Group #18077489 — subgroups

Subgroups — #18077489

Order 4864.

416 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 103 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	#14572706	C₂	1	1	yes	maximal, characteristic
2	#14562297	C₂	1	1	yes	maximal, characteristic
3	#14562108	C₂	1	2	yes	maximal, complemented
4	#14561534	C₂	1	2	yes	maximal, complemented
5	C₁₉×#3135	C₂	1	1	yes	maximal, characteristic
6	#11926616	—	2	4	no	—
7	#11926601	C₂²	1	2	yes	—
8	#11926597	C₂²	1	2	yes	—
9	#11926511	—	2	2	no	—
10	#11926293	—	2	1	no	—
11	#11926288	—	2	1	no	—
12	Q₈⋊(C₁₉⋊C₈)	—	2	4	no	—
13	C₁₉×#533	C₂²	1	1	yes	characteristic
14	C₁₉×#532	C₂²	1	1	yes	characteristic
15	C₁₉×#491	—	2	1	no	—
16	C₁₉×#490	—	2	1	no	—
17	C₁₉×#489	C₄	1	4	yes	—
18	C₁₉×#416	—	2	2	no	—
19	C₂×C₁₉×#97	C₂²	1	1	yes	characteristic
20	C₂×#41×C₁₉	D₄	1	2	yes	—
21	C₂²×C₈×C₁₉	—	2	1	no	—
22	C₁₉×C₄⋊Q₈	C₂×C₄	1	2	yes	—
23	C₁₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
24	C₁₉×#127	D₄	1	1	yes	characteristic
25	C₁₉×#125	—	2	1	no	—
26	C₁₉×#124	C₂×C₄	1	2	yes	—
27	C₁₉×C₂²⋊Q₈	—	2	2	no	—
28	C₁₉×#121	—	2	2	no	—
29	C₄×Q₈×C₁₉	—	2	2	no	—
30	C₄×D₄×C₁₉	—	2	2	no	—
31	C₁₉×#117	—	2	1	no	—
32	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
33	C₂×#64520	—	2	2	no	—
34	C₂²×C₁₉⋊C₈	—	2	2	no	—
35	C₁₉×#103	—	2	4	no	—
36	C₁₉×#102	—	2	4	no	—
37	C₁₉×#97	—	2	1	no	—
38	C₁₉×#97	C₂×C₄	1	2	yes	—
39	C₂²⋊(C₁₉⋊C₈)	—	2	4	no	—
40	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
41	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
42	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
43	C₄⋊(C₁₉⋊C₈)	D₄	1	4	yes	—
44	#10597260	—	2	2	no	—
45	C₄×C₁₉⋊C₈	—	2	2	no	—
46	#41×C₁₉	—	4	2	no	—
47	#41×C₁₉	—	4	2	no	—
48	C₂×Q₈×C₁₉	—	2	2	no	—
49	C₂×Q₈×C₁₉	C₂²⋊C₄	1	2	yes	—
50	C₂×D₄×C₁₉	—	2	2	no	—
51	C₂×D₄×C₁₉	—	2	2	no	—
52	C₂×D₄×C₁₉	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₁₉	—	2	2	no	—
54	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₉	#41	1	2	yes	—
56	C₂×C₈×C₁₉	—	2	2	no	—
57	C₂×C₈×C₁₉	—	4	2	no	—
58	C₄⋊C₄×C₁₉	—	2	2	no	—
59	C₄⋊C₄×C₁₉	—	2	2	no	—
60	C₄⋊C₄×C₁₉	—	2	2	no	—
61	C₄⋊C₄×C₁₉	—	4	2	no	—
62	C₂²⋊C₄×C₁₉	—	2	4	no	—
63	C₂²⋊C₄×C₁₉	—	4	1	no	—
64	C₄²×C₁₉	—	4	1	no	—
65	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
66	C₄²×C₁₉	C₂×D₄	1	2	yes	—
67	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
68	#64520	—	8	2	no	—
69	C₂×C₁₉⋊C₈	—	2	4	no	—
70	C₂×C₁₉⋊C₈	—	4	2	no	—
71	C₂×C₁₉⋊C₈	—	8	2	no	—
72	#9005	—	19	1	no	maximal, Sylow
73	C₂³×C₁₉	—	2	2	no	—
74	C₂³×C₁₉	#123	1	1	yes	characteristic
75	Q₈×C₁₉	—	4	2	no	—
76	Q₈×C₁₉	—	4	4	no	—
77	D₄×C₁₉	—	4	2	no	—
78	D₄×C₁₉	—	8	2	no	—
79	D₄×C₁₉	—	8	2	no	—
80	C₂×C₄×C₁₉	—	2	2	no	—
81	C₂×C₄×C₁₉	—	2	2	no	—
82	C₂×C₄×C₁₉	—	4	2	no	—
83	C₂×C₄×C₁₉	—	4	2	no	—
84	C₂×C₄×C₁₉	—	4	2	no	—
85	C₂×C₄×C₁₉	—	4	2	no	—
86	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
88	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
89	C₂×C₄×C₁₉	#121	1	2	yes	—
90	C₈×C₁₉	—	4	2	no	—
91	C₁₉⋊C₈	—	4	4	no	—
92	C₁₉⋊C₈	—	8	2	no	—
93	#3135	D₁₉	1	1	yes	characteristic
94	#2980	—	19	1	no	—
95	#1543	—	19	1	no	—
96	#1497	—	19	2	no	—
97	#1302	—	19	2	no	—
98	C₂²×C₁₉	—	2	2	no	—
99	C₂²×C₁₉	—	2	1	no	—
100	C₂²×C₁₉	—	4	4	no	—
101	C₂²×C₁₉	—	4	2	no	—
102	C₂²×C₁₉	#386	1	1	yes	characteristic
103	C₄×C₁₉	—	2	2	no	—
104	C₄×C₁₉	—	2	2	no	—
105	C₄×C₁₉	—	2	1	no	—
106	C₄×C₁₉	—	4	2	no	—
107	C₄×C₁₉	—	8	2	no	—
108	C₄×C₁₉	#455	1	2	yes	—
109	#533	C₂×D₁₉	1	1	yes	characteristic
110	#532	C₂×D₁₉	1	1	yes	characteristic
111	#491	—	2	1	no	—
112	#490	—	2	1	no	—
113	#489	Dic₁₉	1	4	yes	—
114	#435	—	38	4	no	—
115	#432	—	19	2	no	—
116	#431	—	19	2	no	—
117	#416	—	2	2	no	—
118	#408	—	38	2	no	—
119	C₂×#97	C₂×D₁₉	1	1	yes	characteristic
120	#333	—	38	1	no	—
121	#331	—	38	1	no	—
122	Q₈⋊C₈	—	38	4	no	—
123	C₂×C₁₉	—	4	1	no	—
124	C₂×C₁₉	—	8	2	no	—
125	C₂×C₁₉	#1881	1	2	yes	—
126	C₂×C₁₉	#1885	1	1	yes	characteristic
127	C₂×#41	C₂²⋊D₁₉	1	2	yes	—
128	C₂×M₁₆	—	38	2	no	—
129	C₂²×C₈	—	2	1	no	—
130	C₂²×C₈	—	38	2	no	—
131	C₄⋊Q₈	C₂×Dic₁₉	1	2	yes	—
132	C₄⋊Q₈	C₂²⋊D₁₉	1	1	yes	characteristic
133	#127	C₂²⋊D₁₉	1	1	yes	characteristic
134	#125	—	2	1	no	—
135	#124	C₂×Dic₁₉	1	2	yes	—
136	C₂²⋊Q₈	—	2	2	no	—
137	#121	—	2	2	no	—
138	C₄×Q₈	—	2	2	no	—
139	C₄×D₄	—	2	2	no	—
140	#117	—	2	1	no	—
141	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
142	C₄⋊C₈	—	19	4	no	—
143	C₄⋊C₈	—	38	2	no	—
144	#103	—	2	4	no	—
145	#102	—	2	4	no	—
146	C₂²⋊C₈	—	38	4	no	—
147	C₂²⋊C₈	—	38	2	no	—
148	C₂²⋊C₈	—	38	2	no	—
149	#97	—	2	1	no	—
150	#97	—	38	2	no	—
151	#97	C₂×Dic₁₉	1	2	yes	—
152	C₄×C₈	—	38	2	no	—
153	C₁₉	#9005	1	1	yes	characteristic, complemented, Sylow
154	#41	—	4	2	no	—
155	#41	—	4	2	no	—
156	C₂×Q₈	—	2	2	no	—
157	C₂×Q₈	C₂²⋊Dic₁₉	1	2	yes	—
158	C₂×D₄	—	2	2	no	—
159	C₂×D₄	—	2	2	no	—
160	C₂×D₄	C₂²⋊Dic₁₉	1	2	yes	—
161	C₂²×C₄	—	2	2	no	—
162	C₂²×C₄	#64543	1	2	yes	—
163	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
164	M₁₆	—	152	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	4	2	no	—
167	C₂×C₈	—	38	4	no	—
168	C₂×C₈	—	76	2	no	—
169	C₂×C₈	—	152	2	no	—
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₂²⋊C₄	—	2	4	no	—
175	C₂²⋊C₄	—	4	1	no	—
176	C₄²	—	4	1	no	—
177	C₄²	D₄×D₁₉	1	2	yes	—
178	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
179	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
180	C₂³	—	2	2	no	—
181	C₂³	#10597381	1	1	yes	characteristic
182	Q₈	—	4	2	no	—
183	Q₈	—	4	4	no	—
184	D₄	—	4	2	no	—
185	D₄	—	8	2	no	—
186	D₄	—	8	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	D₄×Dic₁₉	1	2	yes	—
194	C₂×C₄	#10597383	1	1	yes	characteristic
195	C₂×C₄	#10597385	1	2	yes	—
196	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
197	C₈	—	4	2	no	—
198	C₈	—	76	4	no	—
199	C₈	—	152	2	no	—
200	C₂²	—	2	2	no	—
201	C₂²	—	2	1	no	—
202	C₂²	—	4	4	no	—
203	C₂²	—	4	2	no	—
204	C₂²	#11926424	1	1	yes	characteristic
205	C₄	—	2	2	no	—
206	C₄	—	2	2	no	—
207	C₄	—	2	1	no	—
208	C₄	—	4	2	no	—
209	C₄	—	8	2	no	—
210	C₄	#11926721	1	2	yes	—
211	C₂	—	4	1	no	—
212	C₂	—	8	2	no	—
213	C₂	#14563999	1	2	yes	—
214	C₂	#14564024	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14572706

C₂

yes

maximal, characteristic

#14562297

C₂

yes

maximal, characteristic

#14562108

C₂

yes

maximal, complemented

#14561534

C₂

yes

maximal, complemented

C₁₉×#3135

C₂

yes

maximal, characteristic

#11926616

—

#11926601

C₂²

yes

—

#11926597

C₂²

yes

—

#11926511

—

#11926293

—

#11926288

—

Q₈⋊(C₁₉⋊C₈)

—

C₁₉×#533

C₂²

yes

characteristic

C₁₉×#532

C₂²

yes

characteristic

C₁₉×#491

—

C₁₉×#490

—

C₁₉×#489

C₄

yes

—

C₁₉×#416

—

C₂×C₁₉×#97

C₂²

yes

characteristic

C₂×#41×C₁₉

D₄

yes

—

C₂²×C₈×C₁₉

—

C₁₉×C₄⋊Q₈

C₂×C₄

yes

—

C₁₉×C₄⋊Q₈

D₄

yes

characteristic

C₁₉×#127

D₄

yes

characteristic

C₁₉×#125

—

C₁₉×#124

C₂×C₄

yes

—

C₁₉×C₂²⋊Q₈

—

C₁₉×#121

—

C₄×Q₈×C₁₉

—

C₄×D₄×C₁₉

—

C₁₉×#117

—

C₂×C₄²×C₁₉

C₂³

yes

characteristic

C₂×#64520

—

C₂²×C₁₉⋊C₈

—

C₁₉×#103

—

C₁₉×#102

—

C₁₉×#97

—

C₁₉×#97

C₂×C₄

yes

—

C₂²⋊(C₁₉⋊C₈)

—

C₂²⋊(C₁₉⋊C₈)

—

C₂²⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

D₄

yes

—

#10597260

—

C₄×C₁₉⋊C₈

—

#41×C₁₉

—

#41×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

C₂²⋊C₄

yes

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₉

#41

yes

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

C₂²×C₄

yes

characteristic

C₄²×C₁₉

C₂×D₄

yes

—

C₄²×C₁₉

C₂×D₄

yes

characteristic

#64520

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

#9005

—

maximal, Sylow

C₂³×C₁₉

—

C₂³×C₁₉

#123

yes

characteristic

Q₈×C₁₉

—

Q₈×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₉

C₄×D₄

yes

—

C₂×C₄×C₁₉

#120

yes

characteristic

C₂×C₄×C₁₉

#121

yes

—

C₈×C₁₉

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

#3135

D₁₉

yes

—

—

—

—

C₂²×C₁₉

—

C₂²×C₁₉

—

100

C₂²×C₁₉

—

101

C₂²×C₁₉

—

102

C₂²×C₁₉

#386

yes

characteristic

103

C₄×C₁₉

—

104

C₄×C₁₉

—

105

C₄×C₁₉

—

106

C₄×C₁₉

—

107

C₄×C₁₉

—

108

C₄×C₁₉

#455

yes

—

109

#533

C₂×D₁₉

yes

characteristic

110

#532

C₂×D₁₉

yes

characteristic

111

#491

—

112

#490

—

113

#489

Dic₁₉

yes

—

114

#435

—

115

#432

—

116

#431

—

117

#416

—

118

#408

—

119

C₂×#97

C₂×D₁₉

yes

characteristic

120

#333

—

121

#331

—

122

Q₈⋊C₈

—

123

C₂×C₁₉

—

124

C₂×C₁₉

—

125

C₂×C₁₉

#1881

yes

—

126

C₂×C₁₉

#1885

yes

characteristic

127

C₂×#41

C₂²⋊D₁₉

yes

—

128

C₂×M₁₆

—

129

C₂²×C₈

—

130

C₂²×C₈

—

131

C₄⋊Q₈

C₂×Dic₁₉

yes

—

132

C₄⋊Q₈

C₂²⋊D₁₉

yes

characteristic

133

#127

C₂²⋊D₁₉

yes

characteristic

134

#125

—

135

#124

C₂×Dic₁₉

yes

—

136

C₂²⋊Q₈

—

137

#121

—

138

C₄×Q₈

—

139

C₄×D₄

—

140

#117

—

141

C₂×C₄²

C₂²×D₁₉

yes

characteristic

142

C₄⋊C₈

—

143

C₄⋊C₈

—

144

#103

—

145

#102

—

146

C₂²⋊C₈

—

147

C₂²⋊C₈

—

148

C₂²⋊C₈

—

149

#97

—

150

#97

—

151

#97

C₂×Dic₁₉

yes

—

152

C₄×C₈

—

153

C₁₉

#9005

yes

characteristic, complemented, Sylow

154

#41

—

155

#41

—

156

C₂×Q₈

—

157

C₂×Q₈

C₂²⋊Dic₁₉

yes

—

158

C₂×D₄

—

159

C₂×D₄

—

160

C₂×D₄

C₂²⋊Dic₁₉

yes

—

161

C₂²×C₄

—

162

C₂²×C₄

#64543

yes

—

163

C₂²×C₄

C₂×C₂²⋊D₁₉

yes

characteristic

164

M₁₆

—

152

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

—

168

C₂×C₈

—

169

C₂×C₈

—

152

—

170

C₄⋊C₄

—

171

C₄⋊C₄

—

172

C₄⋊C₄

—

173

C₄⋊C₄

—

174

C₂²⋊C₄

—

175

C₂²⋊C₄

—

176

C₄²

—

177

C₄²

D₄×D₁₉

yes

—

178

C₄²

C₂²×Dic₁₉

yes

characteristic

179

C₄²

C₂×C₂²⋊D₁₉

yes

characteristic

180

C₂³

—

181

C₂³

#10597381

yes

characteristic

182

Q₈

—

183

Q₈

—

184

D₄

—

185

D₄

—

186

D₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

D₄×Dic₁₉

yes

—

194

C₂×C₄

#10597383

yes

characteristic

195

C₂×C₄

#10597385

yes

—

196

C₂×C₄

C₂×C₂²⋊Dic₁₉

yes

characteristic

197

C₈

—

198

C₈

—

199

C₈

—

152

—

200

C₂²

—

201

C₂²

—

202

C₂²

—

203

C₂²

—

204

C₂²

#11926424

yes

characteristic

205

C₄

—

206

C₄

—

207

C₄

—

208

C₄

—

209

C₄

—

210

C₄

#11926721

yes

—

211

C₂

—

212

C₂

—

213

C₂

#14563999

yes

—

214

C₂

#14564024

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#10597383
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#3135	C₂
socle	C₂²×C₁₉	#386
prime radical	C₁₉×#532	C₂²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#3135	D₁₉
hypocenter	C₁₉	#9005
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3135

subgroup

G/H

center

C₂×C₄

#10597383

commutator

C₄²×C₁₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₉

Fitting

C₁₉×#3135

C₂

socle

C₂²×C₁₉

#386

prime radical

C₁₉×#532

C₂²

cosocle

C₂×C₄²×C₁₉

C₂³

D₁₉

C₁₉

G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3135

N	H ≅ G/N	type
3. #14562108	C₂	semidirect
4. #14561534	C₂	semidirect
153. C₁₉	#9005	semidirect

H ≅ G/N

type

3. #14562108

C₂

semidirect

4. #14561534

C₂

semidirect

153. C₁₉

#9005

semidirect

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

subgroup	index
#14572706	2
#14562297	2
#14562108	2
#14561534	2
C₁₉×#3135	2
#9005	19

subgroup

index

C₁₉×#3135

Subgroups — #18077489

Conjugacy classes of subgroups

Complements

Maximal subgroups