Group #19156262 — subgroups

Subgroups — C₂₃×#8636 #19156262

Order 5888.

404 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 138 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×#2931	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1566	C₂	1	2	yes	maximal, complemented
3	C₂₃×#1560	C₂	1	2	yes	maximal
4	C₂₃×#1309	C₂	1	2	yes	maximal, complemented
5	C₂×C₂₃×#131	C₄	1	2	yes	—
6	C₂₃×#532	C₂²	1	1	yes	characteristic
7	C₂₃×#436	—	2	2	no	—
8	C₈×D₄×C₂₃	—	2	2	no	—
9	C₂₃×#423	C₂²	1	4	yes	—
10	C₂₃×#423	C₂²	1	1	yes	characteristic
11	C₂₃×#408	C₄	1	2	yes	—
12	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₂₃×#334	—	2	2	no	—
14	C₂₃×#334	—	2	2	no	—
15	C₂₃×#333	—	2	2	no	—
16	C₂₃×#331	—	2	2	no	—
17	C₂×#41×C₂₃	C₂×C₄	1	2	yes	—
18	C₂₃×#131	—	4	4	no	—
19	C₂×M₁₆×C₂₃	—	2	2	no	—
20	C₂×M₁₆×C₂₃	—	2	4	no	—
21	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₂₃	—	2	2	no	—
23	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
25	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
26	C₂₃×#127	D₄	1	1	yes	characteristic
27	C₂₃×#124	Q₈	1	2	yes	—
28	C₂₃×#121	—	2	2	no	—
29	C₄×D₄×C₂₃	—	2	2	no	—
30	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
31	C₂₃×C₄⋊C₈	—	2	2	no	—
32	C₂₃×C₄⋊C₈	—	2	4	no	—
33	C₂₃×C₄⋊C₈	—	2	4	no	—
34	C₂₃×C₂²⋊C₈	—	2	4	no	—
35	C₄×C₈×C₂₃	D₄	1	2	yes	—
36	C₄×C₈×C₂₃	D₄	1	2	yes	—
37	#41×C₂₃	—	4	2	no	—
38	#41×C₂₃	—	4	2	no	—
39	C₂×Q₈×C₂₃	C₄⋊C₄	1	2	yes	—
40	C₂×D₄×C₂₃	—	2	2	no	—
41	C₂×D₄×C₂₃	—	2	2	no	—
42	C₂×D₄×C₂₃	C₄⋊C₄	1	2	yes	—
43	C₂²×C₄×C₂₃	—	2	2	no	—
44	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₂₃	#41	1	2	yes	—
46	M₁₆×C₂₃	—	4	4	no	—
47	M₁₆×C₂₃	—	4	2	no	—
48	M₁₆×C₂₃	—	8	4	no	—
49	C₂×C₈×C₂₃	—	2	2	no	—
50	C₂×C₈×C₂₃	—	2	2	no	—
51	C₂×C₈×C₂₃	—	4	1	no	—
52	C₂×C₈×C₂₃	—	4	4	no	—
53	C₂×C₈×C₂₃	—	4	4	no	—
54	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
56	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₄⋊C₄×C₂₃	—	2	2	no	—
58	C₂²⋊C₄×C₂₃	—	2	4	no	—
59	C₄²×C₂₃	C₂×D₄	1	2	yes	—
60	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
62	#8636	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂³×C₂₃	—	2	2	no	—
64	C₂³×C₂₃	#117	1	1	yes	characteristic
65	Q₈×C₂₃	—	4	2	no	—
66	D₄×C₂₃	—	4	2	no	—
67	D₄×C₂₃	—	4	4	no	—
68	D₄×C₂₃	—	4	4	no	—
69	C₂×C₄×C₂₃	—	2	1	no	—
70	C₂×C₄×C₂₃	—	2	1	no	—
71	C₂×C₄×C₂₃	—	2	4	no	—
72	C₂×C₄×C₂₃	—	2	2	no	—
73	C₂×C₄×C₂₃	—	4	2	no	—
74	C₂×C₄×C₂₃	—	4	2	no	—
75	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₂₃	#121	1	2	yes	—
78	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
79	C₈×C₂₃	—	2	1	no	—
80	C₈×C₂₃	—	2	2	no	—
81	C₈×C₂₃	—	4	2	no	—
82	C₈×C₂₃	—	8	4	no	—
83	C₈×C₂₃	#100	1	2	yes	—
84	#2931	C₂×C₂₃	1	1	yes	characteristic
85	#1566	C₂×C₂₃	1	2	yes	complemented
86	#1560	C₂×C₂₃	1	2	yes	—
87	#1309	C₂×C₂₃	1	2	yes	complemented
88	C₂²×C₂₃	—	2	1	no	—
89	C₂²×C₂₃	—	2	2	no	—
90	C₂²×C₂₃	—	4	2	no	—
91	C₂²×C₂₃	—	4	4	no	—
92	C₂²×C₂₃	#380	1	1	yes	characteristic
93	C₄×C₂₃	—	2	1	no	—
94	C₄×C₂₃	—	2	2	no	—
95	C₄×C₂₃	—	4	2	no	—
96	C₄×C₂₃	C₂×#100	1	1	yes	characteristic
97	C₄×C₂₃	#413	1	1	yes	characteristic
98	C₄×C₂₃	#471	1	4	yes	—
99	C₂×#131	C₄×C₂₃	1	2	yes	—
100	#532	C₂²×C₂₃	1	1	yes	characteristic
101	#436	—	2	2	no	—
102	C₈×D₄	—	2	2	no	—
103	#423	C₂²×C₂₃	1	4	yes	—
104	#423	C₂²×C₂₃	1	1	yes	characteristic
105	#408	C₄×C₂₃	1	2	yes	—
106	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
107	#334	—	2	2	no	—
108	#334	—	2	2	no	—
109	#333	—	2	2	no	—
110	#331	—	2	2	no	—
111	C₂×C₂₃	—	4	1	no	—
112	C₂×C₂₃	—	8	2	no	—
113	C₂×C₂₃	#1800	1	1	yes	characteristic
114	C₂×C₂₃	#1815	1	2	yes	—
115	C₂×#41	C₂×C₄×C₂₃	1	2	yes	—
116	#131	—	4	4	no	—
117	C₂×M₁₆	—	2	2	no	—
118	C₂×M₁₆	—	2	4	no	—
119	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
120	C₂²×C₈	—	2	2	no	—
121	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
122	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
123	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
124	#127	D₄×C₂₃	1	1	yes	characteristic
125	#124	Q₈×C₂₃	1	2	yes	—
126	#121	—	2	2	no	—
127	C₄×D₄	—	2	2	no	—
128	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
129	C₄⋊C₈	—	2	2	no	—
130	C₄⋊C₈	—	2	4	no	—
131	C₄⋊C₈	—	2	4	no	—
132	C₂²⋊C₈	—	2	4	no	—
133	C₄×C₈	D₄×C₂₃	1	2	yes	—
134	C₄×C₈	D₄×C₂₃	1	2	yes	—
135	C₂₃	#8636	1	1	yes	characteristic, direct product factor, Sylow
136	#41	—	4	2	no	—
137	#41	—	4	2	no	—
138	C₂×Q₈	C₄⋊C₄×C₂₃	1	2	yes	—
139	C₂×D₄	—	2	2	no	—
140	C₂×D₄	—	2	2	no	—
141	C₂×D₄	C₄⋊C₄×C₂₃	1	2	yes	—
142	C₂²×C₄	—	2	2	no	—
143	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
144	C₂²×C₄	#41×C₂₃	1	2	yes	—
145	M₁₆	—	4	4	no	—
146	M₁₆	—	4	2	no	—
147	M₁₆	—	8	4	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	4	1	no	—
151	C₂×C₈	—	4	4	no	—
152	C₂×C₈	—	4	4	no	—
153	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
154	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
155	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
156	C₄⋊C₄	—	2	2	no	—
157	C₂²⋊C₄	—	2	4	no	—
158	C₄²	C₂×D₄×C₂₃	1	2	yes	—
159	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
160	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
161	C₂³	—	2	2	no	—
162	C₂³	C₂₃×#117	1	1	yes	characteristic
163	Q₈	—	4	2	no	—
164	D₄	—	4	2	no	—
165	D₄	—	4	4	no	—
166	D₄	—	4	4	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
174	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
175	C₂×C₄	C₂₃×#121	1	2	yes	—
176	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
177	C₈	—	2	1	no	—
178	C₈	—	2	2	no	—
179	C₈	—	4	2	no	—
180	C₈	—	8	4	no	—
181	C₈	C₂₃×#100	1	2	yes	—
182	C₂²	—	2	1	no	—
183	C₂²	—	2	2	no	—
184	C₂²	—	4	2	no	—
185	C₂²	—	4	4	no	—
186	C₂²	C₂₃×#380	1	1	yes	characteristic
187	C₄	—	2	1	no	—
188	C₄	—	2	2	no	—
189	C₄	—	4	2	no	—
190	C₄	C₂×C₂₃×#100	1	1	yes	characteristic
191	C₄	C₂₃×#413	1	1	yes	characteristic
192	C₄	C₂₃×#471	1	4	yes	—
193	C₂	—	4	1	no	—
194	C₂	—	8	2	no	—
195	C₂	C₂₃×#1800	1	1	yes	characteristic
196	C₂	C₂₃×#1815	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#2931

C₂

yes

maximal, characteristic

C₂₃×#1566

C₂

yes

maximal, complemented

C₂₃×#1560

C₂

yes

maximal

C₂₃×#1309

C₂

yes

maximal, complemented

C₂×C₂₃×#131

C₄

yes

—

C₂₃×#532

C₂²

yes

characteristic

C₂₃×#436

—

C₈×D₄×C₂₃

—

C₂₃×#423

C₂²

yes

—

C₂₃×#423

C₂²

yes

characteristic

C₂₃×#408

C₄

yes

—

C₂×C₄×C₈×C₂₃

C₂²

yes

characteristic

C₂₃×#334

—

C₂₃×#334

—

C₂₃×#333

—

C₂₃×#331

—

C₂×#41×C₂₃

C₂×C₄

yes

—

C₂₃×#131

—

C₂×M₁₆×C₂₃

—

C₂×M₁₆×C₂₃

—

C₂×M₁₆×C₂₃

C₂×C₄

yes

—

C₂²×C₈×C₂₃

—

C₂²×C₈×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₂₃

C₂×C₄

yes

characteristic

C₂₃×C₄⋊Q₈

D₄

yes

characteristic

C₂₃×#127

D₄

yes

characteristic

C₂₃×#124

Q₈

yes

—

C₂₃×#121

—

C₄×D₄×C₂₃

—

C₂×C₄²×C₂₃

C₂³

yes

characteristic

C₂₃×C₄⋊C₈

—

C₂₃×C₄⋊C₈

—

C₂₃×C₄⋊C₈

—

C₂₃×C₂²⋊C₈

—

C₄×C₈×C₂₃

D₄

yes

—

C₄×C₈×C₂₃

D₄

yes

—

#41×C₂₃

—

#41×C₂₃

—

C₂×Q₈×C₂₃

C₄⋊C₄

yes

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

C₄⋊C₄

yes

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₂₃

#41

yes

—

M₁₆×C₂₃

—

M₁₆×C₂₃

—

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₂₃

C₂²⋊C₄

yes

—

C₂×C₈×C₂₃

C₂²⋊C₄

yes

characteristic

C₄⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

C₂×D₄

yes

—

C₄²×C₂₃

C₂×D₄

yes

characteristic

C₄²×C₂₃

C₂×Q₈

yes

characteristic

#8636

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₃

—

C₂³×C₂₃

#117

yes

characteristic

Q₈×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×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₃

#121

yes

—

C₂×C₄×C₂₃

C₂²⋊Q₈

yes

—

C₈×C₂₃

—

C₈×C₂₃

—

C₈×C₂₃

—

C₈×C₂₃

—

C₈×C₂₃

#100

yes

—

#2931

C₂×C₂₃

yes

characteristic

#1566

C₂×C₂₃

yes

complemented

#1560

C₂×C₂₃

yes

—

#1309

C₂×C₂₃

yes

complemented

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#380

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

—

C₂×#131

C₄×C₂₃

yes

—

100

#532

C₂²×C₂₃

yes

characteristic

101

#436

—

102

C₈×D₄

—

103

#423

C₂²×C₂₃

yes

—

104

#423

C₂²×C₂₃

yes

characteristic

105

#408

C₄×C₂₃

yes

—

106

C₂×C₄×C₈

C₂²×C₂₃

yes

characteristic

107

#334

—

108

#334

—

109

#333

—

110

#331

—

111

C₂×C₂₃

—

112

C₂×C₂₃

—

113

C₂×C₂₃

#1800

yes

characteristic

114

C₂×C₂₃

#1815

yes

—

115

C₂×#41

C₂×C₄×C₂₃

yes

—

116

#131

—

117

C₂×M₁₆

—

118

C₂×M₁₆

—

119

C₂×M₁₆

C₂×C₄×C₂₃

yes

—

120

C₂²×C₈

—

121

C₂²×C₈

C₂×C₄×C₂₃

yes

characteristic

122

C₂²×C₈

C₂×C₄×C₂₃

yes

characteristic

123

C₄⋊Q₈

D₄×C₂₃

yes

characteristic

124

#127

D₄×C₂₃

yes

characteristic

125

#124

Q₈×C₂₃

yes

—

126

#121

—

127

C₄×D₄

—

128

C₂×C₄²

C₂³×C₂₃

yes

characteristic

129

C₄⋊C₈

—

130

C₄⋊C₈

—

131

C₄⋊C₈

—

132

C₂²⋊C₈

—

133

C₄×C₈

D₄×C₂₃

yes

—

134

C₄×C₈

D₄×C₂₃

yes

—

135

C₂₃

#8636

yes

characteristic, direct product factor, Sylow

136

#41

—

137

#41

—

138

C₂×Q₈

C₄⋊C₄×C₂₃

yes

—

139

C₂×D₄

—

140

C₂×D₄

—

141

C₂×D₄

C₄⋊C₄×C₂₃

yes

—

142

C₂²×C₄

—

143

C₂²×C₄

C₂²×C₄×C₂₃

yes

characteristic

144

C₂²×C₄

#41×C₂₃

yes

—

145

M₁₆

—

146

M₁₆

—

147

M₁₆

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

C₂²⋊C₄×C₂₃

yes

—

154

C₂×C₈

C₂²⋊C₄×C₂₃

yes

characteristic

155

C₂×C₈

C₂²⋊C₄×C₂₃

yes

characteristic

156

C₄⋊C₄

—

157

C₂²⋊C₄

—

158

C₄²

C₂×D₄×C₂₃

yes

—

159

C₄²

C₂×D₄×C₂₃

yes

characteristic

160

C₄²

C₂×Q₈×C₂₃

yes

characteristic

161

C₂³

—

162

C₂³

C₂₃×#117

yes

characteristic

163

Q₈

—

164

D₄

—

165

D₄

—

166

D₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

C₂×C₂²⋊C₄×C₂₃

yes

characteristic

174

C₂×C₄

C₂×C₄⋊C₄×C₂₃

yes

characteristic

175

C₂×C₄

C₂₃×#121

yes

—

176

C₂×C₄

C₂₃×C₂²⋊Q₈

yes

—

177

C₈

—

178

C₈

—

179

C₈

—

180

C₈

—

181

C₈

C₂₃×#100

yes

—

182

C₂²

—

183

C₂²

—

184

C₂²

—

185

C₂²

—

186

C₂²

C₂₃×#380

yes

characteristic

187

C₄

—

188

C₄

—

189

C₄

—

190

C₄

C₂×C₂₃×#100

yes

characteristic

191

C₄

C₂₃×#413

yes

characteristic

192

C₄

C₂₃×#471

yes

—

193

C₂

—

194

C₂

—

195

C₂

C₂₃×#1800

yes

characteristic

196

C₂

C₂₃×#1815

yes

—

	subgroup	G/H
center, socle	C₂²×C₂₃	#380
commutator	C₂²×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
prime radical	C₂₃×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₂₃

#380

commutator

C₂²×C₄

C₂²×C₄×C₂₃

Frattini, cosocle

C₂×C₄²

C₂³×C₂₃

prime radical

C₂₃×#532

C₂²

LCS

G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

N	H ≅ G/N	type
2. C₂₃×#1566	C₂	semidirect
4. C₂₃×#1309	C₂	semidirect
62. #8636	C₂₃	direct (N × H)
85. #1566	C₂×C₂₃	semidirect
87. #1309	C₂×C₂₃	semidirect
135. C₂₃	#8636	direct (N × H)

H ≅ G/N

type

2. C₂₃×#1566

C₂

semidirect

4. C₂₃×#1309

C₂

semidirect

62. #8636

C₂₃

direct (N × H)

85. #1566

C₂×C₂₃

semidirect

87. #1309

C₂×C₂₃

semidirect

135. C₂₃

#8636

direct (N × H)

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

subgroup	index
C₂₃×#2931	2
C₂₃×#1566	2
C₂₃×#1560	2
C₂₃×#1309	2
#8636	23

subgroup

index

C₂₃×#2931

C₂₃×#1566

C₂₃×#1560

C₂₃×#1309

#8636

Subgroups — C₂₃×#8636 #19156262

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#8636 #19156262

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8636 #19156262