Group #18081251 — subgroups

Subgroups — #18081251

Order 4864.

290 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 119 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	#14564408	C₂	1	2	yes	maximal
2	C₂×#11926295	C₂	1	1	yes	maximal, characteristic
3	#14561924	C₂	1	1	yes	maximal, characteristic
4	#14561530	C₂	1	2	yes	maximal
5	C₁₉×#1567	C₂	1	1	yes	maximal, characteristic
6	#11926597	C₂²	1	1	yes	characteristic
7	C₂×#10597279	—	2	2	no	—
8	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
9	C₄×#64520	C₂²	1	1	yes	characteristic
10	#11926308	—	2	2	no	—
11	#11926295	C₄	1	2	yes	—
12	#11926295	C₄	1	2	yes	—
13	#11926266	—	2	1	no	—
14	#11926265	—	2	1	no	—
15	C₁₉×#423	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×#335	—	2	1	no	—
18	C₁₉×#335	—	2	1	no	—
19	C₂×M₁₆×C₁₉	—	2	2	no	—
20	C₂²×C₈×C₁₉	Q₈	1	1	yes	characteristic
21	C₂²×C₈×C₁₉	Q₈	1	1	yes	characteristic
22	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
23	C₂×#64520	D₄	1	1	yes	characteristic
24	C₂×#64520	D₄	1	1	yes	characteristic
25	C₂²×C₁₉⋊C₈	—	2	2	no	—
26	C₁₉×C₄⋊C₈	—	2	2	no	—
27	C₁₉×C₄⋊C₈	—	2	2	no	—
28	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
31	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
32	C₂²⋊(C₁₉⋊C₈)	Q₈	1	2	yes	—
33	#10597279	—	4	4	no	—
34	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
35	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
36	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	4	yes	—
37	#10597260	—	2	1	no	—
38	C₄×C₁₉⋊C₈	—	2	1	no	—
39	C₄×C₁₉⋊C₈	—	2	1	no	—
40	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
43	M₁₆×C₁₉	—	4	2	no	—
44	M₁₆×C₁₉	—	4	2	no	—
45	C₂×C₈×C₁₉	—	2	2	no	—
46	C₂×C₈×C₁₉	—	4	1	no	—
47	C₂×C₈×C₁₉	—	4	2	no	—
48	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
52	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
54	C₄²×C₁₉	#41	1	1	yes	characteristic
55	C₄²×C₁₉	#41	1	1	yes	characteristic
56	#64520	—	2	2	no	—
57	#64520	—	2	2	no	—
58	C₂×C₁₉⋊C₈	—	2	4	no	—
59	C₂×C₁₉⋊C₈	—	2	1	no	—
60	C₂×C₁₉⋊C₈	—	2	1	no	—
61	C₂×C₁₉⋊C₈	—	2	2	no	—
62	C₂×C₁₉⋊C₈	—	4	2	no	—
63	C₂×C₁₉⋊C₈	—	4	2	no	—
64	#9626	—	19	1	no	maximal, Sylow
65	C₂³×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₁₉	—	2	1	no	—
67	C₂×C₄×C₁₉	—	2	1	no	—
68	C₂×C₄×C₁₉	—	4	1	no	—
69	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
73	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₁₉	#125	1	1	yes	characteristic
75	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
76	C₂×C₄×C₁₉	#137	1	2	yes	—
77	C₈×C₁₉	—	2	1	no	—
78	C₈×C₁₉	—	2	1	no	—
79	C₈×C₁₉	—	2	1	no	—
80	C₈×C₁₉	—	8	2	no	—
81	C₈×C₁₉	#108	1	2	yes	complemented
82	C₁₉⋊C₈	—	4	2	no	—
83	C₁₉⋊C₈	—	4	4	no	—
84	C₁₉⋊C₈	—	4	1	no	—
85	C₁₉⋊C₈	—	4	1	no	—
86	#1950	—	19	2	no	—
87	#1567	D₁₉	1	1	yes	characteristic
88	C₂×#334	—	19	1	no	—
89	#1451	—	19	1	no	—
90	#1301	—	19	2	no	—
91	C₂²×C₁₉	—	2	1	no	—
92	C₂²×C₁₉	—	2	1	no	—
93	C₂²×C₁₉	#389	1	1	yes	characteristic
94	C₂²×C₁₉	#501	1	2	yes	—
95	C₄×C₁₉	—	2	1	no	—
96	C₄×C₁₉	—	2	1	no	—
97	C₄×C₁₉	—	2	1	no	—
98	C₄×C₁₉	C₂×#108	1	1	yes	characteristic
99	C₄×C₁₉	#430	1	1	yes	characteristic
100	C₄×C₁₉	#440	1	2	yes	—
101	C₄×C₁₉	#469	1	1	yes	characteristic
102	C₄×C₁₉	#470	1	1	yes	characteristic
103	#431	—	19	1	no	—
104	C₂×#108	—	38	2	no	—
105	#423	C₂×D₁₉	1	2	yes	—
106	C₂×C₄⋊C₈	—	19	2	no	—
107	C₄×M₁₆	—	19	1	no	—
108	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
109	#341	—	38	2	no	—
110	#335	—	2	1	no	—
111	#335	—	2	1	no	—
112	#334	—	19	2	no	—
113	#334	—	19	2	no	—
114	#322	—	38	1	no	—
115	#322	—	38	1	no	—
116	C₂×C₁₉	—	2	2	no	—
117	C₂×C₁₉	#1944	1	1	yes	characteristic
118	C₂×C₁₉	#1950	1	1	yes	characteristic
119	C₂×C₁₉	#1953	1	1	yes	characteristic
120	C₂×M₁₆	—	2	2	no	—
121	C₂×M₁₆	—	19	1	no	—
122	C₂×M₁₆	—	19	1	no	—
123	C₂²×C₈	—	38	2	no	—
124	C₂²×C₈	Dic₃₈	1	1	yes	characteristic
125	C₂²×C₈	Dic₃₈	1	1	yes	characteristic
126	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
127	#108	—	76	4	no	—
128	C₄⋊C₈	—	2	2	no	—
129	C₄⋊C₈	—	2	2	no	—
130	C₄⋊C₈	—	19	4	no	—
131	C₄⋊C₈	—	38	2	no	—
132	C₄⋊C₈	—	38	1	no	—
133	C₂²⋊C₈	—	19	2	no	—
134	#97	—	38	1	no	—
135	C₄×C₈	—	38	1	no	—
136	C₄×C₈	—	38	1	no	—
137	C₄×C₈	C₄×D₁₉	1	1	yes	characteristic
138	C₄×C₈	C₄×D₁₉	1	1	yes	characteristic
139	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
140	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
141	C₁₉	#9626	1	1	yes	characteristic, complemented, Sylow
142	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
143	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
144	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
145	M₁₆	—	4	2	no	—
146	M₁₆	—	4	2	no	—
147	M₁₆	—	38	2	no	—
148	M₁₆	—	38	2	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	4	1	no	—
151	C₂×C₈	—	4	2	no	—
152	C₂×C₈	—	38	1	no	—
153	C₂×C₈	—	38	1	no	—
154	C₂×C₈	—	38	2	no	—
155	C₂×C₈	—	38	4	no	—
156	C₂×C₈	—	76	2	no	—
157	C₂×C₈	—	76	2	no	—
158	C₂×C₈	Dic₁₉⋊C₄	1	1	yes	characteristic
159	C₂×C₈	Dic₁₉⋊C₄	1	1	yes	characteristic
160	C₂×C₈	Dic₁₉⋊C₄	1	1	yes	characteristic
161	C₂×C₈	Dic₁₉⋊C₄	1	1	yes	characteristic
162	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
163	C₄²	#64543	1	1	yes	characteristic
164	C₄²	#64545	1	1	yes	characteristic
165	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
166	C₂³	#10597334	1	1	yes	characteristic
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	4	1	no	—
170	C₂×C₄	#10597333	1	1	yes	characteristic
171	C₂×C₄	#10597336	1	1	yes	characteristic
172	C₂×C₄	#10597339	1	1	yes	characteristic
173	C₂×C₄	#10597357	1	2	yes	—
174	C₂×C₄	#10597364	1	2	yes	—
175	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
176	C₂×C₄	#10597384	1	1	yes	characteristic
177	C₂×C₄	#10597392	1	1	yes	characteristic
178	C₈	—	2	1	no	—
179	C₈	—	2	1	no	—
180	C₈	—	2	1	no	—
181	C₈	—	8	2	no	—
182	C₈	—	76	2	no	—
183	C₈	—	76	1	no	—
184	C₈	—	76	1	no	—
185	C₈	—	76	4	no	—
186	C₈	#10597279	1	2	yes	complemented
187	C₂²	—	2	1	no	—
188	C₂²	—	2	1	no	—
189	C₂²	#11926440	1	1	yes	characteristic
190	C₂²	#11926952	1	2	yes	—
191	C₄	—	2	1	no	—
192	C₄	—	2	1	no	—
193	C₄	—	2	1	no	—
194	C₄	C₂×#10597279	1	1	yes	characteristic
195	C₄	#11926592	1	1	yes	characteristic
196	C₄	#11926643	1	2	yes	—
197	C₄	#11926794	1	1	yes	characteristic
198	C₄	#11926799	1	1	yes	characteristic
199	C₂	—	2	2	no	—
200	C₂	#14564376	1	1	yes	characteristic
201	C₂	#14564408	1	1	yes	characteristic
202	C₂	#14564423	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14564408

C₂

yes

maximal

C₂×#11926295

C₂

yes

maximal, characteristic

#14561924

C₂

yes

maximal, characteristic

#14561530

C₂

yes

maximal

C₁₉×#1567

C₂

yes

maximal, characteristic

#11926597

C₂²

yes

characteristic

C₂×#10597279

—

C₂×C₄⋊(C₁₉⋊C₈)

C₂²

yes

—

C₄×#64520

C₂²

yes

characteristic

#11926308

—

#11926295

C₄

yes

—

#11926295

C₄

yes

—

#11926266

—

#11926265

—

C₁₉×#423

C₂²

yes

—

C₂×C₄×C₈×C₁₉

C₂²

yes

characteristic

C₁₉×#335

—

C₁₉×#335

—

C₂×M₁₆×C₁₉

—

C₂²×C₈×C₁₉

Q₈

yes

characteristic

C₂²×C₈×C₁₉

Q₈

yes

characteristic

C₂×C₄²×C₁₉

C₂³

yes

characteristic

C₂×#64520

D₄

yes

characteristic

C₂×#64520

D₄

yes

characteristic

C₂²×C₁₉⋊C₈

—

C₁₉×C₄⋊C₈

—

C₁₉×C₄⋊C₈

—

C₄×C₈×C₁₉

C₂×C₄

yes

characteristic

C₄×C₈×C₁₉

C₂×C₄

yes

characteristic

C₄×C₈×C₁₉

D₄

yes

characteristic

C₄×C₈×C₁₉

D₄

yes

characteristic

C₂²⋊(C₁₉⋊C₈)

Q₈

yes

—

#10597279

—

C₄⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

C₂×C₄

yes

—

#10597260

—

C₄×C₁₉⋊C₈

—

C₄×C₁₉⋊C₈

—

C₂²×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₉

C₂×Q₈

yes

characteristic

C₂²×C₄×C₁₉

C₂×Q₈

yes

characteristic

M₁₆×C₁₉

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₉

C₄⋊C₄

yes

characteristic

C₄²×C₁₉

C₂²×C₄

yes

characteristic

C₄²×C₁₉

C₂×D₄

yes

C₄²×C₁₉

yes

C₄²×C₁₉

yes

—

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

#9626

—

maximal, Sylow

C₂³×C₁₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₉

C₄×D₄

yes

characteristic

C₂×C₄×C₁₉

C₄×Q₈

yes

characteristic

C₂×C₄×C₁₉

#121

yes

characteristic

C₂×C₄×C₁₉

C₂²⋊Q₈

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

—

C₂×C₄×C₁₉

#137

yes

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

#108

yes

complemented

C₁₉⋊C₈

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

#1950

—

#1567

D₁₉

yes

—

—

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#389

yes

characteristic

C₂²×C₁₉

#501

yes

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

yes

C₄×C₁₉

yes

100

C₄×C₁₉

#440

yes

—

101

C₄×C₁₉

#469

yes

characteristic

102

C₄×C₁₉

#470

yes

characteristic

103

#431

—

104

C₂×#108

—

105

#423

C₂×D₁₉

yes

—

106

C₂×C₄⋊C₈

—

107

C₄×M₁₆

—

108

C₂×C₄×C₈

C₂×D₁₉

yes

characteristic

109

#341

—

110

#335

—

111

#335

—

112

#334

—

113

#334

—

114

#322

—

115

#322

—

116

C₂×C₁₉

—

117

C₂×C₁₉

#1944

yes

characteristic

118

C₂×C₁₉

#1950

yes

characteristic

119

C₂×C₁₉

#1953

yes

characteristic

120

C₂×M₁₆

—

121

C₂×M₁₆

—

122

C₂×M₁₆

—

123

C₂²×C₈

—

124

C₂²×C₈

Dic₃₈

yes

characteristic

125

C₂²×C₈

Dic₃₈

yes

characteristic

126

C₂×C₄²

C₂²×D₁₉

yes

characteristic

127

#108

—

128

C₄⋊C₈

—

129

C₄⋊C₈

—

130

C₄⋊C₈

—

131

C₄⋊C₈

—

132

C₄⋊C₈

—

133

C₂²⋊C₈

—

134

#97

—

135

C₄×C₈

—

136

C₄×C₈

—

137

C₄×C₈

C₄×D₁₉

yes

characteristic

138

C₄×C₈

C₄×D₁₉

yes

characteristic

139

C₄×C₈

C₂²⋊D₁₉

yes

characteristic

140

C₄×C₈

C₂²⋊D₁₉

yes

characteristic

141

C₁₉

#9626

yes

characteristic, complemented, Sylow

142

C₂²×C₄

C₂×Dic₃₈

yes

characteristic

143

C₂²×C₄

D₄×D₁₉

yes

characteristic

144

C₂²×C₄

Q₈×D₁₉

yes

characteristic

145

M₁₆

—

146

M₁₆

—

147

M₁₆

—

148

M₁₆

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

Dic₁₉⋊C₄

yes

characteristic

159

C₂×C₈

Dic₁₉⋊C₄

yes

characteristic

160

C₂×C₈

Dic₁₉⋊C₄

yes

characteristic

161

C₂×C₈

Dic₁₉⋊C₄

yes

characteristic

162

C₄²

C₂×C₄×D₁₉

yes

characteristic

163

C₄²

#64543

yes

characteristic

164

C₄²

#64545

yes

characteristic

165

C₄²

C₂×C₂²⋊D₁₉

yes

characteristic

166

C₂³

#10597334

yes

characteristic

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

#10597333

yes

characteristic

171

C₂×C₄

#10597336

yes

characteristic

172

C₂×C₄

#10597339

yes

characteristic

173

C₂×C₄

#10597357

yes

—

174

C₂×C₄

#10597364

yes

—

175

C₂×C₄

C₂×Dic₁₉⋊C₄

yes

characteristic

176

C₂×C₄

#10597384

yes

characteristic

177

C₂×C₄

#10597392

yes

characteristic

178

C₈

—

179

C₈

—

180

C₈

—

181

C₈

—

182

C₈

—

183

C₈

—

184

C₈

—

185

C₈

—

186

C₈

#10597279

yes

complemented

187

C₂²

—

188

C₂²

—

189

C₂²

#11926440

yes

characteristic

190

C₂²

#11926952

yes

—

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

C₂×#10597279

yes

characteristic

195

C₄

#11926592

yes

characteristic

196

C₄

#11926643

yes

—

197

C₄

#11926794

yes

characteristic

198

C₄

#11926799

yes

characteristic

199

C₂

—

200

C₂

#14564376

yes

characteristic

201

C₂

#14564408

yes

characteristic

202

C₂

#14564423

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#10597384
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1567	C₂
socle	C₂²×C₁₉	#389
prime radical	C₂³×C₁₉	C₄⋊Q₈
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1567	D₁₉
hypocenter	C₁₉	#9626
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

subgroup

G/H

center

C₂×C₄

#10597384

commutator

C₄²×C₁₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₉

Fitting

C₁₉×#1567

C₂

socle

C₂²×C₁₉

#389

prime radical

C₂³×C₁₉

C₄⋊Q₈

cosocle

C₂×C₄²×C₁₉

C₂³

D₁₉

C₁₉

G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

N	H ≅ G/N	type
81. C₈×C₁₉	#108	semidirect
141. C₁₉	#9626	semidirect
186. C₈	#10597279	semidirect

H ≅ G/N

type

81. C₈×C₁₉

#108

semidirect

141. C₁₉

#9626

semidirect

186. C₈

#10597279

semidirect

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

subgroup	index
#14564408	2
C₂×#11926295	2
#14561924	2
#14561530	2
C₁₉×#1567	2
#9626	19

subgroup

index

C₁₉×#1567

Subgroups — #18081251

Conjugacy classes of subgroups

Complements

Maximal subgroups