Group #14574593 — subgroups

Subgroups — #14574593

Order 2432.

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

normal subgroups only

This group has 394 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	#11927101	C₂	1	1	yes	maximal, characteristic, complemented
2	#11927094	C₂	1	1	yes	maximal, characteristic
3	#11926924	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926917	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926895	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926891	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926767	C₂	1	4	yes	maximal, complemented
8	#11926708	C₂	1	4	yes	maximal, complemented
9	C₁₉×#431	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#64541	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
12	#10597376	—	2	1	no	—
13	#10597376	—	2	1	no	—
14	#10597372	—	2	1	no	—
15	#10597370	—	2	1	no	—
16	#10597370	C₂²	1	2	yes	—
17	C₂×Dic₇₆	C₂²	1	4	yes	—
18	C₂×#64516	C₂²	1	4	yes	—
19	#10597321	C₂²	1	2	yes	—
20	#10597320	C₂²	1	1	yes	characteristic
21	C₄×D₇₆	—	2	1	no	—
22	C₄×D₇₆	—	2	1	no	—
23	#10597316	—	2	1	no	—
24	#10597315	C₂²	1	1	yes	characteristic
25	#10597315	C₂²	1	1	yes	characteristic
26	#10597315	C₂²	1	1	yes	characteristic
27	C₄×Dic₃₈	—	2	1	no	—
28	C₄×Dic₃₈	C₂²	1	2	yes	—
29	C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
30	C₁₉×C₂²⋊C₈	C₂²	1	2	yes	—
31	C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
32	C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
33	#10597278	C₂²	1	4	yes	—
34	#10597273	C₂²	1	4	yes	—
35	C₂²×C₄×C₁₉	D₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₉	D₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
38	C₂×C₂²⋊D₁₉	—	2	2	no	—
39	#64541	—	4	4	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×Dic₃₈	—	2	1	no	—
45	C₂×Dic₃₈	C₂³	1	2	yes	—
46	C₂×Dic₃₈	C₂³	1	2	yes	—
47	C₂×C₈×C₁₉	C₂³	1	2	yes	—
48	C₂×C₈×C₁₉	C₂³	1	2	yes	—
49	C₄²×C₁₉	D₄	1	1	yes	characteristic
50	C₄²×C₁₉	D₄	1	1	yes	characteristic
51	C₄²×C₁₉	C₂³	1	1	yes	characteristic
52	C₄²×C₁₉	C₂³	1	1	yes	characteristic
53	C₂²⋊Dic₁₉	—	2	2	no	—
54	(C₂×D₁₉)⋊C₄	—	2	2	no	—
55	(C₂×D₁₉)⋊C₄	—	2	2	no	—
56	C₄⋊Dic₁₉	—	2	1	no	—
57	C₄⋊Dic₁₉	—	2	1	no	—
58	C₄⋊Dic₁₉	—	2	1	no	—
59	C₄⋊Dic₁₉	—	2	1	no	—
60	C₄⋊Dic₁₉	—	2	1	no	—
61	C₄⋊Dic₁₉	C₂³	1	2	yes	—
62	Dic₁₉⋊C₄	—	2	2	no	—
63	Dic₁₉⋊C₄	—	2	2	no	—
64	C₄×Dic₁₉	—	2	2	no	—
65	Dic₇₆	—	4	4	no	—
66	#64516	—	4	4	no	—
67	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×D₁₉	—	2	2	no	—
69	C₂×C₄×C₁₉	—	2	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	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
78	C₂²⋊D₁₉	—	4	4	no	—
79	C₂×Dic₁₉	—	2	2	no	—
80	C₂×Dic₁₉	—	2	2	no	—
81	C₂×Dic₁₉	—	2	2	no	—
82	D₇₆	—	2	4	no	—
83	D₇₆	—	4	2	no	—
84	D₇₆	—	4	2	no	—
85	C₄×D₁₉	—	4	4	no	—
86	Dic₃₈	—	2	4	no	—
87	Dic₃₈	—	2	4	no	—
88	Dic₃₈	—	4	2	no	—
89	C₈×C₁₉	—	2	2	no	—
90	C₈×C₁₉	—	2	2	no	—
91	#3210	—	19	1	no	maximal, Sylow
92	C₂²×C₁₉	—	2	1	no	—
93	C₂²×C₁₉	—	2	1	no	—
94	C₂²×C₁₉	—	2	1	no	—
95	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
96	C₂×D₁₉	—	4	2	no	—
97	C₂×D₁₉	—	4	2	no	—
98	C₂×D₁₉	—	4	2	no	—
99	C₄×C₁₉	—	2	1	no	—
100	C₄×C₁₉	—	2	1	no	—
101	C₄×C₁₉	—	2	1	no	—
102	C₄×C₁₉	#135	1	2	yes	—
103	C₄×C₁₉	#137	1	2	yes	—
104	C₄×C₁₉	H(5,𝔽₂)	1	2	yes	—
105	Dic₁₉	—	4	2	no	—
106	Dic₁₉	—	4	2	no	—
107	Dic₁₉	—	4	2	no	—
108	#533	—	19	1	no	—
109	#532	—	19	1	no	—
110	#494	—	19	1	no	—
111	#492	—	19	1	no	—
112	#487	—	19	1	no	—
113	#486	—	19	1	no	—
114	#464	—	19	4	no	—
115	#452	—	19	4	no	—
116	#431	D₁₉	1	1	yes	characteristic, complemented
117	C₂×C₁₉	—	4	1	no	—
118	C₂×C₁₉	#535	1	1	yes	characteristic
119	C₂×C₁₉	C₂×#135	1	1	yes	characteristic
120	C₂×C₁₉	C₂×#137	1	1	yes	characteristic
121	D₁₉	—	8	2	no	—
122	C₂×#41	—	19	2	no	—
123	C₂×Q₁₆	—	19	4	no	—
124	C₂×SD₁₆	—	19	4	no	—
125	C₄⋊Q₈	—	19	1	no	—
126	C₄⋊Q₈	—	19	1	no	—
127	C₄⋊Q₈	—	19	1	no	—
128	#127	—	19	1	no	—
129	#125	—	38	1	no	—
130	#124	—	19	2	no	—
131	C₂²⋊Q₈	—	19	2	no	—
132	C₂²⋊Q₈	—	38	1	no	—
133	#121	—	38	1	no	—
134	#121	—	38	1	no	—
135	C₄×Q₈	—	19	2	no	—
136	C₄×Q₈	—	38	1	no	—
137	C₄×D₄	—	38	1	no	—
138	C₄×D₄	—	38	1	no	—
139	#117	—	38	1	no	—
140	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
141	C₄⋊C₈	C₂×D₁₉	1	2	yes	—
142	#103	—	19	4	no	—
143	#102	—	19	4	no	—
144	C₂²⋊C₈	C₂×D₁₉	1	2	yes	—
145	C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
146	C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
147	C₁₉	#3210	1	1	yes	characteristic, complemented, Sylow
148	#41	—	76	4	no	—
149	C₂×Q₈	—	19	2	no	—
150	C₂×Q₈	—	19	2	no	—
151	C₂×Q₈	—	38	1	no	—
152	C₂×D₄	—	19	2	no	—
153	C₂×D₄	—	38	2	no	—
154	C₂×D₄	—	38	1	no	—
155	C₂×D₄	—	38	1	no	—
156	C₂²×C₄	—	38	2	no	—
157	C₂²×C₄	D₇₆	1	1	yes	characteristic
158	C₂²×C₄	D₇₆	1	1	yes	characteristic
159	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
160	Q₁₆	—	76	4	no	—
161	SD₁₆	—	76	4	no	—
162	C₂×C₈	C₂²×D₁₉	1	2	yes	—
163	C₂×C₈	C₂²×D₁₉	1	2	yes	—
164	C₄⋊C₄	—	19	2	no	—
165	C₄⋊C₄	—	38	2	no	—
166	C₄⋊C₄	—	38	2	no	—
167	C₄⋊C₄	—	38	1	no	—
168	C₄⋊C₄	—	38	1	no	—
169	C₄⋊C₄	—	38	1	no	—
170	C₄⋊C₄	—	38	1	no	—
171	C₄⋊C₄	—	38	1	no	—
172	C₂²⋊C₄	—	38	2	no	—
173	C₂²⋊C₄	—	38	2	no	—
174	C₂²⋊C₄	—	38	2	no	—
175	C₄²	—	38	2	no	—
176	C₄²	D₇₆	1	1	yes	characteristic
177	C₄²	D₇₆	1	1	yes	characteristic
178	C₄²	C₂²×D₁₉	1	1	yes	characteristic
179	C₄²	C₂²×D₁₉	1	1	yes	characteristic
180	C₂³	—	38	2	no	—
181	C₂³	C₂×D₇₆	1	1	yes	characteristic
182	Q₈	—	38	4	no	—
183	Q₈	—	38	4	no	—
184	Q₈	—	76	2	no	—
185	D₄	—	38	4	no	—
186	D₄	—	76	4	no	—
187	D₄	—	76	2	no	—
188	D₄	—	76	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	38	2	no	—
193	C₂×C₄	—	38	2	no	—
194	C₂×C₄	—	38	2	no	—
195	C₂×C₄	—	76	4	no	—
196	C₂×C₄	C₂×D₇₆	1	1	yes	characteristic
197	C₂×C₄	C₂×D₇₆	1	1	yes	characteristic
198	C₂×C₄	C₂×D₇₆	1	1	yes	characteristic
199	C₂×C₄	C₂×D₇₆	1	1	yes	characteristic
200	C₂×C₄	C₂×D₇₆	1	1	yes	characteristic
201	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
202	C₈	—	2	2	no	—
203	C₈	—	2	2	no	—
204	C₂²	—	2	1	no	—
205	C₂²	—	2	1	no	—
206	C₂²	—	2	1	no	—
207	C₂²	—	76	2	no	—
208	C₂²	—	76	2	no	—
209	C₂²	—	76	2	no	—
210	C₂²	C₂²×D₇₆	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	76	2	no	—
215	C₄	—	76	2	no	—
216	C₄	—	76	2	no	—
217	C₄	#10597350	1	2	yes	—
218	C₄	#10597355	1	2	yes	—
219	C₄	#10597435	1	2	yes	—
220	C₂	—	4	1	no	—
221	C₂	—	152	2	no	—
222	C₂	#11927113	1	1	yes	characteristic
223	C₂	C₂×#10597350	1	1	yes	characteristic
224	C₂	C₂×#10597355	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11927101

C₂

yes

maximal, characteristic, complemented

#11927094

C₂

yes

maximal, characteristic

#11926924

C₂

yes

maximal, characteristic, complemented

#11926917

C₂

yes

maximal, characteristic, complemented

#11926895

C₂

yes

maximal, characteristic, complemented

#11926891

C₂

yes

maximal, characteristic, complemented

#11926767

C₂

yes

maximal, complemented

#11926708

C₂

yes

maximal, complemented

C₁₉×#431

C₂

yes

maximal, characteristic, complemented

C₂×#64541

C₂²

yes

—

C₂×C₄²×C₁₉

C₂²

yes

—

—

—

—

C₂²

yes

—

C₂×Dic₇₆

C₂²

yes

—

C₂×#64516

C₂²

yes

—

#10597321

C₂²

yes

—

#10597320

C₂²

yes

characteristic

C₄×D₇₆

—

C₄×D₇₆

—

#10597316

—

#10597315

C₂²

yes

characteristic

#10597315

C₂²

yes

characteristic

#10597315

C₂²

yes

characteristic

C₄×Dic₃₈

—

C₄×Dic₃₈

C₂²

yes

—

C₁₉×C₄⋊C₈

C₂²

yes

—

C₁₉×C₂²⋊C₈

C₂²

yes

—

C₄×C₈×C₁₉

C₂²

yes

characteristic

C₄×C₈×C₁₉

C₂²

yes

characteristic

#10597278

C₂²

yes

—

#10597273

C₂²

yes

—

C₂²×C₄×C₁₉

D₄

yes

characteristic

C₂²×C₄×C₁₉

D₄

yes

characteristic

C₂²×C₄×C₁₉

C₂³

yes

characteristic

C₂×C₂²⋊D₁₉

—

#64541

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

C₂³

yes

—

C₂×C₄×D₁₉

—

C₂×Dic₃₈

—

C₂×Dic₃₈

C₂³

yes

—

C₂×Dic₃₈

C₂³

yes

—

C₂×C₈×C₁₉

C₂³

yes

—

C₂×C₈×C₁₉

C₂³

yes

—

C₄²×C₁₉

D₄

yes

characteristic

C₄²×C₁₉

D₄

yes

characteristic

C₄²×C₁₉

C₂³

yes

characteristic

C₄²×C₁₉

C₂³

yes

characteristic

C₂²⋊Dic₁₉

—

(C₂×D₁₉)⋊C₄

—

(C₂×D₁₉)⋊C₄

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

C₂³

yes

—

Dic₁₉⋊C₄

—

Dic₁₉⋊C₄

—

C₄×Dic₁₉

—

Dic₇₆

—

#64516

—

C₂³×C₁₉

C₂×D₄

yes

characteristic

C₂²×D₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉

C₂×D₄

yes

characteristic

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

—

D₇₆

—

D₇₆

—

D₇₆

—

C₄×D₁₉

—

Dic₃₈

—

Dic₃₈

—

Dic₃₈

—

C₈×C₁₉

—

C₈×C₁₉

—

#3210

—

maximal, Sylow

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

C₂²×D₄

yes

characteristic

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

—

C₄×C₁₉

—

100

C₄×C₁₉

—

101

C₄×C₁₉

—

102

C₄×C₁₉

#135

yes

—

103

C₄×C₁₉

#137

yes

—

104

C₄×C₁₉

H(5,𝔽₂)

yes

—

105

Dic₁₉

—

106

Dic₁₉

—

107

Dic₁₉

—

108

#533

—

109

#532

—

110

#494

—

111

#492

—

112

#487

—

113

#486

—

114

#464

—

115

#452

—

116

#431

D₁₉

yes

characteristic, complemented

117

C₂×C₁₉

—

118

C₂×C₁₉

#535

yes

characteristic

119

C₂×C₁₉

C₂×#135

yes

characteristic

120

C₂×C₁₉

C₂×#137

yes

characteristic

121

D₁₉

—

122

C₂×#41

—

123

C₂×Q₁₆

—

124

C₂×SD₁₆

—

125

C₄⋊Q₈

—

126

C₄⋊Q₈

—

127

C₄⋊Q₈

—

128

#127

—

129

#125

—

130

#124

—

131

C₂²⋊Q₈

—

132

C₂²⋊Q₈

—

133

#121

—

134

#121

—

135

C₄×Q₈

—

136

C₄×Q₈

—

137

C₄×D₄

—

138

C₄×D₄

—

139

#117

—

140

C₂×C₄²

C₂×D₁₉

yes

characteristic

141

C₄⋊C₈

C₂×D₁₉

yes

—

142

#103

—

143

#102

—

144

C₂²⋊C₈

C₂×D₁₉

yes

—

145

C₄×C₈

C₂×D₁₉

yes

characteristic

146

C₄×C₈

C₂×D₁₉

yes

characteristic

147

C₁₉

#3210

yes

characteristic, complemented, Sylow

148

#41

—

149

C₂×Q₈

—

150

C₂×Q₈

—

151

C₂×Q₈

—

152

C₂×D₄

—

153

C₂×D₄

—

154

C₂×D₄

—

155

C₂×D₄

—

156

C₂²×C₄

—

157

C₂²×C₄

D₇₆

yes

characteristic

158

C₂²×C₄

D₇₆

yes

characteristic

159

C₂²×C₄

C₂²×D₁₉

yes

characteristic

160

Q₁₆

—

161

SD₁₆

—

162

C₂×C₈

C₂²×D₁₉

yes

—

163

C₂×C₈

C₂²×D₁₉

yes

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

168

C₄⋊C₄

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

171

C₄⋊C₄

—

172

C₂²⋊C₄

—

173

C₂²⋊C₄

—

174

C₂²⋊C₄

—

175

C₄²

—

176

C₄²

D₇₆

yes

characteristic

177

C₄²

D₇₆

yes

characteristic

178

C₄²

C₂²×D₁₉

yes

characteristic

179

C₄²

C₂²×D₁₉

yes

characteristic

180

C₂³

—

181

C₂³

C₂×D₇₆

yes

characteristic

182

Q₈

—

183

Q₈

—

184

Q₈

—

185

D₄

—

186

D₄

—

187

D₄

—

188

D₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

C₂×D₇₆

yes

characteristic

197

C₂×C₄

C₂×D₇₆

yes

characteristic

198

C₂×C₄

C₂×D₇₆

yes

characteristic

199

C₂×C₄

C₂×D₇₆

yes

characteristic

200

C₂×C₄

C₂×D₇₆

yes

characteristic

201

C₂×C₄

C₂³×D₁₉

yes

characteristic

202

C₈

—

203

C₈

—

204

C₂²

—

205

C₂²

—

206

C₂²

—

207

C₂²

—

208

C₂²

—

209

C₂²

—

210

C₂²

C₂²×D₇₆

yes

characteristic

211

C₄

—

212

C₄

—

213

C₄

—

214

C₄

—

215

C₄

—

216

C₄

—

217

C₄

#10597350

yes

—

218

C₄

#10597355

yes

—

219

C₄

#10597435

yes

—

220

C₂

—

221

C₂

—

152

—

222

C₂

#11927113

yes

characteristic

223

C₂

C₂×#10597350

yes

characteristic

224

C₂

C₂×#10597355

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₇₆
commutator, cosocle	C₂×C₄×C₁₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉
Fitting	C₁₉×#431	C₂
socle	C₂²×C₁₉	C₂²×D₄
prime radical	#11927094	C₂
hypercenter	#431	D₁₉
hypocenter	C₁₉	#3210
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂²

C₂²×D₇₆

commutator, cosocle

C₂×C₄×C₁₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₉

Fitting

C₁₉×#431

C₂

socle

C₂²×C₁₉

C₂²×D₄

C₂

D₁₉

C₁₉

G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
1. #11927101	C₂	semidirect
3. #11926924	C₂	semidirect
4. #11926917	C₂	semidirect
5. #11926895	C₂	semidirect
6. #11926891	C₂	semidirect
7. #11926767	C₂	semidirect
8. #11926708	C₂	semidirect
9. C₁₉×#431	C₂	semidirect
116. #431	D₁₉	semidirect
147. C₁₉	#3210	semidirect

H ≅ G/N

type

1. #11927101

C₂

semidirect

3. #11926924

C₂

semidirect

4. #11926917

C₂

semidirect

5. #11926895

C₂

semidirect

6. #11926891

C₂

semidirect

7. #11926767

C₂

semidirect

8. #11926708

C₂

semidirect

9. C₁₉×#431

C₂

semidirect

116. #431

D₁₉

semidirect

147. C₁₉

#3210

semidirect

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

subgroup	index
#11927101	2
#11927094	2
#11926924	2
#11926917	2
#11926895	2
#11926891	2
#11926767	2
#11926708	2
C₁₉×#431	2
#3210	19

subgroup

index

C₁₉×#431

Subgroups — #14574593

Conjugacy classes of subgroups

Complements

Maximal subgroups