1	no	—
40	C₄⋊C₄×C₄₃	—	2	1	no	—
41	C₄⋊C₄×C₄₃	—	2	1	no	—
42	C₄⋊C₄×C₄₃	D₄	1	2	yes	—
43	C₄⋊Dic₄₃	—	2	1	no	—
44	C₄⋊Dic₄₃	—	2	1	no	—
45	C₄⋊Dic₄₃	—	4	2	no	—
46	C₄×Dic₄₃	—	2	1	no	—
47	C₄×Dic₄₃	—	2	1	no	—
48	C₂×C₄₃⋊C₈	—	2	1	no	—
49	C₂×C₄₃⋊C₈	—	2	1	no	—
50	C₂×C₄₃⋊C₈	—	2	1	no	—
51	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₄₃	—	2	2	no	—
53	C₂×C₄×C₄₃	—	4	1	no	—
54	C₂×C₄×C₄₃	—	4	1	no	—
55	C₂×C₄×C₄₃	—	4	1	no	—
56	C₂×C₄×C₄₃	—	4	1	no	—
57	C₂×C₄×C₄₃	—	4	1	no	—
58	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
62	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
63	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
64	C₂×Dic₄₃	—	2	1	no	—
65	C₂×Dic₄₃	—	2	1	no	—
66	C₂×Dic₄₃	—	2	1	no	—
67	C₂×Dic₄₃	—	2	2	no	—
68	C₂×Dic₄₃	—	2	1	no	—
69	C₂×Dic₄₃	—	2	1	no	—
70	C₂×Dic₄₃	—	4	1	no	—
71	C₂×Dic₄₃	—	4	1	no	—
72	C₂×Dic₄₃	—	4	1	no	—
73	C₂×Dic₄₃	—	4	1	no	—
74	C₂×Dic₄₃	—	4	1	no	—
75	C₂×Dic₄₃	Q₁₆	1	2	yes	—
76	Dic₈₆	—	2	2	no	—
77	Dic₈₆	—	2	2	no	—
78	Dic₈₆	—	4	1	no	—
79	Dic₈₆	—	4	1	no	—
80	C₈×C₄₃	—	2	2	no	—
81	C₄₃⋊C₈	—	4	1	no	—
82	C₂²×C₄₃	#121	1	1	yes	characteristic
83	C₂²×C₄₃	#121	1	1	yes	characteristic
84	C₂²×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
85	C₂²×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
86	C₂²×C₄₃	#135	1	1	yes	characteristic
87	C₂²×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
88	C₂²×C₄₃	#137	1	1	yes	characteristic
89	C₄×C₄₃	—	4	2	no	—
90	C₄×C₄₃	—	8	1	no	—
91	C₄×C₄₃	#121	1	1	yes	characteristic
92	C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
93	C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
94	C₄×C₄₃	#126	1	1	yes	characteristic
95	Dic₄₃	—	2	2	no	—
96	Dic₄₃	—	4	2	no	—
97	Dic₄₃	—	4	1	no	—
98	Dic₄₃	—	8	1	no	—
99	#2068	—	43	1	no	maximal, Sylow
100	C₂×C₄₃	#396	1	1	yes	characteristic
101	C₂×C₄₃	#462	1	1	yes	characteristic
102	C₂×C₄₃	#463	1	1	yes	characteristic
103	C₂×C₄₃	#467	1	1	yes	characteristic
104	C₂×C₄₃	#468	1	1	yes	characteristic
105	C₂×C₄₃	#477	1	1	yes	characteristic
106	C₂×C₄₃	#479	1	1	yes	characteristic
107	C₂×#107	D₄₃	1	1	yes	characteristic
108	C₂×C₄⋊C₈	—	43	1	no	—
109	C₂×#103	—	43	1	no	—
110	C₂×#103	—	43	1	no	—
111	#391	—	43	1	no	—
112	#389	—	43	1	no	—
113	#340	—	43	1	no	—
114	C₄₃	#2068	1	1	yes	characteristic, complemented, Sylow
115	C₂²×Q₈	—	43	1	no	—
116	C₂²×C₈	—	43	1	no	—
117	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
118	C₂×C₄⋊C₄	—	43	1	no	—
119	C₂×C₄⋊C₄	—	86	1	no	—
120	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
121	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
122	C₂×C₄²	—	43	1	no	—
123	#107	—	2	2	no	—
124	C₄⋊C₈	—	86	2	no	—
125	#103	—	86	2	no	—
126	#103	—	86	2	no	—
127	#95	—	86	1	no	—
128	#95	—	86	1	no	—
129	#95	—	86	1	no	—
130	C₂×Q₈	—	43	2	no	—
131	C₂×Q₈	—	86	1	no	—
132	C₂×Q₈	—	86	1	no	—
133	C₂×Q₈	—	86	1	no	—
134	C₂×Q₈	—	86	1	no	—
135	C₂×Q₈	—	86	1	no	—
136	C₂²×C₄	—	2	1	no	—
137	C₂²×C₄	—	2	1	no	—
138	C₂²×C₄	—	43	1	no	—
139	C₂²×C₄	—	43	1	no	—
140	C₂²×C₄	—	86	1	no	—
141	C₂²×C₄	—	86	1	no	—
142	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	86	1	no	—
146	C₂×C₈	—	86	1	no	—
147	C₂×C₈	—	86	1	no	—
148	C₂×C₈	D₁₇₂	1	2	yes	—
149	C₄⋊C₄	—	2	1	no	—
150	C₄⋊C₄	—	2	1	no	—
151	C₄⋊C₄	—	2	1	no	—
152	C₄⋊C₄	—	86	1	no	—
153	C₄⋊C₄	—	86	1	no	—
154	C₄⋊C₄	—	172	2	no	—
155	C₄⋊C₄	C₂²⋊D₄₃	1	2	yes	—
156	C₄²	—	86	1	no	—
157	C₄²	—	86	1	no	—
158	C₂³	D₄×D₄₃	1	1	yes	characteristic
159	Q₈	—	86	2	no	—
160	Q₈	—	86	2	no	—
161	Q₈	—	172	1	no	—
162	Q₈	—	172	1	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	4	1	no	—
165	C₂×C₄	—	4	1	no	—
166	C₂×C₄	—	4	1	no	—
167	C₂×C₄	—	4	1	no	—
168	C₂×C₄	—	4	1	no	—
169	C₂×C₄	—	43	2	no	—
170	C₂×C₄	—	86	1	no	—
171	C₂×C₄	—	86	1	no	—
172	C₂×C₄	—	86	1	no	—
173	C₂×C₄	—	86	2	no	—
174	C₂×C₄	—	86	1	no	—
175	C₂×C₄	—	86	1	no	—
176	C₂×C₄	—	172	1	no	—
177	C₂×C₄	—	172	1	no	—
178	C₂×C₄	—	172	1	no	—
179	C₂×C₄	—	172	1	no	—
180	C₂×C₄	—	172	1	no	—
181	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
182	C₂×C₄	#10621898	1	1	yes	characteristic
183	C₂×C₄	#10621900	1	1	yes	characteristic
184	C₂×C₄	Q₈×D₄₃	1	1	yes	characteristic
185	C₂×C₄	#10621902	1	1	yes	characteristic
186	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
187	C₈	—	2	2	no	—
188	C₈	—	172	1	no	—
189	C₂²	#13063297	1	1	yes	characteristic
190	C₂²	#13063300	1	1	yes	characteristic
191	C₂²	#13063303	1	1	yes	characteristic
192	C₂²	Q₁₆×D₄₃	1	1	yes	characteristic
193	C₂²	#13063344	1	1	yes	characteristic
194	C₂²	#13063349	1	1	yes	characteristic
195	C₂²	#13063357	1	1	yes	characteristic
196	C₄	—	4	2	no	—
197	C₄	—	8	1	no	—
198	C₄	—	86	2	no	—
199	C₄	—	344	1	no	—
200	C₄	—	172	2	no	—
201	C₄	—	172	1	no	—
202	C₄	#13063299	1	1	yes	characteristic
203	C₄	#13063301	1	1	yes	characteristic
204	C₄	#13063305	1	1	yes	characteristic
205	C₄	#13063320	1	1	yes	characteristic
206	C₂	#14582821	1	1	yes	characteristic
207	C₂	#14583107	1	1	yes	characteristic
208	C₂	#14583112	1	1	yes	characteristic
209	C₂	#14583141	1	1	yes	characteristic
210	C₂	#14583146	1	1	yes	characteristic
211	C₂	#14583193	1	1	yes	characteristic
212	C₂	#14583205	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₄₃⋊C₈

C₂

yes

maximal, characteristic

C₂×#13063250

C₂

yes

maximal, characteristic

C₂×#13063248

C₂

yes

maximal, characteristic

#14582804

C₂

yes

maximal, characteristic

#14582798

C₂

yes

maximal, characteristic

#14582661

C₂

yes

maximal, characteristic

C₂×#107×C₄₃

C₂

yes

maximal, characteristic

C₂²×Dic₈₆

C₂²

yes

characteristic

C₂²×C₄₃⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

C₂²

yes

characteristic

C₂×C₄×Dic₄₃

C₂²

yes

characteristic

Dic₄₃⋊C₈

—

#13063250

—

#13063248

—

C₄₃⋊#95

—

C₄₃⋊#95

—

C₄₃⋊#95

—

C₂²×C₈×C₄₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₃

C₂²

yes

characteristic

#107×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂²×Dic₄₃

D₄

yes

characteristic

C₂²×Dic₄₃

D₄

yes

characteristic

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

Q₈

yes

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

D₄

yes

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

D₄

yes

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄×Dic₄₃

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×Q₈

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

Q₁₆

yes

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₄₃

C₂×Q₁₆

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#121

yes

characteristic

C₄×C₄₃

C₂²⋊Q₈

yes

characteristic

C₄×C₄₃

C₂²⋊Q₈

yes

characteristic

C₄×C₄₃

#126

yes

characteristic

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

#2068

—

maximal, Sylow

100

C₂×C₄₃

#396

yes

characteristic

101

C₂×C₄₃

#462

yes

characteristic

102

C₂×C₄₃

#463

yes

characteristic

103

C₂×C₄₃

#467

yes

characteristic

104

C₂×C₄₃

#468

yes

characteristic

105

C₂×C₄₃

#477

yes

characteristic

106

C₂×C₄₃

#479

yes

characteristic

107

C₂×#107

D₄₃

yes

characteristic

108

C₂×C₄⋊C₈

—

109

C₂×#103

—

110

C₂×#103

—

111

#391

—

112

#389

—

113

#340

—

114

C₄₃

#2068

yes

characteristic, complemented, Sylow

115

C₂²×Q₈

—

116

C₂²×C₈

—

117

C₂²×C₈

C₂×D₄₃

yes

characteristic

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

C₂×D₄₃

yes

characteristic

121

C₂×C₄⋊C₄

C₂×D₄₃

yes

characteristic

122

C₂×C₄²

—

123

#107

—

124

C₄⋊C₈

—

125

#103

—

126

#103

—

127

#95

—

128

#95

—

129

#95

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

—

140

C₂²×C₄

—

141

C₂²×C₄

—

142

C₂²×C₄

C₂²×D₄₃

yes

characteristic

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

D₁₇₂

yes

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

172

—

155

C₄⋊C₄

C₂²⋊D₄₃

yes

—

156

C₄²

—

157

C₄²

—

158

C₂³

D₄×D₄₃

yes

characteristic

159

Q₈

—

160

Q₈

—

161

Q₈

—

172

—

162

Q₈

—

172

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

172

—

177

C₂×C₄

—

172

—

178

C₂×C₄

—

172

—

179

C₂×C₄

—

172

—

180

C₂×C₄

—

172

—

181

C₂×C₄

C₂×D₁₇₂

yes

characteristic

182

C₂×C₄

#10621898

yes

characteristic

183

C₂×C₄

#10621900

yes

characteristic

184

C₂×C₄

Q₈×D₄₃

yes

characteristic

185

C₂×C₄

#10621902

yes

characteristic

186

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

187

C₈

—

188

C₈

—

172

—

189

C₂²

#13063297

yes

characteristic

190

C₂²

#13063300

yes

characteristic

191

C₂²

#13063303

yes

characteristic

192

C₂²

Q₁₆×D₄₃

yes

characteristic

193

C₂²

#13063344

yes

characteristic

194

C₂²

#13063349

yes

characteristic

195

C₂²

#13063357

yes

characteristic

196

C₄

—

197

C₄

—

198

C₄

—

199

C₄

—

344

—

200

C₄

—

172

—

201

C₄

—

172

—

202

C₄

#13063299

yes

characteristic

203

C₄

#13063301

yes

characteristic

204

C₄

#13063305

yes

characteristic

205

C₄

#13063320

yes

characteristic

206

C₂

#14582821

yes

characteristic

207

C₂

#14583107

yes

characteristic

208

C₂

#14583112

yes

characteristic

209

C₂

#14583141

yes

characteristic

210

C₂

#14583146

yes

characteristic

211

C₂

#14583193

yes

characteristic

212

C₂

#14583205

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₄₃
commutator	C₂²×C₄×C₄₃	C₂³
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₂×#107×C₄₃	C₂
socle	C₂³×C₄₃	C₂×D₄
hypercenter	C₂×#107	D₄₃
hypocenter	C₄₃	#2068
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₄₃

commutator

C₂²×C₄×C₄₃

C₂³

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₂×#107×C₄₃

C₂

socle

C₂³×C₄₃

C₂×D₄

D₄₃

C₄₃

G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
114. C₄₃	#2068	semidirect

H ≅ G/N

type

114. C₄₃

#2068

semidirect

Subgroups — #19135985

Conjugacy classes of subgroups

Complements