Group #19158156 — subgroups

Subgroups — C₂₃×#10530 #19158156

Order 5888.

402 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 98 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	C₂₃×#3253	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1848	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#1560	C₂	1	2	yes	maximal, complemented
4	C₂₃×#1492	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1309	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#532	C₂²	1	2	yes	—
7	C₂₃×#476	—	2	2	no	—
8	C₂₃×#469	—	2	2	no	—
9	C₂₃×#459	—	2	2	no	—
10	C₂×C₂₃×#108	—	2	2	no	—
11	C₂₃×#423	C₂²	1	2	yes	—
12	C₂₃×#423	C₂²	1	1	yes	characteristic
13	C₂₃×#423	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×#333	—	2	2	no	—
16	C₂₃×#331	—	2	2	no	—
17	C₂₃×D₄⋊C₈	—	2	4	no	—
18	C₂×#41×C₂₃	—	2	2	no	—
19	C₂×D₈×C₂₃	—	2	2	no	—
20	C₂×M₁₆×C₂₃	—	2	1	no	—
21	C₂×M₁₆×C₂₃	—	2	2	no	—
22	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
23	C₂²×C₈×C₂₃	—	2	1	no	—
24	C₂₃×C₄⋊Q₈	D₄	1	2	yes	—
25	C₂₃×#127	D₄	1	2	yes	—
26	C₂₃×#124	—	2	2	no	—
27	C₂₃×#121	—	2	2	no	—
28	C₂₃×#121	—	2	2	no	—
29	C₄×D₄×C₂₃	—	2	2	no	—
30	C₄×D₄×C₂₃	—	2	2	no	—
31	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
32	C₂₃×#108	—	4	4	no	—
33	C₂₃×#106	—	2	2	no	—
34	C₂₃×C₄⋊C₈	—	2	2	no	—
35	C₂₃×C₄⋊C₈	—	2	2	no	—
36	C₂₃×C₄⋊C₈	—	2	2	no	—
37	C₂₃×C₄⋊C₈	—	2	2	no	—
38	C₂₃×#102	—	2	4	no	—
39	C₄×C₈×C₂₃	—	2	1	no	—
40	C₄×C₈×C₂₃	D₄	1	2	yes	—
41	#41×C₂₃	—	8	2	no	—
42	#41×C₂₃	—	8	2	no	—
43	C₂×Q₈×C₂₃	—	2	2	no	—
44	C₂×D₄×C₂₃	—	2	2	no	—
45	C₂×D₄×C₂₃	—	2	2	no	—
46	C₂×D₄×C₂₃	—	2	2	no	—
47	C₂×D₄×C₂₃	—	4	2	no	—
48	C₂²×C₄×C₂₃	—	4	2	no	—
49	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₂₃	#41	1	2	yes	—
51	D₈×C₂₃	—	8	2	no	—
52	M₁₆×C₂₃	—	4	2	no	—
53	M₁₆×C₂₃	—	4	2	no	—
54	M₁₆×C₂₃	—	8	2	no	—
55	C₂×C₈×C₂₃	—	2	2	no	—
56	C₂×C₈×C₂₃	—	2	2	no	—
57	C₂×C₈×C₂₃	—	2	2	no	—
58	C₂×C₈×C₂₃	—	2	2	no	—
59	C₂×C₈×C₂₃	—	4	1	no	—
60	C₂×C₈×C₂₃	—	4	2	no	—
61	C₄⋊C₄×C₂₃	—	2	2	no	—
62	C₄⋊C₄×C₂₃	—	2	2	no	—
63	C₂²⋊C₄×C₂₃	—	4	2	no	—
64	C₂²⋊C₄×C₂₃	—	4	2	no	—
65	C₄²×C₂₃	C₂×D₄	1	2	yes	—
66	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₂₃	#41	1	1	yes	characteristic
68	#10530	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂³×C₂₃	—	4	2	no	—
70	C₂³×C₂₃	#124	1	1	yes	characteristic
71	Q₈×C₂₃	—	8	2	no	—
72	D₄×C₂₃	—	4	2	no	—
73	D₄×C₂₃	—	4	2	no	—
74	D₄×C₂₃	—	4	2	no	—
75	D₄×C₂₃	—	4	2	no	—
76	D₄×C₂₃	—	8	2	no	—
77	D₄×C₂₃	—	8	4	no	—
78	C₂×C₄×C₂₃	—	2	1	no	—
79	C₂×C₄×C₂₃	—	2	1	no	—
80	C₂×C₄×C₂₃	—	2	2	no	—
81	C₂×C₄×C₂₃	—	2	2	no	—
82	C₂×C₄×C₂₃	—	4	2	no	—
83	C₂×C₄×C₂₃	—	8	2	no	—
84	C₂×C₄×C₂₃	—	8	2	no	—
85	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
86	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
87	C₂×C₄×C₂₃	#121	1	2	yes	—
88	C₂×C₄×C₂₃	#123	1	2	yes	—
89	C₈×C₂₃	—	2	4	no	—
90	C₈×C₂₃	—	4	2	no	—
91	C₈×C₂₃	—	8	1	no	—
92	C₈×C₂₃	—	8	2	no	—
93	#3253	C₂×C₂₃	1	1	yes	characteristic
94	#1848	C₂×C₂₃	1	1	yes	characteristic, complemented
95	#1560	C₂×C₂₃	1	2	yes	complemented
96	#1492	C₂×C₂₃	1	2	yes	complemented
97	#1309	C₂×C₂₃	1	1	yes	characteristic, complemented
98	C₂²×C₂₃	—	2	2	no	—
99	C₂²×C₂₃	—	2	1	no	—
100	C₂²×C₂₃	—	8	2	no	—
101	C₂²×C₂₃	—	8	2	no	—
102	C₂²×C₂₃	—	8	2	no	—
103	C₂²×C₂₃	#394	1	1	yes	characteristic
104	C₄×C₂₃	—	2	2	no	—
105	C₄×C₂₃	—	2	1	no	—
106	C₄×C₂₃	—	8	2	no	—
107	C₄×C₂₃	#453	1	1	yes	characteristic
108	C₄×C₂₃	#456	1	1	yes	characteristic
109	C₄×C₂₃	#471	1	2	yes	—
110	C₄×C₂₃	#472	1	2	yes	—
111	#532	C₂²×C₂₃	1	2	yes	—
112	#476	—	2	2	no	—
113	#469	—	2	2	no	—
114	#459	—	2	2	no	—
115	C₂×#108	—	2	2	no	—
116	#423	C₂²×C₂₃	1	1	yes	characteristic
117	#423	C₂²×C₂₃	1	2	yes	—
118	#423	C₂²×C₂₃	1	1	yes	characteristic
119	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
120	#333	—	2	2	no	—
121	#331	—	2	2	no	—
122	D₄⋊C₈	—	2	4	no	—
123	C₂×C₂₃	—	4	1	no	—
124	C₂×C₂₃	—	16	2	no	—
125	C₂×C₂₃	#2039	1	1	yes	characteristic
126	C₂×C₂₃	#2051	1	2	yes	—
127	C₂×#41	—	2	2	no	—
128	C₂×D₈	—	2	2	no	—
129	C₂×M₁₆	—	2	1	no	—
130	C₂×M₁₆	—	2	2	no	—
131	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
132	C₂²×C₈	—	2	1	no	—
133	C₄⋊Q₈	D₄×C₂₃	1	2	yes	—
134	#127	D₄×C₂₃	1	2	yes	—
135	#124	—	2	2	no	—
136	#121	—	2	2	no	—
137	#121	—	2	2	no	—
138	C₄×D₄	—	2	2	no	—
139	C₄×D₄	—	2	2	no	—
140	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
141	#108	—	4	4	no	—
142	#106	—	2	2	no	—
143	C₄⋊C₈	—	2	2	no	—
144	C₄⋊C₈	—	2	2	no	—
145	C₄⋊C₈	—	2	2	no	—
146	C₄⋊C₈	—	2	2	no	—
147	#102	—	2	4	no	—
148	C₄×C₈	—	2	1	no	—
149	C₄×C₈	D₄×C₂₃	1	2	yes	—
150	C₂₃	#10530	1	1	yes	characteristic, direct product factor, Sylow
151	#41	—	8	2	no	—
152	#41	—	8	2	no	—
153	C₂×Q₈	—	2	2	no	—
154	C₂×D₄	—	2	2	no	—
155	C₂×D₄	—	2	2	no	—
156	C₂×D₄	—	2	2	no	—
157	C₂×D₄	—	4	2	no	—
158	C₂²×C₄	—	4	2	no	—
159	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
160	C₂²×C₄	#41×C₂₃	1	2	yes	—
161	D₈	—	8	2	no	—
162	M₁₆	—	4	2	no	—
163	M₁₆	—	4	2	no	—
164	M₁₆	—	8	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	4	1	no	—
170	C₂×C₈	—	4	2	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₂²⋊C₄	—	4	2	no	—
174	C₂²⋊C₄	—	4	2	no	—
175	C₄²	C₂×D₄×C₂₃	1	2	yes	—
176	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
177	C₄²	#41×C₂₃	1	1	yes	characteristic
178	C₂³	—	4	2	no	—
179	C₂³	C₂₃×#124	1	1	yes	characteristic
180	Q₈	—	8	2	no	—
181	D₄	—	4	2	no	—
182	D₄	—	4	2	no	—
183	D₄	—	4	2	no	—
184	D₄	—	4	2	no	—
185	D₄	—	8	2	no	—
186	D₄	—	8	4	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	8	2	no	—
193	C₂×C₄	—	8	2	no	—
194	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
195	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
196	C₂×C₄	C₂₃×#121	1	2	yes	—
197	C₂×C₄	C₂₃×#123	1	2	yes	—
198	C₈	—	2	4	no	—
199	C₈	—	4	2	no	—
200	C₈	—	8	1	no	—
201	C₈	—	8	2	no	—
202	C₂²	—	2	1	no	—
203	C₂²	—	2	2	no	—
204	C₂²	—	8	2	no	—
205	C₂²	—	8	2	no	—
206	C₂²	—	8	2	no	—
207	C₂²	C₂₃×#394	1	1	yes	characteristic
208	C₄	—	2	1	no	—
209	C₄	—	2	2	no	—
210	C₄	—	8	2	no	—
211	C₄	C₂₃×#453	1	1	yes	characteristic
212	C₄	C₂₃×#456	1	1	yes	characteristic
213	C₄	C₂₃×#471	1	2	yes	—
214	C₄	C₂₃×#472	1	2	yes	—
215	C₂	—	4	1	no	—
216	C₂	—	16	2	no	—
217	C₂	C₂₃×#2039	1	1	yes	characteristic
218	C₂	C₂₃×#2051	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#3253

C₂

yes

maximal, characteristic

C₂₃×#1848

C₂

yes

maximal, characteristic, complemented

C₂₃×#1560

C₂

yes

maximal, complemented

C₂₃×#1492

C₂

yes

maximal, complemented

C₂₃×#1309

C₂

yes

maximal, characteristic, complemented

C₂₃×#532

C₂²

yes

—

C₂₃×#476

—

C₂₃×#469

—

C₂₃×#459

—

C₂×C₂₃×#108

—

C₂₃×#423

C₂²

yes

—

C₂₃×#423

C₂²

yes

characteristic

C₂₃×#423

C₂²

yes

characteristic

C₂×C₄×C₈×C₂₃

C₂²

yes

characteristic

C₂₃×#333

—

C₂₃×#331

—

C₂₃×D₄⋊C₈

—

C₂×#41×C₂₃

—

C₂×D₈×C₂₃

—

C₂×M₁₆×C₂₃

—

C₂×M₁₆×C₂₃

—

C₂×M₁₆×C₂₃

D₄

yes

—

C₂²×C₈×C₂₃

—

C₂₃×C₄⋊Q₈

D₄

yes

—

C₂₃×#127

D₄

yes

—

C₂₃×#124

—

C₂₃×#121

—

C₂₃×#121

—

C₄×D₄×C₂₃

—

C₄×D₄×C₂₃

—

C₂×C₄²×C₂₃

C₂³

yes

characteristic

C₂₃×#108

—

C₂₃×#106

—

C₂₃×C₄⋊C₈

—

C₂₃×C₄⋊C₈

—

C₂₃×C₄⋊C₈

—

C₂₃×C₄⋊C₈

—

C₂₃×#102

—

C₄×C₈×C₂₃

—

C₄×C₈×C₂₃

D₄

yes

—

#41×C₂₃

—

#41×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₂₃

#41

yes

—

D₈×C₂₃

—

M₁₆×C₂₃

—

M₁₆×C₂₃

—

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

C₂×D₄

yes

—

C₄²×C₂₃

C₂×D₄

yes

C₄²×C₂₃

yes

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₃

—

C₂³×C₂₃

#124

yes

characteristic

Q₈×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×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

yes

C₂×C₄×C₂₃

yes

C₂×C₄×C₂₃

yes

—

C₂×C₄×C₂₃

#123

yes

—

C₈×C₂₃

—

C₈×C₂₃

—

C₈×C₂₃

—

C₈×C₂₃

—

#3253

C₂×C₂₃

yes

characteristic

#1848

C₂×C₂₃

yes

characteristic, complemented

#1560

C₂×C₂₃

yes

complemented

#1492

C₂×C₂₃

yes

complemented

#1309

C₂×C₂₃

yes

characteristic, complemented

C₂²×C₂₃

—

C₂²×C₂₃

—

100

C₂²×C₂₃

—

101

C₂²×C₂₃

—

102

C₂²×C₂₃

—

103

C₂²×C₂₃

#394

yes

characteristic

104

C₄×C₂₃

—

105

C₄×C₂₃

—

106

C₄×C₂₃

—

107

C₄×C₂₃

#453

yes

characteristic

108

C₄×C₂₃

#456

yes

characteristic

109

C₄×C₂₃

#471

yes

—

110

C₄×C₂₃

#472

yes

—

111

#532

C₂²×C₂₃

yes

—

112

#476

—

113

#469

—

114

#459

—

115

C₂×#108

—

116

#423

C₂²×C₂₃

yes

characteristic

117

#423

C₂²×C₂₃

yes

—

118

#423

C₂²×C₂₃

yes

characteristic

119

C₂×C₄×C₈

C₂²×C₂₃

yes

characteristic

120

#333

—

121

#331

—

122

D₄⋊C₈

—

123

C₂×C₂₃

—

124

C₂×C₂₃

—

125

C₂×C₂₃

#2039

yes

characteristic

126

C₂×C₂₃

#2051

yes

—

127

C₂×#41

—

128

C₂×D₈

—

129

C₂×M₁₆

—

130

C₂×M₁₆

—

131

C₂×M₁₆

D₄×C₂₃

yes

—

132

C₂²×C₈

—

133

C₄⋊Q₈

D₄×C₂₃

yes

—

134

#127

D₄×C₂₃

yes

—

135

#124

—

136

#121

—

137

#121

—

138

C₄×D₄

—

139

C₄×D₄

—

140

C₂×C₄²

C₂³×C₂₃

yes

characteristic

141

#108

—

142

#106

—

143

C₄⋊C₈

—

144

C₄⋊C₈

—

145

C₄⋊C₈

—

146

C₄⋊C₈

—

147

#102

—

148

C₄×C₈

—

149

C₄×C₈

D₄×C₂₃

yes

—

150

C₂₃

#10530

yes

characteristic, direct product factor, Sylow

151

#41

—

152

#41

—

153

C₂×Q₈

—

154

C₂×D₄

—

155

C₂×D₄

—

156

C₂×D₄

—

157

C₂×D₄

—

158

C₂²×C₄

—

159

C₂²×C₄

C₂×D₄×C₂₃

yes

characteristic

160

C₂²×C₄

#41×C₂₃

yes

—

161

D₈

—

162

M₁₆

—

163

M₁₆

—

164

M₁₆

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

—

168

C₂×C₈

—

169

C₂×C₈

—

170

C₂×C₈

—

171

C₄⋊C₄

—

172

C₄⋊C₄

—

173

C₂²⋊C₄

—

174

C₂²⋊C₄

—

175

C₄²

C₂×D₄×C₂₃

yes

—

176

C₄²

C₂×D₄×C₂₃

yes

characteristic

177

C₄²

#41×C₂₃

yes

characteristic

178

C₂³

—

179

C₂³

C₂₃×#124

yes

characteristic

180

Q₈

—

181

D₄

—

182

D₄

—

183

D₄

—

184

D₄

—

185

D₄

—

186

D₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

C₂₃×#120

yes

characteristic

195

C₂×C₄

C₂₃×#121

yes

characteristic

196

C₂×C₄

C₂₃×#121

yes

—

197

C₂×C₄

C₂₃×#123

yes

—

198

C₈

—

199

C₈

—

200

C₈

—

201

C₈

—

202

C₂²

—

203

C₂²

—

204

C₂²

—

205

C₂²

—

206

C₂²

—

207

C₂²

C₂₃×#394

yes

characteristic

208

C₄

—

209

C₄

—

210

C₄

—

211

C₄

C₂₃×#453

yes

characteristic

212

C₄

C₂₃×#456

yes

characteristic

213

C₄

C₂₃×#471

yes

—

214

C₄

C₂₃×#472

yes

—

215

C₂

—

216

C₂

—

217

C₂

C₂₃×#2039

yes

characteristic

218

C₂

C₂₃×#2051

yes

—

	subgroup	G/H
center, socle	C₂²×C₂₃	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
prime radical	C₂₃×#3253	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₂₃

#394

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₂₃

prime radical

C₂₃×#3253

C₂

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

N	H ≅ G/N	type
2. C₂₃×#1848	C₂	semidirect
3. C₂₃×#1560	C₂	semidirect
4. C₂₃×#1492	C₂	semidirect
5. C₂₃×#1309	C₂	semidirect
68. #10530	C₂₃	direct (N × H)
94. #1848	C₂×C₂₃	semidirect
95. #1560	C₂×C₂₃	semidirect
96. #1492	C₂×C₂₃	semidirect
97. #1309	C₂×C₂₃	semidirect
150. C₂₃	#10530	direct (N × H)

H ≅ G/N

type

2. C₂₃×#1848

C₂

semidirect

3. C₂₃×#1560

C₂

semidirect

4. C₂₃×#1492

C₂

semidirect

5. C₂₃×#1309

C₂

semidirect

68. #10530

C₂₃

direct (N × H)

94. #1848

C₂×C₂₃

semidirect

95. #1560

C₂×C₂₃

semidirect

96. #1492

C₂×C₂₃

semidirect

97. #1309

C₂×C₂₃

semidirect

150. C₂₃

#10530

direct (N × H)

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

subgroup	index
C₂₃×#3253	2
C₂₃×#1848	2
C₂₃×#1560	2
C₂₃×#1492	2
C₂₃×#1309	2
#10530	23

subgroup

index

C₂₃×#3253

C₂₃×#1848

C₂₃×#1560

C₂₃×#1492

C₂₃×#1309

#10530

Subgroups — C₂₃×#10530 #19158156

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#10530 #19158156

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#10530 #19158156