Group #19212334 — subgroups

Subgroups — #19212334

Order 5888.

440 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 143 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	#15673540	C₂	1	2	yes	maximal
2	#15671223	C₂	1	4	yes	maximal
3	C₄×C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
4	#13092172	—	2	2	no	—
5	#13092172	—	2	2	no	—
6	#13092172	C₂²	1	4	yes	—
7	C₂×C₂₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
8	C₂×C₄×Q₈×C₂₃	C₄	1	2	yes	—
9	C₂₃×#391	C₂²	1	2	yes	—
10	C₂₃×#389	—	2	2	no	—
11	C₂₃×#384	C₄	1	1	yes	characteristic
12	C₄×C₄⋊C₄×C₂₃	—	2	2	no	—
13	C₄³×C₂₃	C₂²	1	1	yes	characteristic
14	C₂²×Q₈×C₂₃	C₂×C₄	1	2	yes	—
15	C₂₃×C₄⋊Q₈	—	2	2	no	—
16	C₂₃×C₄⋊Q₈	C₈	1	4	yes	complemented
17	C₄×Q₈×C₂₃	—	4	4	no	—
18	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
19	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
20	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
21	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₂₃	—	2	2	no	—
23	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₂₃	D₄	1	2	yes	—
26	C₂×C₄²×C₂₃	D₄	1	2	yes	—
27	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
28	C₂²×C₂₃⋊C₈	—	2	4	no	—
29	C₂₃×#95	—	2	4	no	—
30	C₂×Q₈×C₂₃	—	2	2	no	—
31	C₂×Q₈×C₂₃	—	4	2	no	—
32	C₂×Q₈×C₂₃	—	4	2	no	—
33	C₂×Q₈×C₂₃	C₂×C₈	1	4	yes	—
34	C₂²×C₄×C₂₃	—	2	2	no	—
35	C₂²×C₄×C₂₃	—	2	2	no	—
36	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
38	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
41	C₄⋊C₄×C₂₃	—	2	8	no	—
42	C₄⋊C₄×C₂₃	—	4	2	no	—
43	C₄⋊C₄×C₂₃	—	4	4	no	—
44	C₄⋊C₄×C₂₃	—	4	4	no	—
45	C₄²×C₂₃	—	2	2	no	—
46	C₄²×C₂₃	—	2	4	no	—
47	C₄²×C₂₃	—	4	2	no	—
48	C₄²×C₂₃	—	4	1	no	—
49	C₄²×C₂₃	—	4	4	no	—
50	C₄²×C₂₃	C₂×C₈	1	2	yes	—
51	C₄²×C₂₃	M₁₆	1	2	yes	—
52	C₂×C₂₃⋊C₈	—	4	4	no	—
53	C₂×C₂₃⋊C₈	—	4	4	no	—
54	C₂×C₂₃⋊C₈	—	4	4	no	—
55	#7605	—	23	1	no	maximal, Sylow
56	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	Q₈×C₂₃	—	4	4	no	—
58	Q₈×C₂₃	—	4	4	no	—
59	C₂×C₄×C₂₃	—	2	2	no	—
60	C₂×C₄×C₂₃	—	2	1	no	—
61	C₂×C₄×C₂₃	—	2	1	no	—
62	C₂×C₄×C₂₃	—	2	2	no	—
63	C₂×C₄×C₂₃	—	2	2	no	—
64	C₂×C₄×C₂₃	—	2	2	no	—
65	C₂×C₄×C₂₃	—	2	2	no	—
66	C₂×C₄×C₂₃	—	2	4	no	—
67	C₂×C₄×C₂₃	—	2	1	no	—
68	C₂×C₄×C₂₃	—	2	2	no	—
69	C₂×C₄×C₂₃	—	4	2	no	—
70	C₂×C₄×C₂₃	—	4	2	no	—
71	C₂×C₄×C₂₃	—	4	2	no	—
72	C₂×C₄×C₂₃	—	4	4	no	—
73	C₂×C₄×C₂₃	C₂²⋊C₈	1	4	yes	—
74	C₂×C₄×C₂₃	#100	1	2	yes	—
75	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₄×C₂₃	#137	1	4	yes	—
78	C₂₃⋊C₈	—	8	4	no	—
79	C₄×C₄⋊Q₈	D₂₃	1	1	yes	characteristic
80	#1839	—	23	2	no	—
81	#1272	—	23	4	no	—
82	C₂²×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
83	C₂²×C₂₃	#410	1	1	yes	characteristic
84	C₂²×C₂₃	C₂×#100	1	1	yes	characteristic
85	C₂²×C₂₃	#419	1	2	yes	—
86	C₂²×C₂₃	#421	1	2	yes	—
87	C₄×C₂₃	—	2	2	no	—
88	C₄×C₂₃	—	2	1	no	—
89	C₄×C₂₃	—	2	1	no	—
90	C₄×C₂₃	—	2	4	no	—
91	C₄×C₂₃	—	2	4	no	—
92	C₄×C₂₃	—	4	1	no	—
93	C₄×C₂₃	—	4	4	no	—
94	C₄×C₂₃	—	8	2	no	—
95	C₂×C₄⋊Q₈	Dic₂₃	1	1	yes	characteristic
96	C₂×C₄×Q₈	Dic₂₃	1	2	yes	—
97	#391	C₂×D₂₃	1	2	yes	—
98	#389	—	2	2	no	—
99	#384	Dic₂₃	1	1	yes	characteristic
100	C₄×C₄⋊C₄	—	2	2	no	—
101	C₄³	C₂×D₂₃	1	1	yes	characteristic
102	#336	—	23	4	no	—
103	#336	—	46	2	no	—
104	#336	—	46	2	no	—
105	C₂×C₂₃	#1460	1	1	yes	characteristic
106	C₂×C₂₃	#1479	1	2	yes	—
107	C₂×C₂₃	#1515	1	2	yes	—
108	C₂×C₂₃	#1536	1	1	yes	characteristic
109	C₂×C₂₃	#1558	1	1	yes	characteristic
110	C₂²×Q₈	C₂×Dic₂₃	1	2	yes	—
111	C₂²×C₈	—	46	4	no	—
112	C₄⋊Q₈	—	2	2	no	—
113	C₄⋊Q₈	C₂₃⋊C₈	1	4	yes	complemented
114	C₄×Q₈	—	4	4	no	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	—	2	2	no	—
117	C₂×C₄⋊C₄	—	2	2	no	—
118	C₂×C₄⋊C₄	C₂×Dic₂₃	1	2	yes	—
119	C₂×C₄²	—	2	2	no	—
120	C₂×C₄²	C₂×Dic₂₃	1	1	yes	characteristic
121	C₂×C₄²	C₂×Dic₂₃	1	1	yes	characteristic
122	C₂×C₄²	C₂²⋊D₂₃	1	2	yes	—
123	C₂×C₄²	C₂²⋊D₂₃	1	2	yes	—
124	C₂×C₄²	C₂²×D₂₃	1	1	yes	characteristic
125	#95	—	2	4	no	—
126	C₂₃	#7605	1	1	yes	characteristic, complemented, Sylow
127	C₂×Q₈	—	2	2	no	—
128	C₂×Q₈	—	4	2	no	—
129	C₂×Q₈	—	4	2	no	—
130	C₂×Q₈	C₂×C₂₃⋊C₈	1	4	yes	—
131	C₂²×C₄	—	2	2	no	—
132	C₂²×C₄	—	2	2	no	—
133	C₂²×C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
134	C₂²×C₄	C₂²⋊Dic₂₃	1	2	yes	—
135	C₂²×C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
136	C₂²×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
137	C₂²×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
138	C₂×C₈	—	92	4	no	—
139	C₂×C₈	—	92	4	no	—
140	C₂×C₈	—	92	4	no	—
141	C₄⋊C₄	—	2	8	no	—
142	C₄⋊C₄	—	4	2	no	—
143	C₄⋊C₄	—	4	4	no	—
144	C₄⋊C₄	—	4	4	no	—
145	C₄²	—	2	4	no	—
146	C₄²	—	2	2	no	—
147	C₄²	—	4	2	no	—
148	C₄²	—	4	1	no	—
149	C₄²	—	4	4	no	—
150	C₄²	C₂×C₂₃⋊C₈	1	2	yes	—
151	C₄²	#67267	1	2	yes	—
152	C₂³	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic
153	Q₈	—	4	4	no	—
154	Q₈	—	4	4	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	4	2	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	4	4	no	—
169	C₂×C₄	C₂²⋊(C₂₃⋊C₈)	1	4	yes	—
170	C₂×C₄	#10625015	1	2	yes	—
171	C₂×C₄	C₂²×C₂₃⋊C₈	1	1	yes	characteristic
172	C₂×C₄	C₂×#67267	1	1	yes	characteristic
173	C₂×C₄	#10625113	1	4	yes	—
174	C₈	—	184	4	no	—
175	C₂²	C₂×C₂²⋊(C₂₃⋊C₈)	1	1	yes	characteristic
176	C₂²	#13092391	1	1	yes	characteristic
177	C₂²	C₂×#10625015	1	1	yes	characteristic
178	C₂²	#13092426	1	2	yes	—
179	C₂²	#13092434	1	2	yes	—
180	C₄	—	2	2	no	—
181	C₄	—	2	4	no	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	—	2	4	no	—
185	C₄	—	4	4	no	—
186	C₄	—	4	1	no	—
187	C₄	—	8	2	no	—
188	C₂	#15671718	1	1	yes	characteristic
189	C₂	#15671792	1	2	yes	—
190	C₂	#15671948	1	2	yes	—
191	C₂	#15672032	1	1	yes	characteristic
192	C₂	#15672123	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15673540

C₂

yes

maximal

#15671223

C₂

yes

maximal

C₄×C₂₃×C₄⋊Q₈

C₂

yes

maximal, characteristic

#13092172

—

#13092172

—

#13092172

C₂²

yes

—

C₂×C₂₃×C₄⋊Q₈

C₄

yes

characteristic

C₂×C₄×Q₈×C₂₃

C₄

yes

—

C₂₃×#391

C₂²

yes

—

C₂₃×#389

—

C₂₃×#384

C₄

yes

characteristic

C₄×C₄⋊C₄×C₂₃

—

C₄³×C₂₃

C₂²

yes

characteristic

C₂²×Q₈×C₂₃

C₂×C₄

yes

—

C₂₃×C₄⋊Q₈

—

C₂₃×C₄⋊Q₈

C₈

yes

complemented

C₄×Q₈×C₂₃

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

C₂×C₄

yes

—

C₂×C₄²×C₂₃

—

C₂×C₄²×C₂₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₂₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₂₃

D₄

yes

—

C₂×C₄²×C₂₃

D₄

yes

—

C₂×C₄²×C₂₃

C₂³

yes

characteristic

C₂²×C₂₃⋊C₈

—

C₂₃×#95

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

C₂×C₈

yes

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₂₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₂₃

C₂×D₄

yes

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

C₂×C₈

yes

—

C₄²×C₂₃

M₁₆

yes

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

#7605

—

maximal, Sylow

C₂³×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

Q₈×C₂₃

—

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊C₈

yes

—

C₂×C₄×C₂₃

#100

yes

—

C₂×C₄×C₂₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₂₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₂₃

#137

yes

—

C₂₃⋊C₈

—

C₄×C₄⋊Q₈

D₂₃

yes

characteristic

#1839

—

#1272

—

C₂²×C₂₃

C₂×C₂²⋊C₈

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

—

C₂²×C₂₃

#421

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₂×C₄⋊Q₈

Dic₂₃

yes

characteristic

C₂×C₄×Q₈

Dic₂₃

yes

—

#391

C₂×D₂₃

yes

—

#389

—

#384

Dic₂₃

yes

characteristic

100

C₄×C₄⋊C₄

—

101

C₄³

C₂×D₂₃

yes

characteristic

102

#336

—

103

#336

—

104

#336

—

105

C₂×C₂₃

#1460

yes

characteristic

106

C₂×C₂₃

#1479

yes

—

107

C₂×C₂₃

#1515

yes

—

108

C₂×C₂₃

#1536

yes

characteristic

109

C₂×C₂₃

#1558

yes

characteristic

110

C₂²×Q₈

C₂×Dic₂₃

yes

—

111

C₂²×C₈

—

112

C₄⋊Q₈

—

113

C₄⋊Q₈

C₂₃⋊C₈

yes

complemented

114

C₄×Q₈

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄⋊C₄

—

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

C₂×Dic₂₃

yes

—

119

C₂×C₄²

—

120

C₂×C₄²

C₂×Dic₂₃

yes

characteristic

121

C₂×C₄²

C₂×Dic₂₃

yes

characteristic

122

C₂×C₄²

C₂²⋊D₂₃

yes

—

123

C₂×C₄²

C₂²⋊D₂₃

yes

—

124

C₂×C₄²

C₂²×D₂₃

yes

characteristic

125

#95

—

126

C₂₃

#7605

yes

characteristic, complemented, Sylow

127

C₂×Q₈

—

128

C₂×Q₈

—

129

C₂×Q₈

—

130

C₂×Q₈

C₂×C₂₃⋊C₈

yes

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

C₂²⋊Dic₂₃

yes

characteristic

134

C₂²×C₄

C₂²⋊Dic₂₃

yes

—

135

C₂²×C₄

C₂²⋊Dic₂₃

yes

characteristic

136

C₂²×C₄

C₂²×Dic₂₃

yes

characteristic

137

C₂²×C₄

C₂×C₂²⋊D₂₃

yes

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄²

—

146

C₄²

—

147

C₄²

—

148

C₄²

—

149

C₄²

—

150

C₄²

C₂×C₂₃⋊C₈

yes

—

151

C₄²

#67267

yes

—

152

C₂³

C₂×C₂²⋊Dic₂₃

yes

characteristic

153

Q₈

—

154

Q₈

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

C₂²⋊(C₂₃⋊C₈)

yes

—

170

C₂×C₄

#10625015

yes

—

171

C₂×C₄

C₂²×C₂₃⋊C₈

yes

characteristic

172

C₂×C₄

C₂×#67267

yes

characteristic

173

C₂×C₄

#10625113

yes

—

174

C₈

—

184

—

175

C₂²

C₂×C₂²⋊(C₂₃⋊C₈)

yes

characteristic

176

C₂²

#13092391

yes

characteristic

177

C₂²

C₂×#10625015

yes

characteristic

178

C₂²

#13092426

yes

—

179

C₂²

#13092434

yes

—

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

—

187

C₄

—

188

C₂

#15671718

yes

characteristic

189

C₂

#15671792

yes

—

190

C₂

#15671948

yes

—

191

C₂

#15672032

yes

characteristic

192

C₂

#15672123

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₂₃
commutator	C₂×C₄×C₂₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₂₃
Fitting	C₄×C₂₃×C₄⋊Q₈	C₂
socle	C₂³×C₂₃	C₂×C₂²⋊C₄
hypercenter	C₄×C₄⋊Q₈	D₂₃
hypocenter	C₂₃	#7605
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₂₃

commutator

C₂×C₄×C₂₃

C₂²×C₈

Frattini

C₂×C₄²

C₂²×D₂₃

Fitting

C₄×C₂₃×C₄⋊Q₈

C₂

socle

C₂³×C₂₃

C₂×C₂²⋊C₄

hypercenter

C₄×C₄⋊Q₈

D₂₃

hypocenter

C₂₃

#7605

LCS

G ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

N	H ≅ G/N	type
16. C₂₃×C₄⋊Q₈	C₈	semidirect
113. C₄⋊Q₈	C₂₃⋊C₈	semidirect
126. C₂₃	#7605	semidirect

H ≅ G/N

type

16. C₂₃×C₄⋊Q₈

C₈

semidirect

113. C₄⋊Q₈

C₂₃⋊C₈

semidirect

126. C₂₃

#7605

semidirect

Subgroups — #19212334

Conjugacy classes of subgroups

Complements