Group #18083409 — subgroups

Subgroups — #18083409

Order 4864.

528 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 105 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	#14573674	C₂	1	2	yes	maximal
2	#14564616	C₂	1	1	yes	maximal, characteristic
3	#14562064	C₂	1	2	yes	maximal
4	#14561464	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#2399	C₂	1	1	yes	maximal, characteristic
6	#11927101	C₂²	1	1	yes	characteristic
7	#11926879	—	2	2	no	—
8	#11926839	—	2	2	no	—
9	#11926754	—	2	2	no	—
10	#11926700	—	2	2	no	—
11	C₄×#64528	—	2	4	no	—
12	#11926597	C₂²	1	2	yes	—
13	#11926568	C₂²	1	1	yes	characteristic
14	#11926304	—	2	2	no	—
15	Q₈⋊(C₁₉⋊C₈)	—	2	4	no	—
16	C₂×C₄×Q₈×C₁₉	C₂²	1	2	yes	—
17	C₁₉×#398	—	2	2	no	—
18	C₁₉×#395	—	2	2	no	—
19	C₄×C₄⋊C₄×C₁₉	—	2	2	no	—
20	C₁₉×#376	C₂²	1	1	yes	characteristic
21	C₂²×Q₈×C₁₉	D₄	1	2	yes	—
22	C₄×Q₈×C₁₉	—	2	2	no	—
23	C₄×Q₈×C₁₉	—	2	2	no	—
24	C₄×Q₈×C₁₉	—	2	2	no	—
25	C₄×Q₈×C₁₉	—	2	2	no	—
26	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
27	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
28	C₂×C₄⋊C₄×C₁₉	D₄	1	2	yes	—
29	C₂×C₄²×C₁₉	—	2	2	no	—
30	C₂×C₄²×C₁₉	D₄	1	2	yes	—
31	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
32	C₂×#64528	—	2	4	no	—
33	#10597372	—	2	1	no	—
34	#10597370	—	2	2	no	—
35	C₂×#64520	—	2	1	no	—
36	#10597316	—	2	1	no	—
37	#10597315	D₄	1	2	yes	—
38	C₄×Dic₃₈	—	2	2	no	—
39	C₁₉×#95	—	2	2	no	—
40	C₁₉×#95	—	2	2	no	—
41	C₁₉×#95	—	2	2	no	—
42	#10597292	—	2	4	no	—
43	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
44	#10597267	—	2	4	no	—
45	#10597264	—	2	4	no	—
46	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
47	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
48	C₄×C₁₉⋊C₈	D₄	1	4	yes	—
49	C₂×Q₈×C₁₉	—	2	2	no	—
50	C₂×Q₈×C₁₉	—	2	2	no	—
51	C₂×Q₈×C₁₉	—	4	4	no	—
52	C₂²×C₄×C₁₉	—	2	2	no	—
53	C₂²×C₄×C₁₉	—	2	2	no	—
54	C₂²×C₄×C₁₉	—	2	2	no	—
55	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
57	C₂×Dic₃₈	—	2	2	no	—
58	C₄⋊C₄×C₁₉	—	2	2	no	—
59	C₄⋊C₄×C₁₉	—	2	2	no	—
60	C₄⋊C₄×C₁₉	—	4	8	no	—
61	C₄⋊C₄×C₁₉	—	4	2	no	—
62	C₄⋊C₄×C₁₉	—	4	2	no	—
63	C₄²×C₁₉	—	2	2	no	—
64	C₄²×C₁₉	—	4	2	no	—
65	C₄²×C₁₉	—	4	2	no	—
66	C₄²×C₁₉	Q₁₆	1	4	yes	complemented
67	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₁₉	C₂×D₄	1	2	yes	—
69	C₄²×C₁₉	#41	1	1	yes	characteristic
70	C₂²⋊Dic₁₉	—	4	1	no	—
71	#64528	—	4	8	no	—
72	C₄⋊Dic₁₉	—	2	2	no	—
73	C₄⋊Dic₁₉	—	2	2	no	—
74	Dic₁₉⋊C₄	—	4	2	no	—
75	C₄×Dic₁₉	—	4	1	no	—
76	#64520	—	4	2	no	—
77	C₂×C₁₉⋊C₈	—	2	4	no	—
78	C₂×C₁₉⋊C₈	—	4	1	no	—
79	#10017	—	19	1	no	maximal, Sylow
80	C₂³×C₁₉	#120	1	1	yes	characteristic
81	Q₈×C₁₉	—	2	4	no	—
82	Q₈×C₁₉	—	2	4	no	—
83	Q₈×C₁₉	—	4	2	no	—
84	Q₈×C₁₉	—	4	2	no	—
85	C₂×C₄×C₁₉	—	2	4	no	—
86	C₂×C₄×C₁₉	—	2	4	no	—
87	C₂×C₄×C₁₉	—	4	2	no	—
88	C₂×C₄×C₁₉	—	4	4	no	—
89	C₂×C₄×C₁₉	—	4	2	no	—
90	C₂×C₄×C₁₉	—	4	2	no	—
91	C₂×C₄×C₁₉	—	4	4	no	—
92	C₂×C₄×C₁₉	—	4	2	no	—
93	C₂×C₄×C₁₉	#120	1	2	yes	—
94	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
95	C₂×C₄×C₁₉	#121	1	2	yes	—
96	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
97	C₂×C₄×C₁₉	C₂×Q₁₆	1	2	yes	—
98	C₂×C₄×C₁₉	#137	1	2	yes	—
99	C₂×Dic₁₉	—	4	2	no	—
100	Dic₃₈	—	4	4	no	—
101	C₁₉⋊C₈	—	4	4	no	—
102	C₁₉⋊C₈	—	8	1	no	—
103	#3104	—	19	2	no	—
104	#2399	D₁₉	1	1	yes	characteristic
105	#1995	—	19	1	no	—
106	#1487	—	19	2	no	—
107	#1274	—	19	1	no	—
108	C₂²×C₁₉	—	2	2	no	—
109	C₂²×C₁₉	#392	1	1	yes	characteristic
110	C₂²×C₁₉	#451	1	2	yes	—
111	C₄×C₁₉	—	2	2	no	—
112	C₄×C₁₉	—	2	2	no	—
113	C₄×C₁₉	—	4	4	no	—
114	C₄×C₁₉	—	4	4	no	—
115	C₄×C₁₉	—	8	2	no	—
116	C₄×C₁₉	#462	1	2	yes	—
117	C₄×C₁₉	#494	1	1	yes	characteristic
118	C₄×C₁₉	#497	1	1	yes	characteristic
119	Dic₁₉	—	8	2	no	—
120	#533	—	19	1	no	—
121	C₂×C₄×Q₈	C₂×D₁₉	1	2	yes	—
122	#484	—	38	2	no	—
123	#477	—	38	2	no	—
124	#462	—	38	2	no	—
125	#451	—	38	2	no	—
126	C₄×Q₁₆	—	38	4	no	—
127	#431	—	19	2	no	—
128	#423	—	19	1	no	—
129	#398	—	2	2	no	—
130	#395	—	2	2	no	—
131	C₄×C₄⋊C₄	—	2	2	no	—
132	#376	C₂×D₁₉	1	1	yes	characteristic
133	#339	—	38	2	no	—
134	Q₈⋊C₈	—	38	4	no	—
135	C₂×C₁₉	—	2	2	no	—
136	C₂×C₁₉	#2010	1	2	yes	—
137	C₂×C₁₉	#2016	1	1	yes	characteristic
138	C₂²×Q₈	C₂²⋊D₁₉	1	2	yes	—
139	C₂×Q₁₆	—	38	4	no	—
140	C₂×M₁₆	—	38	1	no	—
141	C₄⋊Q₈	—	19	2	no	—
142	#125	—	38	1	no	—
143	C₂²⋊Q₈	—	38	2	no	—
144	C₄×Q₈	—	2	2	no	—
145	C₄×Q₈	—	2	2	no	—
146	C₄×Q₈	—	2	2	no	—
147	C₄×Q₈	—	2	2	no	—
148	C₄×Q₈	—	38	2	no	—
149	#117	—	38	1	no	—
150	C₂×C₄⋊C₄	—	2	4	no	—
151	C₂×C₄⋊C₄	—	2	2	no	—
152	C₂×C₄⋊C₄	C₂²⋊D₁₉	1	2	yes	—
153	C₂×C₄²	—	2	2	no	—
154	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
155	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
156	#107	—	38	4	no	—
157	C₄⋊C₈	—	38	2	no	—
158	C₄⋊C₈	—	38	2	no	—
159	#103	—	38	4	no	—
160	#103	—	38	4	no	—
161	C₂²⋊C₈	—	38	2	no	—
162	C₄×C₈	—	19	4	no	—
163	#95	—	2	2	no	—
164	#95	—	2	2	no	—
165	#95	—	2	2	no	—
166	C₁₉	#10017	1	1	yes	characteristic, complemented, Sylow
167	C₂×Q₈	—	2	2	no	—
168	C₂×Q₈	—	2	2	no	—
169	C₂×Q₈	—	4	4	no	—
170	C₂×Q₈	—	38	2	no	—
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
175	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
176	Q₁₆	—	76	8	no	—
177	M₁₆	—	76	2	no	—
178	C₂×C₈	—	38	4	no	—
179	C₂×C₈	—	76	1	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	—	4	8	no	—
183	C₄⋊C₄	—	4	2	no	—
184	C₄⋊C₄	—	4	2	no	—
185	C₄⋊C₄	—	38	2	no	—
186	C₄⋊C₄	—	38	2	no	—
187	C₄⋊C₄	—	76	2	no	—
188	C₂²⋊C₄	—	76	1	no	—
189	C₄²	—	2	2	no	—
190	C₄²	—	4	2	no	—
191	C₄²	—	4	2	no	—
192	C₄²	—	76	1	no	—
193	C₄²	#64528	1	4	yes	complemented
194	C₄²	D₄×D₁₉	1	1	yes	characteristic
195	C₄²	D₄×D₁₉	1	2	yes	—
196	C₄²	#64543	1	1	yes	characteristic
197	C₂³	#10597399	1	1	yes	characteristic
198	Q₈	—	2	4	no	—
199	Q₈	—	2	4	no	—
200	Q₈	—	4	2	no	—
201	Q₈	—	4	2	no	—
202	Q₈	—	76	4	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	76	2	no	—
212	C₂×C₄	#10597383	1	2	yes	—
213	C₂×C₄	#10597384	1	1	yes	characteristic
214	C₂×C₄	#10597385	1	2	yes	—
215	C₂×C₄	#10597386	1	1	yes	characteristic
216	C₂×C₄	#10597388	1	2	yes	—
217	C₂×C₄	C₂×#64528	1	2	yes	—
218	C₈	—	76	4	no	—
219	C₈	—	152	1	no	—
220	C₂²	—	2	2	no	—
221	C₂²	#11926452	1	1	yes	characteristic
222	C₂²	#11926700	1	2	yes	—
223	C₄	—	2	2	no	—
224	C₄	—	2	2	no	—
225	C₄	—	4	4	no	—
226	C₄	—	4	4	no	—
227	C₄	—	8	2	no	—
228	C₄	—	152	2	no	—
229	C₄	#11926751	1	2	yes	—
230	C₄	#11926923	1	1	yes	characteristic
231	C₄	#11926936	1	1	yes	characteristic
232	C₂	—	2	2	no	—
233	C₂	#14564700	1	2	yes	—
234	C₂	#14564735	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14573674

C₂

yes

maximal

#14564616

C₂

yes

maximal, characteristic

#14562064

C₂

yes

maximal

#14561464

C₂

yes

maximal, characteristic

C₁₉×#2399

C₂

yes

maximal, characteristic

#11927101

C₂²

yes

—

—

—

—

—

C₂²

yes

—

#11926568

C₂²

yes

characteristic

#11926304

—

Q₈⋊(C₁₉⋊C₈)

—

C₂×C₄×Q₈×C₁₉

C₂²

yes

—

C₁₉×#398

—

C₁₉×#395

—

C₄×C₄⋊C₄×C₁₉

—

C₁₉×#376

C₂²

yes

characteristic

C₂²×Q₈×C₁₉

D₄

yes

—

C₄×Q₈×C₁₉

—

C₄×Q₈×C₁₉

—

C₄×Q₈×C₁₉

—

C₄×Q₈×C₁₉

—

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄⋊C₄×C₁₉

D₄

yes

—

C₂×C₄²×C₁₉

—

C₂×C₄²×C₁₉

D₄

yes

—

C₂×C₄²×C₁₉

C₂³

yes

—

—

—

—

—

D₄

yes

—

C₄×Dic₃₈

—

C₁₉×#95

—

C₁₉×#95

—

C₁₉×#95

—

#10597292

—

C₂²⋊(C₁₉⋊C₈)

—

#10597267

—

#10597264

—

C₄⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

—

C₄×C₁₉⋊C₈

D₄

yes

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂×D₄

yes

—

C₂²×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂×Dic₃₈

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

Q₁₆

yes

complemented

C₄²×C₁₉

C₂×D₄

yes

characteristic

C₄²×C₁₉

C₂×D₄

yes

—

C₄²×C₁₉

#41

yes

characteristic

C₂²⋊Dic₁₉

—

#64528

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

Dic₁₉⋊C₄

—

C₄×Dic₁₉

—

#64520

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

—

C₂³×C₁₉

yes

Q₈×C₁₉

—

Q₈×C₁₉

—

Q₈×C₁₉

—

Q₈×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

#120

yes

—

C₂×C₄×C₁₉

#121

yes

characteristic

C₂×C₄×C₁₉

#121

yes

—

C₂×C₄×C₁₉

#127

yes

characteristic

C₂×C₄×C₁₉

C₂×Q₁₆

yes

—

C₂×C₄×C₁₉

#137

yes

—

C₂×Dic₁₉

—

100

Dic₃₈

—

101

C₁₉⋊C₈

—

102

C₁₉⋊C₈

—

103

#3104

—

104

#2399

D₁₉

yes

characteristic

105

#1995

—

106

#1487

—

107

#1274

—

108

C₂²×C₁₉

—

109

C₂²×C₁₉

#392

yes

characteristic

110

C₂²×C₁₉

#451

yes

—

111

C₄×C₁₉

—

112

C₄×C₁₉

—

113

C₄×C₁₉

—

114

C₄×C₁₉

—

115

C₄×C₁₉

—

116

C₄×C₁₉

#462

yes

—

117

C₄×C₁₉

#494

yes

characteristic

118

C₄×C₁₉

#497

yes

characteristic

119

Dic₁₉

—

120

#533

—

121

C₂×C₄×Q₈

C₂×D₁₉

yes

—

122

#484

—

123

#477

—

124

#462

—

125

#451

—

126

C₄×Q₁₆

—

127

#431

—

128

#423

—

129

#398

—

130

#395

—

131

C₄×C₄⋊C₄

—

132

#376

C₂×D₁₉

yes

characteristic

133

#339

—

134

Q₈⋊C₈

—

135

C₂×C₁₉

—

136

C₂×C₁₉

#2010

yes

—

137

C₂×C₁₉

#2016

yes

characteristic

138

C₂²×Q₈

C₂²⋊D₁₉

yes

—

139

C₂×Q₁₆

—

140

C₂×M₁₆

—

141

C₄⋊Q₈

—

142

#125

—

143

C₂²⋊Q₈

—

144

C₄×Q₈

—

145

C₄×Q₈

—

146

C₄×Q₈

—

147

C₄×Q₈

—

148

C₄×Q₈

—

149

#117

—

150

C₂×C₄⋊C₄

—

151

C₂×C₄⋊C₄

—

152

C₂×C₄⋊C₄

C₂²⋊D₁₉

yes

—

153

C₂×C₄²

—

154

C₂×C₄²

C₂²⋊D₁₉

yes

—

155

C₂×C₄²

C₂²×D₁₉

yes

characteristic

156

#107

—

157

C₄⋊C₈

—

158

C₄⋊C₈

—

159

#103

—

160

#103

—

161

C₂²⋊C₈

—

162

C₄×C₈

—

163

#95

—

164

#95

—

165

#95

—

166

C₁₉

#10017

yes

characteristic, complemented, Sylow

167

C₂×Q₈

—

168

C₂×Q₈

—

169

C₂×Q₈

—

170

C₂×Q₈

—

171

C₂²×C₄

—

172

C₂²×C₄

—

173

C₂²×C₄

—

174

C₂²×C₄

C₂×C₂²⋊D₁₉

yes

—

175

C₂²×C₄

C₂×C₂²⋊D₁₉

yes

characteristic

176

Q₁₆

—

177

M₁₆

—

178

C₂×C₈

—

179

C₂×C₈

—

180

C₄⋊C₄

—

181

C₄⋊C₄

—

182

C₄⋊C₄

—

183

C₄⋊C₄

—

184

C₄⋊C₄

—

185

C₄⋊C₄

—

186

C₄⋊C₄

—

187

C₄⋊C₄

—

188

C₂²⋊C₄

—

189

C₄²

—

190

C₄²

—

191

C₄²

—

192

C₄²

—

193

C₄²

#64528

yes

complemented

194

C₄²

D₄×D₁₉

yes

characteristic

195

C₄²

D₄×D₁₉

yes

—

196

C₄²

#64543

yes

characteristic

197

C₂³

#10597399

yes

characteristic

198

Q₈

—

199

Q₈

—

200

Q₈

—

201

Q₈

—

202

Q₈

—

203

C₂×C₄

—

204

C₂×C₄

—

205

C₂×C₄

—

206

C₂×C₄

—

207

C₂×C₄

—

208

C₂×C₄

—

209

C₂×C₄

—

210

C₂×C₄

—

211

C₂×C₄

—

212

C₂×C₄

#10597383

yes

—

213

C₂×C₄

#10597384

yes

characteristic

214

C₂×C₄

#10597385

yes

—

215

C₂×C₄

#10597386

yes

characteristic

216

C₂×C₄

#10597388

yes

—

217

C₂×C₄

C₂×#64528

yes

—

218

C₈

—

219

C₈

—

152

—

220

C₂²

—

221

C₂²

#11926452

yes

characteristic

222

C₂²

#11926700

yes

—

223

C₄

—

224

C₄

—

225

C₄

—

226

C₄

—

227

C₄

—

228

C₄

—

152

—

229

C₄

#11926751

yes

—

230

C₄

#11926923

yes

characteristic

231

C₄

#11926936

yes

characteristic

232

C₂

—

233

C₂

#14564700

yes

—

234

C₂

#14564735

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#10597386
commutator, cosocle	C₂×C₄²×C₁₉	C₂³
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#2399	C₂
socle	C₂²×C₁₉	#392
prime radical	C₂³×C₁₉	#120
hypercenter	#2399	D₁₉
hypocenter	C₁₉	#10017
LCS	G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #2399

subgroup

G/H

center

C₂×C₄

#10597386

commutator, cosocle

C₂×C₄²×C₁₉

C₂³

Frattini

C₂×C₄²

C₂²×D₁₉

Fitting

C₁₉×#2399

C₂

C₂²×C₁₉

C₂³×C₁₉

D₁₉

C₁₉

G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #2399

N	H ≅ G/N	type
66. C₄²×C₁₉	Q₁₆	semidirect
166. C₁₉	#10017	semidirect
193. C₄²	#64528	semidirect

H ≅ G/N

type

66. C₄²×C₁₉

Q₁₆

semidirect

166. C₁₉

#10017

semidirect

193. C₄²

#64528

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#14573674	2
#14564616	2
#14562064	2
#14561464	2
C₁₉×#2399	2
#10017	19

subgroup

index

C₁₉×#2399

Subgroups — #18083409

Conjugacy classes of subgroups

Complements

Maximal subgroups