Group #19144990 — subgroups

Subgroups — #19144990

Order 5504.

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

normal subgroups only

This group has 424 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	#14583450	C₂	1	1	yes	maximal, characteristic
2	C₄⋊Q₈×D₄₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583260	C₂	1	4	yes	maximal, complemented
4	#14583249	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583247	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583246	C₂	1	1	yes	maximal, characteristic, complemented
7	#14582899	C₂	1	4	yes	maximal, complemented
8	C₄×#10621872	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₃×#487	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#10621902	C₂²	1	2	yes	—
11	C₂×Q₈×D₄₃	—	2	1	no	—
12	C₂×#10621872	C₂²	1	2	yes	—
13	#13063325	C₂²	1	1	yes	characteristic
14	#13063324	—	2	1	no	—
15	#13063324	C₂²	1	2	yes	—
16	#13063323	—	2	1	no	—
17	#13063322	C₂²	1	1	yes	characteristic
18	#13063314	C₂²	1	2	yes	—
19	#13063300	—	2	1	no	—
20	#13063300	—	2	1	no	—
21	#13063287	—	2	1	no	—
22	#13063287	—	2	1	no	—
23	C₄⋊C₄×D₄₃	—	2	1	no	—
24	C₄⋊C₄×D₄₃	C₂²	1	2	yes	—
25	C₄²×D₄₃	C₂²	1	1	yes	characteristic
26	#13063249	C₂²	1	4	yes	—
27	#13063247	C₂²	1	4	yes	—
28	#13063245	C₂²	1	4	yes	—
29	#13063235	C₂²	1	2	yes	—
30	C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
31	C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
32	C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
33	#103×C₄₃	C₂²	1	4	yes	—
34	C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
35	C₂×Q₈×C₄₃	—	2	1	no	—
36	C₂×Q₈×C₄₃	C₂³	1	2	yes	—
37	#10621902	—	2	8	no	—
38	Q₈×D₄₃	—	4	4	no	—
39	C₂×D₁₇₂	—	2	1	no	—
40	C₂×D₁₇₂	—	2	1	no	—
41	C₂×D₁₇₂	—	2	2	no	—
42	C₂×D₁₇₂	C₂³	1	2	yes	—
43	C₂×C₄×D₄₃	—	2	2	no	—
44	C₂×C₄×D₄₃	—	2	2	no	—
45	C₂×C₄×D₄₃	D₄	1	1	yes	characteristic
46	C₂×C₄×D₄₃	D₄	1	1	yes	characteristic
47	C₂×C₄×D₄₃	C₂³	1	1	yes	characteristic
48	C₂×Dic₈₆	—	2	1	no	—
49	C₂×Dic₈₆	—	2	2	no	—
50	C₂×C₈×C₄₃	C₂³	1	2	yes	—
51	C₄⋊C₄×C₄₃	—	2	1	no	—
52	C₄⋊C₄×C₄₃	—	2	1	no	—
53	C₄⋊C₄×C₄₃	—	2	1	no	—
54	C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
55	C₄²×C₄₃	C₂³	1	1	yes	characteristic
56	(C₂×D₄₃)⋊C₄	—	2	2	no	—
57	(C₂×D₄₃)⋊C₄	—	2	2	no	—
58	C₄⋊Dic₄₃	—	2	1	no	—
59	C₄⋊Dic₄₃	C₂³	1	2	yes	—
60	Dic₄₃⋊C₄	—	2	2	no	—
61	Dic₄₃⋊C₄	—	2	2	no	—
62	C₄×Dic₄₃	D₄	1	1	yes	characteristic
63	C₄×Dic₄₃	D₄	1	1	yes	characteristic
64	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
65	C₂×C₄₃⋊C₈	C₂³	1	2	yes	—
66	#10621872	—	2	4	no	—
67	C₂²×D₄₃	—	2	2	no	—
68	C₂²×D₄₃	C₂×D₄	1	1	yes	characteristic
69	Q₈×C₄₃	—	2	4	no	—
70	Q₈×C₄₃	—	4	2	no	—
71	C₂×C₄×C₄₃	—	2	2	no	—
72	C₂×C₄×C₄₃	—	2	2	no	—
73	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
76	C₂×Dic₄₃	—	2	2	no	—
77	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
78	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
79	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
80	D₁₇₂	—	2	4	no	—
81	D₁₇₂	—	4	4	no	—
82	D₁₇₂	—	4	2	no	—
83	D₁₇₂	—	4	2	no	—
84	C₄×D₄₃	—	2	2	no	—
85	C₄×D₄₃	—	2	2	no	—
86	C₄×D₄₃	—	4	4	no	—
87	C₄×D₄₃	—	4	4	no	—
88	C₄×D₄₃	#41	1	4	yes	—
89	Dic₈₆	—	4	4	no	—
90	Dic₈₆	—	4	2	no	—
91	C₈×C₄₃	—	2	2	no	—
92	C₄₃⋊C₈	—	2	2	no	—
93	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
94	C₂×D₄₃	—	2	1	no	—
95	C₂×D₄₃	—	2	1	no	—
96	C₂×D₄₃	—	4	2	no	—
97	C₂×D₄₃	—	4	2	no	—
98	C₂×D₄₃	—	4	2	no	—
99	C₂×D₄₃	#124	1	2	yes	—
100	C₄×C₄₃	—	4	2	no	—
101	C₄×C₄₃	—	4	2	no	—
102	C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
103	C₄×C₄₃	#137	1	2	yes	—
104	C₄×C₄₃	C₂×#41	1	2	yes	—
105	Dic₄₃	—	2	1	no	—
106	Dic₄₃	—	2	1	no	—
107	Dic₄₃	—	4	2	no	—
108	Dic₄₃	#124	1	2	yes	—
109	#3141	—	43	1	no	maximal, Sylow
110	C₂×C₄₃	C₂×#124	1	1	yes	characteristic
111	C₂×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
112	C₂×C₄₃	C₂×#137	1	1	yes	characteristic
113	D₄₃	—	2	2	no	—
114	D₄₃	—	8	2	no	—
115	#532	—	43	1	no	—
116	C₂×C₄⋊Q₈	—	43	1	no	—
117	#489	—	43	4	no	—
118	#487	—	43	1	no	—
119	#487	D₄₃	1	1	yes	characteristic, complemented
120	#486	—	43	1	no	—
121	#486	—	43	1	no	—
122	#417	—	43	4	no	—
123	C₄×M₁₆	—	43	1	no	—
124	C₄₃	#3141	1	1	yes	characteristic, complemented, Sylow
125	C₂×#41	—	43	2	no	—
126	C₂²×Q₈	—	86	1	no	—
127	C₂×M₁₆	—	43	2	no	—
128	C₄⋊Q₈	—	43	2	no	—
129	C₄⋊Q₈	—	43	1	no	—
130	C₄⋊Q₈	—	86	1	no	—
131	C₄⋊Q₈	—	86	1	no	—
132	C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
133	C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
134	#127	—	43	1	no	—
135	#124	—	43	2	no	—
136	#121	—	86	1	no	—
137	#121	—	86	1	no	—
138	C₄×D₄	—	86	1	no	—
139	C₄×D₄	—	86	1	no	—
140	C₂×C₄⋊C₄	—	43	2	no	—
141	C₂×C₄⋊C₄	—	86	1	no	—
142	C₂×C₄²	—	43	1	no	—
143	#103	—	43	4	no	—
144	#103	C₂×D₄₃	1	4	yes	—
145	#102	—	43	4	no	—
146	#102	—	43	4	no	—
147	#97	—	43	2	no	—
148	C₄×C₈	—	43	1	no	—
149	C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
150	#41	—	86	8	no	—
151	C₂×Q₈	—	2	1	no	—
152	C₂×Q₈	—	86	2	no	—
153	C₂×Q₈	—	86	1	no	—
154	C₂×Q₈	—	172	4	no	—
155	C₂×Q₈	C₂²×D₄₃	1	2	yes	—
156	C₂×D₄	—	43	2	no	—
157	C₂×D₄	—	86	2	no	—
158	C₂×D₄	—	86	1	no	—
159	C₂×D₄	—	86	1	no	—
160	C₂²×C₄	—	43	1	no	—
161	C₂²×C₄	—	43	1	no	—
162	C₂²×C₄	—	43	1	no	—
163	C₂²×C₄	—	86	2	no	—
164	C₂²×C₄	—	86	2	no	—
165	M₁₆	—	86	4	no	—
166	C₂×C₈	—	43	2	no	—
167	C₂×C₈	C₂²×D₄₃	1	2	yes	—
168	C₄⋊C₄	—	2	1	no	—
169	C₄⋊C₄	—	2	1	no	—
170	C₄⋊C₄	—	2	1	no	—
171	C₄⋊C₄	—	43	2	no	—
172	C₄⋊C₄	—	86	2	no	—
173	C₄⋊C₄	—	86	2	no	—
174	C₄⋊C₄	—	86	1	no	—
175	C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
176	C₂²⋊C₄	—	86	2	no	—
177	C₂²⋊C₄	—	86	2	no	—
178	C₄²	—	43	1	no	—
179	C₄²	—	43	1	no	—
180	C₄²	—	43	1	no	—
181	C₄²	C₂²×D₄₃	1	1	yes	characteristic
182	C₂³	—	43	1	no	—
183	C₂³	—	86	2	no	—
184	Q₈	—	2	4	no	—
185	Q₈	—	4	2	no	—
186	Q₈	—	172	4	no	—
187	Q₈	—	172	2	no	—
188	D₄	—	86	4	no	—
189	D₄	—	172	4	no	—
190	D₄	—	172	2	no	—
191	D₄	—	172	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	43	1	no	—
195	C₂×C₄	—	43	4	no	—
196	C₂×C₄	—	43	1	no	—
197	C₂×C₄	—	43	1	no	—
198	C₂×C₄	—	86	2	no	—
199	C₂×C₄	—	86	2	no	—
200	C₂×C₄	—	86	2	no	—
201	C₂×C₄	—	172	4	no	—
202	C₂×C₄	—	172	4	no	—
203	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
204	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
205	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
206	C₈	—	2	2	no	—
207	C₈	—	86	2	no	—
208	C₂²	—	43	2	no	—
209	C₂²	—	86	1	no	—
210	C₂²	—	86	1	no	—
211	C₂²	—	172	2	no	—
212	C₂²	—	172	2	no	—
213	C₂²	—	172	2	no	—
214	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
215	C₄	—	4	2	no	—
216	C₄	—	4	2	no	—
217	C₄	—	43	2	no	—
218	C₄	—	86	1	no	—
219	C₄	—	86	1	no	—
220	C₄	—	172	2	no	—
221	C₄	#13063352	1	2	yes	—
222	C₄	#13063357	1	2	yes	—
223	C₄	#41×D₄₃	1	2	yes	—
224	C₂	—	86	2	no	—
225	C₂	—	344	2	no	—
226	C₂	#124×D₄₃	1	1	yes	characteristic
227	C₂	C₂×#13063352	1	1	yes	characteristic
228	C₂	C₂×#13063357	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14583450

C₂

yes

maximal, characteristic

C₄⋊Q₈×D₄₃

C₂

yes

maximal, characteristic, complemented

#14583260

C₂

yes

maximal, complemented

#14583249

C₂

yes

maximal, characteristic, complemented

#14583247

C₂

yes

maximal, characteristic, complemented

#14583246

C₂

yes

maximal, characteristic, complemented

#14582899

C₂

yes

maximal, complemented

C₄×#10621872

C₂

yes

maximal, characteristic, complemented

C₄₃×#487

C₂

yes

maximal, characteristic, complemented

C₂×#10621902

C₂²

yes

—

C₂×Q₈×D₄₃

—

C₂×#10621872

C₂²

yes

—

#13063325

C₂²

yes

characteristic

#13063324

—

#13063324

C₂²

yes

—

#13063323

—

#13063322

C₂²

yes

characteristic

#13063314

C₂²

yes

—

—

—

—

—

C₄⋊C₄×D₄₃

—

C₄⋊C₄×D₄₃

C₂²

yes

—

C₄²×D₄₃

C₂²

yes

characteristic

#13063249

C₂²

yes

—

#13063247

C₂²

yes

—

#13063245

C₂²

yes

—

#13063235

C₂²

yes

—

C₄×C₄₃⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₄₃

C₂²

yes

characteristic

C₄⋊Q₈×C₄₃

C₂²

yes

characteristic

#103×C₄₃

C₂²

yes

—

C₄×C₈×C₄₃

C₂²

yes

characteristic

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

C₂³

yes

—

#10621902

—

Q₈×D₄₃

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

C₂³

yes

—

C₂×C₄×D₄₃

—

C₂×C₄×D₄₃

—

C₂×C₄×D₄₃

D₄

yes

characteristic

C₂×C₄×D₄₃

D₄

yes

characteristic

C₂×C₄×D₄₃

C₂³

yes

characteristic

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

C₂³

yes

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

C₂³

yes

—

C₄²×C₄₃

C₂³

yes

characteristic

(C₂×D₄₃)⋊C₄

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

C₂³

yes

—

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

D₄

yes

characteristic

C₄×Dic₄₃

D₄

yes

characteristic

C₄×Dic₄₃

C₂³

yes

characteristic

C₂×C₄₃⋊C₈

C₂³

yes

—

#10621872

—

C₂²×D₄₃

—

C₂²×D₄₃

C₂×D₄

yes

characteristic

Q₈×C₄₃

—

Q₈×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×D₄

yes

characteristic

C₂×Dic₄₃

C₂×D₄

yes

characteristic

C₂×Dic₄₃

C₂×D₄

yes

characteristic

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

C₄×D₄₃

—

C₄×D₄₃

—

C₄×D₄₃

—

C₄×D₄₃

—

C₄×D₄₃

#41

yes

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₂²×C₄₃

C₂²×D₄

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

#124

yes

—

100

C₄×C₄₃

—

101

C₄×C₄₃

—

102

C₄×C₄₃

AGL(1,ℤ/8)

yes

—

103

C₄×C₄₃

#137

yes

—

104

C₄×C₄₃

C₂×#41

yes

—

105

Dic₄₃

—

106

Dic₄₃

—

107

Dic₄₃

—

108

Dic₄₃

#124

yes

—

109

#3141

—

maximal, Sylow

110

C₂×C₄₃

C₂×#124

yes

characteristic

111

C₂×C₄₃

C₂×AGL(1,ℤ/8)

yes

characteristic

112

C₂×C₄₃

C₂×#137

yes

characteristic

113

D₄₃

—

114

D₄₃

—

115

#532

—

116

C₂×C₄⋊Q₈

—

117

#489

—

118

#487

—

119

#487

D₄₃

yes

characteristic, complemented

120

#486

—

121

#486

—

122

#417

—

123

C₄×M₁₆

—

124

C₄₃

#3141

yes

characteristic, complemented, Sylow

125

C₂×#41

—

126

C₂²×Q₈

—

127

C₂×M₁₆

—

128

C₄⋊Q₈

—

129

C₄⋊Q₈

—

130

C₄⋊Q₈

—

131

C₄⋊Q₈

—

132

C₄⋊Q₈

C₂×D₄₃

yes

characteristic

133

C₄⋊Q₈

C₂×D₄₃

yes

characteristic

134

#127

—

135

#124

—

136

#121

—

137

#121

—

138

C₄×D₄

—

139

C₄×D₄

—

140

C₂×C₄⋊C₄

—

141

C₂×C₄⋊C₄

—

142

C₂×C₄²

—

143

#103

—

144

#103

C₂×D₄₃

yes

—

145

#102

—

146

#102

—

147

#97

—

148

C₄×C₈

—

149

C₄×C₈

C₂×D₄₃

yes

characteristic

150

#41

—

151

C₂×Q₈

—

152

C₂×Q₈

—

153

C₂×Q₈

—

154

C₂×Q₈

—

172

—

155

C₂×Q₈

C₂²×D₄₃

yes

—

156

C₂×D₄

—

157

C₂×D₄

—

158

C₂×D₄

—

159

C₂×D₄

—

160

C₂²×C₄

—

161

C₂²×C₄

—

162

C₂²×C₄

—

163

C₂²×C₄

—

164

C₂²×C₄

—

165

M₁₆

—

166

C₂×C₈

—

167

C₂×C₈

C₂²×D₄₃

yes

—

168

C₄⋊C₄

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

171

C₄⋊C₄

—

172

C₄⋊C₄

—

173

C₄⋊C₄

—

174

C₄⋊C₄

—

175

C₄⋊C₄

C₂²×D₄₃

yes

—

176

C₂²⋊C₄

—

177

C₂²⋊C₄

—

178

C₄²

—

179

C₄²

—

180

C₄²

—

181

C₄²

C₂²×D₄₃

yes

characteristic

182

C₂³

—

183

C₂³

—

184

Q₈

—

185

Q₈

—

186

Q₈

—

172

—

187

Q₈

—

172

—

188

D₄

—

189

D₄

—

172

—

190

D₄

—

172

—

191

D₄

—

172

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

172

—

202

C₂×C₄

—

172

—

203

C₂×C₄

D₄×D₄₃

yes

characteristic

204

C₂×C₄

D₄×D₄₃

yes

characteristic

205

C₂×C₄

C₂³×D₄₃

yes

characteristic

206

C₈

—

207

C₈

—

208

C₂²

—

209

C₂²

—

210

C₂²

—

211

C₂²

—

172

—

212

C₂²

—

172

—

213

C₂²

—

172

—

214

C₂²

C₂×D₄×D₄₃

yes

characteristic

215

C₄

—

216

C₄

—

217

C₄

—

218

C₄

—

219

C₄

—

220

C₄

—

172

—

221

C₄

#13063352

yes

—

222

C₄

#13063357

yes

—

223

C₄

#41×D₄₃

yes

—

224

C₂

—

225

C₂

—

344

—

226

C₂

#124×D₄₃

yes

characteristic

227

C₂

C₂×#13063352

yes

characteristic

228

C₂

C₂×#13063357

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#487	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#487	D₄₃
hypocenter	C₄₃	#3141
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #487

subgroup

G/H

center

C₂²

C₂×D₄×D₄₃

commutator

C₂×C₄×C₄₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₃

Fitting

C₄₃×#487

C₂

socle

C₂²×C₄₃

C₂²×D₄

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #487

N	H ≅ G/N	type
2. C₄⋊Q₈×D₄₃	C₂	semidirect
3. #14583260	C₂	semidirect
4. #14583249	C₂	semidirect
5. #14583247	C₂	semidirect
6. #14583246	C₂	semidirect
7. #14582899	C₂	semidirect
8. C₄×#10621872	C₂	semidirect
9. C₄₃×#487	C₂	semidirect
119. #487	D₄₃	semidirect
124. C₄₃	#3141	semidirect

H ≅ G/N

type

2. C₄⋊Q₈×D₄₃

C₂

semidirect

3. #14583260

C₂

semidirect

4. #14583249

C₂

semidirect

5. #14583247

C₂

semidirect

6. #14583246

C₂

semidirect

7. #14582899

C₂

semidirect

8. C₄×#10621872

C₂

semidirect

9. C₄₃×#487

C₂

semidirect

119. #487

D₄₃

semidirect

124. C₄₃

#3141

semidirect

Subgroups — #19144990

Conjugacy classes of subgroups

Complements