Group #18069092 — subgroups

Subgroups — #18069092

Order 4864.

442 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 135 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×C₂²⋊(C₁₉⋊C₈)	C₂	1	2	yes	maximal
2	C₁₉×#1728	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
4	C₂×C₄×C₁₉⋊C₈	—	2	2	no	—
5	#11926299	C₄	1	4	yes	—
6	C₂²×M₁₆×C₁₉	C₄	1	2	yes	—
7	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
8	C₁₉×#336	—	2	2	no	—
9	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
10	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
11	C₂³×C₄×C₁₉	Q₈	1	1	yes	characteristic
12	C₂×M₁₆×C₁₉	—	2	2	no	—
13	C₂×M₁₆×C₁₉	—	4	2	no	—
14	C₂×M₁₆×C₁₉	—	4	2	no	—
15	C₂×M₁₆×C₁₉	C₈	1	4	yes	complemented
16	C₂²×C₈×C₁₉	—	2	2	no	—
17	C₂×C₄²×C₁₉	—	4	2	no	—
18	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₁₉	D₄	1	2	yes	—
20	C₂²×C₁₉⋊C₈	—	2	4	no	—
21	C₂²⋊(C₁₉⋊C₈)	—	2	8	no	—
22	C₄×C₁₉⋊C₈	—	4	4	no	—
23	C₂⁴×C₁₉	C₄⋊C₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₉	—	2	2	no	—
25	C₂²×C₄×C₁₉	—	2	1	no	—
26	C₂²×C₄×C₁₉	—	2	1	no	—
27	C₂²×C₄×C₁₉	—	2	4	no	—
28	C₂²×C₄×C₁₉	—	2	4	no	—
29	C₂²×C₄×C₁₉	—	4	1	no	—
30	C₂²×C₄×C₁₉	—	4	1	no	—
31	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
32	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
33	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
35	C₂²×C₄×C₁₉	C₂×C₈	1	2	yes	—
36	C₂²×C₄×C₁₉	M₁₆	1	2	yes	—
37	M₁₆×C₁₉	—	4	4	no	—
38	M₁₆×C₁₉	—	4	4	no	—
39	C₂×C₈×C₁₉	—	2	4	no	—
40	C₂×C₈×C₁₉	—	4	2	no	—
41	C₂×C₈×C₁₉	—	4	2	no	—
42	C₄²×C₁₉	—	2	4	no	—
43	C₄²×C₁₉	—	4	2	no	—
44	C₂×C₁₉⋊C₈	—	2	8	no	—
45	C₂×C₁₉⋊C₈	—	4	4	no	—
46	C₂×C₁₉⋊C₈	—	4	4	no	—
47	#7026	—	19	1	no	maximal, Sylow
48	C₂³×C₁₉	—	2	2	no	—
49	C₂³×C₁₉	—	2	2	no	—
50	C₂³×C₁₉	—	2	1	no	—
51	C₂³×C₁₉	—	2	1	no	—
52	C₂³×C₁₉	#95	1	1	yes	characteristic
53	C₂³×C₁₉	C₄⋊C₈	1	2	yes	—
54	C₂×C₄×C₁₉	—	2	2	no	—
55	C₂×C₄×C₁₉	—	2	2	no	—
56	C₂×C₄×C₁₉	—	2	2	no	—
57	C₂×C₄×C₁₉	—	2	2	no	—
58	C₂×C₄×C₁₉	—	2	1	no	—
59	C₂×C₄×C₁₉	—	2	1	no	—
60	C₂×C₄×C₁₉	—	2	2	no	—
61	C₂×C₄×C₁₉	—	2	2	no	—
62	C₂×C₄×C₁₉	—	2	2	no	—
63	C₂×C₄×C₁₉	—	2	2	no	—
64	C₂×C₄×C₁₉	—	4	4	no	—
65	C₂×C₄×C₁₉	—	4	4	no	—
66	C₂×C₄×C₁₉	C₄×C₈	1	1	yes	characteristic
67	C₂×C₄×C₁₉	#97	1	1	yes	characteristic
68	C₂×C₄×C₁₉	C₂²⋊C₈	1	2	yes	—
69	C₂×C₄×C₁₉	#99	1	2	yes	—
70	C₂×C₄×C₁₉	#100	1	1	yes	characteristic
71	C₂×C₄×C₁₉	#101	1	1	yes	characteristic
72	C₂×C₄×C₁₉	#104	1	4	yes	—
73	C₂×C₄×C₁₉	#108	1	4	yes	—
74	C₈×C₁₉	—	4	4	no	—
75	C₁₉⋊C₈	—	4	8	no	—
76	C₄×C₂²⋊C₈	—	19	2	no	—
77	#1728	D₁₉	1	1	yes	characteristic
78	C₂²×C₁₉	—	2	2	no	—
79	C₂²×C₁₉	—	2	4	no	—
80	C₂²×C₁₉	—	4	1	no	—
81	C₂²×C₁₉	—	4	1	no	—
82	C₂²×C₁₉	—	4	2	no	—
83	C₂²×C₁₉	#336	1	1	yes	characteristic
84	C₂²×C₁₉	#339	1	2	yes	—
85	C₂²×C₁₉	#341	1	2	yes	—
86	C₂²×C₁₉	#342	1	1	yes	characteristic
87	C₂²×C₁₉	#343	1	1	yes	characteristic
88	C₄×C₁₉	—	2	1	no	—
89	C₄×C₁₉	—	2	1	no	—
90	C₄×C₁₉	—	2	4	no	—
91	C₄×C₁₉	—	2	4	no	—
92	C₄×C₁₉	#323	1	2	yes	—
93	C₄×C₁₉	#324	1	2	yes	—
94	C₂²×M₁₆	Dic₁₉	1	2	yes	—
95	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
96	C₂×C₂²⋊C₈	—	38	2	no	—
97	C₂×C₄×C₈	—	38	2	no	—
98	#336	—	2	2	no	—
99	#336	—	19	4	no	—
100	C₂×C₁₉	—	4	2	no	—
101	C₂×C₁₉	#1278	1	2	yes	—
102	C₂×C₁₉	#1280	1	1	yes	characteristic
103	C₂×C₁₉	#1297	1	1	yes	characteristic
104	C₂×C₁₉	#1298	1	1	yes	characteristic
105	C₂×C₁₉	#1299	1	2	yes	—
106	C₂³×C₄	Dic₃₈	1	1	yes	characteristic
107	C₂³×C₄	D₇₆	1	1	yes	characteristic
108	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
109	C₂×M₁₆	—	2	2	no	—
110	C₂×M₁₆	—	4	2	no	—
111	C₂×M₁₆	—	4	2	no	—
112	C₂×M₁₆	C₁₉⋊C₈	1	4	yes	complemented
113	C₂²×C₈	—	2	2	no	—
114	C₂²×C₈	—	38	4	no	—
115	C₂×C₄²	—	4	2	no	—
116	C₂×C₄²	C₄×D₁₉	1	2	yes	—
117	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
118	C₂²⋊C₈	—	38	8	no	—
119	C₄×C₈	—	76	4	no	—
120	C₁₉	#7026	1	1	yes	characteristic, complemented, Sylow
121	C₂⁴	C₄⋊Dic₁₉	1	1	yes	characteristic
122	C₂²×C₄	—	2	1	no	—
123	C₂²×C₄	—	2	2	no	—
124	C₂²×C₄	—	2	1	no	—
125	C₂²×C₄	—	2	4	no	—
126	C₂²×C₄	—	2	4	no	—
127	C₂²×C₄	—	4	1	no	—
128	C₂²×C₄	—	4	1	no	—
129	C₂²×C₄	C₂×C₁₉⋊C₈	1	2	yes	—
130	C₂²×C₄	#64520	1	2	yes	—
131	C₂²×C₄	C₄×Dic₁₉	1	1	yes	characteristic
132	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
133	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
134	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
135	M₁₆	—	4	4	no	—
136	M₁₆	—	4	4	no	—
137	C₂×C₈	—	2	4	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	4	2	no	—
140	C₂×C₈	—	38	8	no	—
141	C₂×C₈	—	76	4	no	—
142	C₂×C₈	—	76	4	no	—
143	C₄²	—	2	4	no	—
144	C₄²	—	4	2	no	—
145	C₂³	—	2	2	no	—
146	C₂³	—	2	2	no	—
147	C₂³	—	2	1	no	—
148	C₂³	—	2	1	no	—
149	C₂³	C₄⋊(C₁₉⋊C₈)	1	2	yes	—
150	C₂³	C₁₉⋊#95	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	—	4	4	no	—
163	C₂×C₄	C₄×C₁₉⋊C₈	1	1	yes	characteristic
164	C₂×C₄	#10597260	1	1	yes	characteristic
165	C₂×C₄	#10597279	1	4	yes	—
166	C₂×C₄	#10597282	1	4	yes	—
167	C₂×C₄	C₂²⋊(C₁₉⋊C₈)	1	2	yes	—
168	C₂×C₄	#10597290	1	1	yes	characteristic
169	C₂×C₄	#10597291	1	2	yes	—
170	C₂×C₄	#10597293	1	1	yes	characteristic
171	C₈	—	4	4	no	—
172	C₈	—	76	8	no	—
173	C₂²	—	2	4	no	—
174	C₂²	—	2	2	no	—
175	C₂²	—	4	1	no	—
176	C₂²	—	4	1	no	—
177	C₂²	—	4	2	no	—
178	C₂²	#11926299	1	1	yes	characteristic
179	C₂²	#11926305	1	2	yes	—
180	C₂²	#11926308	1	2	yes	—
181	C₂²	#11926311	1	1	yes	characteristic
182	C₂²	#11926313	1	1	yes	characteristic
183	C₄	—	2	1	no	—
184	C₄	—	2	1	no	—
185	C₄	—	2	4	no	—
186	C₄	—	2	4	no	—
187	C₄	#11926268	1	2	yes	—
188	C₄	#11926270	1	2	yes	—
189	C₂	—	4	2	no	—
190	C₂	#14561473	1	2	yes	—
191	C₂	#14561478	1	1	yes	characteristic
192	C₂	#14561520	1	1	yes	characteristic
193	C₂	#14561522	1	1	yes	characteristic
194	C₂	#14561524	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₂²⋊(C₁₉⋊C₈)

C₂

yes

maximal

C₁₉×#1728

C₂

yes

maximal, characteristic

C₂×C₂²⋊(C₁₉⋊C₈)

—

C₂×C₄×C₁₉⋊C₈

—

#11926299

C₄

yes

—

C₂²×M₁₆×C₁₉

C₄

yes

—

C₂²×C₄²×C₁₉

C₂²

yes

characteristic

C₁₉×#336

—

C₂³×C₄×C₁₉

C₂×C₄

yes

characteristic

C₂³×C₄×C₁₉

D₄

yes

characteristic

C₂³×C₄×C₁₉

Q₈

yes

characteristic

C₂×M₁₆×C₁₉

—

C₂×M₁₆×C₁₉

—

C₂×M₁₆×C₁₉

—

C₂×M₁₆×C₁₉

C₈

yes

complemented

C₂²×C₈×C₁₉

—

C₂×C₄²×C₁₉

—

C₂×C₄²×C₁₉

C₂×C₄

yes

—

C₂×C₄²×C₁₉

D₄

yes

—

C₂²×C₁₉⋊C₈

—

C₂²⋊(C₁₉⋊C₈)

—

C₄×C₁₉⋊C₈

—

C₂⁴×C₁₉

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₄²

yes

characteristic

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

C₄⋊C₄

yes

—

C₂²×C₄×C₁₉

C₂×C₈

yes

—

C₂²×C₄×C₁₉

M₁₆

yes

—

M₁₆×C₁₉

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

#7026

—

maximal, Sylow

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

#95

yes

characteristic

C₂³×C₁₉

C₄⋊C₈

yes

—

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

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₄×C₈

yes

characteristic

C₂×C₄×C₁₉

#97

yes

characteristic

C₂×C₄×C₁₉

C₂²⋊C₈

yes

—

C₂×C₄×C₁₉

#99

yes

—

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

—

C₂×C₄×C₁₉

#108

yes

—

C₈×C₁₉

—

C₁₉⋊C₈

—

C₄×C₂²⋊C₈

—

#1728

D₁₉

yes

characteristic

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#336

yes

characteristic

C₂²×C₁₉

#339

yes

—

C₂²×C₁₉

#341

yes

—

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

#323

yes

—

C₄×C₁₉

#324

yes

—

C₂²×M₁₆

Dic₁₉

yes

—

C₂²×C₄²

C₂×D₁₉

yes

characteristic

C₂×C₂²⋊C₈

—

C₂×C₄×C₈

—

#336

—

#336

—

100

C₂×C₁₉

—

101

C₂×C₁₉

#1278

yes

—

102

C₂×C₁₉

#1280

yes

characteristic

103

C₂×C₁₉

#1297

yes

characteristic

104

C₂×C₁₉

#1298

yes

characteristic

105

C₂×C₁₉

#1299

yes

—

106

C₂³×C₄

Dic₃₈

yes

characteristic

107

C₂³×C₄

D₇₆

yes

characteristic

108

C₂³×C₄

C₂×Dic₁₉

yes

characteristic

109

C₂×M₁₆

—

110

C₂×M₁₆

—

111

C₂×M₁₆

—

112

C₂×M₁₆

C₁₉⋊C₈

yes

complemented

113

C₂²×C₈

—

114

C₂²×C₈

—

115

C₂×C₄²

—

116

C₂×C₄²

C₄×D₁₉

yes

—

117

C₂×C₄²

C₂²⋊D₁₉

yes

—

118

C₂²⋊C₈

—

119

C₄×C₈

—

120

C₁₉

#7026

yes

characteristic, complemented, Sylow

121

C₂⁴

C₄⋊Dic₁₉

yes

characteristic

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

—

128

C₂²×C₄

—

129

C₂²×C₄

C₂×C₁₉⋊C₈

yes

—

130

C₂²×C₄

#64520

yes

—

131

C₂²×C₄

C₄×Dic₁₉

yes

characteristic

132

C₂²×C₄

Dic₁₉⋊C₄

yes

—

133

C₂²×C₄

(C₂×D₁₉)⋊C₄

yes

—

134

C₂²×C₄

C₂²⋊Dic₁₉

yes

characteristic

135

M₁₆

—

136

M₁₆

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₄²

—

144

C₄²

—

145

C₂³

—

146

C₂³

—

147

C₂³

—

148

C₂³

—

149

C₂³

C₄⋊(C₁₉⋊C₈)

yes

—

150

C₂³

C₁₉⋊#95

yes

characteristic

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

C₄×C₁₉⋊C₈

yes

characteristic

164

C₂×C₄

#10597260

yes

characteristic

165

C₂×C₄

#10597279

yes

—

166

C₂×C₄

#10597282

yes

—

167

C₂×C₄

C₂²⋊(C₁₉⋊C₈)

yes

—

168

C₂×C₄

#10597290

yes

characteristic

169

C₂×C₄

#10597291

yes

—

170

C₂×C₄

#10597293

yes

characteristic

171

C₈

—

172

C₈

—

173

C₂²

—

174

C₂²

—

175

C₂²

—

176

C₂²

—

177

C₂²

—

178

C₂²

#11926299

yes

characteristic

179

C₂²

#11926305

yes

—

180

C₂²

#11926308

yes

—

181

C₂²

#11926311

yes

characteristic

182

C₂²

#11926313

yes

characteristic

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

—

187

C₄

#11926268

yes

—

188

C₄

#11926270

yes

—

189

C₂

—

190

C₂

#14561473

yes

—

191

C₂

#14561478

yes

characteristic

192

C₂

#14561520

yes

characteristic

193

C₂

#14561522

yes

characteristic

194

C₂

#14561524

yes

—

	subgroup	G/H
center	C₂²×C₄	C₂²⋊Dic₁₉
commutator	C₂×C₄×C₁₉	C₄×C₈
Frattini	C₂²×C₄²	C₂×D₁₉
Fitting	C₁₉×#1728	C₂
socle	C₂³×C₁₉	#95
prime radical	C₂⁴×C₁₉	C₄⋊C₄
cosocle	C₂²×C₄²×C₁₉	C₂²
hypercenter	#1728	D₁₉
hypocenter	C₁₉	#7026
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #1728

subgroup

G/H

center

C₂²×C₄

C₂²⋊Dic₁₉

commutator

C₂×C₄×C₁₉

C₄×C₈

Frattini

C₂²×C₄²

C₂×D₁₉

Fitting

C₁₉×#1728

C₂

socle

C₂³×C₁₉

#95

prime radical

C₂⁴×C₁₉

C₄⋊C₄

cosocle

C₂²×C₄²×C₁₉

C₂²

D₁₉

C₁₉

G ▷ C₂×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #1728

N	H ≅ G/N	type
15. C₂×M₁₆×C₁₉	C₈	semidirect
112. C₂×M₁₆	C₁₉⋊C₈	semidirect
120. C₁₉	#7026	semidirect

H ≅ G/N

type

15. C₂×M₁₆×C₁₉

C₈

semidirect

112. C₂×M₁₆

C₁₉⋊C₈

semidirect

120. C₁₉

#7026

semidirect

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

subgroup	index
C₄×C₂²⋊(C₁₉⋊C₈)	2
C₁₉×#1728	2
#7026	19

subgroup

index

C₄×C₂²⋊(C₁₉⋊C₈)

C₁₉×#1728

#7026

Subgroups — #18069092

Conjugacy classes of subgroups

Complements

Maximal subgroups