Group #19135228 — subgroups

Subgroups — #19135228

Order 5504.

254 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 93 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×#13063250	C₂	1	1	yes	maximal, characteristic
2	C₂×#13063248	C₂	1	1	yes	maximal, characteristic
3	#14582804	C₂	1	1	yes	maximal, characteristic
4	C₄⋊C₄×Dic₄₃	C₂	1	1	yes	maximal, characteristic
5	#14582661	C₂	1	1	yes	maximal, characteristic
6	#14582653	C₂	1	1	yes	maximal, characteristic
7	C₂×#106×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	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₄₃	—	2	1	no	—
12	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
13	#13063250	C₄	1	4	yes	complemented
14	#13063248	—	2	2	no	—
15	C₄₃⋊#95	—	2	1	no	—
16	C₄₃⋊#95	—	2	1	no	—
17	C₄₃⋊#95	—	2	1	no	—
18	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
21	#106×C₄₃	—	2	2	no	—
22	C₂²×C₄×C₄₃	—	2	1	no	—
23	C₂²×C₄×C₄₃	—	2	1	no	—
24	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
25	C₂²×Dic₄₃	—	2	1	no	—
26	C₂²×Dic₄₃	—	2	1	no	—
27	C₂²×Dic₄₃	D₄	1	1	yes	characteristic
28	C₂²×Dic₄₃	D₄	1	1	yes	characteristic
29	C₂×Dic₈₆	—	2	1	no	—
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	C₂×C₄	1	2	yes	—
35	C₂×C₈×C₄₃	—	2	1	no	—
36	C₂×C₈×C₄₃	—	2	1	no	—
37	C₂×C₈×C₄₃	C₂×C₄	1	2	yes	—
38	C₄⋊C₄×C₄₃	—	2	1	no	—
39	C₄⋊C₄×C₄₃	—	2	1	no	—
40	C₄⋊C₄×C₄₃	—	2	1	no	—
41	C₄⋊C₄×C₄₃	D₄	1	2	yes	—
42	C₄⋊Dic₄₃	—	2	1	no	—
43	C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
44	C₄×Dic₄₃	—	2	1	no	—
45	C₄×Dic₄₃	—	2	1	no	—
46	C₄×Dic₄₃	—	4	2	no	—
47	C₂×C₄₃⋊C₈	—	2	1	no	—
48	C₂×C₄₃⋊C₈	—	2	1	no	—
49	C₂×C₄₃⋊C₈	—	2	1	no	—
50	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₄₃	—	2	2	no	—
52	C₂×C₄×C₄₃	—	4	1	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	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
60	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
61	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
62	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
63	C₂×Dic₄₃	—	2	1	no	—
64	C₂×Dic₄₃	—	2	1	no	—
65	C₂×Dic₄₃	—	2	2	no	—
66	C₂×Dic₄₃	—	2	2	no	—
67	C₂×Dic₄₃	—	2	1	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	SD₁₆	1	2	yes	—
75	Dic₈₆	—	2	2	no	—
76	Dic₈₆	—	2	2	no	—
77	Dic₈₆	—	4	1	no	—
78	Dic₈₆	—	4	1	no	—
79	C₈×C₄₃	—	2	2	no	—
80	C₄₃⋊C₈	—	4	1	no	—
81	C₂²×C₄₃	C₄×D₄	1	1	yes	characteristic
82	C₂²×C₄₃	#121	1	1	yes	characteristic
83	C₂²×C₄₃	#123	1	1	yes	characteristic
84	C₂²×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
85	C₂²×C₄₃	#135	1	1	yes	characteristic
86	C₂²×C₄₃	#137	1	1	yes	characteristic
87	C₂²×C₄₃	#137	1	1	yes	characteristic
88	C₄×C₄₃	—	4	2	no	—
89	C₄×C₄₃	—	8	1	no	—
90	C₄×C₄₃	#117	1	1	yes	characteristic
91	C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
92	C₄×C₄₃	#124	1	1	yes	characteristic
93	C₄×C₄₃	#126	1	1	yes	characteristic
94	Dic₄₃	—	2	2	no	—
95	Dic₄₃	—	4	2	no	—
96	Dic₄₃	—	4	2	no	—
97	Dic₄₃	—	4	1	no	—
98	#1927	—	43	1	no	maximal, Sylow
99	C₂×C₄₃	#388	1	1	yes	characteristic
100	C₂×C₄₃	C₄×SD₁₆	1	1	yes	characteristic
101	C₂×C₄₃	#441	1	1	yes	characteristic
102	C₂×C₄₃	#461	1	1	yes	characteristic
103	C₂×C₄₃	#464	1	1	yes	characteristic
104	C₂×C₄₃	#483	1	1	yes	characteristic
105	C₂×C₄₃	#485	1	1	yes	characteristic
106	C₂×#106	D₄₃	1	1	yes	characteristic
107	C₂×#103	—	43	1	no	—
108	C₂×#103	—	43	1	no	—
109	#391	—	43	1	no	—
110	C₄×C₄⋊C₄	—	43	1	no	—
111	#340	—	43	1	no	—
112	#336	—	43	1	no	—
113	C₄₃	#1927	1	1	yes	characteristic, complemented, Sylow
114	C₂²×Q₈	—	43	1	no	—
115	C₂²×C₈	—	43	1	no	—
116	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
117	C₂×C₄⋊C₄	—	43	1	no	—
118	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
119	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
120	C₂×C₄²	—	43	1	no	—
121	C₂×C₄²	—	86	1	no	—
122	#106	—	2	2	no	—
123	#103	—	43	4	no	—
124	#103	—	86	2	no	—
125	#95	—	86	1	no	—
126	#95	—	86	1	no	—
127	#95	—	86	1	no	—
128	C₂×Q₈	—	43	2	no	—
129	C₂×Q₈	—	86	1	no	—
130	C₂×Q₈	—	86	1	no	—
131	C₂×Q₈	—	86	1	no	—
132	C₂×Q₈	—	86	1	no	—
133	C₂×Q₈	—	86	1	no	—
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	—	43	1	no	—
137	C₂²×C₄	—	43	1	no	—
138	C₂²×C₄	—	86	1	no	—
139	C₂²×C₄	—	86	1	no	—
140	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	86	1	no	—
144	C₂×C₈	—	86	1	no	—
145	C₂×C₈	—	86	1	no	—
146	C₂×C₈	C₄×D₄₃	1	2	yes	—
147	C₄⋊C₄	—	2	1	no	—
148	C₄⋊C₄	—	2	1	no	—
149	C₄⋊C₄	—	2	1	no	—
150	C₄⋊C₄	—	43	2	no	—
151	C₄⋊C₄	—	86	1	no	—
152	C₄⋊C₄	C₂²⋊D₄₃	1	2	yes	—
153	C₄²	—	86	1	no	—
154	C₄²	—	86	1	no	—
155	C₄²	—	172	2	no	—
156	C₂³	D₄×D₄₃	1	1	yes	characteristic
157	Q₈	—	86	2	no	—
158	Q₈	—	86	2	no	—
159	Q₈	—	172	1	no	—
160	Q₈	—	172	1	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	4	1	no	—
163	C₂×C₄	—	4	1	no	—
164	C₂×C₄	—	4	1	no	—
165	C₂×C₄	—	4	1	no	—
166	C₂×C₄	—	4	1	no	—
167	C₂×C₄	—	43	2	no	—
168	C₂×C₄	—	86	1	no	—
169	C₂×C₄	—	86	1	no	—
170	C₂×C₄	—	86	2	no	—
171	C₂×C₄	—	86	2	no	—
172	C₂×C₄	—	86	1	no	—
173	C₂×C₄	—	86	1	no	—
174	C₂×C₄	—	86	1	no	—
175	C₂×C₄	—	172	1	no	—
176	C₂×C₄	—	172	1	no	—
177	C₂×C₄	—	172	1	no	—
178	C₂×C₄	—	172	1	no	—
179	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
180	C₂×C₄	#10621898	1	1	yes	characteristic
181	C₂×C₄	#10621900	1	1	yes	characteristic
182	C₂×C₄	#10621902	1	1	yes	characteristic
183	C₂×C₄	#10621902	1	1	yes	characteristic
184	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
185	C₈	—	2	2	no	—
186	C₈	—	172	1	no	—
187	C₂²	#13063287	1	1	yes	characteristic
188	C₂²	#13063297	1	1	yes	characteristic
189	C₂²	#13063310	1	1	yes	characteristic
190	C₂²	SD₁₆×D₄₃	1	1	yes	characteristic
191	C₂²	#13063346	1	1	yes	characteristic
192	C₂²	#13063354	1	1	yes	characteristic
193	C₂²	#13063354	1	1	yes	characteristic
194	C₄	—	4	2	no	—
195	C₄	—	8	1	no	—
196	C₄	—	86	2	no	—
197	C₄	—	172	2	no	—
198	C₄	—	172	2	no	—
199	C₄	—	172	1	no	—
200	C₄	#13063283	1	1	yes	characteristic
201	C₄	C₄×C₂²⋊D₄₃	1	1	yes	characteristic
202	C₄	#13063314	1	1	yes	characteristic
203	C₄	#13063320	1	1	yes	characteristic
204	C₂	#14582789	1	1	yes	characteristic
205	C₂	#14582986	1	1	yes	characteristic
206	C₂	#14583005	1	1	yes	characteristic
207	C₂	#14583102	1	1	yes	characteristic
208	C₂	#14583119	1	1	yes	characteristic
209	C₂	#14583231	1	1	yes	characteristic
210	C₂	#14583243	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13063250

C₂

yes

maximal, characteristic

C₂×#13063248

C₂

yes

maximal, characteristic

#14582804

C₂

yes

maximal, characteristic

C₄⋊C₄×Dic₄₃

C₂

yes

maximal, characteristic

#14582661

C₂

yes

maximal, characteristic

#14582653

C₂

yes

maximal, characteristic

C₂×#106×C₄₃

C₂

yes

maximal, characteristic

C₂²×Dic₈₆

C₂²

yes

characteristic

C₂²×C₄₃⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₃

C₂²

yes

characteristic

C₂×C₄×Dic₄₃

—

C₂×C₄×Dic₄₃

C₂²

yes

characteristic

#13063250

C₄

yes

complemented

#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

#106×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

C₂×C₄

yes

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂×C₄

yes

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

D₄

yes

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

C₂×C₄

yes

—

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

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

C₂×C₄×C₄₃

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

SD₁₆

yes

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₂²×C₄₃

C₄×D₄

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

C₂×SD₁₆

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#117

yes

characteristic

C₄×C₄₃

C₄×D₄

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

—

C₂×C₄₃

yes

100

C₂×C₄₃

C₄×SD₁₆

yes

characteristic

101

C₂×C₄₃

#441

yes

characteristic

102

C₂×C₄₃

#461

yes

characteristic

103

C₂×C₄₃

#464

yes

characteristic

104

C₂×C₄₃

#483

yes

characteristic

105

C₂×C₄₃

#485

yes

characteristic

106

C₂×#106

D₄₃

yes

characteristic

107

C₂×#103

—

108

C₂×#103

—

109

#391

—

110

C₄×C₄⋊C₄

—

111

#340

—

112

#336

—

113

C₄₃

#1927

yes

characteristic, complemented, Sylow

114

C₂²×Q₈

—

115

C₂²×C₈

—

116

C₂²×C₈

C₂×D₄₃

yes

characteristic

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

C₂×D₄₃

yes

characteristic

119

C₂×C₄⋊C₄

C₂×D₄₃

yes

characteristic

120

C₂×C₄²

—

121

C₂×C₄²

—

122

#106

—

123

#103

—

124

#103

—

125

#95

—

126

#95

—

127

#95

—

128

C₂×Q₈

—

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

—

140

C₂²×C₄

C₂²×D₄₃

yes

characteristic

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

C₄×D₄₃

yes

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

C₂²⋊D₄₃

yes

—

153

C₄²

—

154

C₄²

—

155

C₄²

—

172

—

156

C₂³

D₄×D₄₃

yes

characteristic

157

Q₈

—

158

Q₈

—

159

Q₈

—

172

—

160

Q₈

—

172

—

161

C₂×C₄

—

162

C₂×C₄

—

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

—

172

—

176

C₂×C₄

—

172

—

177

C₂×C₄

—

172

—

178

C₂×C₄

—

172

—

179

C₂×C₄

C₂×C₄×D₄₃

yes

characteristic

180

C₂×C₄

#10621898

yes

characteristic

181

C₂×C₄

#10621900

yes

characteristic

182

C₂×C₄

#10621902

yes

characteristic

183

C₂×C₄

#10621902

yes

characteristic

184

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

185

C₈

—

186

C₈

—

172

—

187

C₂²

#13063287

yes

characteristic

188

C₂²

#13063297

yes

characteristic

189

C₂²

#13063310

yes

characteristic

190

C₂²

SD₁₆×D₄₃

yes

characteristic

191

C₂²

#13063346

yes

characteristic

192

C₂²

#13063354

yes

characteristic

193

C₂²

#13063354

yes

characteristic

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

—

172

—

198

C₄

—

172

—

199

C₄

—

172

—

200

C₄

#13063283

yes

characteristic

201

C₄

C₄×C₂²⋊D₄₃

yes

characteristic

202

C₄

#13063314

yes

characteristic

203

C₄

#13063320

yes

characteristic

204

C₂

#14582789

yes

characteristic

205

C₂

#14582986

yes

characteristic

206

C₂

#14583005

yes

characteristic

207

C₂

#14583102

yes

characteristic

208

C₂

#14583119

yes

characteristic

209

C₂

#14583231

yes

characteristic

210

C₂

#14583243

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₂×#106×C₄₃	C₂
socle	C₂³×C₄₃	C₂×D₄
hypercenter	C₂×#106	D₄₃
hypocenter	C₄₃	#1927
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₄₃

commutator

C₂×C₄×C₄₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₂×#106×C₄₃

C₂

socle

C₂³×C₄₃

C₂×D₄

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
13. #13063250	C₄	semidirect
113. C₄₃	#1927	semidirect

H ≅ G/N

type

13. #13063250

C₄

semidirect

113. C₄₃

#1927

semidirect

Subgroups — #19135228

Conjugacy classes of subgroups

Complements