Group #14574764 — subgroups

Subgroups — #14574764

Order 2432.

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

normal subgroups only

This group has 404 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	#11927095	C₂	1	1	yes	maximal, characteristic
2	C₄⋊Q₈×D₁₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#11926940	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926892	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926891	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926864	C₂	1	4	yes	maximal, complemented
7	#11926833	C₂	1	4	yes	maximal, complemented
8	#11926595	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₉×#498	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#64545	—	2	1	no	—
11	C₂×Q₈×D₁₉	—	2	1	no	—
12	C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	#10597393	—	2	1	no	—
15	#10597390	—	2	1	no	—
16	#10597341	—	2	1	no	—
17	#10597341	C₂²	1	2	yes	—
18	#10597339	—	2	1	no	—
19	#10597339	C₂²	1	2	yes	—
20	C₄⋊C₄×D₁₉	—	2	1	no	—
21	C₄⋊C₄×D₁₉	C₂²	1	2	yes	—
22	#10597334	—	2	1	no	—
23	#10597334	C₂²	1	2	yes	—
24	#10597320	C₂²	1	1	yes	characteristic
25	C₄²×D₁₉	C₂²	1	1	yes	characteristic
26	#10597315	C₂²	1	1	yes	characteristic
27	C₁₉×#106	C₂²	1	4	yes	—
28	C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
29	#10597278	C₂²	1	4	yes	—
30	(C₂×D₁₉)⋊C₈	C₂²	1	2	yes	—
31	Dic₁₉⋊C₈	C₂²	1	2	yes	—
32	#10597266	C₂²	1	4	yes	—
33	#10597265	C₂²	1	4	yes	—
34	C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
35	C₂×Q₈×C₁₉	—	2	1	no	—
36	C₂×Q₈×C₁₉	—	2	1	no	—
37	#64545	—	4	4	no	—
38	Q₈×D₁₉	—	4	4	no	—
39	C₂×D₇₆	—	2	2	no	—
40	C₂×D₇₆	—	2	1	no	—
41	C₂×D₇₆	—	2	1	no	—
42	C₂×D₇₆	C₂³	1	2	yes	—
43	C₂×C₄×D₁₉	—	2	2	no	—
44	C₂×C₄×D₁₉	—	2	2	no	—
45	C₂×C₄×D₁₉	D₄	1	1	yes	characteristic
46	C₂×C₄×D₁₉	D₄	1	1	yes	characteristic
47	C₂×C₄×D₁₉	C₂³	1	1	yes	characteristic
48	C₂×Dic₃₈	—	2	1	no	—
49	C₂×Dic₃₈	—	2	2	no	—
50	C₂×C₈×C₁₉	C₂³	1	2	yes	—
51	C₄⋊C₄×C₁₉	—	2	1	no	—
52	C₄⋊C₄×C₁₉	—	2	1	no	—
53	C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
54	C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
55	C₄²×C₁₉	C₂³	1	1	yes	characteristic
56	(C₂×D₁₉)⋊C₄	—	2	2	no	—
57	(C₂×D₁₉)⋊C₄	—	2	2	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	C₂²×D₁₉	—	2	2	no	—
67	C₂²×D₁₉	C₂×D₄	1	1	yes	characteristic
68	Q₈×C₁₉	—	4	2	no	—
69	Q₈×C₁₉	—	4	2	no	—
70	C₂×C₄×C₁₉	—	2	2	no	—
71	C₂×C₄×C₁₉	—	2	2	no	—
72	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
75	C₂×Dic₁₉	—	2	2	no	—
76	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
77	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
78	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
79	D₇₆	—	2	4	no	—
80	D₇₆	—	4	4	no	—
81	D₇₆	—	4	2	no	—
82	D₇₆	—	4	2	no	—
83	C₄×D₁₉	—	2	2	no	—
84	C₄×D₁₉	—	2	2	no	—
85	C₄×D₁₉	—	4	4	no	—
86	C₄×D₁₉	—	4	4	no	—
87	C₄×D₁₉	SD₁₆	1	4	yes	—
88	Dic₃₈	—	4	4	no	—
89	Dic₃₈	—	4	2	no	—
90	C₈×C₁₉	—	2	2	no	—
91	C₁₉⋊C₈	—	2	2	no	—
92	#3227	—	19	1	no	maximal, Sylow
93	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
94	C₂×D₁₉	—	2	1	no	—
95	C₂×D₁₉	—	2	1	no	—
96	C₂×D₁₉	—	4	2	no	—
97	C₂×D₁₉	—	4	2	no	—
98	C₂×D₁₉	—	4	2	no	—
99	C₂×D₁₉	C₂×SD₁₆	1	2	yes	—
100	C₄×C₁₉	—	4	2	no	—
101	C₄×C₁₉	—	4	2	no	—
102	C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
103	C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
104	C₄×C₁₉	2₋¹⁺⁴	1	2	yes	—
105	Dic₁₉	—	2	1	no	—
106	Dic₁₉	—	2	1	no	—
107	Dic₁₉	—	4	2	no	—
108	Dic₁₉	C₂×SD₁₆	1	2	yes	—
109	#532	—	19	1	no	—
110	C₂×C₄⋊Q₈	—	19	1	no	—
111	#498	—	19	1	no	—
112	#498	D₁₉	1	1	yes	characteristic, complemented
113	#486	—	19	1	no	—
114	#486	—	19	1	no	—
115	#481	—	19	4	no	—
116	#476	—	19	4	no	—
117	#431	—	19	1	no	—
118	C₂×C₁₉	#536	1	1	yes	characteristic
119	C₂×C₁₉	C₂²×SD₁₆	1	1	yes	characteristic
120	C₂×C₁₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
121	D₁₉	—	2	2	no	—
122	D₁₉	—	8	2	no	—
123	C₂×#41	—	38	1	no	—
124	C₂²×Q₈	—	38	1	no	—
125	C₄⋊Q₈	—	19	2	no	—
126	C₄⋊Q₈	—	19	1	no	—
127	C₄⋊Q₈	—	38	1	no	—
128	C₄⋊Q₈	—	38	1	no	—
129	C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
130	C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
131	#127	—	19	1	no	—
132	#124	—	38	1	no	—
133	#121	—	19	2	no	—
134	#121	—	38	1	no	—
135	C₄×D₄	—	19	2	no	—
136	C₄×D₄	—	38	1	no	—
137	C₂×C₄⋊C₄	—	19	2	no	—
138	C₂×C₄⋊C₄	—	38	1	no	—
139	C₂×C₄²	—	19	1	no	—
140	#106	—	19	4	no	—
141	#106	C₂×D₁₉	1	4	yes	—
142	C₄⋊C₈	—	19	2	no	—
143	#102	—	19	4	no	—
144	#102	—	19	4	no	—
145	C₂²⋊C₈	—	19	2	no	—
146	C₄×C₈	—	19	1	no	—
147	C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
148	C₁₉	#3227	1	1	yes	characteristic, complemented, Sylow
149	#41	—	76	4	no	—
150	C₂×Q₈	—	2	1	no	—
151	C₂×Q₈	—	2	1	no	—
152	C₂×Q₈	—	38	2	no	—
153	C₂×Q₈	—	38	1	no	—
154	C₂×Q₈	—	76	4	no	—
155	C₂×D₄	—	19	2	no	—
156	C₂×D₄	—	38	2	no	—
157	C₂×D₄	—	38	1	no	—
158	C₂×D₄	—	38	1	no	—
159	C₂²×C₄	—	19	1	no	—
160	C₂²×C₄	—	19	1	no	—
161	C₂²×C₄	—	19	1	no	—
162	C₂²×C₄	—	38	2	no	—
163	C₂²×C₄	—	38	2	no	—
164	C₂×C₈	—	19	2	no	—
165	C₂×C₈	C₂²×D₁₉	1	2	yes	—
166	C₄⋊C₄	—	2	1	no	—
167	C₄⋊C₄	—	2	1	no	—
168	C₄⋊C₄	—	19	2	no	—
169	C₄⋊C₄	—	38	2	no	—
170	C₄⋊C₄	—	38	2	no	—
171	C₄⋊C₄	—	38	1	no	—
172	C₄⋊C₄	C₂²×D₁₉	1	2	yes	—
173	C₄⋊C₄	C₂²×D₁₉	1	2	yes	—
174	C₂²⋊C₄	—	38	2	no	—
175	C₂²⋊C₄	—	38	2	no	—
176	C₄²	—	19	1	no	—
177	C₄²	—	19	1	no	—
178	C₄²	—	19	1	no	—
179	C₄²	C₂²×D₁₉	1	1	yes	characteristic
180	C₂³	—	19	1	no	—
181	C₂³	—	38	2	no	—
182	Q₈	—	4	2	no	—
183	Q₈	—	4	2	no	—
184	Q₈	—	76	4	no	—
185	Q₈	—	76	2	no	—
186	D₄	—	38	4	no	—
187	D₄	—	76	4	no	—
188	D₄	—	76	2	no	—
189	D₄	—	76	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	19	1	no	—
193	C₂×C₄	—	19	4	no	—
194	C₂×C₄	—	19	1	no	—
195	C₂×C₄	—	19	1	no	—
196	C₂×C₄	—	38	2	no	—
197	C₂×C₄	—	38	2	no	—
198	C₂×C₄	—	38	2	no	—
199	C₂×C₄	—	76	4	no	—
200	C₂×C₄	—	76	4	no	—
201	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
202	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
203	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
204	C₈	—	2	2	no	—
205	C₈	—	38	2	no	—
206	C₂²	—	19	2	no	—
207	C₂²	—	38	1	no	—
208	C₂²	—	38	1	no	—
209	C₂²	—	76	2	no	—
210	C₂²	—	76	2	no	—
211	C₂²	—	76	2	no	—
212	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
213	C₄	—	4	2	no	—
214	C₄	—	4	2	no	—
215	C₄	—	19	2	no	—
216	C₄	—	38	1	no	—
217	C₄	—	38	1	no	—
218	C₄	—	76	2	no	—
219	C₄	SD₁₆×D₁₉	1	2	yes	—
220	C₄	#10597360	1	2	yes	—
221	C₄	#10597433	1	2	yes	—
222	C₂	—	38	2	no	—
223	C₂	—	152	2	no	—
224	C₂	#11927119	1	1	yes	characteristic
225	C₂	C₂×SD₁₆×D₁₉	1	1	yes	characteristic
226	C₂	C₂×#10597360	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11927095

C₂

yes

maximal, characteristic

C₄⋊Q₈×D₁₉

C₂

yes

maximal, characteristic, complemented

#11926940

C₂

yes

maximal, characteristic, complemented

#11926892

C₂

yes

maximal, characteristic, complemented

#11926891

C₂

yes

maximal, characteristic, complemented

#11926864

C₂

yes

maximal, complemented

#11926833

C₂

yes

maximal, complemented

#11926595

C₂

yes

maximal, characteristic, complemented

C₁₉×#498

C₂

yes

maximal, characteristic, complemented

C₂×#64545

—

C₂×Q₈×D₁₉

—

C₁₉×C₄⋊Q₈

C₂²

yes

characteristic

C₁₉×C₄⋊Q₈

C₂²

yes

—

—

—

C₂²

yes

—

#10597339

—

#10597339

C₂²

yes

—

C₄⋊C₄×D₁₉

—

C₄⋊C₄×D₁₉

C₂²

yes

—

#10597334

—

#10597334

C₂²

yes

—

#10597320

C₂²

yes

characteristic

C₄²×D₁₉

C₂²

yes

characteristic

#10597315

C₂²

yes

characteristic

C₁₉×#106

C₂²

yes

—

C₄×C₈×C₁₉

C₂²

yes

characteristic

#10597278

C₂²

yes

—

(C₂×D₁₉)⋊C₈

C₂²

yes

—

Dic₁₉⋊C₈

C₂²

yes

—

#10597266

C₂²

yes

—

#10597265

C₂²

yes

—

C₄×C₁₉⋊C₈

C₂²

yes

characteristic

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

#64545

—

Q₈×D₁₉

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

C₂³

yes

—

C₂×C₄×D₁₉

—

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×C₈×C₁₉

C₂³

yes

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

C₂³

yes

—

C₄⋊C₄×C₁₉

C₂³

yes

—

C₄²×C₁₉

C₂³

yes

characteristic

(C₂×D₁₉)⋊C₄

—

(C₂×D₁₉)⋊C₄

—

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

—

C₂²×D₁₉

—

C₂²×D₁₉

C₂×D₄

yes

characteristic

Q₈×C₁₉

—

Q₈×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉

C₂⁴

yes

characteristic

C₂×Dic₁₉

—

C₂×Dic₁₉

C₂×D₄

yes

characteristic

C₂×Dic₁₉

C₂×D₄

yes

characteristic

C₂×Dic₁₉

C₂×D₄

yes

characteristic

D₇₆

—

D₇₆

—

D₇₆

—

D₇₆

—

C₄×D₁₉

—

C₄×D₁₉

—

C₄×D₁₉

—

C₄×D₁₉

—

C₄×D₁₉

SD₁₆

yes

—

Dic₃₈

—

Dic₃₈

—

C₈×C₁₉

—

C₁₉⋊C₈

—

#3227

—

maximal, Sylow

C₂²×C₁₉

C₂²×D₄

yes

characteristic

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

C₂×SD₁₆

yes

—

100

C₄×C₁₉

—

101

C₄×C₁₉

—

102

C₄×C₁₉

C₂×SD₁₆

yes

—

103

C₄×C₁₉

AGL(1,ℤ/8)

yes

—

104

C₄×C₁₉

2₋¹⁺⁴

yes

—

105

Dic₁₉

—

106

Dic₁₉

—

107

Dic₁₉

—

108

Dic₁₉

C₂×SD₁₆

yes

—

109

#532

—

110

C₂×C₄⋊Q₈

—

111

#498

—

112

#498

D₁₉

yes

characteristic, complemented

113

#486

—

114

#486

—

115

#481

—

116

#476

—

117

#431

—

118

C₂×C₁₉

#536

yes

characteristic

119

C₂×C₁₉

C₂²×SD₁₆

yes

characteristic

120

C₂×C₁₉

C₂×AGL(1,ℤ/8)

yes

characteristic

121

D₁₉

—

122

D₁₉

—

123

C₂×#41

—

124

C₂²×Q₈

—

125

C₄⋊Q₈

—

126

C₄⋊Q₈

—

127

C₄⋊Q₈

—

128

C₄⋊Q₈

—

129

C₄⋊Q₈

C₂×D₁₉

yes

characteristic

130

C₄⋊Q₈

C₂×D₁₉

yes

characteristic

131

#127

—

132

#124

—

133

#121

—

134

#121

—

135

C₄×D₄

—

136

C₄×D₄

—

137

C₂×C₄⋊C₄

—

138

C₂×C₄⋊C₄

—

139

C₂×C₄²

—

140

#106

—

141

#106

C₂×D₁₉

yes

—

142

C₄⋊C₈

—

143

#102

—

144

#102

—

145

C₂²⋊C₈

—

146

C₄×C₈

—

147

C₄×C₈

C₂×D₁₉

yes

characteristic

148

C₁₉

#3227

yes

characteristic, complemented, Sylow

149

#41

—

150

C₂×Q₈

—

151

C₂×Q₈

—

152

C₂×Q₈

—

153

C₂×Q₈

—

154

C₂×Q₈

—

155

C₂×D₄

—

156

C₂×D₄

—

157

C₂×D₄

—

158

C₂×D₄

—

159

C₂²×C₄

—

160

C₂²×C₄

—

161

C₂²×C₄

—

162

C₂²×C₄

—

163

C₂²×C₄

—

164

C₂×C₈

—

165

C₂×C₈

C₂²×D₁₉

yes

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

168

C₄⋊C₄

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

171

C₄⋊C₄

—

172

C₄⋊C₄

C₂²×D₁₉

yes

—

173

C₄⋊C₄

C₂²×D₁₉

yes

—

174

C₂²⋊C₄

—

175

C₂²⋊C₄

—

176

C₄²

—

177

C₄²

—

178

C₄²

—

179

C₄²

C₂²×D₁₉

yes

characteristic

180

C₂³

—

181

C₂³

—

182

Q₈

—

183

Q₈

—

184

Q₈

—

185

Q₈

—

186

D₄

—

187

D₄

—

188

D₄

—

189

D₄

—

190

C₂×C₄

—

191

C₂×C₄

—

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

D₄×D₁₉

yes

characteristic

202

C₂×C₄

D₄×D₁₉

yes

characteristic

203

C₂×C₄

C₂³×D₁₉

yes

characteristic

204

C₈

—

205

C₈

—

206

C₂²

—

207

C₂²

—

208

C₂²

—

209

C₂²

—

210

C₂²

—

211

C₂²

—

212

C₂²

C₂×D₄×D₁₉

yes

characteristic

213

C₄

—

214

C₄

—

215

C₄

—

216

C₄

—

217

C₄

—

218

C₄

—

219

C₄

SD₁₆×D₁₉

yes

—

220

C₄

#10597360

yes

—

221

C₄

#10597433

yes

—

222

C₂

—

223

C₂

—

152

—

224

C₂

#11927119

yes

characteristic

225

C₂

C₂×SD₁₆×D₁₉

yes

characteristic

226

C₂

C₂×#10597360

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₉
commutator, cosocle	C₂×C₄×C₁₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉
Fitting	C₁₉×#498	C₂
socle	C₂²×C₁₉	C₂²×D₄
prime radical	#11927095	C₂
hypercenter	#498	D₁₉
hypocenter	C₁₉	#3227
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

C₂×D₄×D₁₉

commutator, cosocle

C₂×C₄×C₁₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₉

Fitting

C₁₉×#498

C₂

socle

C₂²×C₁₉

C₂²×D₄

C₂

D₁₉

C₁₉

G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
2. C₄⋊Q₈×D₁₉	C₂	semidirect
3. #11926940	C₂	semidirect
4. #11926892	C₂	semidirect
5. #11926891	C₂	semidirect
6. #11926864	C₂	semidirect
7. #11926833	C₂	semidirect
8. #11926595	C₂	semidirect
9. C₁₉×#498	C₂	semidirect
112. #498	D₁₉	semidirect
148. C₁₉	#3227	semidirect

H ≅ G/N

type

2. C₄⋊Q₈×D₁₉

C₂

semidirect

3. #11926940

C₂

semidirect

4. #11926892

C₂

semidirect

5. #11926891

C₂

semidirect

6. #11926864

C₂

semidirect

7. #11926833

C₂

semidirect

8. #11926595

C₂

semidirect

9. C₁₉×#498

C₂

semidirect

112. #498

D₁₉

semidirect

148. C₁₉

#3227

semidirect

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

subgroup	index
#11927095	2
C₄⋊Q₈×D₁₉	2
#11926940	2
#11926892	2
#11926891	2
#11926864	2
#11926833	2
#11926595	2
C₁₉×#498	2
#3227	19

subgroup

index

C₄⋊Q₈×D₁₉

C₁₉×#498

Subgroups — #14574764

Conjugacy classes of subgroups

Complements

Maximal subgroups