Group #19135405 — subgroups

Subgroups — #19135405

Order 5504.

336 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 99 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	C₂×#13063312	C₂	1	1	yes	maximal, characteristic
2	C₂×#13063250	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#13063248	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#13063246	C₂	1	1	yes	maximal, characteristic
5	C₂×C₈×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#14582764	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#102×C₄₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₈₆	C₂²	1	1	yes	characteristic
9	C₂²×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
10	#13063312	—	2	2	no	—
11	#13063312	C₄	1	4	yes	complemented
12	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊Dic₄₃	—	2	2	no	—
14	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
15	#13063250	—	2	2	no	—
16	#13063248	—	2	2	no	—
17	#13063246	—	2	2	no	—
18	C₈×Dic₄₃	—	2	2	no	—
19	C₄₃⋊#95	—	2	2	no	—
20	C₂²×D₄×C₄₃	C₂²	1	1	yes	characteristic
21	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
23	#102×C₄₃	—	2	2	no	—
24	C₂⁴×C₄₃	—	2	1	no	—
25	C₂×D₄×C₄₃	—	2	1	no	—
26	C₂×D₄×C₄₃	—	2	1	no	—
27	C₂×D₄×C₄₃	—	2	1	no	—
28	C₂×D₄×C₄₃	—	2	1	no	—
29	C₂×D₄×C₄₃	—	2	1	no	—
30	C₂×D₄×C₄₃	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₄₃	—	2	1	no	—
32	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
33	C₂²×Dic₄₃	—	2	1	no	—
34	C₂²×Dic₄₃	—	2	1	no	—
35	C₂²×Dic₄₃	D₄	1	2	yes	—
36	C₂×Dic₈₆	—	2	1	no	—
37	C₂×Dic₈₆	—	2	1	no	—
38	C₂×Dic₈₆	—	2	1	no	—
39	C₂×Dic₈₆	—	2	1	no	—
40	C₂×Dic₈₆	—	2	1	no	—
41	C₂×Dic₈₆	C₂×C₄	1	2	yes	—
42	C₂×C₈×C₄₃	—	2	1	no	—
43	C₂×C₈×C₄₃	—	2	1	no	—
44	C₂×C₈×C₄₃	D₄	1	2	yes	—
45	C₄⋊C₄×C₄₃	—	2	1	no	—
46	C₄⋊C₄×C₄₃	—	2	1	no	—
47	C₂²⋊Dic₄₃	—	2	8	no	—
48	C₄⋊Dic₄₃	—	2	1	no	—
49	C₄⋊Dic₄₃	—	2	1	no	—
50	C₄×Dic₄₃	C₂×C₄	1	2	yes	—
51	C₄×Dic₄₃	D₄	1	2	yes	—
52	C₂×C₄₃⋊C₈	—	2	1	no	—
53	C₂×C₄₃⋊C₈	—	2	1	no	—
54	C₂×C₄₃⋊C₈	D₄	1	2	yes	—
55	C₂³×C₄₃	—	2	2	no	—
56	C₂³×C₄₃	—	4	1	no	—
57	C₂³×C₄₃	—	4	1	no	—
58	C₂³×C₄₃	—	4	2	no	—
59	C₂³×C₄₃	—	4	2	no	—
60	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
61	D₄×C₄₃	—	2	2	no	—
62	D₄×C₄₃	—	2	2	no	—
63	D₄×C₄₃	—	4	1	no	—
64	D₄×C₄₃	—	4	1	no	—
65	C₂×C₄×C₄₃	—	4	1	no	—
66	C₂×C₄×C₄₃	—	4	2	no	—
67	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
72	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
73	C₂×Dic₄₃	—	2	2	no	—
74	C₂×Dic₄₃	—	2	2	no	—
75	C₂×Dic₄₃	—	2	2	no	—
76	C₂×Dic₄₃	—	4	1	no	—
77	C₂×Dic₄₃	—	4	1	no	—
78	C₂×Dic₄₃	—	4	1	no	—
79	C₂×Dic₄₃	—	4	2	no	—
80	C₂×Dic₄₃	C₂²⋊C₄	1	4	yes	—
81	Dic₈₆	—	2	2	no	—
82	Dic₈₆	—	2	2	no	—
83	Dic₈₆	—	4	1	no	—
84	Dic₈₆	—	4	1	no	—
85	C₈×C₄₃	—	2	2	no	—
86	C₄₃⋊C₈	—	2	2	no	—
87	C₂²×C₄₃	—	4	2	no	—
88	C₂²×C₄₃	—	4	4	no	—
89	C₂²×C₄₃	—	8	2	no	—
90	C₂²×C₄₃	—	8	1	no	—
91	C₂²×C₄₃	—	8	1	no	—
92	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂²×C₄₃	#124	1	1	yes	characteristic
94	C₂²×C₄₃	#127	1	1	yes	characteristic
95	C₂²×C₄₃	#135	1	2	yes	—
96	C₂²×C₄₃	#135	1	2	yes	—
97	C₄×C₄₃	—	8	1	no	—
98	C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
99	C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
100	C₄×C₄₃	#121	1	1	yes	characteristic
101	C₄×C₄₃	#121	1	1	yes	characteristic
102	Dic₄₃	—	2	4	no	—
103	Dic₄₃	—	4	2	no	—
104	Dic₄₃	—	8	1	no	—
105	#1958	—	43	1	no	maximal, Sylow
106	C₂×C₄₃	—	8	2	no	—
107	C₂×C₄₃	#390	1	1	yes	characteristic
108	C₂×C₄₃	#416	1	2	yes	—
109	C₂×C₄₃	#488	1	2	yes	—
110	C₂×C₄₃	#495	1	1	yes	characteristic
111	C₂×C₄₃	#495	1	1	yes	characteristic
112	C₂×#124	—	43	1	no	—
113	C₂×#103	—	43	1	no	—
114	C₂×#103	—	43	1	no	—
115	C₂×#102	—	43	1	no	—
116	C₂×#102	D₄₃	1	1	yes	characteristic
117	C₂×C₄×C₈	—	43	1	no	—
118	#382	—	43	1	no	—
119	C₄₃	#1958	1	1	yes	characteristic, complemented, Sylow
120	C₂²×Q₈	—	43	1	no	—
121	C₂²×D₄	C₂×D₄₃	1	1	yes	characteristic
122	C₂²×C₈	—	43	1	no	—
123	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
124	#124	—	43	4	no	—
125	#124	—	86	2	no	—
126	C₂×C₄⋊C₄	—	43	1	no	—
127	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
128	C₂×C₂²⋊C₄	—	86	2	no	—
129	C₂×C₄²	—	43	1	no	—
130	#103	—	86	2	no	—
131	#103	—	86	2	no	—
132	#102	—	2	2	no	—
133	#102	—	86	2	no	—
134	C₄×C₈	—	86	2	no	—
135	#95	—	86	2	no	—
136	C₂⁴	—	2	1	no	—
137	C₂×Q₈	—	43	2	no	—
138	C₂×Q₈	—	86	1	no	—
139	C₂×Q₈	—	86	1	no	—
140	C₂×Q₈	—	86	1	no	—
141	C₂×Q₈	—	86	1	no	—
142	C₂×Q₈	—	86	1	no	—
143	C₂×D₄	—	2	1	no	—
144	C₂×D₄	—	2	1	no	—
145	C₂×D₄	—	2	1	no	—
146	C₂×D₄	—	2	1	no	—
147	C₂×D₄	—	2	1	no	—
148	C₂×D₄	C₄×D₄₃	1	2	yes	—
149	C₂²×C₄	—	2	1	no	—
150	C₂²×C₄	—	43	2	no	—
151	C₂²×C₄	—	86	1	no	—
152	C₂²×C₄	—	86	1	no	—
153	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	43	2	no	—
157	C₂×C₈	—	86	1	no	—
158	C₂×C₈	—	86	1	no	—
159	C₂×C₈	C₂²⋊D₄₃	1	2	yes	—
160	C₄⋊C₄	—	2	1	no	—
161	C₄⋊C₄	—	2	1	no	—
162	C₄⋊C₄	—	86	1	no	—
163	C₄⋊C₄	—	86	1	no	—
164	C₂²⋊C₄	—	86	8	no	—
165	C₄²	—	43	2	no	—
166	C₄²	—	43	2	no	—
167	C₂³	—	2	2	no	—
168	C₂³	—	4	1	no	—
169	C₂³	—	4	1	no	—
170	C₂³	—	4	2	no	—
171	C₂³	—	4	2	no	—
172	C₂³	D₄×D₄₃	1	1	yes	characteristic
173	Q₈	—	86	2	no	—
174	Q₈	—	86	2	no	—
175	Q₈	—	172	1	no	—
176	Q₈	—	172	1	no	—
177	D₄	—	2	2	no	—
178	D₄	—	2	2	no	—
179	D₄	—	4	1	no	—
180	D₄	—	4	1	no	—
181	C₂×C₄	—	4	1	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	43	4	no	—
184	C₂×C₄	—	86	2	no	—
185	C₂×C₄	—	86	2	no	—
186	C₂×C₄	—	86	2	no	—
187	C₂×C₄	—	172	1	no	—
188	C₂×C₄	—	172	2	no	—
189	C₂×C₄	—	172	1	no	—
190	C₂×C₄	—	172	1	no	—
191	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
192	C₂×C₄	#10621898	1	1	yes	characteristic
193	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
194	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
195	C₂×C₄	#10621900	1	1	yes	characteristic
196	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
197	C₈	—	2	2	no	—
198	C₈	—	86	2	no	—
199	C₂²	—	4	2	no	—
200	C₂²	—	4	4	no	—
201	C₂²	—	8	2	no	—
202	C₂²	—	8	1	no	—
203	C₂²	—	8	1	no	—
204	C₂²	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
205	C₂²	#13063311	1	1	yes	characteristic
206	C₂²	#13063321	1	1	yes	characteristic
207	C₂²	#13063344	1	2	yes	—
208	C₂²	#13063346	1	2	yes	—
209	C₄	—	8	1	no	—
210	C₄	—	86	4	no	—
211	C₄	—	344	1	no	—
212	C₄	—	172	2	no	—
213	C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
214	C₄	C₄×C₂²⋊D₄₃	1	1	yes	characteristic
215	C₄	#13063297	1	1	yes	characteristic
216	C₄	#13063298	1	1	yes	characteristic
217	C₂	—	8	2	no	—
218	C₂	#14582799	1	1	yes	characteristic
219	C₂	#14582892	1	2	yes	—
220	C₂	#14583252	1	2	yes	—
221	C₂	#14583280	1	1	yes	characteristic
222	C₂	#14583282	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13063312

C₂

yes

maximal, characteristic

C₂×#13063250

C₂

yes

maximal, characteristic, complemented

C₂×#13063248

C₂

yes

maximal, characteristic, complemented

C₂×#13063246

C₂

yes

maximal, characteristic

C₂×C₈×Dic₄₃

C₂

yes

maximal, characteristic, complemented

#14582764

C₂

yes

maximal, characteristic, complemented

C₂×#102×C₄₃

C₂

yes

maximal, characteristic

C₂²×Dic₈₆

C₂²

yes

characteristic

C₂²×C₄₃⋊C₈

C₂²

yes

characteristic

#13063312

—

#13063312

C₄

yes

complemented

C₂×C₄⋊Dic₄₃

C₂²

yes

characteristic

C₂×C₂²⋊Dic₄₃

—

C₂×C₄×Dic₄₃

C₂²

yes

—

—

—

C₈×Dic₄₃

—

C₄₃⋊#95

—

C₂²×D₄×C₄₃

C₂²

yes

characteristic

C₂²×C₈×C₄₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₃

C₂²

yes

characteristic

#102×C₄₃

—

C₂⁴×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

C₂×C₄

yes

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂²×Dic₄₃

D₄

yes

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

C₂×C₄

yes

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

D₄

yes

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₂²⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄×Dic₄₃

C₂×C₄

yes

—

C₄×Dic₄₃

D₄

yes

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

D₄

yes

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

C₂×D₄

yes

characteristic

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

C₂²⋊C₄

yes

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂×C₂²⋊C₄

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

—

C₂²×C₄₃

#135

yes

—

C₄×C₄₃

—

C₄×C₄₃

C₄×D₄

yes

characteristic

C₄×C₄₃

C₄×D₄

yes

characteristic

100

C₄×C₄₃

#121

yes

characteristic

101

C₄×C₄₃

#121

yes

characteristic

102

Dic₄₃

—

103

Dic₄₃

—

104

Dic₄₃

—

105

#1958

—

maximal, Sylow

106

C₂×C₄₃

—

107

C₂×C₄₃

#390

yes

characteristic

108

C₂×C₄₃

#416

yes

—

109

C₂×C₄₃

#488

yes

—

110

C₂×C₄₃

#495

yes

characteristic

111

C₂×C₄₃

#495

yes

characteristic

112

C₂×#124

—

113

C₂×#103

—

114

C₂×#103

—

115

C₂×#102

—

116

C₂×#102

D₄₃

yes

characteristic

117

C₂×C₄×C₈

—

118

#382

—

119

C₄₃

#1958

yes

characteristic, complemented, Sylow

120

C₂²×Q₈

—

121

C₂²×D₄

C₂×D₄₃

yes

characteristic

122

C₂²×C₈

—

123

C₂²×C₈

C₂×D₄₃

yes

characteristic

124

#124

—

125

#124

—

126

C₂×C₄⋊C₄

—

127

C₂×C₄⋊C₄

C₂×D₄₃

yes

characteristic

128

C₂×C₂²⋊C₄

—

129

C₂×C₄²

—

130

#103

—

131

#103

—

132

#102

—

133

#102

—

134

C₄×C₈

—

135

#95

—

136

C₂⁴

—

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂×Q₈

—

142

C₂×Q₈

—

143

C₂×D₄

—

144

C₂×D₄

—

145

C₂×D₄

—

146

C₂×D₄

—

147

C₂×D₄

—

148

C₂×D₄

C₄×D₄₃

yes

—

149

C₂²×C₄

—

150

C₂²×C₄

—

151

C₂²×C₄

—

152

C₂²×C₄

—

153

C₂²×C₄

C₂²×D₄₃

yes

characteristic

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

C₂²⋊D₄₃

yes

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₂²⋊C₄

—

165

C₄²

—

166

C₄²

—

167

C₂³

—

168

C₂³

—

169

C₂³

—

170

C₂³

—

171

C₂³

—

172

C₂³

D₄×D₄₃

yes

characteristic

173

Q₈

—

174

Q₈

—

175

Q₈

—

172

—

176

Q₈

—

172

—

177

D₄

—

178

D₄

—

179

D₄

—

180

D₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

172

—

188

C₂×C₄

—

172

—

189

C₂×C₄

—

172

—

190

C₂×C₄

—

172

—

191

C₂×C₄

C₂×C₄×D₄₃

yes

characteristic

192

C₂×C₄

#10621898

yes

characteristic

193

C₂×C₄

D₄×D₄₃

yes

characteristic

194

C₂×C₄

D₄×D₄₃

yes

characteristic

195

C₂×C₄

#10621900

yes

characteristic

196

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

197

C₈

—

198

C₈

—

199

C₂²

—

200

C₂²

—

201

C₂²

—

202

C₂²

—

203

C₂²

—

204

C₂²

C₂²⋊C₄×D₄₃

yes

characteristic

205

C₂²

#13063311

yes

characteristic

206

C₂²

#13063321

yes

characteristic

207

C₂²

#13063344

yes

—

208

C₂²

#13063346

yes

—

209

C₄

—

210

C₄

—

211

C₄

—

344

—

212

C₄

—

172

—

213

C₄

(C₂²⋊D₄₃)⋊C₄

yes

characteristic

214

C₄

C₄×C₂²⋊D₄₃

yes

characteristic

215

C₄

#13063297

yes

characteristic

216

C₄

#13063298

yes

characteristic

217

C₂

—

218

C₂

#14582799

yes

characteristic

219

C₂

#14582892

yes

—

220

C₂

#14583252

yes

—

221

C₂

#14583280

yes

characteristic

222

C₂

#14583282

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₂×#102×C₄₃	C₂
socle	C₂³×C₄₃	C₂×D₄
hypercenter	C₂×#102	D₄₃
hypocenter	C₄₃	#1958
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#102

subgroup

G/H

center

C₂³

D₄×D₄₃

commutator

C₂×C₄×C₄₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₂×#102×C₄₃

C₂

socle

C₂³×C₄₃

C₂×D₄

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#102

N	H ≅ G/N	type
2. C₂×#13063250	C₂	semidirect
3. C₂×#13063248	C₂	semidirect
5. C₂×C₈×Dic₄₃	C₂	semidirect
6. #14582764	C₂	semidirect
11. #13063312	C₄	semidirect
119. C₄₃	#1958	semidirect

H ≅ G/N

type

2. C₂×#13063250

C₂

semidirect

3. C₂×#13063248

C₂

semidirect

5. C₂×C₈×Dic₄₃

C₂

semidirect

6. #14582764

C₂

semidirect

11. #13063312

C₄

semidirect

119. C₄₃

#1958

semidirect

Subgroups — #19135405

Conjugacy classes of subgroups

Complements