Group #19157925 — subgroups

Subgroups — C₂₃×#10299 #19157925

Order 5888.

576 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 130 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×#3229	C₂	1	2	yes	maximal
2	C₂₃×#2868	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1994	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1546	C₂	1	2	yes	maximal
5	C₂₃×#1276	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#533	C₂²	1	1	yes	characteristic
7	C₂×C₂₃×C₄⋊Q₈	—	2	2	no	—
8	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₂₃×#498	—	2	2	no	—
10	C₂₃×#487	—	2	2	no	—
11	C₂₃×#483	—	2	4	no	—
12	C₂₃×#475	—	2	4	no	—
13	C₂₃×#431	C₂²	1	2	yes	—
14	C₂₃×#423	C₂²	1	1	yes	characteristic
15	C₂₃×#384	—	2	4	no	—
16	C₄³×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×#339	—	2	2	no	—
18	C₂₃×#332	—	2	4	no	—
19	C₂²×Q₈×C₂₃	—	2	2	no	—
20	C₂×M₁₆×C₂₃	—	2	1	no	—
21	C₂₃×C₄⋊Q₈	—	2	2	no	—
22	C₂₃×C₄⋊Q₈	—	2	2	no	—
23	C₂₃×C₄⋊Q₈	—	2	2	no	—
24	C₂₃×C₄⋊Q₈	—	2	2	no	—
25	C₂₃×C₄⋊Q₈	—	4	4	no	—
26	C₂₃×C₄⋊Q₈	—	4	4	no	—
27	C₂₃×C₄⋊Q₈	D₄	1	2	yes	—
28	C₂₃×#125	—	2	1	no	—
29	C₂₃×C₂²⋊Q₈	—	2	2	no	—
30	C₄×Q₈×C₂₃	—	2	2	no	—
31	C₂₃×#117	—	2	1	no	—
32	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
33	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
34	C₂×C₄⋊C₄×C₂₃	D₄	1	4	yes	—
35	C₂×C₄²×C₂₃	—	2	2	no	—
36	C₂×C₄²×C₂₃	D₄	1	2	yes	—
37	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
38	C₂₃×#106	—	2	8	no	—
39	C₂₃×C₄⋊C₈	—	2	2	no	—
40	C₂₃×C₄⋊C₈	D₄	1	4	yes	—
41	C₂₃×#103	—	2	8	no	—
42	C₂₃×C₂²⋊C₈	—	2	2	no	—
43	C₄×C₈×C₂₃	—	2	2	no	—
44	C₂×Q₈×C₂₃	—	2	2	no	—
45	C₂×Q₈×C₂₃	—	4	2	no	—
46	C₂×Q₈×C₂₃	—	4	2	no	—
47	C₂×Q₈×C₂₃	—	4	4	no	—
48	C₂×Q₈×C₂₃	—	4	4	no	—
49	C₂²×C₄×C₂₃	—	2	2	no	—
50	C₂²×C₄×C₂₃	—	2	4	no	—
51	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
53	M₁₆×C₂₃	—	4	2	no	—
54	C₂×C₈×C₂₃	—	2	4	no	—
55	C₂×C₈×C₂₃	—	4	1	no	—
56	C₄⋊C₄×C₂₃	—	2	4	no	—
57	C₄⋊C₄×C₂₃	—	2	2	no	—
58	C₄⋊C₄×C₂₃	—	2	4	no	—
59	C₄⋊C₄×C₂₃	—	2	2	no	—
60	C₄⋊C₄×C₂₃	—	4	16	no	—
61	C₄⋊C₄×C₂₃	—	4	2	no	—
62	C₄⋊C₄×C₂₃	—	4	2	no	—
63	C₄⋊C₄×C₂₃	—	4	2	no	—
64	C₂²⋊C₄×C₂₃	—	4	1	no	—
65	C₄²×C₂₃	—	2	8	no	—
66	C₄²×C₂₃	—	2	2	no	—
67	C₄²×C₂₃	—	4	1	no	—
68	C₄²×C₂₃	SD₁₆	1	4	yes	—
69	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₂₃	C₂×D₄	1	2	yes	—
71	C₄²×C₂₃	#41	1	1	yes	characteristic
72	#10299	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂³×C₂₃	#120	1	1	yes	characteristic
74	Q₈×C₂₃	—	4	4	no	—
75	Q₈×C₂₃	—	8	4	no	—
76	Q₈×C₂₃	—	8	2	no	—
77	Q₈×C₂₃	—	8	2	no	—
78	C₂×C₄×C₂₃	—	2	8	no	—
79	C₂×C₄×C₂₃	—	2	4	no	—
80	C₂×C₄×C₂₃	—	2	4	no	—
81	C₂×C₄×C₂₃	—	4	2	no	—
82	C₂×C₄×C₂₃	—	4	4	no	—
83	C₂×C₄×C₂₃	—	4	8	no	—
84	C₂×C₄×C₂₃	#120	1	2	yes	—
85	C₂×C₄×C₂₃	#121	1	2	yes	—
86	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
87	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
88	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
89	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
90	C₈×C₂₃	—	4	4	no	—
91	C₈×C₂₃	—	8	1	no	—
92	#3229	C₂×C₂₃	1	2	yes	—
93	#2868	C₂×C₂₃	1	1	yes	characteristic
94	#1994	C₂×C₂₃	1	1	yes	characteristic
95	#1546	C₂×C₂₃	1	2	yes	—
96	#1276	C₂×C₂₃	1	1	yes	characteristic
97	C₂²×C₂₃	—	2	2	no	—
98	C₂²×C₂₃	#392	1	1	yes	characteristic
99	C₂²×C₂₃	#450	1	2	yes	—
100	C₄×C₂₃	—	2	8	no	—
101	C₄×C₂₃	—	2	2	no	—
102	C₄×C₂₃	—	8	2	no	—
103	C₄×C₂₃	—	8	4	no	—
104	C₄×C₂₃	#456	1	4	yes	—
105	C₄×C₂₃	#460	1	2	yes	—
106	C₄×C₂₃	#492	1	1	yes	characteristic
107	C₄×C₂₃	#496	1	1	yes	characteristic
108	#533	C₂²×C₂₃	1	1	yes	characteristic
109	C₂×C₄⋊Q₈	—	2	2	no	—
110	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
111	#498	—	2	2	no	—
112	#487	—	2	2	no	—
113	#483	—	2	4	no	—
114	#475	—	2	4	no	—
115	#431	C₂²×C₂₃	1	2	yes	—
116	#423	C₂²×C₂₃	1	1	yes	characteristic
117	#384	—	2	4	no	—
118	C₄³	C₂²×C₂₃	1	1	yes	characteristic
119	#339	—	2	2	no	—
120	#332	—	2	4	no	—
121	C₂×C₂₃	—	2	2	no	—
122	C₂×C₂₃	#2005	1	2	yes	—
123	C₂×C₂₃	#2014	1	1	yes	characteristic
124	C₂²×Q₈	—	2	2	no	—
125	C₂×M₁₆	—	2	1	no	—
126	C₄⋊Q₈	—	2	2	no	—
127	C₄⋊Q₈	—	2	2	no	—
128	C₄⋊Q₈	—	2	2	no	—
129	C₄⋊Q₈	—	2	2	no	—
130	C₄⋊Q₈	—	4	4	no	—
131	C₄⋊Q₈	—	4	4	no	—
132	C₄⋊Q₈	D₄×C₂₃	1	2	yes	—
133	#125	—	2	1	no	—
134	C₂²⋊Q₈	—	2	2	no	—
135	C₄×Q₈	—	2	2	no	—
136	#117	—	2	1	no	—
137	C₂×C₄⋊C₄	—	2	8	no	—
138	C₂×C₄⋊C₄	—	2	2	no	—
139	C₂×C₄⋊C₄	D₄×C₂₃	1	4	yes	—
140	C₂×C₄²	—	2	2	no	—
141	C₂×C₄²	D₄×C₂₃	1	2	yes	—
142	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
143	#106	—	2	8	no	—
144	C₄⋊C₈	—	2	2	no	—
145	C₄⋊C₈	D₄×C₂₃	1	4	yes	—
146	#103	—	2	8	no	—
147	C₂²⋊C₈	—	2	2	no	—
148	C₄×C₈	—	2	2	no	—
149	C₂₃	#10299	1	1	yes	characteristic, direct product factor, Sylow
150	C₂×Q₈	—	2	2	no	—
151	C₂×Q₈	—	4	2	no	—
152	C₂×Q₈	—	4	2	no	—
153	C₂×Q₈	—	4	4	no	—
154	C₂×Q₈	—	4	4	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	2	4	no	—
157	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
158	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
159	M₁₆	—	4	2	no	—
160	C₂×C₈	—	2	4	no	—
161	C₂×C₈	—	4	1	no	—
162	C₄⋊C₄	—	2	4	no	—
163	C₄⋊C₄	—	2	2	no	—
164	C₄⋊C₄	—	2	4	no	—
165	C₄⋊C₄	—	2	2	no	—
166	C₄⋊C₄	—	4	16	no	—
167	C₄⋊C₄	—	4	2	no	—
168	C₄⋊C₄	—	4	2	no	—
169	C₄⋊C₄	—	4	2	no	—
170	C₂²⋊C₄	—	4	1	no	—
171	C₄²	—	2	8	no	—
172	C₄²	—	2	2	no	—
173	C₄²	—	4	1	no	—
174	C₄²	SD₁₆×C₂₃	1	4	yes	—
175	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
176	C₄²	C₂×D₄×C₂₃	1	2	yes	—
177	C₄²	#41×C₂₃	1	1	yes	characteristic
178	C₂³	C₂₃×#120	1	1	yes	characteristic
179	Q₈	—	4	4	no	—
180	Q₈	—	8	4	no	—
181	Q₈	—	8	2	no	—
182	Q₈	—	8	2	no	—
183	C₂×C₄	—	2	8	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	4	no	—
188	C₂×C₄	—	4	8	no	—
189	C₂×C₄	C₂₃×#120	1	2	yes	—
190	C₂×C₄	C₂₃×#121	1	2	yes	—
191	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
192	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
193	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
194	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
195	C₈	—	4	4	no	—
196	C₈	—	8	1	no	—
197	C₂²	—	2	2	no	—
198	C₂²	C₂₃×#392	1	1	yes	characteristic
199	C₂²	C₂₃×#450	1	2	yes	—
200	C₄	—	2	8	no	—
201	C₄	—	2	2	no	—
202	C₄	—	8	2	no	—
203	C₄	—	8	4	no	—
204	C₄	C₂₃×#456	1	4	yes	—
205	C₄	C₂₃×#460	1	2	yes	—
206	C₄	C₂₃×#492	1	1	yes	characteristic
207	C₄	C₂₃×#496	1	1	yes	characteristic
208	C₂	—	2	2	no	—
209	C₂	C₂₃×#2005	1	2	yes	—
210	C₂	C₂₃×#2014	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#3229

C₂

yes

maximal

C₂₃×#2868

C₂

yes

maximal, characteristic

C₂₃×#1994

C₂

yes

maximal, characteristic

C₂₃×#1546

C₂

yes

maximal

C₂₃×#1276

C₂

yes

maximal, characteristic

C₂₃×#533

C₂²

yes

characteristic

C₂×C₂₃×C₄⋊Q₈

—

C₂×C₂₃×C₄⋊Q₈

C₂²

yes

—

C₂₃×#498

—

C₂₃×#487

—

C₂₃×#483

—

C₂₃×#475

—

C₂₃×#431

C₂²

yes

—

C₂₃×#423

C₂²

yes

characteristic

C₂₃×#384

—

C₄³×C₂₃

C₂²

yes

characteristic

C₂₃×#339

—

C₂₃×#332

—

C₂²×Q₈×C₂₃

—

C₂×M₁₆×C₂₃

—

C₂₃×C₄⋊Q₈

—

C₂₃×C₄⋊Q₈

—

C₂₃×C₄⋊Q₈

—

C₂₃×C₄⋊Q₈

—

C₂₃×C₄⋊Q₈

—

C₂₃×C₄⋊Q₈

—

C₂₃×C₄⋊Q₈

D₄

yes

—

C₂₃×#125

—

C₂₃×C₂²⋊Q₈

—

C₄×Q₈×C₂₃

—

C₂₃×#117

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

D₄

yes

—

C₂×C₄²×C₂₃

—

C₂×C₄²×C₂₃

D₄

yes

—

C₂×C₄²×C₂₃

C₂³

yes

characteristic

C₂₃×#106

—

C₂₃×C₄⋊C₈

—

C₂₃×C₄⋊C₈

D₄

yes

—

C₂₃×#103

—

C₂₃×C₂²⋊C₈

—

C₄×C₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂×D₄

yes

—

C₂²×C₄×C₂₃

C₂×D₄

yes

characteristic

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

SD₁₆

yes

—

C₄²×C₂₃

C₂×D₄

yes

characteristic

C₄²×C₂₃

C₂×D₄

yes

—

C₄²×C₂₃

#41

yes

characteristic

#10299

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₃

#120

yes

characteristic

Q₈×C₂₃

—

Q₈×C₂₃

—

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

#120

yes

—

C₂×C₄×C₂₃

#121

yes

—

C₂×C₄×C₂₃

yes

C₂×C₄×C₂₃

yes

C₂×C₄×C₂₃

C₂×SD₁₆

yes

—

C₂×C₄×C₂₃

AGL(1,ℤ/8)

yes

—

C₈×C₂₃

—

C₈×C₂₃

—

#3229

C₂×C₂₃

yes

—

#2868

C₂×C₂₃

yes

characteristic

#1994

C₂×C₂₃

yes

characteristic

#1546

C₂×C₂₃

yes

—

#1276

C₂×C₂₃

yes

characteristic

C₂²×C₂₃

—

C₂²×C₂₃

#392

yes

characteristic

C₂²×C₂₃

#450

yes

—

100

C₄×C₂₃

—

101

C₄×C₂₃

—

102

C₄×C₂₃

—

103

C₄×C₂₃

—

104

C₄×C₂₃

#456

yes

—

105

C₄×C₂₃

#460

yes

—

106

C₄×C₂₃

#492

yes

characteristic

107

C₄×C₂₃

#496

yes

characteristic

108

#533

C₂²×C₂₃

yes

characteristic

109

C₂×C₄⋊Q₈

—

110

C₂×C₄⋊Q₈

C₂²×C₂₃

yes

—

111

#498

—

112

#487

—

113

#483

—

114

#475

—

115

#431

C₂²×C₂₃

yes

—

116

#423

C₂²×C₂₃

yes

characteristic

117

#384

—

118

C₄³

C₂²×C₂₃

yes

characteristic

119

#339

—

120

#332

—

121

C₂×C₂₃

—

122

C₂×C₂₃

#2005

yes

—

123

C₂×C₂₃

#2014

yes

characteristic

124

C₂²×Q₈

—

125

C₂×M₁₆

—

126

C₄⋊Q₈

—

127

C₄⋊Q₈

—

128

C₄⋊Q₈

—

129

C₄⋊Q₈

—

130

C₄⋊Q₈

—

131

C₄⋊Q₈

—

132

C₄⋊Q₈

D₄×C₂₃

yes

—

133

#125

—

134

C₂²⋊Q₈

—

135

C₄×Q₈

—

136

#117

—

137

C₂×C₄⋊C₄

—

138

C₂×C₄⋊C₄

—

139

C₂×C₄⋊C₄

D₄×C₂₃

yes

—

140

C₂×C₄²

—

141

C₂×C₄²

D₄×C₂₃

yes

—

142

C₂×C₄²

C₂³×C₂₃

yes

characteristic

143

#106

—

144

C₄⋊C₈

—

145

C₄⋊C₈

D₄×C₂₃

yes

—

146

#103

—

147

C₂²⋊C₈

—

148

C₄×C₈

—

149

C₂₃

#10299

yes

characteristic, direct product factor, Sylow

150

C₂×Q₈

—

151

C₂×Q₈

—

152

C₂×Q₈

—

153

C₂×Q₈

—

154

C₂×Q₈

—

155

C₂²×C₄

—

156

C₂²×C₄

—

157

C₂²×C₄

C₂×D₄×C₂₃

yes

—

158

C₂²×C₄

C₂×D₄×C₂₃

yes

characteristic

159

M₁₆

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

168

C₄⋊C₄

—

169

C₄⋊C₄

—

170

C₂²⋊C₄

—

171

C₄²

—

172

C₄²

—

173

C₄²

—

174

C₄²

SD₁₆×C₂₃

yes

—

175

C₄²

C₂×D₄×C₂₃

yes

characteristic

176

C₄²

C₂×D₄×C₂₃

yes

—

177

C₄²

#41×C₂₃

yes

characteristic

178

C₂³

C₂₃×#120

yes

characteristic

179

Q₈

—

180

Q₈

—

181

Q₈

—

182

Q₈

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

C₂₃×#120

yes

—

190

C₂×C₄

C₂₃×#121

yes

—

191

C₂×C₄

C₂₃×#121

yes

characteristic

192

C₂×C₄

C₂₃×#127

yes

characteristic

193

C₂×C₄

C₂×SD₁₆×C₂₃

yes

—

194

C₂×C₄

C₂₃×AGL(1,ℤ/8)

yes

—

195

C₈

—

196

C₈

—

197

C₂²

—

198

C₂²

C₂₃×#392

yes

characteristic

199

C₂²

C₂₃×#450

yes

—

200

C₄

—

201

C₄

—

202

C₄

—

203

C₄

—

204

C₄

C₂₃×#456

yes

—

205

C₄

C₂₃×#460

yes

—

206

C₄

C₂₃×#492

yes

characteristic

207

C₄

C₂₃×#496

yes

characteristic

208

C₂

—

209

C₂

C₂₃×#2005

yes

—

210

C₂

C₂₃×#2014

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
prime radical	C₂³×C₂₃	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₂₃

#392

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₂₃

prime radical

C₂³×C₂₃

#120

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

N	H ≅ G/N	type
72. #10299	C₂₃	direct (N × H)
149. C₂₃	#10299	direct (N × H)

H ≅ G/N

type

72. #10299

C₂₃

direct (N × H)

149. C₂₃

#10299

direct (N × H)

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

subgroup	index
C₂₃×#3229	2
C₂₃×#2868	2
C₂₃×#1994	2
C₂₃×#1546	2
C₂₃×#1276	2
#10299	23

subgroup

index

C₂₃×#3229

C₂₃×#2868

C₂₃×#1994

C₂₃×#1546

C₂₃×#1276

#10299

Subgroups — C₂₃×#10299 #19157925

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#10299 #19157925

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#10299 #19157925