Group #19144718 — subgroups

Subgroups — #19144718

Order 5504.

424 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 129 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	#14583449	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₈₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583213	C₂	1	2	yes	maximal, complemented
4	#14583195	C₂	1	2	yes	maximal, complemented
5	#14583100	C₂	1	2	yes	maximal, complemented
6	#14583040	C₂	1	2	yes	maximal, complemented
7	#14583001	C₂	1	4	yes	maximal, complemented
8	C₄₃×#432	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#10621898	—	2	1	no	—
10	C₂²×Dic₈₆	C₂²	1	1	yes	characteristic
11	C₂×#10621873	C₂²	1	4	yes	—
12	#13063325	C₂²	1	1	yes	characteristic
13	#13063322	C₂²	1	1	yes	characteristic
14	#13063313	—	2	1	no	—
15	#13063299	—	2	1	no	—
16	#13063299	C₂²	1	2	yes	—
17	C₄×Dic₈₆	—	2	1	no	—
18	C₄×Dic₈₆	—	2	1	no	—
19	C₄×Dic₈₆	C₂²	1	2	yes	—
20	C₄×Dic₈₆	C₂²	1	2	yes	—
21	C₄×D₁₇₂	—	2	1	no	—
22	C₄×D₁₇₂	C₂²	1	2	yes	—
23	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
24	C₂×Dic₄₃⋊C₄	—	2	1	no	—
25	C₂×C₄×Dic₄₃	—	2	1	no	—
26	#13063259	C₂²	1	4	yes	—
27	#13063250	C₂²	1	4	yes	—
28	#13063247	C₂²	1	4	yes	—
29	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
30	C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
31	C₂²⋊C₈×C₄₃	C₂²	1	2	yes	—
32	#97×C₄₃	C₂²	1	2	yes	—
33	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
36	C₂×C₂²⋊D₄₃	—	2	2	no	—
37	C₂²×Dic₄₃	—	2	1	no	—
38	C₂²×Dic₄₃	—	2	1	no	—
39	#10621898	—	4	4	no	—
40	C₂×D₁₇₂	—	2	1	no	—
41	C₂×D₁₇₂	—	2	1	no	—
42	C₂×D₁₇₂	C₂³	1	2	yes	—
43	C₂×C₄×D₄₃	—	2	2	no	—
44	C₂×Dic₈₆	—	2	1	no	—
45	C₂×Dic₈₆	—	2	4	no	—
46	C₂×Dic₈₆	—	2	1	no	—
47	C₂×Dic₈₆	C₂³	1	2	yes	—
48	C₂×C₈×C₄₃	C₂³	1	4	yes	—
49	C₄²×C₄₃	D₄	1	1	yes	characteristic
50	C₄²×C₄₃	D₄	1	1	yes	characteristic
51	C₄²×C₄₃	C₂³	1	1	yes	characteristic
52	C₄²×C₄₃	C₂³	1	1	yes	characteristic
53	(C₂×D₄₃)⋊C₄	—	2	2	no	—
54	(C₂×D₄₃)⋊C₄	—	2	2	no	—
55	C₄⋊Dic₄₃	—	2	1	no	—
56	C₄⋊Dic₄₃	—	2	1	no	—
57	C₄⋊Dic₄₃	C₂³	1	2	yes	—
58	C₄⋊Dic₄₃	C₂³	1	2	yes	—
59	Dic₄₃⋊C₄	—	2	2	no	—
60	Dic₄₃⋊C₄	—	2	2	no	—
61	C₄×Dic₄₃	—	2	2	no	—
62	C₄×Dic₄₃	—	2	2	no	—
63	#10621873	—	2	8	no	—
64	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×D₄₃	—	2	2	no	—
66	C₂×C₄×C₄₃	—	2	2	no	—
67	C₂×C₄×C₄₃	—	2	2	no	—
68	C₂×C₄×C₄₃	—	2	2	no	—
69	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
75	C₂²⋊D₄₃	—	4	4	no	—
76	C₂×Dic₄₃	—	2	4	no	—
77	C₂×Dic₄₃	—	2	2	no	—
78	C₂×Dic₄₃	—	2	2	no	—
79	C₂×Dic₄₃	—	2	2	no	—
80	C₂×Dic₄₃	—	4	2	no	—
81	D₁₇₂	—	2	4	no	—
82	D₁₇₂	—	4	2	no	—
83	D₁₇₂	—	4	2	no	—
84	C₄×D₄₃	—	4	4	no	—
85	Dic₈₆	—	2	4	no	—
86	Dic₈₆	—	2	2	no	—
87	Dic₈₆	—	4	2	no	—
88	Dic₈₆	#41	1	4	yes	complemented
89	C₈×C₄₃	—	2	4	no	—
90	C₂²×C₄₃	—	2	1	no	—
91	C₂²×C₄₃	—	2	1	no	—
92	C₂²×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
93	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
94	C₂×D₄₃	—	4	2	no	—
95	C₂×D₄₃	—	4	2	no	—
96	C₂×D₄₃	—	4	2	no	—
97	C₄×C₄₃	—	2	1	no	—
98	C₄×C₄₃	—	2	1	no	—
99	C₄×C₄₃	—	2	1	no	—
100	C₄×C₄₃	—	2	1	no	—
101	C₄×C₄₃	#137	1	2	yes	—
102	C₄×C₄₃	C₂×#41	1	2	yes	—
103	Dic₄₃	—	2	4	no	—
104	Dic₄₃	—	4	2	no	—
105	Dic₄₃	—	4	2	no	—
106	#3112	—	43	1	no	maximal, Sylow
107	C₂×C₄₃	—	2	2	no	—
108	C₂×C₄₃	#525	1	1	yes	characteristic
109	C₂×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
110	C₂×C₄₃	C₂×#137	1	1	yes	characteristic
111	D₄₃	—	8	2	no	—
112	#532	—	43	1	no	—
113	C₂×C₄×Q₈	—	43	1	no	—
114	#480	—	43	2	no	—
115	#477	—	43	2	no	—
116	#460	—	43	2	no	—
117	#448	—	43	2	no	—
118	#440	—	43	4	no	—
119	#432	D₄₃	1	1	yes	characteristic, complemented
120	C₄₃	#3112	1	1	yes	characteristic, complemented, Sylow
121	C₂×#41	—	86	1	no	—
122	C₂²×Q₈	—	43	1	no	—
123	C₂×SD₁₆	—	43	4	no	—
124	C₄⋊Q₈	—	43	1	no	—
125	#127	—	43	1	no	—
126	#124	—	86	1	no	—
127	#121	—	43	2	no	—
128	#121	—	86	1	no	—
129	C₄×Q₈	—	43	2	no	—
130	C₄×Q₈	—	43	2	no	—
131	C₄×Q₈	—	86	1	no	—
132	C₄×Q₈	—	86	1	no	—
133	C₄×D₄	—	43	2	no	—
134	C₄×D₄	—	86	1	no	—
135	C₂×C₄⋊C₄	—	43	1	no	—
136	C₂×C₄⋊C₄	—	86	1	no	—
137	C₂×C₄²	—	86	1	no	—
138	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
139	#107	—	43	4	no	—
140	C₄⋊C₈	C₂×D₄₃	1	2	yes	—
141	#103	—	43	4	no	—
142	#102	—	43	4	no	—
143	C₂²⋊C₈	C₂×D₄₃	1	2	yes	—
144	#97	C₂×D₄₃	1	2	yes	—
145	#41	—	172	4	no	—
146	C₂×Q₈	—	43	2	no	—
147	C₂×Q₈	—	86	4	no	—
148	C₂×Q₈	—	86	1	no	—
149	C₂×Q₈	—	86	1	no	—
150	C₂×D₄	—	43	2	no	—
151	C₂×D₄	—	86	2	no	—
152	C₂×D₄	—	86	1	no	—
153	C₂×D₄	—	86	1	no	—
154	C₂²×C₄	—	86	1	no	—
155	C₂²×C₄	—	86	1	no	—
156	C₂²×C₄	—	86	2	no	—
157	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
158	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
159	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
160	SD₁₆	—	86	8	no	—
161	C₂×C₈	C₂²×D₄₃	1	4	yes	—
162	C₄⋊C₄	—	43	2	no	—
163	C₄⋊C₄	—	43	2	no	—
164	C₄⋊C₄	—	86	1	no	—
165	C₄⋊C₄	—	86	2	no	—
166	C₄⋊C₄	—	86	2	no	—
167	C₄⋊C₄	—	86	1	no	—
168	C₂²⋊C₄	—	86	2	no	—
169	C₂²⋊C₄	—	86	2	no	—
170	C₄²	—	86	2	no	—
171	C₄²	—	86	2	no	—
172	C₄²	D₁₇₂	1	1	yes	characteristic
173	C₄²	D₁₇₂	1	1	yes	characteristic
174	C₄²	C₂²×D₄₃	1	1	yes	characteristic
175	C₄²	C₂²×D₄₃	1	1	yes	characteristic
176	C₂³	—	86	2	no	—
177	C₂³	C₂×D₁₇₂	1	1	yes	characteristic
178	Q₈	—	43	4	no	—
179	Q₈	—	86	4	no	—
180	Q₈	—	86	2	no	—
181	Q₈	—	172	2	no	—
182	D₄	—	86	4	no	—
183	D₄	—	172	2	no	—
184	D₄	—	172	4	no	—
185	D₄	—	172	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	86	4	no	—
190	C₂×C₄	—	86	2	no	—
191	C₂×C₄	—	86	2	no	—
192	C₂×C₄	—	86	2	no	—
193	C₂×C₄	—	172	4	no	—
194	C₂×C₄	—	172	2	no	—
195	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
196	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
197	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
198	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
199	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
200	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
201	C₈	—	2	4	no	—
202	C₂²	—	2	1	no	—
203	C₂²	—	2	1	no	—
204	C₂²	—	172	2	no	—
205	C₂²	—	172	2	no	—
206	C₂²	—	172	2	no	—
207	C₂²	#13063353	1	2	yes	—
208	C₂²	C₂²×D₁₇₂	1	1	yes	characteristic
209	C₄	—	2	1	no	—
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	86	4	no	—
214	C₄	—	172	2	no	—
215	C₄	—	172	2	no	—
216	C₄	#13063358	1	2	yes	—
217	C₄	#41×D₄₃	1	2	yes	—
218	C₂	—	2	2	no	—
219	C₂	—	344	2	no	—
220	C₂	#14583411	1	1	yes	characteristic
221	C₂	C₂×#13063353	1	1	yes	characteristic
222	C₂	C₂×#13063358	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14583449

C₂

yes

maximal, characteristic

C₂×C₄×Dic₈₆

C₂

yes

maximal, characteristic, complemented

#14583213

C₂

yes

maximal, complemented

#14583195

C₂

yes

maximal, complemented

#14583100

C₂

yes

maximal, complemented

#14583040

C₂

yes

maximal, complemented

#14583001

C₂

yes

maximal, complemented

C₄₃×#432

C₂

yes

maximal, characteristic, complemented

C₂×#10621898

—

C₂²×Dic₈₆

C₂²

yes

characteristic

C₂×#10621873

C₂²

yes

—

#13063325

C₂²

yes

characteristic

#13063322

C₂²

yes

—

—

C₂²

yes

—

C₄×Dic₈₆

—

C₄×Dic₈₆

—

C₄×Dic₈₆

C₂²

yes

—

C₄×Dic₈₆

C₂²

yes

—

C₄×D₁₇₂

—

C₄×D₁₇₂

C₂²

yes

—

C₂×C₄⋊Dic₄₃

C₂²

yes

characteristic

C₂×Dic₄₃⋊C₄

—

C₂×C₄×Dic₄₃

—

#13063259

C₂²

yes

—

#13063250

C₂²

yes

—

#13063247

C₂²

yes

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₄⋊C₈×C₄₃

C₂²

yes

—

C₂²⋊C₈×C₄₃

C₂²

yes

—

#97×C₄₃

C₂²

yes

—

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂×C₂²⋊D₄₃

—

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

#10621898

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

C₂³

yes

—

C₂×C₄×D₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

C₂³

yes

—

C₂×C₈×C₄₃

C₂³

yes

—

C₄²×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

C₂³

yes

characteristic

C₄²×C₄₃

C₂³

yes

characteristic

(C₂×D₄₃)⋊C₄

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

C₂³

yes

—

C₄⋊Dic₄₃

C₂³

yes

—

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₄×Dic₄₃

—

#10621873

—

C₂³×C₄₃

C₂×D₄

yes

characteristic

C₂²×D₄₃

—

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

C₂×D₄

yes

characteristic

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×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

C₄×D₄₃

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

#41

yes

complemented

C₈×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

AGL(1,ℤ/8)

yes

—

C₂²×C₄₃

C₂²×D₄

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

100

C₄×C₄₃

—

101

C₄×C₄₃

#137

yes

—

102

C₄×C₄₃

C₂×#41

yes

—

103

Dic₄₃

—

104

Dic₄₃

—

105

Dic₄₃

—

106

#3112

—

maximal, Sylow

107

C₂×C₄₃

—

108

C₂×C₄₃

#525

yes

characteristic

109

C₂×C₄₃

C₂×AGL(1,ℤ/8)

yes

characteristic

110

C₂×C₄₃

C₂×#137

yes

characteristic

111

D₄₃

—

112

#532

—

113

C₂×C₄×Q₈

—

114

#480

—

115

#477

—

116

#460

—

117

#448

—

118

#440

—

119

#432

D₄₃

yes

characteristic, complemented

120

C₄₃

#3112

yes

characteristic, complemented, Sylow

121

C₂×#41

—

122

C₂²×Q₈

—

123

C₂×SD₁₆

—

124

C₄⋊Q₈

—

125

#127

—

126

#124

—

127

#121

—

128

#121

—

129

C₄×Q₈

—

130

C₄×Q₈

—

131

C₄×Q₈

—

132

C₄×Q₈

—

133

C₄×D₄

—

134

C₄×D₄

—

135

C₂×C₄⋊C₄

—

136

C₂×C₄⋊C₄

—

137

C₂×C₄²

—

138

C₂×C₄²

C₂×D₄₃

yes

characteristic

139

#107

—

140

C₄⋊C₈

C₂×D₄₃

yes

—

141

#103

—

142

#102

—

143

C₂²⋊C₈

C₂×D₄₃

yes

—

144

#97

C₂×D₄₃

yes

—

145

#41

—

172

—

146

C₂×Q₈

—

147

C₂×Q₈

—

148

C₂×Q₈

—

149

C₂×Q₈

—

150

C₂×D₄

—

151

C₂×D₄

—

152

C₂×D₄

—

153

C₂×D₄

—

154

C₂²×C₄

—

155

C₂²×C₄

—

156

C₂²×C₄

—

157

C₂²×C₄

D₁₇₂

yes

characteristic

158

C₂²×C₄

D₁₇₂

yes

characteristic

159

C₂²×C₄

C₂²×D₄₃

yes

characteristic

160

SD₁₆

—

161

C₂×C₈

C₂²×D₄₃

yes

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

168

C₂²⋊C₄

—

169

C₂²⋊C₄

—

170

C₄²

—

171

C₄²

—

172

C₄²

D₁₇₂

yes

characteristic

173

C₄²

D₁₇₂

yes

characteristic

174

C₄²

C₂²×D₄₃

yes

characteristic

175

C₄²

C₂²×D₄₃

yes

characteristic

176

C₂³

—

177

C₂³

C₂×D₁₇₂

yes

characteristic

178

Q₈

—

179

Q₈

—

180

Q₈

—

181

Q₈

—

172

—

182

D₄

—

183

D₄

—

172

—

184

D₄

—

172

—

185

D₄

—

172

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

172

—

194

C₂×C₄

—

172

—

195

C₂×C₄

C₂×D₁₇₂

yes

characteristic

196

C₂×C₄

C₂×D₁₇₂

yes

characteristic

197

C₂×C₄

C₂×D₁₇₂

yes

characteristic

198

C₂×C₄

C₂×D₁₇₂

yes

characteristic

199

C₂×C₄

C₂×D₁₇₂

yes

characteristic

200

C₂×C₄

C₂³×D₄₃

yes

characteristic

201

C₈

—

202

C₂²

—

203

C₂²

—

204

C₂²

—

172

—

205

C₂²

—

172

—

206

C₂²

—

172

—

207

C₂²

#13063353

yes

—

208

C₂²

C₂²×D₁₇₂

yes

characteristic

209

C₄

—

210

C₄

—

211

C₄

—

212

C₄

—

213

C₄

—

214

C₄

—

172

—

215

C₄

—

172

—

216

C₄

#13063358

yes

—

217

C₄

#41×D₄₃

yes

—

218

C₂

—

219

C₂

—

344

—

220

C₂

#14583411

yes

characteristic

221

C₂

C₂×#13063353

yes

characteristic

222

C₂

C₂×#13063358

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#432	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#432	D₄₃
hypocenter	C₄₃	#3112
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

subgroup

G/H

center

C₂²

C₂²×D₁₇₂

commutator

C₂×C₄×C₄₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₃

Fitting

C₄₃×#432

C₂

socle

C₂²×C₄₃

C₂²×D₄

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

N	H ≅ G/N	type
2. C₂×C₄×Dic₈₆	C₂	semidirect
3. #14583213	C₂	semidirect
4. #14583195	C₂	semidirect
5. #14583100	C₂	semidirect
6. #14583040	C₂	semidirect
7. #14583001	C₂	semidirect
8. C₄₃×#432	C₂	semidirect
88. Dic₈₆	#41	semidirect
119. #432	D₄₃	semidirect
120. C₄₃	#3112	semidirect

H ≅ G/N

type

2. C₂×C₄×Dic₈₆

C₂

semidirect

3. #14583213

C₂

semidirect

4. #14583195

C₂

semidirect

5. #14583100

C₂

semidirect

6. #14583040

C₂

semidirect

7. #14583001

C₂

semidirect

8. C₄₃×#432

C₂

semidirect

88. Dic₈₆

#41

semidirect

119. #432

D₄₃

semidirect

120. C₄₃

#3112

semidirect

Subgroups — #19144718

Conjugacy classes of subgroups

Complements