Group #19145997 — subgroups

Subgroups — #19145997

Order 5504.

354 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 109 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	#14583435	C₂	1	2	yes	maximal, complemented
2	#14583204	C₂	1	2	yes	maximal, complemented
3	#14583198	C₂	1	2	yes	maximal, complemented
4	#14583152	C₂	1	1	yes	maximal, characteristic
5	#14583147	C₂	1	2	yes	maximal, complemented
6	#14583123	C₂	1	2	yes	maximal, complemented
7	#14583117	C₂	1	2	yes	maximal, complemented
8	#14582922	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₃×#470	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#10621884	C₂²	1	2	yes	—
11	C₂×D₃₄₄	C₂²	1	1	yes	characteristic
12	C₂×#10621877	C₂²	1	1	yes	characteristic
13	#13063320	—	2	2	no	—
14	#13063316	C₂²	1	2	yes	—
15	#13063314	C₂²	1	2	yes	—
16	#13063309	—	2	2	no	—
17	#13063299	C₂²	1	2	yes	—
18	Q₈×Dic₄₃	C₂²	1	2	yes	—
19	#13063287	C₂²	1	2	yes	—
20	(C₂²⋊D₄₃)⋊C₄	—	2	2	no	—
21	#13063283	—	2	2	no	—
22	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
23	#13063258	C₂²	1	2	yes	—
24	#13063257	C₂²	1	1	yes	characteristic
25	Dic₄₃⋊C₈	C₂²	1	2	yes	—
26	Dic₄₃⋊C₈	C₂²	1	2	yes	—
27	#13063249	C₂²	1	2	yes	—
28	#13063247	C₂²	1	2	yes	—
29	#13063245	C₂²	1	2	yes	—
30	C₂×Q₁₆×C₄₃	C₂²	1	1	yes	characteristic
31	C₂×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
32	C₂²⋊Q₈×C₄₃	C₂²	1	2	yes	—
33	#106×C₄₃	C₂²	1	1	yes	characteristic
34	#103×C₄₃	C₂²	1	2	yes	—
35	C₂×Q₈×C₄₃	C₂³	1	2	yes	—
36	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
37	C₂×C₂²⋊D₄₃	—	2	2	no	—
38	C₂²×Dic₄₃	D₄	1	2	yes	—
39	C₂×D₁₇₂	C₂³	1	2	yes	—
40	C₂×C₄×D₄₃	—	2	2	no	—
41	Q₁₆×C₄₃	—	4	1	no	—
42	M₁₆×C₄₃	—	4	1	no	—
43	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
45	C₄⋊C₄×C₄₃	—	2	2	no	—
46	C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
47	C₂²⋊C₄×C₄₃	—	2	2	no	—
48	#10621884	—	4	2	no	—
49	(C₂×D₄₃)⋊C₄	—	2	2	no	—
50	(C₂×D₄₃)⋊C₄	—	2	2	no	—
51	(C₂×D₄₃)⋊C₄	—	2	2	no	—
52	(C₂×D₄₃)⋊C₄	—	2	2	no	—
53	C₄⋊Dic₄₃	—	2	2	no	—
54	C₄⋊Dic₄₃	—	2	2	no	—
55	C₄⋊Dic₄₃	C₂³	1	2	yes	—
56	Dic₄₃⋊C₄	—	2	2	no	—
57	C₄×Dic₄₃	—	2	2	no	—
58	C₄×Dic₄₃	D₄	1	1	yes	characteristic
59	C₄×Dic₄₃	D₄	1	1	yes	characteristic
60	C₄×Dic₄₃	C₂³	1	2	yes	—
61	#10621877	—	4	1	no	—
62	C₂×C₄₃⋊C₈	C₂³	1	2	yes	—
63	D₃₄₄	—	4	1	no	—
64	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×D₄₃	—	2	2	no	—
66	Q₈×C₄₃	—	2	2	no	—
67	Q₈×C₄₃	—	2	2	no	—
68	C₂×C₄×C₄₃	—	2	1	no	—
69	C₂×C₄×C₄₃	—	2	1	no	—
70	C₂×C₄×C₄₃	—	2	2	no	—
71	C₂×C₄×C₄₃	—	2	2	no	—
72	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
74	C₂²⋊D₄₃	—	4	4	no	—
75	C₂×Dic₄₃	—	2	2	no	—
76	C₂×Dic₄₃	—	4	2	no	—
77	C₂×Dic₄₃	C₂×D₄	1	2	yes	—
78	C₂×Dic₄₃	C₂×D₄	1	2	yes	—
79	D₁₇₂	—	2	2	no	—
80	D₁₇₂	—	2	2	no	—
81	C₄×D₄₃	—	4	4	no	—
82	C₈×C₄₃	—	2	1	no	—
83	C₈×C₄₃	—	2	1	no	—
84	C₄₃⋊C₈	—	2	2	no	—
85	C₂²×C₄₃	—	2	1	no	—
86	C₂²×C₄₃	—	2	2	no	—
87	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
88	C₂×D₄₃	—	4	2	no	—
89	C₂×D₄₃	—	4	2	no	—
90	C₂×D₄₃	—	4	2	no	—
91	C₄×C₄₃	—	2	1	no	—
92	C₄×C₄₃	—	4	2	no	—
93	C₄×C₄₃	—	4	2	no	—
94	C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
95	C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
96	Dic₄₃	—	2	2	no	—
97	Dic₄₃	—	2	2	no	—
98	Dic₄₃	—	4	2	no	—
99	#3256	—	43	1	no	maximal, Sylow
100	C₂×C₄₃	—	4	1	no	—
101	C₂×C₄₃	#537	1	1	yes	characteristic
102	C₂×C₄₃	#575	1	2	yes	—
103	D₄₃	—	8	2	no	—
104	#529	—	43	2	no	—
105	#479	—	43	2	no	—
106	#478	—	43	2	no	—
107	#470	D₄₃	1	1	yes	characteristic, complemented
108	#469	—	43	1	no	—
109	#468	—	43	2	no	—
110	#464	—	43	2	no	—
111	#463	—	43	2	no	—
112	#423	—	43	1	no	—
113	C₄₃	#3256	1	1	yes	characteristic, complemented, Sylow
114	C₂×Q₁₆	C₂×D₄₃	1	1	yes	characteristic
115	C₂×SD₁₆	—	43	2	no	—
116	C₂×D₈	—	43	1	no	—
117	C₂×M₁₆	—	43	1	no	—
118	C₂×M₁₆	C₂×D₄₃	1	1	yes	characteristic
119	#126	—	86	2	no	—
120	#125	—	43	2	no	—
121	#124	—	43	2	no	—
122	#123	—	86	2	no	—
123	C₂²⋊Q₈	C₂×D₄₃	1	2	yes	—
124	#121	—	43	2	no	—
125	C₄×Q₈	—	43	2	no	—
126	C₄×D₄	—	43	2	no	—
127	C₄×D₄	—	86	2	no	—
128	#117	—	86	2	no	—
129	C₂×C₄²	—	43	1	no	—
130	#107	—	43	2	no	—
131	#106	—	43	1	no	—
132	#106	C₂×D₄₃	1	1	yes	characteristic
133	C₄⋊C₈	—	43	2	no	—
134	C₄⋊C₈	—	43	2	no	—
135	#103	—	43	2	no	—
136	#103	C₂×D₄₃	1	2	yes	—
137	#102	—	43	2	no	—
138	#102	—	43	2	no	—
139	C₂×Q₈	C₂²×D₄₃	1	2	yes	—
140	C₂×D₄	—	43	2	no	—
141	C₂×D₄	—	86	2	no	—
142	C₂²×C₄	—	43	2	no	—
143	C₂²×C₄	—	86	2	no	—
144	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
145	Q₁₆	—	4	1	no	—
146	SD₁₆	—	172	2	no	—
147	D₈	—	172	1	no	—
148	M₁₆	—	4	1	no	—
149	M₁₆	—	172	1	no	—
150	C₂×C₈	—	43	2	no	—
151	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
152	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
153	C₄⋊C₄	—	2	2	no	—
154	C₄⋊C₄	—	43	2	no	—
155	C₄⋊C₄	—	86	2	no	—
156	C₄⋊C₄	—	86	2	no	—
157	C₄⋊C₄	—	86	2	no	—
158	C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
159	C₂²⋊C₄	—	2	2	no	—
160	C₂²⋊C₄	—	86	2	no	—
161	C₂²⋊C₄	—	86	2	no	—
162	C₂²⋊C₄	—	86	2	no	—
163	C₂²⋊C₄	—	86	2	no	—
164	C₄²	—	43	1	no	—
165	C₄²	—	43	1	no	—
166	C₄²	—	43	2	no	—
167	C₄²	—	86	2	no	—
168	C₂³	—	86	2	no	—
169	C₂³	D₄×D₄₃	1	1	yes	characteristic
170	Q₈	—	2	2	no	—
171	Q₈	—	2	2	no	—
172	D₄	—	86	2	no	—
173	D₄	—	86	2	no	—
174	D₄	—	172	4	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	43	2	no	—
180	C₂×C₄	—	43	2	no	—
181	C₂×C₄	—	86	2	no	—
182	C₂×C₄	—	172	4	no	—
183	C₂×C₄	—	172	2	no	—
184	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
185	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
186	C₈	—	2	1	no	—
187	C₈	—	2	1	no	—
188	C₈	—	86	2	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	2	1	no	—
191	C₂²	—	172	2	no	—
192	C₂²	—	172	2	no	—
193	C₂²	—	172	2	no	—
194	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
195	C₄	—	2	1	no	—
196	C₄	—	4	2	no	—
197	C₄	—	4	2	no	—
198	C₄	—	86	2	no	—
199	C₄	—	86	2	no	—
200	C₄	—	172	2	no	—
201	C₄	#13063370	1	1	yes	characteristic
202	C₄	#13063372	1	1	yes	characteristic
203	C₂	—	4	1	no	—
204	C₂	—	344	2	no	—
205	C₂	#14583476	1	1	yes	characteristic
206	C₂	#14583698	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14583435

C₂

yes

maximal, complemented

#14583204

C₂

yes

maximal, complemented

#14583198

C₂

yes

maximal, complemented

#14583152

C₂

yes

maximal, characteristic

#14583147

C₂

yes

maximal, complemented

#14583123

C₂

yes

maximal, complemented

#14583117

C₂

yes

maximal, complemented

#14582922

C₂

yes

maximal, characteristic, complemented

C₄₃×#470

C₂

yes

maximal, characteristic, complemented

C₂×#10621884

C₂²

yes

—

C₂×D₃₄₄

C₂²

yes

characteristic

C₂×#10621877

C₂²

yes

characteristic

#13063320

—

#13063316

C₂²

yes

—

#13063314

C₂²

yes

—

#13063309

—

#13063299

C₂²

yes

—

Q₈×Dic₄₃

C₂²

yes

—

#13063287

C₂²

yes

—

(C₂²⋊D₄₃)⋊C₄

—

#13063283

—

C₂×C₄×Dic₄₃

C₂²

yes

characteristic

#13063258

C₂²

yes

—

#13063257

C₂²

yes

characteristic

Dic₄₃⋊C₈

C₂²

yes

—

Dic₄₃⋊C₈

C₂²

yes

—

#13063249

C₂²

yes

—

#13063247

C₂²

yes

—

#13063245

C₂²

yes

—

C₂×Q₁₆×C₄₃

C₂²

yes

characteristic

C₂×M₁₆×C₄₃

C₂²

yes

characteristic

C₂²⋊Q₈×C₄₃

C₂²

yes

—

#106×C₄₃

C₂²

yes

characteristic

#103×C₄₃

C₂²

yes

—

C₂×Q₈×C₄₃

C₂³

yes

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂×C₂²⋊D₄₃

—

C₂²×Dic₄₃

D₄

yes

—

C₂×D₁₇₂

C₂³

yes

—

C₂×C₄×D₄₃

—

Q₁₆×C₄₃

—

M₁₆×C₄₃

—

C₂×C₈×C₄₃

C₂³

yes

characteristic

C₂×C₈×C₄₃

C₂³

yes

characteristic

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

C₂³

yes

—

C₂²⋊C₄×C₄₃

—

#10621884

—

(C₂×D₄₃)⋊C₄

—

(C₂×D₄₃)⋊C₄

—

(C₂×D₄₃)⋊C₄

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

C₂³

yes

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₄×Dic₄₃

D₄

yes

characteristic

C₄×Dic₄₃

D₄

yes

characteristic

C₄×Dic₄₃

C₂³

yes

—

#10621877

—

C₂×C₄₃⋊C₈

C₂³

yes

—

D₃₄₄

—

C₂³×C₄₃

C₂×D₄

yes

characteristic

C₂²×D₄₃

—

Q₈×C₄₃

—

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂⁴

yes

characteristic

C₂²⋊D₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

C₂×D₄

yes

—

C₂×Dic₄₃

C₂×D₄

yes

—

D₁₇₂

—

D₁₇₂

—

C₄×D₄₃

—

C₈×C₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂²×D₄

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

H(5,𝔽₂)

yes

characteristic

C₄×C₄₃

2₋¹⁺⁴

yes

characteristic

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

#3256

—

maximal, Sylow

100

C₂×C₄₃

—

101

C₂×C₄₃

#537

yes

characteristic

102

C₂×C₄₃

#575

yes

—

103

D₄₃

—

104

#529

—

105

#479

—

106

#478

—

107

#470

D₄₃

yes

characteristic, complemented

108

#469

—

109

#468

—

110

#464

—

111

#463

—

112

#423

—

113

C₄₃

#3256

yes

characteristic, complemented, Sylow

114

C₂×Q₁₆

C₂×D₄₃

yes

characteristic

115

C₂×SD₁₆

—

116

C₂×D₈

—

117

C₂×M₁₆

—

118

C₂×M₁₆

C₂×D₄₃

yes

characteristic

119

#126

—

120

#125

—

121

#124

—

122

#123

—

123

C₂²⋊Q₈

C₂×D₄₃

yes

—

124

#121

—

125

C₄×Q₈

—

126

C₄×D₄

—

127

C₄×D₄

—

128

#117

—

129

C₂×C₄²

—

130

#107

—

131

#106

—

132

#106

C₂×D₄₃

yes

characteristic

133

C₄⋊C₈

—

134

C₄⋊C₈

—

135

#103

—

136

#103

C₂×D₄₃

yes

—

137

#102

—

138

#102

—

139

C₂×Q₈

C₂²×D₄₃

yes

—

140

C₂×D₄

—

141

C₂×D₄

—

142

C₂²×C₄

—

143

C₂²×C₄

—

144

C₂²×C₄

C₂²×D₄₃

yes

characteristic

145

Q₁₆

—

146

SD₁₆

—

172

—

147

D₈

—

172

—

148

M₁₆

—

149

M₁₆

—

172

—

150

C₂×C₈

—

151

C₂×C₈

C₂²×D₄₃

yes

characteristic

152

C₂×C₈

C₂²×D₄₃

yes

characteristic

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

C₂²×D₄₃

yes

—

159

C₂²⋊C₄

—

160

C₂²⋊C₄

—

161

C₂²⋊C₄

—

162

C₂²⋊C₄

—

163

C₂²⋊C₄

—

164

C₄²

—

165

C₄²

—

166

C₄²

—

167

C₄²

—

168

C₂³

—

169

C₂³

D₄×D₄₃

yes

characteristic

170

Q₈

—

171

Q₈

—

172

D₄

—

173

D₄

—

174

D₄

—

172

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

172

—

183

C₂×C₄

—

172

—

184

C₂×C₄

D₄×D₄₃

yes

characteristic

185

C₂×C₄

C₂³×D₄₃

yes

characteristic

186

C₈

—

187

C₈

—

188

C₈

—

189

C₂²

—

190

C₂²

—

191

C₂²

—

172

—

192

C₂²

—

172

—

193

C₂²

—

172

—

194

C₂²

C₂×D₄×D₄₃

yes

characteristic

195

C₄

—

196

C₄

—

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

—

172

—

201

C₄

#13063370

yes

characteristic

202

C₄

#13063372

yes

characteristic

203

C₂

—

204

C₂

—

344

—

205

C₂

#14583476

yes

characteristic

206

C₂

#14583698

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#470	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#470	D₄₃
hypocenter	C₄₃	#3256
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #470

subgroup

G/H

center

C₂²

C₂×D₄×D₄₃

commutator

C₂×C₄×C₄₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₃

Fitting

C₄₃×#470

C₂

socle

C₂²×C₄₃

C₂²×D₄

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #470

N	H ≅ G/N	type
1. #14583435	C₂	semidirect
2. #14583204	C₂	semidirect
3. #14583198	C₂	semidirect
5. #14583147	C₂	semidirect
6. #14583123	C₂	semidirect
7. #14583117	C₂	semidirect
8. #14582922	C₂	semidirect
9. C₄₃×#470	C₂	semidirect
107. #470	D₄₃	semidirect
113. C₄₃	#3256	semidirect

H ≅ G/N

type

1. #14583435

C₂

semidirect

2. #14583204

C₂

semidirect

3. #14583198

C₂

semidirect

5. #14583147

C₂

semidirect

6. #14583123

C₂

semidirect

7. #14583117

C₂

semidirect

8. #14582922

C₂

semidirect

9. C₄₃×#470

C₂

semidirect

107. #470

D₄₃

semidirect

113. C₄₃

#3256

semidirect

Subgroups — #19145997

Conjugacy classes of subgroups

Complements