Group #14574712 — subgroups

Subgroups — #14574712

Order 2432.

364 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 117 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	#11927120	C₂	1	1	yes	maximal, characteristic
2	#11926936	C₂	1	1	yes	maximal, characteristic, complemented
3	#11926923	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926914	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926901	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926766	C₂	1	2	yes	maximal, complemented
7	#11926754	C₂	1	2	yes	maximal, complemented
8	#11926704	C₂	1	2	yes	maximal
9	#11926699	C₂	1	2	yes	maximal, complemented
10	#11926605	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₉×#541	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#64541	C₂²	1	1	yes	characteristic
13	C₂²×Dic₃₈	C₂²	1	1	yes	characteristic
14	C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₁₉×#126	—	2	2	no	—
16	C₁₉×#125	—	2	2	no	—
17	C₁₉×#125	C₂²	1	1	yes	characteristic
18	C₁₉×C₂²⋊Q₈	C₂²	1	2	yes	—
19	C₄×Q₈×C₁₉	C₂²	1	2	yes	—
20	C₁₉×#117	C₂²	1	1	yes	characteristic
21	C₂×#64528	C₂²	1	2	yes	—
22	C₂×#64528	C₂²	1	2	yes	—
23	C₂×#64528	C₂²	1	2	yes	—
24	C₂×#64527	C₂²	1	2	yes	—
25	#10597343	—	2	2	no	—
26	#10597332	—	2	2	no	—
27	#10597321	C₂²	1	1	yes	characteristic
28	#10597321	C₂²	1	1	yes	characteristic
29	#10597315	C₂²	1	1	yes	characteristic
30	#10597315	C₂²	1	1	yes	characteristic
31	C₂²⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
32	#10597267	C₂²	1	2	yes	—
33	#10597267	C₂²	1	2	yes	—
34	#10597267	C₂²	1	2	yes	—
35	#10597266	C₂²	1	2	yes	—
36	C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
37	#10597260	C₂²	1	1	yes	characteristic
38	C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
39	C₂×Q₈×C₁₉	C₂³	1	2	yes	—
40	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
41	C₂×C₂²⋊D₁₉	—	2	1	no	—
42	C₂²×Dic₁₉	—	2	1	no	—
43	#64541	—	4	2	no	—
44	C₂×D₇₆	C₂³	1	1	yes	characteristic
45	C₂×C₄×D₁₉	—	2	1	no	—
46	C₂×Dic₃₈	—	2	1	no	—
47	C₂×Dic₃₈	—	4	2	no	—
48	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
49	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
50	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
51	C₄⋊C₄×C₁₉	—	2	2	no	—
52	C₄⋊C₄×C₁₉	—	2	2	no	—
53	C₄⋊C₄×C₁₉	—	2	2	no	—
54	C₄⋊C₄×C₁₉	—	2	2	no	—
55	C₄⋊C₄×C₁₉	D₄	1	2	yes	—
56	C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
57	C₂²⋊C₄×C₁₉	—	2	2	no	—
58	C₂²⋊C₄×C₁₉	D₄	1	2	yes	—
59	C₄²×C₁₉	—	2	2	no	—
60	C₄²×C₁₉	C₂³	1	1	yes	characteristic
61	C₄²×C₁₉	C₂³	1	1	yes	characteristic
62	#64528	—	4	2	no	—
63	#64528	—	4	2	no	—
64	#64528	—	4	2	no	—
65	#64527	—	4	2	no	—
66	(C₂×D₁₉)⋊C₄	—	2	2	no	—
67	(C₂×D₁₉)⋊C₄	—	2	2	no	—
68	C₄⋊Dic₁₉	—	2	2	no	—
69	C₄⋊Dic₁₉	—	2	2	no	—
70	C₂×C₁₉⋊C₈	C₂³	1	2	yes	—
71	C₂×C₁₉⋊C₈	C₂³	1	2	yes	—
72	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
73	C₂²×D₁₉	—	2	1	no	—
74	Q₈×C₁₉	—	2	2	no	—
75	Q₈×C₁₉	—	2	2	no	—
76	C₂×C₄×C₁₉	—	2	2	no	—
77	C₂×C₄×C₁₉	—	2	2	no	—
78	C₂×C₄×C₁₉	—	2	2	no	—
79	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
81	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
83	C₂²⋊D₁₉	—	4	2	no	—
84	C₂×Dic₁₉	—	2	1	no	—
85	C₂×Dic₁₉	—	2	1	no	—
86	C₂×Dic₁₉	—	2	1	no	—
87	C₂×Dic₁₉	—	4	2	no	—
88	D₇₆	—	2	2	no	—
89	C₄×D₁₉	—	4	2	no	—
90	Dic₃₈	—	2	2	no	—
91	Dic₃₈	—	2	2	no	—
92	Dic₃₈	—	2	2	no	—
93	Dic₃₈	—	4	2	no	—
94	C₁₉⋊C₈	—	2	2	no	—
95	C₁₉⋊C₈	—	2	2	no	—
96	#3223	—	19	1	no	maximal, Sylow
97	C₂²×C₁₉	—	2	1	no	—
98	C₂²×C₁₉	—	2	2	no	—
99	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
100	C₂×D₁₉	—	4	1	no	—
101	C₂×D₁₉	—	4	2	no	—
102	C₄×C₁₉	—	2	1	no	—
103	C₄×C₁₉	—	2	2	no	—
104	C₄×C₁₉	—	2	2	no	—
105	C₄×C₁₉	—	4	2	no	—
106	C₄×C₁₉	—	4	2	no	—
107	C₄×C₁₉	H(5,𝔽₂)	1	2	yes	—
108	Dic₁₉	—	4	1	no	—
109	Dic₁₉	—	4	1	no	—
110	Dic₁₉	—	4	1	no	—
111	#541	D₁₉	1	1	yes	characteristic, complemented
112	#536	—	19	1	no	—
113	#497	—	19	1	no	—
114	#494	—	19	1	no	—
115	#491	—	19	1	no	—
116	#488	—	19	1	no	—
117	#464	—	19	2	no	—
118	#462	—	19	2	no	—
119	#452	—	19	2	no	—
120	#451	—	19	2	no	—
121	#433	—	19	1	no	—
122	C₂×C₁₉	—	4	1	no	—
123	C₂×C₁₉	#535	1	1	yes	characteristic
124	C₂×C₁₉	#578	1	2	yes	—
125	D₁₉	—	8	1	no	—
126	C₂×#41	—	19	1	no	—
127	C₂²×Q₈	—	19	1	no	—
128	C₂×Q₁₆	—	19	2	no	—
129	C₂×Q₁₆	—	19	2	no	—
130	C₂×Q₁₆	—	19	2	no	—
131	C₂×SD₁₆	—	19	2	no	—
132	C₄⋊Q₈	—	19	1	no	—
133	C₄⋊Q₈	—	19	1	no	—
134	C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
135	#126	—	2	2	no	—
136	#125	—	2	2	no	—
137	#125	C₂×D₁₉	1	1	yes	characteristic
138	#124	—	19	1	no	—
139	#124	—	19	1	no	—
140	#123	—	38	2	no	—
141	C₂²⋊Q₈	—	38	2	no	—
142	C₂²⋊Q₈	C₂×D₁₉	1	2	yes	—
143	C₄×Q₈	C₂×D₁₉	1	2	yes	—
144	#117	C₂×D₁₉	1	1	yes	characteristic
145	C₄⋊C₈	—	19	2	no	—
146	#103	—	19	2	no	—
147	#103	—	19	2	no	—
148	#103	—	19	2	no	—
149	#102	—	19	2	no	—
150	C₂²⋊C₈	—	19	2	no	—
151	#97	—	19	1	no	—
152	C₄×C₈	—	19	1	no	—
153	C₁₉	#3223	1	1	yes	characteristic, complemented, Sylow
154	#41	—	76	2	no	—
155	C₂×Q₈	—	19	1	no	—
156	C₂×Q₈	—	19	1	no	—
157	C₂×Q₈	—	19	1	no	—
158	C₂×Q₈	—	38	1	no	—
159	C₂×Q₈	—	76	2	no	—
160	C₂×Q₈	C₂²×D₁₉	1	2	yes	—
161	C₂×D₄	—	19	1	no	—
162	C₂×D₄	—	38	1	no	—
163	C₂²×C₄	—	38	1	no	—
164	C₂²×C₄	—	38	1	no	—
165	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
166	Q₁₆	—	76	2	no	—
167	Q₁₆	—	76	2	no	—
168	Q₁₆	—	76	2	no	—
169	SD₁₆	—	76	2	no	—
170	C₂×C₈	—	19	2	no	—
171	C₂×C₈	—	19	2	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	38	2	no	—
177	C₄⋊C₄	—	38	2	no	—
178	C₄⋊C₄	C₂²⋊D₁₉	1	2	yes	—
179	C₄⋊C₄	C₂²×D₁₉	1	2	yes	—
180	C₂²⋊C₄	—	2	2	no	—
181	C₂²⋊C₄	—	38	2	no	—
182	C₂²⋊C₄	—	38	2	no	—
183	C₂²⋊C₄	C₂²⋊D₁₉	1	2	yes	—
184	C₄²	—	2	2	no	—
185	C₄²	C₂²×D₁₉	1	1	yes	characteristic
186	C₄²	C₂²×D₁₉	1	1	yes	characteristic
187	C₂³	—	38	1	no	—
188	C₂³	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
189	Q₈	—	2	2	no	—
190	Q₈	—	2	2	no	—
191	Q₈	—	38	2	no	—
192	Q₈	—	38	2	no	—
193	Q₈	—	38	2	no	—
194	Q₈	—	76	2	no	—
195	D₄	—	38	2	no	—
196	D₄	—	76	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	38	1	no	—
201	C₂×C₄	—	38	1	no	—
202	C₂×C₄	—	38	1	no	—
203	C₂×C₄	—	76	2	no	—
204	C₂×C₄	—	76	2	no	—
205	C₂×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
206	C₂×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
207	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
208	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
209	C₈	—	38	2	no	—
210	C₈	—	38	2	no	—
211	C₂²	—	2	2	no	—
212	C₂²	—	2	1	no	—
213	C₂²	—	76	2	no	—
214	C₂²	—	76	1	no	—
215	C₂²	C₂²×C₂²⋊D₁₉	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	2	2	no	—
218	C₄	—	2	1	no	—
219	C₄	—	4	2	no	—
220	C₄	—	4	2	no	—
221	C₄	—	76	1	no	—
222	C₄	—	76	1	no	—
223	C₄	—	76	1	no	—
224	C₄	#10597435	1	2	yes	—
225	C₂	—	4	1	no	—
226	C₂	—	152	1	no	—
227	C₂	#11927108	1	1	yes	characteristic
228	C₂	#11927359	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11927120

C₂

yes

maximal, characteristic

#11926936

C₂

yes

maximal, characteristic, complemented

#11926923

C₂

yes

maximal, characteristic, complemented

#11926914

C₂

yes

maximal, characteristic, complemented

#11926901

C₂

yes

maximal, characteristic, complemented

#11926766

C₂

yes

maximal, complemented

#11926754

C₂

yes

maximal, complemented

#11926704

C₂

yes

maximal

#11926699

C₂

yes

maximal, complemented

#11926605

C₂

yes

maximal, characteristic, complemented

C₁₉×#541

C₂

yes

maximal, characteristic, complemented

C₂×#64541

C₂²

yes

characteristic

C₂²×Dic₃₈

C₂²

yes

characteristic

C₁₉×C₄⋊Q₈

C₂²

yes

characteristic

C₁₉×#126

—

C₁₉×#125

—

C₁₉×#125

C₂²

yes

characteristic

C₁₉×C₂²⋊Q₈

C₂²

yes

—

C₄×Q₈×C₁₉

C₂²

yes

—

C₁₉×#117

C₂²

yes

characteristic

C₂×#64528

C₂²

yes

—

C₂×#64528

C₂²

yes

—

C₂×#64528

C₂²

yes

—

C₂×#64527

C₂²

yes

—

#10597343

—

#10597332

—

#10597321

C₂²

yes

characteristic

#10597321

C₂²

yes

characteristic

#10597315

C₂²

yes

characteristic

#10597315

C₂²

yes

characteristic

C₂²⋊(C₁₉⋊C₈)

C₂²

yes

—

#10597267

C₂²

yes

—

#10597267

C₂²

yes

—

#10597267

C₂²

yes

—

#10597266

C₂²

yes

—

C₄⋊(C₁₉⋊C₈)

C₂²

yes

—

#10597260

C₂²

yes

characteristic

C₄×C₁₉⋊C₈

C₂²

yes

characteristic

C₂×Q₈×C₁₉

C₂³

yes

—

C₂²×C₄×C₁₉

C₂³

yes

characteristic

C₂×C₂²⋊D₁₉

—

C₂²×Dic₁₉

—

#64541

—

C₂×D₇₆

C₂³

yes

characteristic

C₂×C₄×D₁₉

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×Dic₃₈

C₂³

yes

characteristic

C₂×Dic₃₈

C₂³

yes

characteristic

C₂×Dic₃₈

C₂³

yes

characteristic

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

D₄

yes

—

C₄⋊C₄×C₁₉

C₂³

yes

—

C₂²⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

D₄

yes

—

C₄²×C₁₉

—

C₄²×C₁₉

C₂³

yes

characteristic

C₄²×C₁₉

C₂³

yes

—

—

—

—

(C₂×D₁₉)⋊C₄

—

(C₂×D₁₉)⋊C₄

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₂×C₁₉⋊C₈

C₂³

yes

—

C₂×C₁₉⋊C₈

C₂³

yes

—

C₂³×C₁₉

C₂×D₄

yes

characteristic

C₂²×D₁₉

—

Q₈×C₁₉

—

Q₈×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₂×D₄

yes

—

C₂×C₄×C₁₉

C₂×D₄

yes

—

C₂×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉

C₂⁴

yes

characteristic

C₂²⋊D₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

D₇₆

—

C₄×D₁₉

—

Dic₃₈

—

Dic₃₈

—

Dic₃₈

—

Dic₃₈

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

#3223

—

maximal, Sylow

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

C₂²×D₄

yes

characteristic

100

C₂×D₁₉

—

101

C₂×D₁₉

—

102

C₄×C₁₉

—

103

C₄×C₁₉

—

104

C₄×C₁₉

—

105

C₄×C₁₉

—

106

C₄×C₁₉

—

107

C₄×C₁₉

H(5,𝔽₂)

yes

—

108

Dic₁₉

—

109

Dic₁₉

—

110

Dic₁₉

—

111

#541

D₁₉

yes

characteristic, complemented

112

#536

—

113

#497

—

114

#494

—

115

#491

—

116

#488

—

117

#464

—

118

#462

—

119

#452

—

120

#451

—

121

#433

—

122

C₂×C₁₉

—

123

C₂×C₁₉

#535

yes

characteristic

124

C₂×C₁₉

#578

yes

—

125

D₁₉

—

126

C₂×#41

—

127

C₂²×Q₈

—

128

C₂×Q₁₆

—

129

C₂×Q₁₆

—

130

C₂×Q₁₆

—

131

C₂×SD₁₆

—

132

C₄⋊Q₈

—

133

C₄⋊Q₈

—

134

C₄⋊Q₈

C₂×D₁₉

yes

characteristic

135

#126

—

136

#125

—

137

#125

C₂×D₁₉

yes

characteristic

138

#124

—

139

#124

—

140

#123

—

141

C₂²⋊Q₈

—

142

C₂²⋊Q₈

C₂×D₁₉

yes

—

143

C₄×Q₈

C₂×D₁₉

yes

—

144

#117

C₂×D₁₉

yes

characteristic

145

C₄⋊C₈

—

146

#103

—

147

#103

—

148

#103

—

149

#102

—

150

C₂²⋊C₈

—

151

#97

—

152

C₄×C₈

—

153

C₁₉

#3223

yes

characteristic, complemented, Sylow

154

#41

—

155

C₂×Q₈

—

156

C₂×Q₈

—

157

C₂×Q₈

—

158

C₂×Q₈

—

159

C₂×Q₈

—

160

C₂×Q₈

C₂²×D₁₉

yes

—

161

C₂×D₄

—

162

C₂×D₄

—

163

C₂²×C₄

—

164

C₂²×C₄

—

165

C₂²×C₄

C₂²×D₁₉

yes

characteristic

166

Q₁₆

—

167

Q₁₆

—

168

Q₁₆

—

169

SD₁₆

—

170

C₂×C₈

—

171

C₂×C₈

—

172

C₄⋊C₄

—

173

C₄⋊C₄

—

174

C₄⋊C₄

—

175

C₄⋊C₄

—

176

C₄⋊C₄

—

177

C₄⋊C₄

—

178

C₄⋊C₄

C₂²⋊D₁₉

yes

—

179

C₄⋊C₄

C₂²×D₁₉

yes

—

180

C₂²⋊C₄

—

181

C₂²⋊C₄

—

182

C₂²⋊C₄

—

183

C₂²⋊C₄

C₂²⋊D₁₉

yes

—

184

C₄²

—

185

C₄²

C₂²×D₁₉

yes

characteristic

186

C₄²

C₂²×D₁₉

yes

characteristic

187

C₂³

—

188

C₂³

C₂×C₂²⋊D₁₉

yes

characteristic

189

Q₈

—

190

Q₈

—

191

Q₈

—

192

Q₈

—

193

Q₈

—

194

Q₈

—

195

D₄

—

196

D₄

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

202

C₂×C₄

—

203

C₂×C₄

—

204

C₂×C₄

—

205

C₂×C₄

C₂×C₂²⋊D₁₉

yes

—

206

C₂×C₄

C₂×C₂²⋊D₁₉

yes

—

207

C₂×C₄

C₂×C₂²⋊D₁₉

yes

characteristic

208

C₂×C₄

C₂³×D₁₉

yes

characteristic

209

C₈

—

210

C₈

—

211

C₂²

—

212

C₂²

—

213

C₂²

—

214

C₂²

—

215

C₂²

C₂²×C₂²⋊D₁₉

yes

characteristic

216

C₄

—

217

C₄

—

218

C₄

—

219

C₄

—

220

C₄

—

221

C₄

—

222

C₄

—

223

C₄

—

224

C₄

#10597435

yes

—

225

C₂

—

226

C₂

—

152

—

227

C₂

#11927108

yes

characteristic

228

C₂

#11927359

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₉
commutator, cosocle	C₂×C₄×C₁₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉
Fitting	C₁₉×#541	C₂
socle	C₂²×C₁₉	C₂²×D₄
prime radical	C₂×#64541	C₂²
hypercenter	#541	D₁₉
hypocenter	C₁₉	#3223
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #541

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₉

commutator, cosocle

C₂×C₄×C₁₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₉

Fitting

C₁₉×#541

C₂

socle

C₂²×C₁₉

C₂²×D₄

C₂²

D₁₉

C₁₉

G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂² ◁ #117 ◁ #541

N	H ≅ G/N	type
2. #11926936	C₂	semidirect
3. #11926923	C₂	semidirect
4. #11926914	C₂	semidirect
5. #11926901	C₂	semidirect
6. #11926766	C₂	semidirect
7. #11926754	C₂	semidirect
9. #11926699	C₂	semidirect
10. #11926605	C₂	semidirect
11. C₁₉×#541	C₂	semidirect
111. #541	D₁₉	semidirect
153. C₁₉	#3223	semidirect

H ≅ G/N

type

2. #11926936

C₂

semidirect

3. #11926923

C₂

semidirect

4. #11926914

C₂

semidirect

5. #11926901

C₂

semidirect

6. #11926766

C₂

semidirect

7. #11926754

C₂

semidirect

9. #11926699

C₂

semidirect

10. #11926605

C₂

semidirect

11. C₁₉×#541

C₂

semidirect

111. #541

D₁₉

semidirect

153. C₁₉

#3223

semidirect

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

subgroup	index
#11927120	2
#11926936	2
#11926923	2
#11926914	2
#11926901	2
#11926766	2
#11926754	2
#11926704	2
#11926699	2
#11926605	2
C₁₉×#541	2
#3223	19

subgroup

index

C₁₉×#541

Subgroups — #14574712

Conjugacy classes of subgroups

Complements

Maximal subgroups