Group #19147469 — subgroups

Subgroups — #19147469

Order 5504.

338 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 125 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	#14583628	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583610	C₂	1	1	yes	maximal, characteristic
3	#14583300	C₂	1	2	yes	maximal, complemented
4	#14583295	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583208	C₂	1	2	yes	maximal, complemented
6	#14583202	C₂	1	2	yes	maximal
7	#14583180	C₂	1	1	yes	maximal, characteristic
8	#14583023	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#10621884	C₂	1	1	yes	maximal, characteristic
10	#14582984	C₂	1	2	yes	maximal
11	C₄₃×#477	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10621884	C₂²	1	1	yes	characteristic
13	#13063325	—	2	1	no	—
14	#13063325	C₂²	1	1	yes	characteristic
15	#13063316	—	2	2	no	—
16	#13063316	C₂²	1	2	yes	—
17	#13063316	C₂²	1	2	yes	—
18	#13063305	—	2	1	no	—
19	#13063301	—	2	2	no	—
20	Q₈×Dic₄₃	C₂²	1	2	yes	—
21	C₄×Dic₈₆	—	2	1	no	—
22	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
23	C₄×D₁₇₂	C₂²	1	1	yes	characteristic
24	#13063287	C₂²	1	2	yes	—
25	C₄⋊C₄×D₄₃	—	2	1	no	—
26	#13063259	C₂²	1	2	yes	—
27	#13063256	C₂²	1	1	yes	characteristic
28	#13063256	C₂²	1	2	yes	—
29	#13063256	C₂²	1	2	yes	—
30	Dic₄₃⋊C₈	C₂²	1	2	yes	—
31	#13063249	C₂²	1	1	yes	characteristic
32	#13063247	C₂²	1	2	yes	—
33	#13063245	C₂²	1	1	yes	characteristic
34	C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
35	C₈×Dic₄₃	C₂²	1	2	yes	—
36	C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
37	C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
38	#107×C₄₃	C₂²	1	2	yes	—
39	C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
40	#103×C₄₃	C₂²	1	2	yes	—
41	C₂×Q₈×C₄₃	—	2	1	no	—
42	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
43	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
44	C₂×C₄×D₄₃	—	2	1	no	—
45	C₂×C₄×D₄₃	—	2	2	no	—
46	C₂×Dic₈₆	—	2	1	no	—
47	C₂×Dic₈₆	D₄	1	1	yes	characteristic
48	C₂×C₈×C₄₃	C₂³	1	2	yes	—
49	C₄⋊C₄×C₄₃	—	2	1	no	—
50	C₄⋊C₄×C₄₃	—	2	1	no	—
51	C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
52	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
53	C₄²×C₄₃	—	2	1	no	—
54	C₄²×C₄₃	C₂³	1	1	yes	characteristic
55	#10621884	Q₈	1	4	yes	complemented
56	(C₂×D₄₃)⋊C₄	—	2	1	no	—
57	(C₂×D₄₃)⋊C₄	—	2	2	no	—
58	C₄⋊Dic₄₃	—	2	1	no	—
59	C₄⋊Dic₄₃	—	2	2	no	—
60	C₄⋊Dic₄₃	—	2	2	no	—
61	C₄⋊Dic₄₃	—	2	2	no	—
62	C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
63	C₄⋊Dic₄₃	C₂³	1	2	yes	—
64	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
65	Dic₄₃⋊C₄	—	2	2	no	—
66	Dic₄₃⋊C₄	—	2	2	no	—
67	Dic₄₃⋊C₄	D₄	1	2	yes	—
68	C₄×Dic₄₃	—	2	2	no	—
69	C₄×Dic₄₃	C₂³	1	2	yes	—
70	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
71	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
72	C₂²×D₄₃	—	2	1	no	—
73	Q₈×C₄₃	—	2	1	no	—
74	Q₈×C₄₃	—	8	1	no	—
75	Q₈×C₄₃	C₂×Q₈	1	2	yes	—
76	C₂×C₄×C₄₃	—	2	1	no	—
77	C₂×C₄×C₄₃	—	2	1	no	—
78	C₂×C₄×C₄₃	—	2	2	no	—
79	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
82	C₂×Dic₄₃	—	2	2	no	—
83	C₂×Dic₄₃	—	2	1	no	—
84	C₂×Dic₄₃	C₂×D₄	1	2	yes	—
85	C₂×Dic₄₃	C₂×D₄	1	2	yes	—
86	D₁₇₂	—	2	1	no	—
87	D₁₇₂	C₂×Q₈	1	2	yes	—
88	C₄×D₄₃	—	4	2	no	—
89	C₄×D₄₃	—	8	2	no	—
90	Dic₈₆	—	2	2	no	—
91	Dic₈₆	—	4	2	no	—
92	C₈×C₄₃	—	2	2	no	—
93	C₄₃⋊C₈	—	2	1	no	—
94	C₄₃⋊C₈	C₂×Q₈	1	2	yes	—
95	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
96	C₂×D₄₃	—	2	2	no	—
97	C₂×D₄₃	—	4	1	no	—
98	C₂×D₄₃	—	4	1	no	—
99	C₄×C₄₃	—	2	2	no	—
100	C₄×C₄₃	—	2	1	no	—
101	C₄×C₄₃	—	4	2	no	—
102	C₄×C₄₃	—	4	1	no	—
103	C₄×C₄₃	#135	1	2	yes	—
104	C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
105	C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
106	Dic₄₃	—	2	2	no	—
107	Dic₄₃	—	2	2	no	—
108	Dic₄₃	—	4	2	no	—
109	Dic₄₃	—	4	1	no	—
110	#3381	—	43	1	no	maximal, Sylow
111	C₂×C₄₃	D₄×Q₈	1	1	yes	characteristic
112	C₂×C₄₃	C₂×#135	1	1	yes	characteristic
113	C₂×C₄₃	#577	1	1	yes	characteristic
114	D₄₃	—	4	2	no	—
115	#557	—	43	1	no	—
116	#554	—	43	1	no	—
117	#499	—	43	2	no	—
118	#498	—	43	1	no	—
119	#479	—	43	2	no	—
120	#478	—	43	2	no	—
121	#477	D₄₃	1	1	yes	characteristic, complemented
122	#474	—	43	1	no	—
123	C₈×Q₈	—	43	1	no	—
124	C₄×SD₁₆	—	43	1	no	—
125	C₄×SD₁₆	—	43	2	no	—
126	C₄₃	#3381	1	1	yes	characteristic, complemented, Sylow
127	C₂×SD₁₆	—	43	1	no	—
128	C₄⋊Q₈	—	43	1	no	—
129	C₄⋊Q₈	—	86	1	no	—
130	C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
131	#125	—	43	2	no	—
132	#125	—	43	2	no	—
133	#125	—	86	2	no	—
134	C₂²⋊Q₈	—	86	1	no	—
135	C₂²⋊Q₈	—	86	2	no	—
136	C₄×Q₈	—	43	1	no	—
137	C₄×Q₈	—	43	2	no	—
138	C₄×Q₈	—	86	1	no	—
139	C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
140	C₄×D₄	—	43	2	no	—
141	C₄×D₄	—	43	1	no	—
142	C₂×C₄⋊C₄	—	86	1	no	—
143	#107	—	43	2	no	—
144	#107	C₂×D₄₃	1	2	yes	—
145	#106	—	43	2	no	—
146	#106	—	43	1	no	—
147	#106	—	43	2	no	—
148	C₄⋊C₈	—	43	2	no	—
149	C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
150	#103	—	43	1	no	—
151	#103	C₂×D₄₃	1	2	yes	—
152	#102	—	43	2	no	—
153	#102	—	43	1	no	—
154	C₄×C₈	—	43	2	no	—
155	C₄×C₈	—	43	1	no	—
156	C₂×Q₈	—	2	1	no	—
157	C₂×Q₈	—	43	1	no	—
158	C₂×Q₈	—	86	1	no	—
159	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
160	C₂×D₄	—	43	1	no	—
161	C₂²×C₄	—	86	2	no	—
162	C₂²×C₄	—	86	1	no	—
163	SD₁₆	—	43	4	no	—
164	C₂×C₈	—	43	1	no	—
165	C₂×C₈	—	43	1	no	—
166	C₂×C₈	C₂²×D₄₃	1	2	yes	—
167	C₄⋊C₄	—	2	1	no	—
168	C₄⋊C₄	—	2	1	no	—
169	C₄⋊C₄	—	43	1	no	—
170	C₄⋊C₄	—	43	2	no	—
171	C₄⋊C₄	—	43	2	no	—
172	C₄⋊C₄	—	43	1	no	—
173	C₄⋊C₄	—	86	2	no	—
174	C₄⋊C₄	—	86	2	no	—
175	C₄⋊C₄	—	86	2	no	—
176	C₄⋊C₄	—	86	2	no	—
177	C₄⋊C₄	—	86	2	no	—
178	C₄⋊C₄	—	86	1	no	—
179	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
180	C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
181	C₂²⋊C₄	—	86	2	no	—
182	C₂²⋊C₄	—	86	1	no	—
183	C₄²	—	2	1	no	—
184	C₄²	—	43	2	no	—
185	C₄²	—	86	2	no	—
186	C₄²	C₂²×D₄₃	1	1	yes	characteristic
187	C₂³	—	86	1	no	—
188	Q₈	—	2	1	no	—
189	Q₈	—	8	1	no	—
190	Q₈	—	86	2	no	—
191	Q₈	—	172	2	no	—
192	Q₈	Q₈×D₄₃	1	2	yes	—
193	D₄	—	43	2	no	—
194	D₄	—	86	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	43	2	no	—
199	C₂×C₄	—	43	2	no	—
200	C₂×C₄	—	86	2	no	—
201	C₂×C₄	—	86	1	no	—
202	C₂×C₄	—	344	2	no	—
203	C₂×C₄	—	172	2	no	—
204	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
205	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
206	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
207	C₈	—	2	2	no	—
208	C₈	—	43	2	no	—
209	C₈	—	86	1	no	—
210	C₂²	—	86	2	no	—
211	C₂²	—	172	1	no	—
212	C₂²	—	172	1	no	—
213	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
214	C₄	—	2	2	no	—
215	C₄	—	2	1	no	—
216	C₄	—	4	2	no	—
217	C₄	—	4	1	no	—
218	C₄	—	86	2	no	—
219	C₄	—	86	2	no	—
220	C₄	—	172	1	no	—
221	C₄	—	172	2	no	—
222	C₄	#13063348	1	2	yes	—
223	C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
224	C₄	#13063372	1	1	yes	characteristic
225	C₂	—	172	2	no	—
226	C₂	#14583562	1	1	yes	characteristic
227	C₂	C₂×#13063348	1	1	yes	characteristic
228	C₂	#14583709	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14583628

C₂

yes

maximal, characteristic, complemented

#14583610

C₂

yes

maximal, characteristic

#14583300

C₂

yes

maximal, complemented

#14583295

C₂

yes

maximal, characteristic, complemented

#14583208

C₂

yes

maximal, complemented

#14583202

C₂

yes

maximal

#14583180

C₂

yes

maximal, characteristic

#14583023

C₂

yes

maximal, characteristic, complemented

C₄×#10621884

C₂

yes

maximal, characteristic

#14582984

C₂

yes

maximal

C₄₃×#477

C₂

yes

maximal, characteristic, complemented

C₂×#10621884

C₂²

yes

characteristic

#13063325

—

#13063325

C₂²

yes

characteristic

#13063316

—

#13063316

C₂²

yes

—

#13063316

C₂²

yes

—

#13063305

—

#13063301

—

Q₈×Dic₄₃

C₂²

yes

—

C₄×Dic₈₆

—

C₄×Dic₈₆

C₂²

yes

characteristic

C₄×D₁₇₂

C₂²

yes

characteristic

#13063287

C₂²

yes

—

C₄⋊C₄×D₄₃

—

#13063259

C₂²

yes

—

#13063256

C₂²

yes

characteristic

#13063256

C₂²

yes

—

#13063256

C₂²

yes

—

Dic₄₃⋊C₈

C₂²

yes

—

#13063249

C₂²

yes

characteristic

#13063247

C₂²

yes

—

#13063245

C₂²

yes

characteristic

C₄×C₄₃⋊C₈

C₂²

yes

characteristic

C₈×Dic₄₃

C₂²

yes

—

C₄⋊Q₈×C₄₃

C₂²

yes

characteristic

C₄×Q₈×C₄₃

C₂²

yes

characteristic

#107×C₄₃

C₂²

yes

—

C₄⋊C₈×C₄₃

C₂²

yes

characteristic

#103×C₄₃

C₂²

yes

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

C₂³

yes

characteristic

C₂×D₁₇₂

C₂³

yes

characteristic

C₂×C₄×D₄₃

—

C₂×C₄×D₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

D₄

yes

characteristic

C₂×C₈×C₄₃

C₂³

yes

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

C₂³

yes

—

C₄⋊C₄×C₄₃

C₂³

yes

characteristic

C₄²×C₄₃

—

C₄²×C₄₃

C₂³

yes

characteristic

#10621884

Q₈

yes

complemented

(C₂×D₄₃)⋊C₄

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

D₄

yes

characteristic

C₄⋊Dic₄₃

C₂³

yes

—

C₄⋊Dic₄₃

C₂³

yes

characteristic

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

D₄

yes

—

C₄×Dic₄₃

—

C₄×Dic₄₃

C₂³

yes

—

C₂×C₄₃⋊C₈

C₂³

yes

characteristic

C₂×C₄₃⋊C₈

C₂³

yes

characteristic

C₂²×D₄₃

—

Q₈×C₄₃

—

Q₈×C₄₃

—

Q₈×C₄₃

C₂×Q₈

yes

—

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

C₂⁴

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

C₂×D₄

yes

—

C₂×Dic₄₃

C₂×D₄

yes

—

D₁₇₂

—

D₁₇₂

C₂×Q₈

yes

—

C₄×D₄₃

—

C₄×D₄₃

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₄₃⋊C₈

C₂×Q₈

yes

—

C₂²×C₄₃

C₂²×D₄

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

100

C₄×C₄₃

—

101

C₄×C₄₃

—

102

C₄×C₄₃

—

103

C₄×C₄₃

#135

yes

—

104

C₄×C₄₃

C₂²×Q₈

yes

characteristic

105

C₄×C₄₃

2₋¹⁺⁴

yes

characteristic

106

Dic₄₃

—

107

Dic₄₃

—

108

Dic₄₃

—

109

Dic₄₃

—

110

#3381

—

maximal, Sylow

111

C₂×C₄₃

D₄×Q₈

yes

characteristic

112

C₂×C₄₃

C₂×#135

yes

characteristic

113

C₂×C₄₃

#577

yes

characteristic

114

D₄₃

—

115

#557

—

116

#554

—

117

#499

—

118

#498

—

119

#479

—

120

#478

—

121

#477

D₄₃

yes

characteristic, complemented

122

#474

—

123

C₈×Q₈

—

124

C₄×SD₁₆

—

125

C₄×SD₁₆

—

126

C₄₃

#3381

yes

characteristic, complemented, Sylow

127

C₂×SD₁₆

—

128

C₄⋊Q₈

—

129

C₄⋊Q₈

—

130

C₄⋊Q₈

C₂×D₄₃

yes

characteristic

131

#125

—

132

#125

—

133

#125

—

134

C₂²⋊Q₈

—

135

C₂²⋊Q₈

—

136

C₄×Q₈

—

137

C₄×Q₈

—

138

C₄×Q₈

—

139

C₄×Q₈

C₂×D₄₃

yes

characteristic

140

C₄×D₄

—

141

C₄×D₄

—

142

C₂×C₄⋊C₄

—

143

#107

—

144

#107

C₂×D₄₃

yes

—

145

#106

—

146

#106

—

147

#106

—

148

C₄⋊C₈

—

149

C₄⋊C₈

C₂×D₄₃

yes

characteristic

150

#103

—

151

#103

C₂×D₄₃

yes

—

152

#102

—

153

#102

—

154

C₄×C₈

—

155

C₄×C₈

—

156

C₂×Q₈

—

157

C₂×Q₈

—

158

C₂×Q₈

—

159

C₂×Q₈

C₂²×D₄₃

yes

characteristic

160

C₂×D₄

—

161

C₂²×C₄

—

162

C₂²×C₄

—

163

SD₁₆

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₂×C₈

C₂²×D₄₃

yes

—

167

C₄⋊C₄

—

168

C₄⋊C₄

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

171

C₄⋊C₄

—

172

C₄⋊C₄

—

173

C₄⋊C₄

—

174

C₄⋊C₄

—

175

C₄⋊C₄

—

176

C₄⋊C₄

—

177

C₄⋊C₄

—

178

C₄⋊C₄

—

179

C₄⋊C₄

C₂²×D₄₃

yes

characteristic

180

C₄⋊C₄

C₂²×D₄₃

yes

—

181

C₂²⋊C₄

—

182

C₂²⋊C₄

—

183

C₄²

—

184

C₄²

—

185

C₄²

—

186

C₄²

C₂²×D₄₃

yes

characteristic

187

C₂³

—

188

Q₈

—

189

Q₈

—

190

Q₈

—

191

Q₈

—

172

—

192

Q₈

Q₈×D₄₃

yes

—

193

D₄

—

194

D₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

202

C₂×C₄

—

344

—

203

C₂×C₄

—

172

—

204

C₂×C₄

D₄×D₄₃

yes

characteristic

205

C₂×C₄

D₄×D₄₃

yes

characteristic

206

C₂×C₄

C₂³×D₄₃

yes

characteristic

207

C₈

—

208

C₈

—

209

C₈

—

210

C₂²

—

211

C₂²

—

172

—

212

C₂²

—

172

—

213

C₂²

C₂×D₄×D₄₃

yes

characteristic

214

C₄

—

215

C₄

—

216

C₄

—

217

C₄

—

218

C₄

—

219

C₄

—

220

C₄

—

172

—

221

C₄

—

172

—

222

C₄

#13063348

yes

—

223

C₄

C₂×Q₈×D₄₃

yes

characteristic

224

C₄

#13063372

yes

characteristic

225

C₂

—

172

—

226

C₂

#14583562

yes

characteristic

227

C₂

C₂×#13063348

yes

characteristic

228

C₂

#14583709

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#477	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#477	D₄₃
hypocenter	C₄₃	#3381
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #477

subgroup

G/H

center

C₂²

C₂×D₄×D₄₃

commutator

C₂×C₄×C₄₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₃

Fitting

C₄₃×#477

C₂

socle

C₂²×C₄₃

C₂²×D₄

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #477

N	H ≅ G/N	type
1. #14583628	C₂	semidirect
3. #14583300	C₂	semidirect
4. #14583295	C₂	semidirect
5. #14583208	C₂	semidirect
8. #14583023	C₂	semidirect
11. C₄₃×#477	C₂	semidirect
55. #10621884	Q₈	semidirect
121. #477	D₄₃	semidirect
126. C₄₃	#3381	semidirect

H ≅ G/N

type

1. #14583628

C₂

semidirect

3. #14583300

C₂

semidirect

4. #14583295

C₂

semidirect

5. #14583208

C₂

semidirect

8. #14583023

C₂

semidirect

11. C₄₃×#477

C₂

semidirect

55. #10621884

Q₈

semidirect

121. #477

D₄₃

semidirect

126. C₄₃

#3381

semidirect

Subgroups — #19147469

Conjugacy classes of subgroups

Complements