Group #19145719 — subgroups

Subgroups — #19145719

Order 5504.

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

normal subgroups only

This group has 404 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	#14583447	C₂	1	1	yes	maximal, characteristic
2	C₄⋊Q₈×D₄₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583297	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583295	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583245	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583216	C₂	1	4	yes	maximal, complemented
7	#14583187	C₂	1	4	yes	maximal, complemented
8	#14582950	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₃×#486	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#10621900	—	2	1	no	—
11	C₂×Q₈×D₄₃	—	2	1	no	—
12	#13063325	C₂²	1	1	yes	characteristic
13	#13063325	C₂²	1	1	yes	characteristic
14	#13063324	—	2	1	no	—
15	#13063324	C₂²	1	2	yes	—
16	#13063323	—	2	1	no	—
17	#13063312	—	2	1	no	—
18	#13063296	—	2	1	no	—
19	#13063296	C₂²	1	2	yes	—
20	D₄×Dic₄₃	—	2	1	no	—
21	D₄×Dic₄₃	C₂²	1	2	yes	—
22	C₄⋊C₄×D₄₃	—	2	1	no	—
23	C₄⋊C₄×D₄₃	C₂²	1	2	yes	—
24	C₄²×D₄₃	C₂²	1	1	yes	characteristic
25	#13063257	C₂²	1	4	yes	—
26	#13063256	C₂²	1	4	yes	—
27	Dic₄₃⋊C₈	C₂²	1	2	yes	—
28	#13063246	C₂²	1	4	yes	—
29	(C₂×D₄₃)⋊C₈	C₂²	1	2	yes	—
30	C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
31	C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
32	#127×C₄₃	C₂²	1	1	yes	characteristic
33	#102×C₄₃	C₂²	1	4	yes	—
34	C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
35	C₂×Q₈×C₄₃	—	2	1	no	—
36	C₂×D₄×C₄₃	—	2	1	no	—
37	C₂×D₄×C₄₃	—	2	1	no	—
38	C₂×D₄×C₄₃	C₂³	1	2	yes	—
39	C₂×C₂²⋊D₄₃	—	2	2	no	—
40	C₂²×Dic₄₃	—	2	2	no	—
41	Q₈×D₄₃	—	4	4	no	—
42	#10621900	—	4	4	no	—
43	C₂×C₄×D₄₃	—	2	2	no	—
44	C₂×C₄×D₄₃	D₄	1	1	yes	characteristic
45	C₂×C₄×D₄₃	D₄	1	1	yes	characteristic
46	C₂×C₄×D₄₃	C₂³	1	1	yes	characteristic
47	C₂×Dic₈₆	—	2	1	no	—
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	C₂³	1	2	yes	—
53	C₄²×C₄₃	C₂³	1	1	yes	characteristic
54	C₂²⋊Dic₄₃	—	2	2	no	—
55	C₂²⋊Dic₄₃	—	2	2	no	—
56	C₄⋊Dic₄₃	—	2	1	no	—
57	C₄⋊Dic₄₃	—	2	1	no	—
58	C₄⋊Dic₄₃	C₂³	1	2	yes	—
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	C₂³×C₄₃	—	2	2	no	—
67	C₂²×D₄₃	C₂×D₄	1	1	yes	characteristic
68	Q₈×C₄₃	—	4	2	no	—
69	D₄×C₄₃	—	2	4	no	—
70	D₄×C₄₃	—	4	2	no	—
71	D₄×C₄₃	—	4	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	C₂²⋊D₄₃	—	4	4	no	—
77	C₂×Dic₄₃	—	2	2	no	—
78	C₂×Dic₄₃	—	2	2	no	—
79	C₂×Dic₄₃	—	4	4	no	—
80	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
81	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
82	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
83	C₄×D₄₃	—	2	2	no	—
84	C₄×D₄₃	—	2	2	no	—
85	C₄×D₄₃	—	4	4	no	—
86	C₄×D₄₃	SD₁₆	1	4	yes	—
87	Dic₈₆	—	4	4	no	—
88	Dic₈₆	—	4	2	no	—
89	Dic₈₆	—	4	2	no	—
90	C₈×C₄₃	—	2	2	no	—
91	C₄₃⋊C₈	—	2	2	no	—
92	C₂²×C₄₃	—	4	2	no	—
93	C₂²×C₄₃	—	4	2	no	—
94	C₂²×C₄₃	—	4	2	no	—
95	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
96	C₂×D₄₃	—	2	1	no	—
97	C₂×D₄₃	—	2	1	no	—
98	C₂×D₄₃	C₂×SD₁₆	1	2	yes	—
99	C₄×C₄₃	—	4	2	no	—
100	C₄×C₄₃	C₂×SD₁₆	1	2	yes	—
101	C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
102	C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
103	Dic₄₃	—	2	1	no	—
104	Dic₄₃	—	2	1	no	—
105	Dic₄₃	—	4	2	no	—
106	Dic₄₃	—	4	2	no	—
107	Dic₄₃	C₂×SD₁₆	1	2	yes	—
108	#3227	—	43	1	no	maximal, Sylow
109	C₂×C₄₃	—	8	2	no	—
110	C₂×C₄₃	#536	1	1	yes	characteristic
111	C₂×C₄₃	C₂²×SD₁₆	1	1	yes	characteristic
112	C₂×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
113	D₄₃	—	2	2	no	—
114	#532	—	43	1	no	—
115	C₂×C₄⋊Q₈	—	43	1	no	—
116	#498	—	43	1	no	—
117	#498	—	43	1	no	—
118	#486	—	43	1	no	—
119	#486	D₄₃	1	1	yes	characteristic, complemented
120	#481	—	43	4	no	—
121	#476	—	43	4	no	—
122	#431	—	43	1	no	—
123	C₄₃	#3227	1	1	yes	characteristic, complemented, Sylow
124	C₂×#41	—	86	1	no	—
125	C₂²×Q₈	—	86	1	no	—
126	C₄⋊Q₈	—	43	2	no	—
127	C₄⋊Q₈	—	43	1	no	—
128	C₄⋊Q₈	—	43	1	no	—
129	C₄⋊Q₈	—	86	1	no	—
130	C₄⋊Q₈	—	86	1	no	—
131	C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
132	#127	C₂×D₄₃	1	1	yes	characteristic
133	#124	—	86	1	no	—
134	#121	—	43	2	no	—
135	#121	—	86	1	no	—
136	C₄×D₄	—	43	2	no	—
137	C₄×D₄	—	86	1	no	—
138	C₂×C₄⋊C₄	—	43	2	no	—
139	C₂×C₄⋊C₄	—	86	1	no	—
140	C₂×C₄²	—	43	1	no	—
141	#106	—	43	4	no	—
142	#106	—	43	4	no	—
143	C₄⋊C₈	—	43	2	no	—
144	#102	—	43	4	no	—
145	#102	C₂×D₄₃	1	4	yes	—
146	C₂²⋊C₈	—	43	2	no	—
147	C₄×C₈	—	43	1	no	—
148	C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
149	#41	—	172	4	no	—
150	C₂×Q₈	—	2	1	no	—
151	C₂×Q₈	—	86	2	no	—
152	C₂×Q₈	—	86	1	no	—
153	C₂×Q₈	—	86	1	no	—
154	C₂×Q₈	—	172	4	no	—
155	C₂×D₄	—	2	1	no	—
156	C₂×D₄	—	2	1	no	—
157	C₂×D₄	—	86	2	no	—
158	C₂×D₄	C₂²×D₄₃	1	2	yes	—
159	C₂²×C₄	—	43	1	no	—
160	C₂²×C₄	—	43	1	no	—
161	C₂²×C₄	—	43	1	no	—
162	C₂²×C₄	—	86	2	no	—
163	C₂²×C₄	—	86	2	no	—
164	C₂×C₈	—	43	2	no	—
165	C₂×C₈	C₂²×D₄₃	1	2	yes	—
166	C₄⋊C₄	—	2	1	no	—
167	C₄⋊C₄	—	43	2	no	—
168	C₄⋊C₄	—	43	2	no	—
169	C₄⋊C₄	—	86	2	no	—
170	C₄⋊C₄	—	86	2	no	—
171	C₄⋊C₄	—	86	1	no	—
172	C₄⋊C₄	—	86	1	no	—
173	C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
174	C₂²⋊C₄	—	86	2	no	—
175	C₂²⋊C₄	—	86	2	no	—
176	C₄²	—	43	1	no	—
177	C₄²	—	43	1	no	—
178	C₄²	—	43	1	no	—
179	C₄²	C₂²×D₄₃	1	1	yes	characteristic
180	C₂³	—	2	2	no	—
181	C₂³	—	43	1	no	—
182	Q₈	—	4	2	no	—
183	Q₈	—	172	4	no	—
184	Q₈	—	172	2	no	—
185	Q₈	—	172	2	no	—
186	D₄	—	2	4	no	—
187	D₄	—	4	2	no	—
188	D₄	—	4	2	no	—
189	D₄	—	172	4	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	43	1	no	—
192	C₂×C₄	—	43	4	no	—
193	C₂×C₄	—	43	1	no	—
194	C₂×C₄	—	43	1	no	—
195	C₂×C₄	—	86	2	no	—
196	C₂×C₄	—	86	2	no	—
197	C₂×C₄	—	86	2	no	—
198	C₂×C₄	—	86	2	no	—
199	C₂×C₄	—	172	4	no	—
200	C₂×C₄	—	172	4	no	—
201	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
202	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
203	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
204	C₈	—	2	2	no	—
205	C₈	—	86	2	no	—
206	C₂²	—	4	2	no	—
207	C₂²	—	4	2	no	—
208	C₂²	—	4	2	no	—
209	C₂²	—	43	2	no	—
210	C₂²	—	86	1	no	—
211	C₂²	—	86	1	no	—
212	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
213	C₄	—	4	2	no	—
214	C₄	—	43	2	no	—
215	C₄	—	86	1	no	—
216	C₄	—	86	1	no	—
217	C₄	—	172	2	no	—
218	C₄	—	172	2	no	—
219	C₄	SD₁₆×D₄₃	1	2	yes	—
220	C₄	#13063349	1	2	yes	—
221	C₄	#13063372	1	2	yes	—
222	C₂	—	8	2	no	—
223	C₂	—	86	2	no	—
224	C₂	#14583469	1	1	yes	characteristic
225	C₂	C₂×SD₁₆×D₄₃	1	1	yes	characteristic
226	C₂	C₂×#13063349	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14583447

C₂

yes

maximal, characteristic

C₄⋊Q₈×D₄₃

C₂

yes

maximal, characteristic, complemented

#14583297

C₂

yes

maximal, characteristic, complemented

#14583295

C₂

yes

maximal, characteristic, complemented

#14583245

C₂

yes

maximal, characteristic, complemented

#14583216

C₂

yes

maximal, complemented

#14583187

C₂

yes

maximal, complemented

#14582950

C₂

yes

maximal, characteristic, complemented

C₄₃×#486

C₂

yes

maximal, characteristic, complemented

C₂×#10621900

—

C₂×Q₈×D₄₃

—

#13063325

C₂²

yes

characteristic

#13063325

C₂²

yes

characteristic

#13063324

—

#13063324

C₂²

yes

—

—

—

—

C₂²

yes

—

D₄×Dic₄₃

—

D₄×Dic₄₃

C₂²

yes

—

C₄⋊C₄×D₄₃

—

C₄⋊C₄×D₄₃

C₂²

yes

—

C₄²×D₄₃

C₂²

yes

characteristic

#13063257

C₂²

yes

—

#13063256

C₂²

yes

—

Dic₄₃⋊C₈

C₂²

yes

—

#13063246

C₂²

yes

—

(C₂×D₄₃)⋊C₈

C₂²

yes

—

C₄×C₄₃⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₄₃

C₂²

yes

characteristic

#127×C₄₃

C₂²

yes

characteristic

#102×C₄₃

C₂²

yes

—

C₄×C₈×C₄₃

C₂²

yes

characteristic

C₂×Q₈×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

C₂³

yes

—

C₂×C₂²⋊D₄₃

—

C₂²×Dic₄₃

—

Q₈×D₄₃

—

#10621900

—

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×Dic₈₆

—

C₂×C₈×C₄₃

C₂³

yes

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

C₂³

yes

—

C₄²×C₄₃

C₂³

yes

characteristic

C₂²⋊Dic₄₃

—

C₂²⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

C₂³

yes

—

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

—

C₂³×C₄₃

—

C₂²×D₄₃

C₂×D₄

yes

characteristic

Q₈×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂⁴

yes

characteristic

C₂²⋊D₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

C₂×D₄

yes

characteristic

C₂×Dic₄₃

C₂×D₄

yes

characteristic

C₂×Dic₄₃

C₂×D₄

yes

characteristic

C₄×D₄₃

—

C₄×D₄₃

—

C₄×D₄₃

—

C₄×D₄₃

SD₁₆

yes

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂²×D₄

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

C₂×SD₁₆

yes

—

C₄×C₄₃

—

100

C₄×C₄₃

C₂×SD₁₆

yes

—

101

C₄×C₄₃

AGL(1,ℤ/8)

yes

—

102

C₄×C₄₃

2₋¹⁺⁴

yes

—

103

Dic₄₃

—

104

Dic₄₃

—

105

Dic₄₃

—

106

Dic₄₃

—

107

Dic₄₃

C₂×SD₁₆

yes

—

108

#3227

—

maximal, Sylow

109

C₂×C₄₃

—

110

C₂×C₄₃

#536

yes

characteristic

111

C₂×C₄₃

C₂²×SD₁₆

yes

characteristic

112

C₂×C₄₃

C₂×AGL(1,ℤ/8)

yes

characteristic

113

D₄₃

—

114

#532

—

115

C₂×C₄⋊Q₈

—

116

#498

—

117

#498

—

118

#486

—

119

#486

D₄₃

yes

characteristic, complemented

120

#481

—

121

#476

—

122

#431

—

123

C₄₃

#3227

yes

characteristic, complemented, Sylow

124

C₂×#41

—

125

C₂²×Q₈

—

126

C₄⋊Q₈

—

127

C₄⋊Q₈

—

128

C₄⋊Q₈

—

129

C₄⋊Q₈

—

130

C₄⋊Q₈

—

131

C₄⋊Q₈

C₂×D₄₃

yes

characteristic

132

#127

C₂×D₄₃

yes

characteristic

133

#124

—

134

#121

—

135

#121

—

136

C₄×D₄

—

137

C₄×D₄

—

138

C₂×C₄⋊C₄

—

139

C₂×C₄⋊C₄

—

140

C₂×C₄²

—

141

#106

—

142

#106

—

143

C₄⋊C₈

—

144

#102

—

145

#102

C₂×D₄₃

yes

—

146

C₂²⋊C₈

—

147

C₄×C₈

—

148

C₄×C₈

C₂×D₄₃

yes

characteristic

149

#41

—

172

—

150

C₂×Q₈

—

151

C₂×Q₈

—

152

C₂×Q₈

—

153

C₂×Q₈

—

154

C₂×Q₈

—

172

—

155

C₂×D₄

—

156

C₂×D₄

—

157

C₂×D₄

—

158

C₂×D₄

C₂²×D₄₃

yes

—

159

C₂²×C₄

—

160

C₂²×C₄

—

161

C₂²×C₄

—

162

C₂²×C₄

—

163

C₂²×C₄

—

164

C₂×C₈

—

165

C₂×C₈

C₂²×D₄₃

yes

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

168

C₄⋊C₄

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

171

C₄⋊C₄

—

172

C₄⋊C₄

—

173

C₄⋊C₄

C₂²×D₄₃

yes

—

174

C₂²⋊C₄

—

175

C₂²⋊C₄

—

176

C₄²

—

177

C₄²

—

178

C₄²

—

179

C₄²

C₂²×D₄₃

yes

characteristic

180

C₂³

—

181

C₂³

—

182

Q₈

—

183

Q₈

—

172

—

184

Q₈

—

172

—

185

Q₈

—

172

—

186

D₄

—

187

D₄

—

188

D₄

—

189

D₄

—

172

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

—

172

—

200

C₂×C₄

—

172

—

201

C₂×C₄

D₄×D₄₃

yes

characteristic

202

C₂×C₄

D₄×D₄₃

yes

characteristic

203

C₂×C₄

C₂³×D₄₃

yes

characteristic

204

C₈

—

205

C₈

—

206

C₂²

—

207

C₂²

—

208

C₂²

—

209

C₂²

—

210

C₂²

—

211

C₂²

—

212

C₂²

C₂×D₄×D₄₃

yes

characteristic

213

C₄

—

214

C₄

—

215

C₄

—

216

C₄

—

217

C₄

—

172

—

218

C₄

—

172

—

219

C₄

SD₁₆×D₄₃

yes

—

220

C₄

#13063349

yes

—

221

C₄

#13063372

yes

—

222

C₂

—

223

C₂

—

224

C₂

#14583469

yes

characteristic

225

C₂

C₂×SD₁₆×D₄₃

yes

characteristic

226

C₂

C₂×#13063349

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#486	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#486	D₄₃
hypocenter	C₄₃	#3227
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #486

subgroup

G/H

center

C₂²

C₂×D₄×D₄₃

commutator

C₂×C₄×C₄₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₃

Fitting

C₄₃×#486

C₂

socle

C₂²×C₄₃

C₂²×D₄

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #486

N	H ≅ G/N	type
2. C₄⋊Q₈×D₄₃	C₂	semidirect
3. #14583297	C₂	semidirect
4. #14583295	C₂	semidirect
5. #14583245	C₂	semidirect
6. #14583216	C₂	semidirect
7. #14583187	C₂	semidirect
8. #14582950	C₂	semidirect
9. C₄₃×#486	C₂	semidirect
119. #486	D₄₃	semidirect
123. C₄₃	#3227	semidirect

H ≅ G/N

type

2. C₄⋊Q₈×D₄₃

C₂

semidirect

3. #14583297

C₂

semidirect

4. #14583295

C₂

semidirect

5. #14583245

C₂

semidirect

6. #14583216

C₂

semidirect

7. #14583187

C₂

semidirect

8. #14582950

C₂

semidirect

9. C₄₃×#486

C₂

semidirect

119. #486

D₄₃

semidirect

123. C₄₃

#3227

semidirect

Subgroups — #19145719

Conjugacy classes of subgroups

Complements