Group #19133035 — subgroups

Subgroups — #19133035

Order 5504.

266 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 81 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	C₄×#10621898	C₂	1	1	yes	maximal, characteristic
2	#14582923	C₂	1	1	yes	maximal, characteristic, complemented
3	#14582631	C₂	1	2	yes	maximal, complemented
4	#14582628	C₂	1	2	yes	maximal, complemented
5	C₄×M₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10621898	C₄	1	1	yes	characteristic
7	C₂×#10621877	—	2	1	no	—
8	C₄×Dic₈₆	C₄	1	1	yes	characteristic
9	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
10	C₄×D₁₇₂	C₄	1	1	yes	characteristic
11	C₄×D₁₇₂	C₂²	1	1	yes	characteristic
12	C₄×C₂²⋊D₄₃	—	2	1	no	—
13	C₄×C₂²⋊D₄₃	—	2	1	no	—
14	#13063284	C₄	1	1	yes	characteristic
15	#13063282	—	2	1	no	—
16	C₄²×D₄₃	—	2	1	no	—
17	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
18	C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
19	C₂×M₁₆×C₄₃	—	2	1	no	—
20	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
21	#97×C₄₃	—	2	1	no	—
22	C₄×C₈×C₄₃	C₂²	1	2	yes	—
23	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
26	C₂×C₂²⋊D₄₃	—	2	1	no	—
27	#10621898	—	2	2	no	—
28	#10621898	—	2	2	no	—
29	C₂×D₁₇₂	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×D₄₃	—	2	1	no	—
31	C₂×C₄×D₄₃	—	2	1	no	—
32	C₂×C₄×D₄₃	—	2	1	no	—
33	C₂×Dic₈₆	C₂×C₄	1	1	yes	characteristic
34	M₁₆×C₄₃	—	2	4	no	—
35	C₂×C₈×C₄₃	—	2	2	no	—
36	C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
37	C₄²×C₄₃	D₄	1	1	yes	characteristic
38	C₄²×C₄₃	D₄	1	1	yes	characteristic
39	C₄²×C₄₃	C₂³	1	1	yes	characteristic
40	C₂²⋊Dic₄₃	—	2	1	no	—
41	(C₂×D₄₃)⋊C₄	—	2	1	no	—
42	(C₂×D₄₃)⋊C₄	—	2	1	no	—
43	C₄⋊Dic₄₃	C₂×C₄	1	1	yes	characteristic
44	C₄⋊Dic₄₃	C₂×C₄	1	1	yes	characteristic
45	Dic₄₃⋊C₄	—	2	1	no	—
46	Dic₄₃⋊C₄	—	2	1	no	—
47	C₄×Dic₄₃	—	2	1	no	—
48	C₄×Dic₄₃	—	2	1	no	—
49	C₄×Dic₄₃	—	2	1	no	—
50	#10621877	—	4	2	no	—
51	C₂×C₄₃⋊C₈	—	2	2	no	—
52	C₂³×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×D₄₃	—	2	1	no	—
54	C₂×C₄×C₄₃	—	2	2	no	—
55	C₂×C₄×C₄₃	—	2	1	no	—
56	C₂×C₄×C₄₃	—	2	1	no	—
57	C₂×C₄×C₄₃	—	2	1	no	—
58	C₂×C₄×C₄₃	—	2	1	no	—
59	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
65	C₂²⋊D₄₃	—	2	4	no	—
66	C₂×Dic₄₃	—	2	1	no	—
67	C₂×Dic₄₃	—	2	1	no	—
68	C₂×Dic₄₃	—	2	1	no	—
69	D₁₇₂	—	2	1	no	—
70	D₁₇₂	M₁₆	1	2	yes	complemented
71	C₄×D₄₃	—	2	4	no	—
72	C₄×D₄₃	—	4	2	no	—
73	C₄×D₄₃	—	4	1	no	—
74	C₄×D₄₃	—	4	1	no	—
75	Dic₈₆	—	2	1	no	—
76	Dic₈₆	M₁₆	1	2	yes	complemented
77	C₈×C₄₃	—	2	4	no	—
78	C₄₃⋊C₈	—	4	2	no	—
79	C₂²×C₄₃	—	2	1	no	—
80	C₂²×C₄₃	—	2	1	no	—
81	C₂²×C₄₃	—	2	1	no	—
82	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×D₄₃	—	2	2	no	—
84	C₂×D₄₃	—	4	1	no	—
85	C₂×D₄₃	—	4	1	no	—
86	C₄×C₄₃	—	2	1	no	—
87	C₄×C₄₃	—	2	1	no	—
88	C₄×C₄₃	—	2	1	no	—
89	C₄×C₄₃	#104	1	2	yes	—
90	C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
91	C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
92	C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
93	C₄×C₄₃	#137	1	1	yes	characteristic
94	Dic₄₃	—	2	2	no	—
95	Dic₄₃	—	4	1	no	—
96	Dic₄₃	—	4	1	no	—
97	#1489	—	43	1	no	maximal, Sylow
98	C₂×C₄₃	—	4	1	no	—
99	C₂×C₄₃	#407	1	1	yes	characteristic
100	C₂×C₄₃	#417	1	1	yes	characteristic
101	C₂×C₄₃	C₂×#104	1	1	yes	characteristic
102	D₄₃	—	4	2	no	—
103	C₄×#41	—	43	1	no	—
104	#423	—	43	1	no	—
105	C₄×M₁₆	D₄₃	1	1	yes	characteristic, complemented
106	Q₈⋊C₈	—	43	2	no	—
107	D₄⋊C₈	—	43	2	no	—
108	C₄₃	#1489	1	1	yes	characteristic, complemented, Sylow
109	C₂×#41	—	43	1	no	—
110	C₂×M₁₆	—	2	1	no	—
111	C₂×M₁₆	—	86	1	no	—
112	C₄×Q₈	—	43	1	no	—
113	C₄×Q₈	—	43	1	no	—
114	C₄×D₄	—	43	1	no	—
115	C₄×D₄	—	43	1	no	—
116	C₄×D₄	—	86	1	no	—
117	C₄×D₄	—	86	1	no	—
118	#117	—	43	1	no	—
119	#117	—	86	1	no	—
120	C₂×C₄²	—	86	1	no	—
121	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
122	C₄⋊C₈	—	43	2	no	—
123	C₄⋊C₈	—	86	1	no	—
124	#97	—	2	1	no	—
125	C₄×C₈	C₂×D₄₃	1	2	yes	—
126	#41	—	86	2	no	—
127	#41	—	86	2	no	—
128	C₂×Q₈	—	43	1	no	—
129	C₂×D₄	—	43	1	no	—
130	C₂×D₄	—	86	1	no	—
131	C₂²×C₄	—	86	1	no	—
132	C₂²×C₄	—	86	1	no	—
133	C₂²×C₄	—	86	1	no	—
134	C₂²×C₄	C₄×D₄₃	1	1	yes	characteristic
135	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
136	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
137	M₁₆	—	2	4	no	—
138	M₁₆	—	172	2	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	86	2	no	—
141	C₄⋊C₄	—	43	1	no	—
142	C₄⋊C₄	—	43	1	no	—
143	C₄⋊C₄	—	86	1	no	—
144	C₄⋊C₄	—	86	1	no	—
145	C₂²⋊C₄	—	86	1	no	—
146	C₂²⋊C₄	—	86	1	no	—
147	C₂²⋊C₄	—	86	1	no	—
148	C₄²	—	86	1	no	—
149	C₄²	—	86	1	no	—
150	C₄²	—	86	1	no	—
151	C₄²	C₄×D₄₃	1	1	yes	characteristic
152	C₄²	D₁₇₂	1	1	yes	characteristic
153	C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
154	C₄²	C₂²×D₄₃	1	1	yes	characteristic
155	C₂³	—	86	1	no	—
156	C₂³	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
157	Q₈	—	43	2	no	—
158	Q₈	—	86	1	no	—
159	D₄	—	43	2	no	—
160	D₄	—	86	4	no	—
161	D₄	—	86	1	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	86	4	no	—
168	C₂×C₄	—	86	1	no	—
169	C₂×C₄	—	86	1	no	—
170	C₂×C₄	—	86	1	no	—
171	C₂×C₄	—	172	2	no	—
172	C₂×C₄	—	172	1	no	—
173	C₂×C₄	—	172	1	no	—
174	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
175	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
176	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
177	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
178	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
179	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
180	C₈	—	2	4	no	—
181	C₈	—	172	2	no	—
182	C₂²	—	2	1	no	—
183	C₂²	—	2	1	no	—
184	C₂²	—	2	1	no	—
185	C₂²	—	86	2	no	—
186	C₂²	—	172	1	no	—
187	C₂²	—	172	1	no	—
188	C₂²	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
189	C₄	—	2	1	no	—
190	C₄	—	2	1	no	—
191	C₄	—	2	1	no	—
192	C₄	—	86	2	no	—
193	C₄	—	172	1	no	—
194	C₄	—	172	1	no	—
195	C₄	#13063253	1	2	yes	—
196	C₄	M₁₆×D₄₃	1	1	yes	characteristic
197	C₄	M₁₆×D₄₃	1	1	yes	characteristic
198	C₄	#13063353	1	1	yes	characteristic
199	C₄	#13063358	1	1	yes	characteristic
200	C₂	—	4	1	no	—
201	C₂	—	172	2	no	—
202	C₂	#14582862	1	1	yes	characteristic
203	C₂	#14582901	1	1	yes	characteristic
204	C₂	C₂×#13063253	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#10621898

C₂

yes

maximal, characteristic

#14582923

C₂

yes

maximal, characteristic, complemented

#14582631

C₂

yes

maximal, complemented

#14582628

C₂

yes

maximal, complemented

C₄×M₁₆×C₄₃

C₂

yes

maximal, characteristic, complemented

C₂×#10621898

C₄

yes

characteristic

C₂×#10621877

—

C₄×Dic₈₆

C₄

yes

characteristic

C₄×Dic₈₆

C₂²

yes

characteristic

C₄×D₁₇₂

C₄

yes

characteristic

C₄×D₁₇₂

C₂²

yes

characteristic

C₄×C₂²⋊D₄₃

—

C₄×C₂²⋊D₄₃

—

#13063284

C₄

yes

characteristic

#13063282

—

C₄²×D₄₃

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

C₂²

yes

—

C₂×M₁₆×C₄₃

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

#97×C₄₃

—

C₄×C₈×C₄₃

C₂²

yes

—

C₂²×C₄×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂×C₂²⋊D₄₃

—

#10621898

—

#10621898

—

C₂×D₁₇₂

C₂×C₄

yes

characteristic

C₂×C₄×D₄₃

—

C₂×C₄×D₄₃

—

C₂×C₄×D₄₃

—

C₂×Dic₈₆

C₂×C₄

yes

characteristic

M₁₆×C₄₃

—

C₂×C₈×C₄₃

—

C₄²×C₄₃

C₂×C₄

yes

characteristic

C₄²×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

C₂³

yes

characteristic

C₂²⋊Dic₄₃

—

(C₂×D₄₃)⋊C₄

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

C₂×C₄

yes

characteristic

C₄⋊Dic₄₃

C₂×C₄

yes

characteristic

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₄×Dic₄₃

—

C₄×Dic₄₃

—

#10621877

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

C₂²⋊C₄

yes

characteristic

C₂²×D₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²⋊D₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

M₁₆

yes

complemented

C₄×D₄₃

—

C₄×D₄₃

—

C₄×D₄₃

—

C₄×D₄₃

—

Dic₈₆

—

Dic₈₆

M₁₆

yes

complemented

C₈×C₄₃

—

C₄₃⋊C₈

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#104

yes

—

C₄×C₄₃

C₂×M₁₆

yes

characteristic

C₄×C₄₃

C₂×M₁₆

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

#1489

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

#407

yes

characteristic

100

C₂×C₄₃

#417

yes

characteristic

101

C₂×C₄₃

C₂×#104

yes

characteristic

102

D₄₃

—

103

C₄×#41

—

104

#423

—

105

C₄×M₁₆

D₄₃

yes

characteristic, complemented

106

Q₈⋊C₈

—

107

D₄⋊C₈

—

108

C₄₃

#1489

yes

characteristic, complemented, Sylow

109

C₂×#41

—

110

C₂×M₁₆

—

111

C₂×M₁₆

—

112

C₄×Q₈

—

113

C₄×Q₈

—

114

C₄×D₄

—

115

C₄×D₄

—

116

C₄×D₄

—

117

C₄×D₄

—

118

#117

—

119

#117

—

120

C₂×C₄²

—

121

C₂×C₄²

C₂×D₄₃

yes

characteristic

122

C₄⋊C₈

—

123

C₄⋊C₈

—

124

#97

—

125

C₄×C₈

C₂×D₄₃

yes

—

126

#41

—

127

#41

—

128

C₂×Q₈

—

129

C₂×D₄

—

130

C₂×D₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

—

134

C₂²×C₄

C₄×D₄₃

yes

characteristic

135

C₂²×C₄

D₁₇₂

yes

characteristic

136

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

137

M₁₆

—

138

M₁₆

—

172

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₂²⋊C₄

—

146

C₂²⋊C₄

—

147

C₂²⋊C₄

—

148

C₄²

—

149

C₄²

—

150

C₄²

—

151

C₄²

C₄×D₄₃

yes

characteristic

152

C₄²

D₁₇₂

yes

characteristic

153

C₄²

C₂²⋊D₄₃

yes

characteristic

154

C₄²

C₂²×D₄₃

yes

characteristic

155

C₂³

—

156

C₂³

(C₂×D₄₃)⋊C₄

yes

characteristic

157

Q₈

—

158

Q₈

—

159

D₄

—

160

D₄

—

161

D₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

—

172

C₂×C₄

—

172

—

173

C₂×C₄

—

172

—

174

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

175

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

176

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

177

C₂×C₄

C₂×C₄×D₄₃

yes

characteristic

178

C₂×C₄

C₂×D₁₇₂

yes

characteristic

179

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

180

C₈

—

181

C₈

—

172

—

182

C₂²

—

183

C₂²

—

184

C₂²

—

185

C₂²

—

186

C₂²

—

172

—

187

C₂²

—

172

—

188

C₂²

C₂×(C₂×D₄₃)⋊C₄

yes

characteristic

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

—

172

—

194

C₄

—

172

—

195

C₄

#13063253

yes

—

196

C₄

M₁₆×D₄₃

yes

characteristic

197

C₄

M₁₆×D₄₃

yes

characteristic

198

C₄

#13063353

yes

characteristic

199

C₄

#13063358

yes

characteristic

200

C₂

—

201

C₂

—

172

—

202

C₂

#14582862

yes

characteristic

203

C₂

#14582901

yes

characteristic

204

C₂

C₂×#13063253

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂²×C₄
Frattini	C₄²	C₂²×D₄₃
Fitting	C₄×M₁₆×C₄₃	C₂
socle	C₂²×C₄₃	C₂×C₂²⋊C₄
hypercenter	C₄×M₁₆	D₄₃
hypocenter	C₄₃	#1489
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂×C₄

C₂×D₁₇₂

commutator

C₂×C₄×C₄₃

C₂²×C₄

Frattini

C₄²

C₂²×D₄₃

Fitting

C₄×M₁₆×C₄₃

C₂

socle

C₂²×C₄₃

C₂×C₂²⋊C₄

hypercenter

C₄×M₁₆

D₄₃

hypocenter

C₄₃

#1489

LCS

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
2. #14582923	C₂	semidirect
3. #14582631	C₂	semidirect
4. #14582628	C₂	semidirect
5. C₄×M₁₆×C₄₃	C₂	semidirect
70. D₁₇₂	M₁₆	semidirect
76. Dic₈₆	M₁₆	semidirect
105. C₄×M₁₆	D₄₃	semidirect
108. C₄₃	#1489	semidirect

H ≅ G/N

type

2. #14582923

C₂

semidirect

3. #14582631

C₂

semidirect

4. #14582628

C₂

semidirect

5. C₄×M₁₆×C₄₃

C₂

semidirect

70. D₁₇₂

M₁₆

semidirect

76. Dic₈₆

M₁₆

semidirect

105. C₄×M₁₆

D₄₃

semidirect

108. C₄₃

#1489

semidirect

Subgroups — #19133035

Conjugacy classes of subgroups

Complements