Group #14574025 — subgroups

Subgroups — #14574025

Order 2432.

424 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 125 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	#11927092	C₂	1	1	yes	maximal, characteristic
2	C₄⋊Q₈×D₁₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#11926902	C₂	1	4	yes	maximal, complemented
4	#11926896	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926895	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926890	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926542	C₂	1	4	yes	maximal, complemented
8	C₄×#64515	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₉×#486	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Q₈×D₁₉	—	2	1	no	—
11	C₂×#64543	C₂²	1	2	yes	—
12	C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₁₉×#127	C₂²	1	1	yes	characteristic
14	#10597390	—	2	1	no	—
15	#10597384	—	2	1	no	—
16	#10597384	—	2	1	no	—
17	#10597382	C₂²	1	2	yes	—
18	D₄×Dic₁₉	—	2	1	no	—
19	D₄×Dic₁₉	—	2	1	no	—
20	C₂×#64515	C₂²	1	2	yes	—
21	C₄⋊C₄×D₁₉	—	2	1	no	—
22	C₄⋊C₄×D₁₉	C₂²	1	2	yes	—
23	#10597334	—	2	1	no	—
24	#10597334	C₂²	1	2	yes	—
25	C₄²×D₁₉	C₂²	1	1	yes	characteristic
26	#10597315	C₂²	1	1	yes	characteristic
27	#10597315	C₂²	1	1	yes	characteristic
28	C₁₉×#102	C₂²	1	4	yes	—
29	C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
30	#10597289	C₂²	1	4	yes	—
31	#10597273	C₂²	1	4	yes	—
32	#10597272	C₂²	1	2	yes	—
33	#10597267	C₂²	1	4	yes	—
34	C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
35	C₂×Q₈×C₁₉	—	2	1	no	—
36	C₂×D₄×C₁₉	—	2	1	no	—
37	C₂×D₄×C₁₉	—	2	1	no	—
38	C₂×D₄×C₁₉	C₂³	1	2	yes	—
39	C₂×C₂²⋊D₁₉	—	2	2	no	—
40	C₂²×Dic₁₉	—	2	2	no	—
41	Q₈×D₁₉	—	4	4	no	—
42	#64543	—	2	8	no	—
43	C₂×C₄×D₁₉	—	2	2	no	—
44	C₂×C₄×D₁₉	D₄	1	1	yes	characteristic
45	C₂×C₄×D₁₉	D₄	1	1	yes	characteristic
46	C₂×C₄×D₁₉	C₂³	1	1	yes	characteristic
47	C₂×Dic₃₈	—	2	1	no	—
48	C₂×Dic₃₈	—	2	2	no	—
49	C₂×Dic₃₈	C₂³	1	2	yes	—
50	C₂×C₈×C₁₉	C₂³	1	2	yes	—
51	C₄⋊C₄×C₁₉	—	2	1	no	—
52	C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
53	C₄²×C₁₉	C₂³	1	1	yes	characteristic
54	C₂²⋊Dic₁₉	—	2	2	no	—
55	C₂²⋊Dic₁₉	—	2	2	no	—
56	C₄⋊Dic₁₉	—	2	1	no	—
57	C₄⋊Dic₁₉	—	2	1	no	—
58	C₄⋊Dic₁₉	—	2	1	no	—
59	C₄⋊Dic₁₉	C₂³	1	2	yes	—
60	Dic₁₉⋊C₄	—	2	2	no	—
61	Dic₁₉⋊C₄	—	2	2	no	—
62	C₄×Dic₁₉	D₄	1	1	yes	characteristic
63	C₄×Dic₁₉	D₄	1	1	yes	characteristic
64	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
65	C₂×C₁₉⋊C₈	C₂³	1	2	yes	—
66	#64515	—	2	4	no	—
67	C₂³×C₁₉	—	2	2	no	—
68	C₂²×D₁₉	C₂×D₄	1	1	yes	characteristic
69	Q₈×C₁₉	—	4	2	no	—
70	D₄×C₁₉	—	2	4	no	—
71	D₄×C₁₉	—	4	2	no	—
72	D₄×C₁₉	—	4	2	no	—
73	C₂×C₄×C₁₉	—	2	2	no	—
74	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
77	C₂²⋊D₁₉	—	4	4	no	—
78	C₂×Dic₁₉	—	2	2	no	—
79	C₂×Dic₁₉	—	2	2	no	—
80	C₂×Dic₁₉	—	4	4	no	—
81	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
82	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
83	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
84	C₄×D₁₉	—	2	2	no	—
85	C₄×D₁₉	—	2	2	no	—
86	C₄×D₁₉	—	4	4	no	—
87	C₄×D₁₉	#41	1	4	yes	—
88	Dic₃₈	—	2	4	no	—
89	Dic₃₈	—	4	4	no	—
90	Dic₃₈	—	4	2	no	—
91	C₈×C₁₉	—	2	2	no	—
92	C₁₉⋊C₈	—	2	2	no	—
93	#3141	—	19	1	no	maximal, Sylow
94	C₂²×C₁₉	—	4	2	no	—
95	C₂²×C₁₉	—	4	2	no	—
96	C₂²×C₁₉	—	4	2	no	—
97	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
98	C₂×D₁₉	—	2	1	no	—
99	C₂×D₁₉	—	2	1	no	—
100	C₂×D₁₉	#124	1	2	yes	—
101	C₄×C₁₉	—	4	2	no	—
102	C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
103	C₄×C₁₉	#137	1	2	yes	—
104	C₄×C₁₉	C₂×#41	1	2	yes	—
105	Dic₁₉	—	2	1	no	—
106	Dic₁₉	—	2	1	no	—
107	Dic₁₉	—	4	2	no	—
108	Dic₁₉	—	4	2	no	—
109	Dic₁₉	#124	1	2	yes	—
110	#532	—	19	1	no	—
111	C₂×C₄⋊Q₈	—	19	1	no	—
112	#489	—	19	4	no	—
113	#487	—	19	1	no	—
114	#487	—	19	1	no	—
115	#486	—	19	1	no	—
116	#486	D₁₉	1	1	yes	characteristic, complemented
117	#417	—	19	4	no	—
118	C₄×M₁₆	—	19	1	no	—
119	C₂×C₁₉	—	8	2	no	—
120	C₂×C₁₉	C₂×#124	1	1	yes	characteristic
121	C₂×C₁₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
122	C₂×C₁₉	C₂×#137	1	1	yes	characteristic
123	D₁₉	—	2	2	no	—
124	C₂×#41	—	19	2	no	—
125	C₂²×Q₈	—	38	1	no	—
126	C₂×M₁₆	—	19	2	no	—
127	C₄⋊Q₈	—	19	2	no	—
128	C₄⋊Q₈	—	19	1	no	—
129	C₄⋊Q₈	—	19	1	no	—
130	C₄⋊Q₈	—	38	1	no	—
131	C₄⋊Q₈	—	38	1	no	—
132	C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
133	#127	C₂×D₁₉	1	1	yes	characteristic
134	#124	—	19	2	no	—
135	#121	—	38	1	no	—
136	#121	—	38	1	no	—
137	C₄×D₄	—	38	1	no	—
138	C₄×D₄	—	38	1	no	—
139	C₂×C₄⋊C₄	—	19	2	no	—
140	C₂×C₄⋊C₄	—	38	1	no	—
141	C₂×C₄²	—	19	1	no	—
142	#103	—	19	4	no	—
143	#103	—	19	4	no	—
144	#102	—	19	4	no	—
145	#102	C₂×D₁₉	1	4	yes	—
146	#97	—	19	2	no	—
147	C₄×C₈	—	19	1	no	—
148	C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
149	C₁₉	#3141	1	1	yes	characteristic, complemented, Sylow
150	#41	—	38	8	no	—
151	C₂×Q₈	—	2	1	no	—
152	C₂×Q₈	—	19	2	no	—
153	C₂×Q₈	—	38	2	no	—
154	C₂×Q₈	—	38	1	no	—
155	C₂×Q₈	—	76	4	no	—
156	C₂×D₄	—	2	1	no	—
157	C₂×D₄	—	2	1	no	—
158	C₂×D₄	—	38	2	no	—
159	C₂×D₄	C₂²×D₁₉	1	2	yes	—
160	C₂²×C₄	—	19	1	no	—
161	C₂²×C₄	—	19	1	no	—
162	C₂²×C₄	—	19	1	no	—
163	C₂²×C₄	—	38	2	no	—
164	C₂²×C₄	—	38	2	no	—
165	M₁₆	—	38	4	no	—
166	C₂×C₈	—	19	2	no	—
167	C₂×C₈	C₂²×D₁₉	1	2	yes	—
168	C₄⋊C₄	—	2	1	no	—
169	C₄⋊C₄	—	19	2	no	—
170	C₄⋊C₄	—	38	2	no	—
171	C₄⋊C₄	—	38	2	no	—
172	C₄⋊C₄	—	38	1	no	—
173	C₄⋊C₄	—	38	1	no	—
174	C₄⋊C₄	—	38	1	no	—
175	C₄⋊C₄	C₂²×D₁₉	1	2	yes	—
176	C₂²⋊C₄	—	38	2	no	—
177	C₂²⋊C₄	—	38	2	no	—
178	C₄²	—	19	1	no	—
179	C₄²	—	19	1	no	—
180	C₄²	—	19	1	no	—
181	C₄²	C₂²×D₁₉	1	1	yes	characteristic
182	C₂³	—	2	2	no	—
183	C₂³	—	19	1	no	—
184	Q₈	—	4	2	no	—
185	Q₈	—	38	4	no	—
186	Q₈	—	76	4	no	—
187	Q₈	—	76	2	no	—
188	D₄	—	2	4	no	—
189	D₄	—	4	2	no	—
190	D₄	—	4	2	no	—
191	D₄	—	76	4	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	19	1	no	—
194	C₂×C₄	—	19	4	no	—
195	C₂×C₄	—	19	1	no	—
196	C₂×C₄	—	19	1	no	—
197	C₂×C₄	—	38	2	no	—
198	C₂×C₄	—	38	2	no	—
199	C₂×C₄	—	38	2	no	—
200	C₂×C₄	—	38	2	no	—
201	C₂×C₄	—	76	4	no	—
202	C₂×C₄	—	76	4	no	—
203	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
204	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
205	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
206	C₈	—	2	2	no	—
207	C₈	—	38	2	no	—
208	C₂²	—	4	2	no	—
209	C₂²	—	4	2	no	—
210	C₂²	—	4	2	no	—
211	C₂²	—	19	2	no	—
212	C₂²	—	38	1	no	—
213	C₂²	—	38	1	no	—
214	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
215	C₄	—	4	2	no	—
216	C₄	—	19	2	no	—
217	C₄	—	38	1	no	—
218	C₄	—	38	1	no	—
219	C₄	—	76	2	no	—
220	C₄	—	76	2	no	—
221	C₄	#10597357	1	2	yes	—
222	C₄	#10597361	1	2	yes	—
223	C₄	#41×D₁₉	1	2	yes	—
224	C₂	—	8	2	no	—
225	C₂	—	38	2	no	—
226	C₂	#124×D₁₉	1	1	yes	characteristic
227	C₂	C₂×#10597357	1	1	yes	characteristic
228	C₂	C₂×#10597361	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11927092

C₂

yes

maximal, characteristic

C₄⋊Q₈×D₁₉

C₂

yes

maximal, characteristic, complemented

#11926902

C₂

yes

maximal, complemented

#11926896

C₂

yes

maximal, characteristic, complemented

#11926895

C₂

yes

maximal, characteristic, complemented

#11926890

C₂

yes

maximal, characteristic, complemented

#11926542

C₂

yes

maximal, complemented

C₄×#64515

C₂

yes

maximal, characteristic, complemented

C₁₉×#486

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₁₉

—

C₂×#64543

C₂²

yes

—

C₁₉×C₄⋊Q₈

C₂²

yes

characteristic

C₁₉×#127

C₂²

yes

—

—

—

C₂²

yes

—

D₄×Dic₁₉

—

D₄×Dic₁₉

—

C₂×#64515

C₂²

yes

—

C₄⋊C₄×D₁₉

—

C₄⋊C₄×D₁₉

C₂²

yes

—

#10597334

—

#10597334

C₂²

yes

—

C₄²×D₁₉

C₂²

yes

characteristic

#10597315

C₂²

yes

characteristic

#10597315

C₂²

yes

characteristic

C₁₉×#102

C₂²

yes

—

C₄×C₈×C₁₉

C₂²

yes

characteristic

#10597289

C₂²

yes

—

#10597273

C₂²

yes

—

#10597272

C₂²

yes

—

#10597267

C₂²

yes

—

C₄×C₁₉⋊C₈

C₂²

yes

characteristic

C₂×Q₈×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

C₂³

yes

—

C₂×C₂²⋊D₁₉

—

C₂²×Dic₁₉

—

Q₈×D₁₉

—

#64543

—

C₂×C₄×D₁₉

—

C₂×C₄×D₁₉

D₄

yes

characteristic

C₂×C₄×D₁₉

D₄

yes

characteristic

C₂×C₄×D₁₉

C₂³

yes

characteristic

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×Dic₃₈

C₂³

yes

—

C₂×C₈×C₁₉

C₂³

yes

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

C₂³

yes

—

C₄²×C₁₉

C₂³

yes

characteristic

C₂²⋊Dic₁₉

—

C₂²⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

C₂³

yes

—

Dic₁₉⋊C₄

—

Dic₁₉⋊C₄

—

C₄×Dic₁₉

D₄

yes

characteristic

C₄×Dic₁₉

D₄

yes

characteristic

C₄×Dic₁₉

C₂³

yes

characteristic

C₂×C₁₉⋊C₈

C₂³

yes

—

#64515

—

C₂³×C₁₉

—

C₂²×D₁₉

C₂×D₄

yes

characteristic

Q₈×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉

C₂⁴

yes

characteristic

C₂²⋊D₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

C₂×D₄

yes

characteristic

C₂×Dic₁₉

C₂×D₄

yes

characteristic

C₂×Dic₁₉

C₂×D₄

yes

characteristic

C₄×D₁₉

—

C₄×D₁₉

—

C₄×D₁₉

—

C₄×D₁₉

#41

yes

—

Dic₃₈

—

Dic₃₈

—

Dic₃₈

—

C₈×C₁₉

—

C₁₉⋊C₈

—

#3141

—

maximal, Sylow

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

C₂²×D₄

yes

characteristic

C₂×D₁₉

—

C₂×D₁₉

—

100

C₂×D₁₉

#124

yes

—

101

C₄×C₁₉

—

102

C₄×C₁₉

AGL(1,ℤ/8)

yes

—

103

C₄×C₁₉

#137

yes

—

104

C₄×C₁₉

C₂×#41

yes

—

105

Dic₁₉

—

106

Dic₁₉

—

107

Dic₁₉

—

108

Dic₁₉

—

109

Dic₁₉

#124

yes

—

110

#532

—

111

C₂×C₄⋊Q₈

—

112

#489

—

113

#487

—

114

#487

—

115

#486

—

116

#486

D₁₉

yes

characteristic, complemented

117

#417

—

118

C₄×M₁₆

—

119

C₂×C₁₉

—

120

C₂×C₁₉

C₂×#124

yes

characteristic

121

C₂×C₁₉

C₂×AGL(1,ℤ/8)

yes

characteristic

122

C₂×C₁₉

C₂×#137

yes

characteristic

123

D₁₉

—

124

C₂×#41

—

125

C₂²×Q₈

—

126

C₂×M₁₆

—

127

C₄⋊Q₈

—

128

C₄⋊Q₈

—

129

C₄⋊Q₈

—

130

C₄⋊Q₈

—

131

C₄⋊Q₈

—

132

C₄⋊Q₈

C₂×D₁₉

yes

characteristic

133

#127

C₂×D₁₉

yes

characteristic

134

#124

—

135

#121

—

136

#121

—

137

C₄×D₄

—

138

C₄×D₄

—

139

C₂×C₄⋊C₄

—

140

C₂×C₄⋊C₄

—

141

C₂×C₄²

—

142

#103

—

143

#103

—

144

#102

—

145

#102

C₂×D₁₉

yes

—

146

#97

—

147

C₄×C₈

—

148

C₄×C₈

C₂×D₁₉

yes

characteristic

149

C₁₉

#3141

yes

characteristic, complemented, Sylow

150

#41

—

151

C₂×Q₈

—

152

C₂×Q₈

—

153

C₂×Q₈

—

154

C₂×Q₈

—

155

C₂×Q₈

—

156

C₂×D₄

—

157

C₂×D₄

—

158

C₂×D₄

—

159

C₂×D₄

C₂²×D₁₉

yes

—

160

C₂²×C₄

—

161

C₂²×C₄

—

162

C₂²×C₄

—

163

C₂²×C₄

—

164

C₂²×C₄

—

165

M₁₆

—

166

C₂×C₈

—

167

C₂×C₈

C₂²×D₁₉

yes

—

168

C₄⋊C₄

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

171

C₄⋊C₄

—

172

C₄⋊C₄

—

173

C₄⋊C₄

—

174

C₄⋊C₄

—

175

C₄⋊C₄

C₂²×D₁₉

yes

—

176

C₂²⋊C₄

—

177

C₂²⋊C₄

—

178

C₄²

—

179

C₄²

—

180

C₄²

—

181

C₄²

C₂²×D₁₉

yes

characteristic

182

C₂³

—

183

C₂³

—

184

Q₈

—

185

Q₈

—

186

Q₈

—

187

Q₈

—

188

D₄

—

189

D₄

—

190

D₄

—

191

D₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

202

C₂×C₄

—

203

C₂×C₄

D₄×D₁₉

yes

characteristic

204

C₂×C₄

D₄×D₁₉

yes

characteristic

205

C₂×C₄

C₂³×D₁₉

yes

characteristic

206

C₈

—

207

C₈

—

208

C₂²

—

209

C₂²

—

210

C₂²

—

211

C₂²

—

212

C₂²

—

213

C₂²

—

214

C₂²

C₂×D₄×D₁₉

yes

characteristic

215

C₄

—

216

C₄

—

217

C₄

—

218

C₄

—

219

C₄

—

220

C₄

—

221

C₄

#10597357

yes

—

222

C₄

#10597361

yes

—

223

C₄

#41×D₁₉

yes

—

224

C₂

—

225

C₂

—

226

C₂

#124×D₁₉

yes

characteristic

227

C₂

C₂×#10597357

yes

characteristic

228

C₂

C₂×#10597361

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₉
commutator, cosocle	C₂×C₄×C₁₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉
Fitting	C₁₉×#486	C₂
socle	C₂²×C₁₉	C₂²×D₄
prime radical	#11927092	C₂
hypercenter	#486	D₁₉
hypocenter	C₁₉	#3141
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #486

subgroup

G/H

center

C₂²

C₂×D₄×D₁₉

commutator, cosocle

C₂×C₄×C₁₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₉

Fitting

C₁₉×#486

C₂

socle

C₂²×C₁₉

C₂²×D₄

C₂

D₁₉

C₁₉

G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #486

N	H ≅ G/N	type
2. C₄⋊Q₈×D₁₉	C₂	semidirect
3. #11926902	C₂	semidirect
4. #11926896	C₂	semidirect
5. #11926895	C₂	semidirect
6. #11926890	C₂	semidirect
7. #11926542	C₂	semidirect
8. C₄×#64515	C₂	semidirect
9. C₁₉×#486	C₂	semidirect
116. #486	D₁₉	semidirect
149. C₁₉	#3141	semidirect

H ≅ G/N

type

2. C₄⋊Q₈×D₁₉

C₂

semidirect

3. #11926902

C₂

semidirect

4. #11926896

C₂

semidirect

5. #11926895

C₂

semidirect

6. #11926890

C₂

semidirect

7. #11926542

C₂

semidirect

8. C₄×#64515

C₂

semidirect

9. C₁₉×#486

C₂

semidirect

116. #486

D₁₉

semidirect

149. C₁₉

#3141

semidirect

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

subgroup	index
#11927092	2
C₄⋊Q₈×D₁₉	2
#11926902	2
#11926896	2
#11926895	2
#11926890	2
#11926542	2
C₄×#64515	2
C₁₉×#486	2
#3141	19

subgroup

index

C₄⋊Q₈×D₁₉

C₁₉×#486

Subgroups — #14574025

Conjugacy classes of subgroups

Complements

Maximal subgroups