Group #19144249 — subgroups

Subgroups — #19144249

Order 5504.

460 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 133 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×#10621885	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13063333	C₂	1	1	yes	maximal, characteristic
3	#14583475	C₂	1	1	yes	maximal, characteristic
4	#14583473	C₂	1	1	yes	maximal, characteristic
5	#14583156	C₂	1	4	yes	maximal, complemented
6	#14583139	C₂	1	4	yes	maximal, complemented
7	#14582935	C₂	1	1	yes	maximal, characteristic, complemented
8	#14582909	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₃×#419	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#10621898	C₂²	1	1	yes	characteristic
11	C₂²×Dic₈₆	C₂²	1	1	yes	characteristic
12	C₂×#10621885	—	2	4	no	—
13	C₂×#10621885	C₂²	1	2	yes	—
14	C₂×#10621885	C₂²	1	2	yes	—
15	#13063333	—	2	4	no	—
16	C₂×#10621872	C₂²	1	2	yes	—
17	C₂²×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₄₃	C₂²	1	2	yes	—
19	#13063325	—	2	1	no	—
20	#13063323	—	2	1	no	—
21	#13063314	—	2	1	no	—
22	#13063313	—	2	1	no	—
23	#13063309	—	2	2	no	—
24	#13063308	—	2	2	no	—
25	#13063305	C₂²	1	4	yes	—
26	#13063301	C₂²	1	4	yes	—
27	#13063284	C₂²	1	1	yes	characteristic
28	#13063258	C₂²	1	2	yes	—
29	#13063256	C₂²	1	2	yes	—
30	#13063250	C₂²	1	4	yes	—
31	C₂²×Q₈×C₄₃	C₂²	1	1	yes	characteristic
32	C₂×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
33	#117×C₄₃	C₂²	1	1	yes	characteristic
34	#103×C₄₃	C₂²	1	4	yes	—
35	C₂×Q₈×C₄₃	—	2	1	no	—
36	C₂×Q₈×C₄₃	—	2	2	no	—
37	C₂×Q₈×C₄₃	—	2	2	no	—
38	C₂×Q₈×C₄₃	C₂³	1	2	yes	—
39	C₂²×C₄×C₄₃	—	2	1	no	—
40	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
41	C₂×C₂²⋊D₄₃	D₄	1	2	yes	—
42	C₂²×Dic₄₃	—	2	1	no	—
43	#10621898	—	2	2	no	—
44	#10621898	—	2	2	no	—
45	C₂×D₁₇₂	D₄	1	1	yes	characteristic
46	C₂×C₄×D₄₃	C₂³	1	2	yes	—
47	C₂×Dic₈₆	—	2	2	no	—
48	C₂×Dic₈₆	—	2	2	no	—
49	C₂×Dic₈₆	—	2	1	no	—
50	C₂×Dic₈₆	D₄	1	1	yes	characteristic
51	C₂×Dic₈₆	C₂³	1	2	yes	—
52	M₁₆×C₄₃	—	2	2	no	—
53	C₂×C₈×C₄₃	C₂³	1	2	yes	—
54	C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
55	C₂²⋊C₄×C₄₃	—	2	1	no	—
56	C₄²×C₄₃	—	2	1	no	—
57	C₂²⋊Dic₄₃	—	2	1	no	—
58	#10621885	—	2	4	no	—
59	#10621885	—	2	4	no	—
60	(C₂×D₄₃)⋊C₄	—	2	4	no	—
61	(C₂×D₄₃)⋊C₄	—	2	4	no	—
62	C₄⋊Dic₄₃	—	2	4	no	—
63	C₄⋊Dic₄₃	C₂³	1	2	yes	—
64	Dic₄₃⋊C₄	—	2	4	no	—
65	C₄×Dic₄₃	—	2	1	no	—
66	C₂×C₄₃⋊C₈	D₄	1	4	yes	—
67	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
68	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
69	#10621872	—	2	4	no	—
70	C₈×D₄₃	—	2	4	no	—
71	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×D₄₃	C₂×D₄	1	2	yes	—
73	Q₈×C₄₃	—	2	2	no	—
74	Q₈×C₄₃	—	2	2	no	—
75	Q₈×C₄₃	—	4	2	no	—
76	C₂×C₄×C₄₃	—	2	2	no	—
77	C₂×C₄×C₄₃	—	2	2	no	—
78	C₂×C₄×C₄₃	—	4	2	no	—
79	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
81	C₂×C₄×C₄₃	#41	1	2	yes	—
82	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
83	C₂²⋊D₄₃	—	2	4	no	—
84	C₂×Dic₄₃	—	2	2	no	—
85	C₂×Dic₄₃	—	2	2	no	—
86	C₂×Dic₄₃	—	4	2	no	—
87	C₂×Dic₄₃	C₂×D₄	1	2	yes	—
88	D₁₇₂	—	2	2	no	—
89	C₄×D₄₃	—	2	2	no	—
90	C₄×D₄₃	—	2	2	no	—
91	Dic₈₆	—	2	2	no	—
92	Dic₈₆	—	2	2	no	—
93	Dic₈₆	—	2	2	no	—
94	Dic₈₆	—	4	2	no	—
95	C₈×C₄₃	—	2	2	no	—
96	C₄₃⋊C₈	C₂×D₄	1	2	yes	—
97	C₄₃⋊C₈	C₂×D₄	1	2	yes	—
98	C₂²×C₄₃	—	2	2	no	—
99	C₂²×C₄₃	#121	1	2	yes	—
100	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
101	C₂×D₄₃	—	2	2	no	—
102	C₂×D₄₃	—	2	4	no	—
103	C₄×C₄₃	—	4	2	no	—
104	C₄×C₄₃	—	4	2	no	—
105	C₄×C₄₃	#121	1	2	yes	—
106	C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
107	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
108	Dic₄₃	—	2	2	no	—
109	Dic₄₃	—	4	2	no	—
110	Dic₄₃	—	4	2	no	—
111	#3059	—	43	1	no	maximal, Sylow
112	C₂×C₄₃	—	2	2	no	—
113	C₂×C₄₃	C₂×#121	1	1	yes	characteristic
114	C₂×C₄₃	#578	1	2	yes	—
115	D₄₃	—	4	2	no	—
116	C₂²×Q₁₆	—	43	1	no	—
117	C₂×#131	—	43	1	no	—
118	#536	—	43	1	no	—
119	#536	—	43	1	no	—
120	#470	—	43	4	no	—
121	#467	—	43	4	no	—
122	#427	—	43	1	no	—
123	#419	—	43	1	no	—
124	#419	D₄₃	1	1	yes	characteristic, complemented
125	C₄₃	#3059	1	1	yes	characteristic, complemented, Sylow
126	C₂×#41	—	43	1	no	—
127	C₂²×Q₈	—	43	1	no	—
128	C₂²×Q₈	C₂×D₄₃	1	1	yes	characteristic
129	C₂×Q₁₆	—	43	2	no	—
130	C₂×Q₁₆	—	43	2	no	—
131	C₂×Q₁₆	—	86	4	no	—
132	#131	—	86	4	no	—
133	C₂×M₁₆	—	43	2	no	—
134	C₂×M₁₆	C₂×D₄₃	1	1	yes	characteristic
135	C₂²×C₈	—	43	1	no	—
136	C₂²×C₈	—	43	2	no	—
137	C₄⋊Q₈	—	86	1	no	—
138	C₄⋊Q₈	—	86	1	no	—
139	#124	—	86	1	no	—
140	#124	—	86	1	no	—
141	#123	—	86	2	no	—
142	#123	—	86	2	no	—
143	C₂²⋊Q₈	—	43	4	no	—
144	C₂²⋊Q₈	—	43	4	no	—
145	#117	—	43	1	no	—
146	#117	C₂×D₄₃	1	1	yes	characteristic
147	#107	—	43	2	no	—
148	#106	—	43	2	no	—
149	#103	—	43	4	no	—
150	#103	C₂×D₄₃	1	4	yes	—
151	#41	—	86	2	no	—
152	#41	—	86	2	no	—
153	C₂×Q₈	—	2	1	no	—
154	C₂×Q₈	—	2	2	no	—
155	C₂×Q₈	—	2	2	no	—
156	C₂×Q₈	—	43	1	no	—
157	C₂×Q₈	—	43	2	no	—
158	C₂×Q₈	—	86	1	no	—
159	C₂×Q₈	—	86	2	no	—
160	C₂×Q₈	—	86	2	no	—
161	C₂×Q₈	C₂²×D₄₃	1	2	yes	—
162	C₂×D₄	—	43	2	no	—
163	C₂×D₄	—	43	1	no	—
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	—	43	2	no	—
166	C₂²×C₄	—	86	1	no	—
167	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
168	Q₁₆	—	86	4	no	—
169	Q₁₆	—	86	4	no	—
170	M₁₆	—	2	2	no	—
171	M₁₆	—	86	4	no	—
172	C₂×C₈	—	43	4	no	—
173	C₂×C₈	—	43	1	no	—
174	C₂×C₈	—	43	1	no	—
175	C₂×C₈	—	86	4	no	—
176	C₂×C₈	C₂²×D₄₃	1	2	yes	—
177	C₄⋊C₄	—	43	2	no	—
178	C₄⋊C₄	—	86	4	no	—
179	C₄⋊C₄	—	86	4	no	—
180	C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
181	C₂²⋊C₄	—	2	1	no	—
182	C₂²⋊C₄	—	86	4	no	—
183	C₂²⋊C₄	—	86	4	no	—
184	C₂²⋊C₄	—	86	1	no	—
185	C₄²	—	2	1	no	—
186	C₄²	—	86	1	no	—
187	C₂³	—	43	2	no	—
188	C₂³	D₄×D₄₃	1	1	yes	characteristic
189	Q₈	—	2	2	no	—
190	Q₈	—	2	2	no	—
191	Q₈	—	4	2	no	—
192	Q₈	—	86	2	no	—
193	Q₈	—	86	2	no	—
194	Q₈	—	86	2	no	—
195	Q₈	—	172	2	no	—
196	D₄	—	86	4	no	—
197	D₄	—	86	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	43	2	no	—
202	C₂×C₄	—	86	2	no	—
203	C₂×C₄	—	86	2	no	—
204	C₂×C₄	—	86	2	no	—
205	C₂×C₄	—	86	2	no	—
206	C₂×C₄	—	172	2	no	—
207	C₂×C₄	#10621898	1	2	yes	—
208	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
209	C₂×C₄	D₄×D₄₃	1	2	yes	—
210	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
211	C₈	—	2	2	no	—
212	C₈	—	43	2	no	—
213	C₈	—	43	2	no	—
214	C₂²	—	2	2	no	—
215	C₂²	—	86	2	no	—
216	C₂²	—	86	4	no	—
217	C₂²	#13063298	1	2	yes	—
218	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
219	C₄	—	4	2	no	—
220	C₄	—	4	2	no	—
221	C₄	—	86	2	no	—
222	C₄	—	172	2	no	—
223	C₄	—	172	2	no	—
224	C₄	#13063298	1	2	yes	—
225	C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
226	C₄	C₂×#10621898	1	1	yes	characteristic
227	C₂	—	2	2	no	—
228	C₂	—	172	2	no	—
229	C₂	C₂×#13063298	1	1	yes	characteristic
230	C₂	#14583715	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#10621885

C₂

yes

maximal, characteristic, complemented

C₂×#13063333

C₂

yes

maximal, characteristic

#14583475

C₂

yes

maximal, characteristic

#14583473

C₂

yes

maximal, characteristic

#14583156

C₂

yes

maximal, complemented

#14583139

C₂

yes

maximal, complemented

#14582935

C₂

yes

maximal, characteristic, complemented

#14582909

C₂

yes

maximal, characteristic, complemented

C₄₃×#419

C₂

yes

maximal, characteristic, complemented

C₂×#10621898

C₂²

yes

characteristic

C₂²×Dic₈₆

C₂²

yes

characteristic

C₂×#10621885

—

C₂×#10621885

C₂²

yes

—

C₂×#10621885

C₂²

yes

—

#13063333

—

C₂×#10621872

C₂²

yes

—

C₂²×C₄₃⋊C₈

C₂²

yes

characteristic

C₂×C₈×D₄₃

C₂²

yes

—

—

—

—

—

—

—

C₂²

yes

—

#13063301

C₂²

yes

—

#13063284

C₂²

yes

characteristic

#13063258

C₂²

yes

—

#13063256

C₂²

yes

—

#13063250

C₂²

yes

—

C₂²×Q₈×C₄₃

C₂²

yes

characteristic

C₂×M₁₆×C₄₃

C₂²

yes

characteristic

#117×C₄₃

C₂²

yes

characteristic

#103×C₄₃

C₂²

yes

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

C₂³

yes

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂×C₂²⋊D₄₃

D₄

yes

—

C₂²×Dic₄₃

—

#10621898

—

#10621898

—

C₂×D₁₇₂

D₄

yes

characteristic

C₂×C₄×D₄₃

C₂³

yes

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

D₄

yes

characteristic

C₂×Dic₈₆

C₂³

yes

—

M₁₆×C₄₃

—

C₂×C₈×C₄₃

C₂³

yes

—

C₄⋊C₄×C₄₃

C₂³

yes

—

C₂²⋊C₄×C₄₃

—

C₄²×C₄₃

—

C₂²⋊Dic₄₃

—

#10621885

—

#10621885

—

(C₂×D₄₃)⋊C₄

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

C₂³

yes

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

D₄

yes

—

C₂×C₄₃⋊C₈

C₂³

yes

characteristic

C₂×C₄₃⋊C₈

C₂³

yes

characteristic

#10621872

—

C₈×D₄₃

—

C₂³×C₄₃

C₂×D₄

yes

characteristic

C₂²×D₄₃

C₂×D₄

yes

—

Q₈×C₄₃

—

Q₈×C₄₃

—

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

#41

yes

—

C₂×C₄×C₄₃

C₂⁴

yes

characteristic

C₂²⋊D₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

C₂×D₄

yes

—

D₁₇₂

—

C₄×D₄₃

—

C₄×D₄₃

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₄₃⋊C₈

C₂×D₄

yes

—

C₄₃⋊C₈

C₂×D₄

yes

—

C₂²×C₄₃

—

C₂²×C₄₃

#121

yes

—

100

C₂²×C₄₃

C₂²×D₄

yes

characteristic

101

C₂×D₄₃

—

102

C₂×D₄₃

—

103

C₄×C₄₃

—

104

C₄×C₄₃

—

105

C₄×C₄₃

#121

yes

—

106

C₄×C₄₃

C₂²×D₄

yes

characteristic

107

C₄×C₄₃

C₂×#41

yes

characteristic

108

Dic₄₃

—

109

Dic₄₃

—

110

Dic₄₃

—

111

#3059

—

maximal, Sylow

112

C₂×C₄₃

—

113

C₂×C₄₃

C₂×#121

yes

characteristic

114

C₂×C₄₃

#578

yes

—

115

D₄₃

—

116

C₂²×Q₁₆

—

117

C₂×#131

—

118

#536

—

119

#536

—

120

#470

—

121

#467

—

122

#427

—

123

#419

—

124

#419

D₄₃

yes

characteristic, complemented

125

C₄₃

#3059

yes

characteristic, complemented, Sylow

126

C₂×#41

—

127

C₂²×Q₈

—

128

C₂²×Q₈

C₂×D₄₃

yes

characteristic

129

C₂×Q₁₆

—

130

C₂×Q₁₆

—

131

C₂×Q₁₆

—

132

#131

—

133

C₂×M₁₆

—

134

C₂×M₁₆

C₂×D₄₃

yes

characteristic

135

C₂²×C₈

—

136

C₂²×C₈

—

137

C₄⋊Q₈

—

138

C₄⋊Q₈

—

139

#124

—

140

#124

—

141

#123

—

142

#123

—

143

C₂²⋊Q₈

—

144

C₂²⋊Q₈

—

145

#117

—

146

#117

C₂×D₄₃

yes

characteristic

147

#107

—

148

#106

—

149

#103

—

150

#103

C₂×D₄₃

yes

—

151

#41

—

152

#41

—

153

C₂×Q₈

—

154

C₂×Q₈

—

155

C₂×Q₈

—

156

C₂×Q₈

—

157

C₂×Q₈

—

158

C₂×Q₈

—

159

C₂×Q₈

—

160

C₂×Q₈

—

161

C₂×Q₈

C₂²×D₄₃

yes

—

162

C₂×D₄

—

163

C₂×D₄

—

164

C₂²×C₄

—

165

C₂²×C₄

—

166

C₂²×C₄

—

167

C₂²×C₄

C₂²×D₄₃

yes

characteristic

168

Q₁₆

—

169

Q₁₆

—

170

M₁₆

—

171

M₁₆

—

172

C₂×C₈

—

173

C₂×C₈

—

174

C₂×C₈

—

175

C₂×C₈

—

176

C₂×C₈

C₂²×D₄₃

yes

—

177

C₄⋊C₄

—

178

C₄⋊C₄

—

179

C₄⋊C₄

—

180

C₄⋊C₄

C₂²×D₄₃

yes

—

181

C₂²⋊C₄

—

182

C₂²⋊C₄

—

183

C₂²⋊C₄

—

184

C₂²⋊C₄

—

185

C₄²

—

186

C₄²

—

187

C₂³

—

188

C₂³

D₄×D₄₃

yes

characteristic

189

Q₈

—

190

Q₈

—

191

Q₈

—

192

Q₈

—

193

Q₈

—

194

Q₈

—

195

Q₈

—

172

—

196

D₄

—

197

D₄

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

202

C₂×C₄

—

203

C₂×C₄

—

204

C₂×C₄

—

205

C₂×C₄

—

206

C₂×C₄

—

172

—

207

C₂×C₄

#10621898

yes

—

208

C₂×C₄

D₄×D₄₃

yes

characteristic

209

C₂×C₄

D₄×D₄₃

yes

—

210

C₂×C₄

C₂³×D₄₃

yes

characteristic

211

C₈

—

212

C₈

—

213

C₈

—

214

C₂²

—

215

C₂²

—

216

C₂²

—

217

C₂²

#13063298

yes

—

218

C₂²

C₂×D₄×D₄₃

yes

characteristic

219

C₄

—

220

C₄

—

221

C₄

—

222

C₄

—

172

—

223

C₄

—

172

—

224

C₄

#13063298

yes

—

225

C₄

C₂×D₄×D₄₃

yes

characteristic

226

C₄

C₂×#10621898

yes

characteristic

227

C₂

—

228

C₂

—

172

—

229

C₂

C₂×#13063298

yes

characteristic

230

C₂

#14583715

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#419	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#419	D₄₃
hypocenter	C₄₃	#3059
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #419

subgroup

G/H

center

C₂²

C₂×D₄×D₄₃

commutator

C₂×C₄×C₄₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₃

Fitting

C₄₃×#419

C₂

socle

C₂²×C₄₃

C₂²×D₄

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #419

N	H ≅ G/N	type
1. C₂²×#10621885	C₂	semidirect
5. #14583156	C₂	semidirect
6. #14583139	C₂	semidirect
7. #14582935	C₂	semidirect
8. #14582909	C₂	semidirect
9. C₄₃×#419	C₂	semidirect
124. #419	D₄₃	semidirect
125. C₄₃	#3059	semidirect

H ≅ G/N

type

1. C₂²×#10621885

C₂

semidirect

5. #14583156

C₂

semidirect

6. #14583139

C₂

semidirect

7. #14582935

C₂

semidirect

8. #14582909

C₂

semidirect

9. C₄₃×#419

C₂

semidirect

124. #419

D₄₃

semidirect

125. C₄₃

#3059

semidirect

Subgroups — #19144249

Conjugacy classes of subgroups

Complements