Group #19136012 — subgroups

Subgroups — #19136012

Order 5504.

258 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 77 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×#13063247	C₂	1	1	yes	maximal, characteristic
2	C₂×#13063245	C₂	1	1	yes	maximal, characteristic
3	#14582800	C₂	1	1	yes	maximal, characteristic
4	#14582764	C₂	1	1	yes	maximal, characteristic, complemented
5	#14582661	C₂	1	1	yes	maximal, characteristic, complemented
6	#14582653	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#340	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₁₇₂	C₂²	1	1	yes	characteristic
9	C₂²×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
11	C₂×(C₂×D₄₃)⋊C₄	—	2	1	no	—
12	C₂×(C₂×D₄₃)⋊C₄	—	2	1	no	—
13	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
14	#13063247	—	2	2	no	—
15	#13063245	—	2	2	no	—
16	C₄₃⋊#95	—	2	1	no	—
17	C₄₃⋊#95	—	2	1	no	—
18	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
21	C₂³×D₄₃	—	2	1	no	—
22	C₂²×C₄×C₄₃	—	2	1	no	—
23	C₂²×C₄×C₄₃	—	2	1	no	—
24	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
25	C₂²×Dic₄₃	—	2	1	no	—
26	C₂²×Dic₄₃	D₄	1	1	yes	characteristic
27	C₂²×Dic₄₃	D₄	1	1	yes	characteristic
28	C₂×D₁₇₂	—	2	1	no	—
29	C₂×D₁₇₂	—	2	1	no	—
30	C₂×D₁₇₂	—	2	1	no	—
31	C₂×D₁₇₂	—	2	1	no	—
32	C₂×D₁₇₂	—	2	1	no	—
33	C₂×D₁₇₂	D₄	1	2	yes	—
34	C₂×C₈×C₄₃	—	2	1	no	—
35	C₂×C₈×C₄₃	—	2	1	no	—
36	C₂×C₈×C₄₃	—	2	1	no	—
37	C₄⋊C₄×C₄₃	—	2	1	no	—
38	C₄⋊C₄×C₄₃	—	2	1	no	—
39	C₄⋊C₄×C₄₃	—	2	1	no	—
40	C₄⋊C₄×C₄₃	—	2	1	no	—
41	(C₂×D₄₃)⋊C₄	—	4	2	no	—
42	(C₂×D₄₃)⋊C₄	—	4	2	no	—
43	C₄⋊Dic₄₃	—	2	1	no	—
44	C₄⋊Dic₄₃	—	2	1	no	—
45	C₄×Dic₄₃	—	2	1	no	—
46	C₄×Dic₄₃	—	2	1	no	—
47	C₂×C₄₃⋊C₈	—	2	1	no	—
48	C₂×C₄₃⋊C₈	—	2	1	no	—
49	C₂×C₄₃⋊C₈	—	2	1	no	—
50	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×D₄₃	—	2	2	no	—
52	C₂²×D₄₃	—	4	1	no	—
53	C₂²×D₄₃	—	4	1	no	—
54	C₂²×D₄₃	—	4	1	no	—
55	C₂²×D₄₃	—	4	1	no	—
56	C₂²×D₄₃	—	4	1	no	—
57	C₂²×D₄₃	—	4	1	no	—
58	C₂×C₄×C₄₃	—	4	1	no	—
59	C₂×C₄×C₄₃	—	4	1	no	—
60	C₂×C₄×C₄₃	—	4	1	no	—
61	C₂×C₄×C₄₃	—	4	1	no	—
62	C₂×C₄×C₄₃	—	4	1	no	—
63	C₂×C₄×C₄₃	—	4	1	no	—
64	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
66	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
67	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
68	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
69	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
70	C₂×Dic₄₃	—	2	1	no	—
71	C₂×Dic₄₃	—	2	1	no	—
72	C₂×Dic₄₃	—	2	1	no	—
73	C₂×Dic₄₃	—	2	1	no	—
74	C₂×Dic₄₃	—	2	1	no	—
75	C₂×Dic₄₃	—	2	1	no	—
76	C₂×Dic₄₃	—	4	1	no	—
77	C₂×Dic₄₃	—	4	1	no	—
78	C₂×Dic₄₃	—	4	1	no	—
79	D₁₇₂	—	2	2	no	—
80	D₁₇₂	—	2	2	no	—
81	D₁₇₂	—	4	1	no	—
82	D₁₇₂	—	4	1	no	—
83	C₈×C₄₃	—	4	1	no	—
84	C₄₃⋊C₈	—	4	1	no	—
85	C₂²×C₄₃	#121	1	1	yes	characteristic
86	C₂²×C₄₃	#123	1	1	yes	characteristic
87	C₂²×C₄₃	#124	1	1	yes	characteristic
88	C₂²×C₄₃	#135	1	1	yes	characteristic
89	C₂²×C₄₃	#135	1	1	yes	characteristic
90	C₂²×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
91	C₂²×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
92	C₂×D₄₃	—	4	2	no	—
93	C₂×D₄₃	—	4	2	no	—
94	C₂×D₄₃	—	4	2	no	—
95	C₂×D₄₃	—	8	1	no	—
96	C₂×D₄₃	—	8	1	no	—
97	C₂×D₄₃	—	8	1	no	—
98	C₂×D₄₃	—	8	1	no	—
99	C₄×C₄₃	—	8	1	no	—
100	C₄×C₄₃	—	8	1	no	—
101	C₄×C₄₃	#123	1	1	yes	characteristic
102	C₄×C₄₃	#123	1	1	yes	characteristic
103	C₄×C₄₃	#126	1	1	yes	characteristic
104	C₄×C₄₃	#126	1	1	yes	characteristic
105	Dic₄₃	—	4	1	no	—
106	Dic₄₃	—	4	1	no	—
107	Dic₄₃	—	8	1	no	—
108	#2073	—	43	1	no	maximal, Sylow
109	C₂×C₄₃	#397	1	1	yes	characteristic
110	C₂×C₄₃	#463	1	1	yes	characteristic
111	C₂×C₄₃	#463	1	1	yes	characteristic
112	C₂×C₄₃	#482	1	1	yes	characteristic
113	C₂×C₄₃	#482	1	1	yes	characteristic
114	C₂×C₄₃	#488	1	1	yes	characteristic
115	C₂×C₄₃	#490	1	1	yes	characteristic
116	D₄₃	—	8	2	no	—
117	C₂×#102	—	43	1	no	—
118	C₂×#102	—	43	1	no	—
119	#390	—	43	1	no	—
120	#382	—	43	1	no	—
121	#340	—	43	1	no	—
122	#340	D₄₃	1	1	yes	characteristic, complemented
123	#336	—	43	1	no	—
124	C₄₃	#2073	1	1	yes	characteristic, complemented, Sylow
125	C₂²×D₄	—	43	1	no	—
126	C₂²×C₈	—	43	1	no	—
127	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
128	C₂×C₄⋊C₄	—	43	1	no	—
129	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
130	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
131	C₂×C₂²⋊C₄	—	86	1	no	—
132	C₂×C₂²⋊C₄	—	86	1	no	—
133	C₂×C₄²	—	43	1	no	—
134	#102	—	86	2	no	—
135	#102	—	86	2	no	—
136	#95	—	86	1	no	—
137	#95	—	86	1	no	—
138	C₂⁴	—	86	1	no	—
139	C₂×D₄	—	43	2	no	—
140	C₂×D₄	—	86	1	no	—
141	C₂×D₄	—	86	1	no	—
142	C₂×D₄	—	86	1	no	—
143	C₂×D₄	—	86	1	no	—
144	C₂×D₄	—	86	1	no	—
145	C₂²×C₄	—	2	1	no	—
146	C₂²×C₄	—	2	1	no	—
147	C₂²×C₄	—	43	1	no	—
148	C₂²×C₄	—	43	1	no	—
149	C₂²×C₄	—	86	1	no	—
150	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	86	1	no	—
155	C₂×C₈	—	86	1	no	—
156	C₂×C₈	—	86	1	no	—
157	C₄⋊C₄	—	2	1	no	—
158	C₄⋊C₄	—	2	1	no	—
159	C₄⋊C₄	—	2	1	no	—
160	C₄⋊C₄	—	2	1	no	—
161	C₄⋊C₄	—	86	1	no	—
162	C₄⋊C₄	—	86	1	no	—
163	C₂²⋊C₄	—	172	2	no	—
164	C₂²⋊C₄	—	172	2	no	—
165	C₄²	—	86	1	no	—
166	C₄²	—	86	1	no	—
167	C₂³	—	86	2	no	—
168	C₂³	—	172	1	no	—
169	C₂³	—	172	1	no	—
170	C₂³	—	172	1	no	—
171	C₂³	—	172	1	no	—
172	C₂³	—	172	1	no	—
173	C₂³	—	172	1	no	—
174	C₂³	D₄×D₄₃	1	1	yes	characteristic
175	D₄	—	86	2	no	—
176	D₄	—	86	2	no	—
177	D₄	—	172	1	no	—
178	D₄	—	172	1	no	—
179	C₂×C₄	—	4	1	no	—
180	C₂×C₄	—	4	1	no	—
181	C₂×C₄	—	4	1	no	—
182	C₂×C₄	—	4	1	no	—
183	C₂×C₄	—	4	1	no	—
184	C₂×C₄	—	4	1	no	—
185	C₂×C₄	—	86	1	no	—
186	C₂×C₄	—	86	1	no	—
187	C₂×C₄	—	86	1	no	—
188	C₂×C₄	—	86	1	no	—
189	C₂×C₄	—	86	1	no	—
190	C₂×C₄	—	86	1	no	—
191	C₂×C₄	—	172	1	no	—
192	C₂×C₄	—	172	1	no	—
193	C₂×C₄	—	172	1	no	—
194	C₂×C₄	#10621898	1	1	yes	characteristic
195	C₂×C₄	#10621898	1	1	yes	characteristic
196	C₂×C₄	#10621898	1	1	yes	characteristic
197	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
198	C₂×C₄	#10621900	1	1	yes	characteristic
199	C₂×C₄	#10621902	1	1	yes	characteristic
200	C₈	—	4	1	no	—
201	C₈	—	172	1	no	—
202	C₂²	—	344	1	no	—
203	C₂²	—	344	1	no	—
204	C₂²	—	344	1	no	—
205	C₂²	—	344	1	no	—
206	C₂²	—	172	2	no	—
207	C₂²	—	172	2	no	—
208	C₂²	—	172	2	no	—
209	C₂²	#13063298	1	1	yes	characteristic
210	C₂²	#13063310	1	1	yes	characteristic
211	C₂²	#13063311	1	1	yes	characteristic
212	C₂²	#13063346	1	1	yes	characteristic
213	C₂²	#13063348	1	1	yes	characteristic
214	C₂²	#13063349	1	1	yes	characteristic
215	C₂²	#13063352	1	1	yes	characteristic
216	C₄	—	8	1	no	—
217	C₄	—	8	1	no	—
218	C₄	—	344	1	no	—
219	C₄	—	172	1	no	—
220	C₄	—	172	1	no	—
221	C₄	#13063307	1	1	yes	characteristic
222	C₄	#13063310	1	1	yes	characteristic
223	C₄	#13063319	1	1	yes	characteristic
224	C₄	#13063320	1	1	yes	characteristic
225	C₂	—	344	2	no	—
226	C₂	#14582828	1	1	yes	characteristic
227	C₂	#14583113	1	1	yes	characteristic
228	C₂	#14583117	1	1	yes	characteristic
229	C₂	#14583225	1	1	yes	characteristic
230	C₂	#14583226	1	1	yes	characteristic
231	C₂	#14583255	1	1	yes	characteristic
232	C₂	#14583262	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13063247

C₂

yes

maximal, characteristic

C₂×#13063245

C₂

yes

maximal, characteristic

#14582800

C₂

yes

maximal, characteristic

#14582764

C₂

yes

maximal, characteristic, complemented

#14582661

C₂

yes

maximal, characteristic, complemented

#14582653

C₂

yes

maximal, characteristic, complemented

C₄₃×#340

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₇₂

C₂²

yes

characteristic

C₂²×C₄₃⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₃

C₂²

yes

characteristic

C₂×(C₂×D₄₃)⋊C₄

—

C₂×(C₂×D₄₃)⋊C₄

—

C₂×C₄×Dic₄₃

C₂²

yes

characteristic

#13063247

—

#13063245

—

C₄₃⋊#95

—

C₄₃⋊#95

—

C₂²×C₈×C₄₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₃

C₂²

yes

characteristic

C₂³×D₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

—

C₂²×Dic₄₃

D₄

yes

characteristic

C₂²×Dic₄₃

D₄

yes

characteristic

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

D₄

yes

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

(C₂×D₄₃)⋊C₄

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄×Dic₄₃

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

C₂×D₄

yes

characteristic

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×D₄

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

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

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

100

C₄×C₄₃

—

101

C₄×C₄₃

#123

yes

characteristic

102

C₄×C₄₃

#123

yes

characteristic

103

C₄×C₄₃

#126

yes

characteristic

104

C₄×C₄₃

#126

yes

characteristic

105

Dic₄₃

—

106

Dic₄₃

—

107

Dic₄₃

—

108

#2073

—

maximal, Sylow

109

C₂×C₄₃

#397

yes

characteristic

110

C₂×C₄₃

#463

yes

characteristic

111

C₂×C₄₃

#463

yes

characteristic

112

C₂×C₄₃

#482

yes

characteristic

113

C₂×C₄₃

#482

yes

characteristic

114

C₂×C₄₃

#488

yes

characteristic

115

C₂×C₄₃

#490

yes

characteristic

116

D₄₃

—

117

C₂×#102

—

118

C₂×#102

—

119

#390

—

120

#382

—

121

#340

—

122

#340

D₄₃

yes

characteristic, complemented

123

#336

—

124

C₄₃

#2073

yes

characteristic, complemented, Sylow

125

C₂²×D₄

—

126

C₂²×C₈

—

127

C₂²×C₈

C₂×D₄₃

yes

characteristic

128

C₂×C₄⋊C₄

—

129

C₂×C₄⋊C₄

C₂×D₄₃

yes

characteristic

130

C₂×C₄⋊C₄

C₂×D₄₃

yes

characteristic

131

C₂×C₂²⋊C₄

—

132

C₂×C₂²⋊C₄

—

133

C₂×C₄²

—

134

#102

—

135

#102

—

136

#95

—

137

#95

—

138

C₂⁴

—

139

C₂×D₄

—

140

C₂×D₄

—

141

C₂×D₄

—

142

C₂×D₄

—

143

C₂×D₄

—

144

C₂×D₄

—

145

C₂²×C₄

—

146

C₂²×C₄

—

147

C₂²×C₄

—

148

C₂²×C₄

—

149

C₂²×C₄

—

150

C₂²×C₄

C₂²×D₄₃

yes

characteristic

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₂²⋊C₄

—

172

—

164

C₂²⋊C₄

—

172

—

165

C₄²

—

166

C₄²

—

167

C₂³

—

168

C₂³

—

172

—

169

C₂³

—

172

—

170

C₂³

—

172

—

171

C₂³

—

172

—

172

C₂³

—

172

—

173

C₂³

—

172

—

174

C₂³

D₄×D₄₃

yes

characteristic

175

D₄

—

176

D₄

—

177

D₄

—

172

—

178

D₄

—

172

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

172

—

192

C₂×C₄

—

172

—

193

C₂×C₄

—

172

—

194

C₂×C₄

#10621898

yes

characteristic

195

C₂×C₄

#10621898

yes

characteristic

196

C₂×C₄

#10621898

yes

characteristic

197

C₂×C₄

D₄×D₄₃

yes

characteristic

198

C₂×C₄

#10621900

yes

characteristic

199

C₂×C₄

#10621902

yes

characteristic

200

C₈

—

201

C₈

—

172

—

202

C₂²

—

344

—

203

C₂²

—

344

—

204

C₂²

—

344

—

205

C₂²

—

344

—

206

C₂²

—

172

—

207

C₂²

—

172

—

208

C₂²

—

172

—

209

C₂²

#13063298

yes

characteristic

210

C₂²

#13063310

yes

characteristic

211

C₂²

#13063311

yes

characteristic

212

C₂²

#13063346

yes

characteristic

213

C₂²

#13063348

yes

characteristic

214

C₂²

#13063349

yes

characteristic

215

C₂²

#13063352

yes

characteristic

216

C₄

—

217

C₄

—

218

C₄

—

344

—

219

C₄

—

172

—

220

C₄

—

172

—

221

C₄

#13063307

yes

characteristic

222

C₄

#13063310

yes

characteristic

223

C₄

#13063319

yes

characteristic

224

C₄

#13063320

yes

characteristic

225

C₂

—

344

—

226

C₂

#14582828

yes

characteristic

227

C₂

#14583113

yes

characteristic

228

C₂

#14583117

yes

characteristic

229

C₂

#14583225

yes

characteristic

230

C₂

#14583226

yes

characteristic

231

C₂

#14583255

yes

characteristic

232

C₂

#14583262

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₄₃
commutator	C₂²×C₄×C₄₃	C₂³
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₄₃×#340	C₂
socle	C₂³×C₄₃	C₂×D₄
hypercenter	#340	D₄₃
hypocenter	C₄₃	#2073
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₄₃

commutator

C₂²×C₄×C₄₃

C₂³

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₄₃×#340

C₂

socle

C₂³×C₄₃

C₂×D₄

D₄₃

C₄₃

G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
4. #14582764	C₂	semidirect
5. #14582661	C₂	semidirect
6. #14582653	C₂	semidirect
7. C₄₃×#340	C₂	semidirect
122. #340	D₄₃	semidirect
124. C₄₃	#2073	semidirect

H ≅ G/N

type

4. #14582764

C₂

semidirect

5. #14582661

C₂

semidirect

6. #14582653

C₂

semidirect

7. C₄₃×#340

C₂

semidirect

122. #340

D₄₃

semidirect

124. C₄₃

#2073

semidirect

Subgroups — #19136012

Conjugacy classes of subgroups

Complements