Group #19135502 — subgroups

Subgroups — #19135502

Order 5504.

316 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 83 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	C₂²×#10621882	C₂	1	1	yes	maximal, characteristic
2	C₂×#13063260	C₂	1	1	yes	maximal, characteristic, complemented
3	#14582905	C₂	1	1	yes	maximal, characteristic, complemented
4	#14582904	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#13063241	C₂	1	1	yes	maximal, characteristic, complemented
6	#14582857	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#100×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₁₇₂	C₂²	1	1	yes	characteristic
9	C₂×#10621882	—	2	1	no	—
10	C₂×#10621882	—	2	1	no	—
11	C₂×#10621882	—	2	2	no	—
12	C₂×#10621882	C₄	1	4	yes	complemented
13	C₂×#10621877	C₂²	1	1	yes	characteristic
14	C₂²×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
15	#13063284	C₂²	1	1	yes	characteristic
16	#13063260	—	2	2	no	—
17	#13063247	—	2	2	no	—
18	#13063246	—	2	2	no	—
19	#13063241	—	2	2	no	—
20	#13063235	—	2	1	no	—
21	C₈×Dic₄₃	—	2	1	no	—
22	C₂²×D₄×C₄₃	C₂²	1	1	yes	characteristic
23	C₂×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
24	C₂×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
25	#100×C₄₃	—	2	2	no	—
26	C₂⁴×C₄₃	—	2	1	no	—
27	C₂³×D₄₃	—	2	1	no	—
28	C₂×D₄×C₄₃	—	2	1	no	—
29	C₂×D₄×C₄₃	—	2	1	no	—
30	C₂×D₄×C₄₃	—	2	1	no	—
31	C₂×D₄×C₄₃	—	2	1	no	—
32	C₂×D₄×C₄₃	—	2	1	no	—
33	C₂×D₄×C₄₃	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
35	C₂×D₁₇₂	—	2	1	no	—
36	C₂×D₁₇₂	—	2	1	no	—
37	C₂×D₁₇₂	—	2	1	no	—
38	C₂×D₁₇₂	—	2	1	no	—
39	C₂×D₁₇₂	—	2	1	no	—
40	C₂×D₁₇₂	C₂×C₄	1	2	yes	—
41	M₁₆×C₄₃	—	2	1	no	—
42	M₁₆×C₄₃	—	2	1	no	—
43	M₁₆×C₄₃	—	2	1	no	—
44	M₁₆×C₄₃	D₄	1	2	yes	—
45	C₂×C₈×C₄₃	—	2	1	no	—
46	C₂×C₈×C₄₃	—	2	1	no	—
47	C₂²⋊Dic₄₃	—	2	1	no	—
48	#10621882	—	2	4	no	—
49	#10621882	—	2	4	no	—
50	C₄⋊Dic₄₃	D₄	1	2	yes	—
51	C₄×Dic₄₃	—	2	1	no	—
52	#10621877	—	2	1	no	—
53	#10621877	—	2	1	no	—
54	C₂×C₄₃⋊C₈	—	2	1	no	—
55	C₂×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
56	C₂×C₄₃⋊C₈	D₄	1	1	yes	characteristic
57	C₂×C₄₃⋊C₈	D₄	1	1	yes	characteristic
58	C₂×C₄₃⋊C₈	D₄	1	2	yes	—
59	C₂³×C₄₃	—	2	2	no	—
60	C₂³×C₄₃	—	4	1	no	—
61	C₂³×C₄₃	—	4	1	no	—
62	C₂³×C₄₃	—	8	2	no	—
63	C₂³×C₄₃	#41	1	1	yes	characteristic
64	C₂²×D₄₃	—	2	2	no	—
65	C₂²×D₄₃	—	4	1	no	—
66	C₂²×D₄₃	—	4	1	no	—
67	C₂²×D₄₃	—	8	2	no	—
68	D₄×C₄₃	—	2	2	no	—
69	D₄×C₄₃	—	2	2	no	—
70	D₄×C₄₃	—	4	1	no	—
71	D₄×C₄₃	—	4	1	no	—
72	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
78	C₂×Dic₄₃	—	2	2	no	—
79	D₁₇₂	—	2	2	no	—
80	D₁₇₂	—	2	2	no	—
81	D₁₇₂	—	4	1	no	—
82	D₁₇₂	—	4	1	no	—
83	C₈×C₄₃	—	2	2	no	—
84	C₈×C₄₃	—	4	1	no	—
85	C₄₃⋊C₈	—	4	1	no	—
86	C₄₃⋊C₈	C₂²⋊C₄	1	2	yes	—
87	C₄₃⋊C₈	C₂²⋊C₄	1	2	yes	—
88	C₂²×C₄₃	—	2	2	no	—
89	C₂²×C₄₃	—	4	2	no	—
90	C₂²×C₄₃	—	8	2	no	—
91	C₂²×C₄₃	—	8	2	no	—
92	C₂²×C₄₃	—	8	2	no	—
93	C₂²×C₄₃	C₄×D₄	1	1	yes	characteristic
94	C₂²×C₄₃	#121	1	1	yes	characteristic
95	C₂²×C₄₃	#124	1	1	yes	characteristic
96	C₂×D₄₃	—	4	2	no	—
97	C₂×D₄₃	—	8	2	no	—
98	C₂×D₄₃	—	8	2	no	—
99	C₂×D₄₃	—	8	2	no	—
100	C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
102	C₄×C₄₃	#121	1	1	yes	characteristic
103	C₄×C₄₃	#127	1	1	yes	characteristic
104	Dic₄₃	—	4	2	no	—
105	#1978	—	43	1	no	maximal, Sylow
106	C₂×C₄₃	—	2	1	no	—
107	C₂×C₄₃	—	2	1	no	—
108	C₂×C₄₃	—	8	2	no	—
109	C₂×C₄₃	#390	1	1	yes	characteristic
110	C₂×C₄₃	#472	1	2	yes	—
111	D₄₃	—	8	2	no	—
112	C₂²×D₈	—	43	1	no	—
113	C₂×#108	—	43	1	no	—
114	#418	—	43	1	no	—
115	#418	—	43	1	no	—
116	C₂×#100	—	43	1	no	—
117	C₂×#100	D₄₃	1	1	yes	characteristic, complemented
118	#405	—	43	1	no	—
119	C₄₃	#1978	1	1	yes	characteristic, complemented, Sylow
120	C₂²×D₄	—	43	1	no	—
121	C₂²×D₄	C₂×D₄₃	1	1	yes	characteristic
122	C₂×D₈	—	43	4	no	—
123	C₂×D₈	—	86	1	no	—
124	C₂×D₈	—	86	1	no	—
125	C₂×D₈	—	86	2	no	—
126	C₂×M₁₆	—	43	1	no	—
127	C₂×M₁₆	C₂×D₄₃	1	1	yes	characteristic
128	C₂×M₁₆	C₂×D₄₃	1	1	yes	characteristic
129	C₂²×C₈	—	43	1	no	—
130	#117	—	43	1	no	—
131	#108	—	86	2	no	—
132	#102	—	86	2	no	—
133	#102	—	86	2	no	—
134	#100	—	2	2	no	—
135	#100	—	86	2	no	—
136	#97	—	86	1	no	—
137	C₄×C₈	—	86	1	no	—
138	C₂⁴	—	2	1	no	—
139	C₂⁴	—	86	1	no	—
140	C₂×D₄	—	2	1	no	—
141	C₂×D₄	—	2	1	no	—
142	C₂×D₄	—	2	1	no	—
143	C₂×D₄	—	2	1	no	—
144	C₂×D₄	—	2	1	no	—
145	C₂×D₄	—	43	2	no	—
146	C₂×D₄	—	86	1	no	—
147	C₂×D₄	—	86	1	no	—
148	C₂×D₄	—	86	1	no	—
149	C₂×D₄	—	86	1	no	—
150	C₂×D₄	—	86	1	no	—
151	C₂×D₄	C₄×D₄₃	1	2	yes	—
152	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
153	D₈	—	86	4	no	—
154	D₈	—	86	4	no	—
155	M₁₆	—	2	1	no	—
156	M₁₆	—	2	1	no	—
157	M₁₆	—	2	1	no	—
158	M₁₆	—	86	1	no	—
159	M₁₆	—	86	1	no	—
160	M₁₆	C₂²⋊D₄₃	1	2	yes	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	43	2	no	—
164	C₂×C₈	—	43	1	no	—
165	C₂×C₈	—	43	1	no	—
166	C₂×C₈	—	43	2	no	—
167	C₂×C₈	—	86	1	no	—
168	C₄⋊C₄	—	43	2	no	—
169	C₂²⋊C₄	—	86	1	no	—
170	C₄²	—	86	1	no	—
171	C₂³	—	2	2	no	—
172	C₂³	—	4	1	no	—
173	C₂³	—	4	1	no	—
174	C₂³	—	8	2	no	—
175	C₂³	—	86	2	no	—
176	C₂³	—	344	2	no	—
177	C₂³	—	172	1	no	—
178	C₂³	—	172	1	no	—
179	C₂³	#10621898	1	1	yes	characteristic
180	D₄	—	2	2	no	—
181	D₄	—	2	2	no	—
182	D₄	—	4	1	no	—
183	D₄	—	4	1	no	—
184	D₄	—	86	2	no	—
185	D₄	—	86	2	no	—
186	D₄	—	172	1	no	—
187	D₄	—	172	1	no	—
188	C₂×C₄	—	86	2	no	—
189	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
190	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
191	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
192	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
193	C₂×C₄	#10621900	1	1	yes	characteristic
194	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
195	C₈	—	2	2	no	—
196	C₈	—	4	1	no	—
197	C₈	—	43	2	no	—
198	C₈	—	43	2	no	—
199	C₈	—	172	1	no	—
200	C₂²	—	2	2	no	—
201	C₂²	—	4	2	no	—
202	C₂²	—	8	2	no	—
203	C₂²	—	8	2	no	—
204	C₂²	—	8	2	no	—
205	C₂²	—	344	2	no	—
206	C₂²	—	344	2	no	—
207	C₂²	—	344	2	no	—
208	C₂²	—	172	2	no	—
209	C₂²	C₄×C₂²⋊D₄₃	1	1	yes	characteristic
210	C₂²	#13063298	1	1	yes	characteristic
211	C₂²	#13063311	1	1	yes	characteristic
212	C₄	—	172	2	no	—
213	C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
214	C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
215	C₄	#13063297	1	1	yes	characteristic
216	C₄	#13063321	1	1	yes	characteristic
217	C₂	—	2	1	no	—
218	C₂	—	2	1	no	—
219	C₂	—	8	2	no	—
220	C₂	—	344	2	no	—
221	C₂	#14582799	1	1	yes	characteristic
222	C₂	#14583168	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#10621882

C₂

yes

maximal, characteristic

C₂×#13063260

C₂

yes

maximal, characteristic, complemented

#14582905

C₂

yes

maximal, characteristic, complemented

#14582904

C₂

yes

maximal, characteristic, complemented

C₂×#13063241

C₂

yes

maximal, characteristic, complemented

#14582857

C₂

yes

maximal, characteristic, complemented

C₂×#100×C₄₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₇₂

C₂²

yes

—

—

—

C₄

yes

complemented

C₂×#10621877

C₂²

yes

characteristic

C₂²×C₄₃⋊C₈

C₂²

yes

characteristic

#13063284

C₂²

yes

—

—

—

—

—

C₈×Dic₄₃

—

C₂²×D₄×C₄₃

C₂²

yes

characteristic

C₂×M₁₆×C₄₃

C₂²

yes

characteristic

C₂×M₁₆×C₄₃

C₂²

yes

characteristic

#100×C₄₃

—

C₂⁴×C₄₃

—

C₂³×D₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

C₂×C₄

yes

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

C₂×C₄

yes

—

M₁₆×C₄₃

—

M₁₆×C₄₃

—

M₁₆×C₄₃

—

M₁₆×C₄₃

D₄

yes

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂²⋊Dic₄₃

—

#10621882

—

#10621882

—

C₄⋊Dic₄₃

D₄

yes

—

C₄×Dic₄₃

—

#10621877

—

#10621877

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

C₂×C₄

yes

—

C₂×C₄₃⋊C₈

D₄

yes

characteristic

C₂×C₄₃⋊C₈

D₄

yes

characteristic

C₂×C₄₃⋊C₈

D₄

yes

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

#41

yes

characteristic

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

C₂×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

#41

yes

characteristic

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

C₈×C₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₄₃⋊C₈

C₂²⋊C₄

yes

—

C₄₃⋊C₈

C₂²⋊C₄

yes

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₄×D₄

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

100

C₄×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

101

C₄×C₄₃

C₄×D₄

yes

characteristic

102

C₄×C₄₃

#121

yes

characteristic

103

C₄×C₄₃

#127

yes

characteristic

104

Dic₄₃

—

105

#1978

—

maximal, Sylow

106

C₂×C₄₃

—

107

C₂×C₄₃

—

108

C₂×C₄₃

—

109

C₂×C₄₃

#390

yes

characteristic

110

C₂×C₄₃

#472

yes

—

111

D₄₃

—

112

C₂²×D₈

—

113

C₂×#108

—

114

#418

—

115

#418

—

116

C₂×#100

—

117

C₂×#100

D₄₃

yes

characteristic, complemented

118

#405

—

119

C₄₃

#1978

yes

characteristic, complemented, Sylow

120

C₂²×D₄

—

121

C₂²×D₄

C₂×D₄₃

yes

characteristic

122

C₂×D₈

—

123

C₂×D₈

—

124

C₂×D₈

—

125

C₂×D₈

—

126

C₂×M₁₆

—

127

C₂×M₁₆

C₂×D₄₃

yes

characteristic

128

C₂×M₁₆

C₂×D₄₃

yes

characteristic

129

C₂²×C₈

—

130

#117

—

131

#108

—

132

#102

—

133

#102

—

134

#100

—

135

#100

—

136

#97

—

137

C₄×C₈

—

138

C₂⁴

—

139

C₂⁴

—

140

C₂×D₄

—

141

C₂×D₄

—

142

C₂×D₄

—

143

C₂×D₄

—

144

C₂×D₄

—

145

C₂×D₄

—

146

C₂×D₄

—

147

C₂×D₄

—

148

C₂×D₄

—

149

C₂×D₄

—

150

C₂×D₄

—

151

C₂×D₄

C₄×D₄₃

yes

—

152

C₂²×C₄

C₂²×D₄₃

yes

characteristic

153

D₈

—

154

D₈

—

155

M₁₆

—

156

M₁₆

—

157

M₁₆

—

158

M₁₆

—

159

M₁₆

—

160

M₁₆

C₂²⋊D₄₃

yes

—

161

C₂×C₈

—

162

C₂×C₈

—

163

C₂×C₈

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

—

168

C₄⋊C₄

—

169

C₂²⋊C₄

—

170

C₄²

—

171

C₂³

—

172

C₂³

—

173

C₂³

—

174

C₂³

—

175

C₂³

—

176

C₂³

—

344

—

177

C₂³

—

172

—

178

C₂³

—

172

—

179

C₂³

#10621898

yes

characteristic

180

D₄

—

181

D₄

—

182

D₄

—

183

D₄

—

184

D₄

—

185

D₄

—

186

D₄

—

172

—

187

D₄

—

172

—

188

C₂×C₄

—

189

C₂×C₄

C₂×C₄×D₄₃

yes

characteristic

190

C₂×C₄

D₄×D₄₃

yes

characteristic

191

C₂×C₄

D₄×D₄₃

yes

characteristic

192

C₂×C₄

D₄×D₄₃

yes

characteristic

193

C₂×C₄

#10621900

yes

characteristic

194

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

195

C₈

—

196

C₈

—

197

C₈

—

198

C₈

—

199

C₈

—

172

—

200

C₂²

—

201

C₂²

—

202

C₂²

—

203

C₂²

—

204

C₂²

—

205

C₂²

—

344

—

206

C₂²

—

344

—

207

C₂²

—

344

—

208

C₂²

—

172

—

209

C₂²

C₄×C₂²⋊D₄₃

yes

characteristic

210

C₂²

#13063298

yes

characteristic

211

C₂²

#13063311

yes

characteristic

212

C₄

—

172

—

213

C₄

C₂²⋊C₄×D₄₃

yes

characteristic

214

C₄

(C₂²⋊D₄₃)⋊C₄

yes

characteristic

215

C₄

#13063297

yes

characteristic

216

C₄

#13063321

yes

characteristic

217

C₂

—

218

C₂

—

219

C₂

—

220

C₂

—

344

—

221

C₂

#14582799

yes

characteristic

222

C₂

#14583168

yes

—

	subgroup	G/H
center	C₂²	#13063298
commutator	C₂×C₄×C₄₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₂×#100×C₄₃	C₂
socle	C₂²×C₄₃	#121
hypercenter	C₂×#100	D₄₃
hypocenter	C₄₃	#1978
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#100

subgroup

G/H

center

C₂²

#13063298

commutator

C₂×C₄×C₄₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₂×#100×C₄₃

C₂

C₂²×C₄₃

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#100

N	H ≅ G/N	type
2. C₂×#13063260	C₂	semidirect
3. #14582905	C₂	semidirect
4. #14582904	C₂	semidirect
5. C₂×#13063241	C₂	semidirect
6. #14582857	C₂	semidirect
7. C₂×#100×C₄₃	C₂	semidirect
12. C₂×#10621882	C₄	semidirect
117. C₂×#100	D₄₃	semidirect
119. C₄₃	#1978	semidirect

H ≅ G/N

type

2. C₂×#13063260

C₂

semidirect

3. #14582905

C₂

semidirect

4. #14582904

C₂

semidirect

5. C₂×#13063241

C₂

semidirect

6. #14582857

C₂

semidirect

7. C₂×#100×C₄₃

C₂

semidirect

12. C₂×#10621882

C₄

semidirect

117. C₂×#100

D₄₃

semidirect

119. C₄₃

#1978

semidirect

Subgroups — #19135502

Conjugacy classes of subgroups

Complements