Group #19158312 — subgroups

Subgroups — C₂₃×#10686 #19158312

Order 5888.

410 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 162 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×#1990	C₂	1	2	yes	maximal
2	C₂₃×#1956	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1844	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1834	C₂	1	2	yes	maximal
5	C₂₃×#1273	C₂	1	1	yes	maximal, characteristic
6	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₂₃×#106	—	2	2	no	—
8	C₂×C₂₃×#103	—	2	2	no	—
9	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
11	C₂₃×#384	C₂²	1	1	yes	characteristic
12	C₂₃×#384	C₂²	1	2	yes	—
13	C₂₃×#340	—	2	4	no	—
14	C₂₃×#340	—	2	4	no	—
15	C₂²×Q₈×C₂₃	—	2	1	no	—
16	C₂²×C₈×C₂₃	D₄	1	4	yes	—
17	C₂²×C₈×C₂₃	D₄	1	2	yes	—
18	C₂₃×C₄⋊Q₈	—	2	2	no	—
19	C₂₃×C₄⋊Q₈	—	2	1	no	—
20	C₂₃×C₄⋊Q₈	Q₈	1	2	yes	—
21	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
22	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
23	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
24	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
25	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
26	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
27	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
28	C₂₃×#106	—	4	4	no	—
29	C₂₃×#103	—	4	4	no	—
30	C₄×C₈×C₂₃	—	2	2	no	—
31	C₄×C₈×C₂₃	—	2	2	no	—
32	C₄×C₈×C₂₃	—	2	2	no	—
33	C₂×Q₈×C₂₃	—	2	2	no	—
34	C₂×Q₈×C₂₃	—	4	1	no	—
35	C₂×Q₈×C₂₃	—	4	2	no	—
36	C₂×Q₈×C₂₃	—	4	2	no	—
37	C₂²×C₄×C₂₃	—	4	1	no	—
38	C₂²×C₄×C₂₃	—	4	1	no	—
39	C₂²×C₄×C₂₃	—	4	2	no	—
40	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
42	C₂×C₈×C₂₃	—	2	2	no	—
43	C₂×C₈×C₂₃	—	2	2	no	—
44	C₂×C₈×C₂₃	—	2	4	no	—
45	C₂×C₈×C₂₃	—	2	4	no	—
46	C₂×C₈×C₂₃	—	2	4	no	—
47	C₂×C₈×C₂₃	SD₁₆	1	4	yes	—
48	C₄⋊C₄×C₂₃	—	2	4	no	—
49	C₄⋊C₄×C₂₃	—	4	2	no	—
50	C₄⋊C₄×C₂₃	—	4	2	no	—
51	C₄⋊C₄×C₂₃	—	4	4	no	—
52	C₄⋊C₄×C₂₃	—	4	4	no	—
53	C₄⋊C₄×C₂₃	—	4	4	no	—
54	C₄⋊C₄×C₂₃	—	4	4	no	—
55	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
56	C₄²×C₂₃	#41	1	1	yes	characteristic
57	C₄²×C₂₃	#41	1	2	yes	—
58	#10686	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂³×C₂₃	#127	1	1	yes	characteristic
60	Q₈×C₂₃	—	4	4	no	—
61	Q₈×C₂₃	—	8	2	no	—
62	C₂×C₄×C₂₃	—	4	2	no	—
63	C₂×C₄×C₂₃	—	8	1	no	—
64	C₂×C₄×C₂₃	—	8	1	no	—
65	C₂×C₄×C₂₃	—	8	1	no	—
66	C₂×C₄×C₂₃	—	8	2	no	—
67	C₂×C₄×C₂₃	—	8	2	no	—
68	C₂×C₄×C₂₃	—	8	2	no	—
69	C₂×C₄×C₂₃	—	8	2	no	—
70	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
72	C₂×C₄×C₂₃	#123	1	1	yes	characteristic
73	C₂×C₄×C₂₃	#123	1	2	yes	—
74	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
75	C₂×C₄×C₂₃	#135	1	4	yes	—
76	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
77	C₂×C₄×C₂₃	#137	1	4	yes	—
78	C₈×C₂₃	—	2	4	no	—
79	C₈×C₂₃	—	4	4	no	—
80	#1990	C₂×C₂₃	1	2	yes	—
81	#1956	C₂×C₂₃	1	1	yes	characteristic
82	#1844	C₂×C₂₃	1	1	yes	characteristic
83	#1834	C₂×C₂₃	1	2	yes	—
84	#1273	C₂×C₂₃	1	1	yes	characteristic
85	C₂²×C₂₃	#399	1	1	yes	characteristic
86	C₂²×C₂₃	#492	1	1	yes	characteristic
87	C₂²×C₂₃	#495	1	2	yes	—
88	C₂²×C₂₃	#496	1	1	yes	characteristic
89	C₂²×C₂₃	#497	1	2	yes	—
90	C₄×C₂₃	—	8	2	no	—
91	C₄×C₂₃	—	16	1	no	—
92	C₄×C₂₃	—	16	2	no	—
93	C₄×C₂₃	#476	1	2	yes	—
94	C₄×C₂₃	#479	1	4	yes	—
95	C₄×C₂₃	#481	1	2	yes	—
96	C₄×C₂₃	#485	1	4	yes	—
97	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
98	C₂×#106	—	2	2	no	—
99	C₂×#103	—	2	2	no	—
100	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
101	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
102	#384	C₂²×C₂₃	1	1	yes	characteristic
103	#384	C₂²×C₂₃	1	2	yes	—
104	#340	—	2	4	no	—
105	#340	—	2	4	no	—
106	C₂×C₂₃	#1708	1	1	yes	characteristic
107	C₂×C₂₃	#1716	1	1	yes	characteristic
108	C₂×C₂₃	#1719	1	2	yes	—
109	C₂×C₂₃	#2094	1	1	yes	characteristic
110	C₂×C₂₃	#2097	1	2	yes	—
111	C₂²×Q₈	—	2	1	no	—
112	C₂²×C₈	D₄×C₂₃	1	4	yes	—
113	C₂²×C₈	D₄×C₂₃	1	2	yes	—
114	C₄⋊Q₈	—	2	2	no	—
115	C₄⋊Q₈	—	2	1	no	—
116	C₄⋊Q₈	Q₈×C₂₃	1	2	yes	—
117	C₂×C₄⋊C₄	—	2	2	no	—
118	C₂×C₄⋊C₄	—	2	2	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	—	2	1	no	—
122	C₂×C₄⋊C₄	—	2	2	no	—
123	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
124	#106	—	4	4	no	—
125	#103	—	4	4	no	—
126	C₄×C₈	—	2	2	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	2	2	no	—
129	C₂₃	#10686	1	1	yes	characteristic, direct product factor, Sylow
130	C₂×Q₈	—	2	2	no	—
131	C₂×Q₈	—	4	1	no	—
132	C₂×Q₈	—	4	2	no	—
133	C₂×Q₈	—	4	2	no	—
134	C₂²×C₄	—	4	1	no	—
135	C₂²×C₄	—	4	1	no	—
136	C₂²×C₄	—	4	2	no	—
137	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
138	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	2	4	no	—
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	SD₁₆×C₂₃	1	4	yes	—
145	C₄⋊C₄	—	2	4	no	—
146	C₄⋊C₄	—	4	2	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄⋊C₄	—	4	4	no	—
149	C₄⋊C₄	—	4	4	no	—
150	C₄⋊C₄	—	4	4	no	—
151	C₄⋊C₄	—	4	4	no	—
152	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
153	C₄²	#41×C₂₃	1	1	yes	characteristic
154	C₄²	#41×C₂₃	1	2	yes	—
155	C₂³	C₂₃×#127	1	1	yes	characteristic
156	Q₈	—	4	4	no	—
157	Q₈	—	8	2	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	8	1	no	—
160	C₂×C₄	—	8	1	no	—
161	C₂×C₄	—	8	1	no	—
162	C₂×C₄	—	8	2	no	—
163	C₂×C₄	—	8	2	no	—
164	C₂×C₄	—	8	2	no	—
165	C₂×C₄	—	8	2	no	—
166	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
167	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
168	C₂×C₄	C₂₃×#123	1	1	yes	characteristic
169	C₂×C₄	C₂₃×#123	1	2	yes	—
170	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
171	C₂×C₄	C₂₃×#135	1	4	yes	—
172	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
173	C₂×C₄	C₂₃×#137	1	4	yes	—
174	C₈	—	2	4	no	—
175	C₈	—	4	4	no	—
176	C₂²	C₂₃×#399	1	1	yes	characteristic
177	C₂²	C₂₃×#492	1	1	yes	characteristic
178	C₂²	C₂₃×#495	1	2	yes	—
179	C₂²	C₂₃×#496	1	1	yes	characteristic
180	C₂²	C₂₃×#497	1	2	yes	—
181	C₄	—	8	2	no	—
182	C₄	—	16	1	no	—
183	C₄	—	16	2	no	—
184	C₄	C₂₃×#476	1	2	yes	—
185	C₄	C₂₃×#479	1	4	yes	—
186	C₄	C₂₃×#481	1	2	yes	—
187	C₄	C₂₃×#485	1	4	yes	—
188	C₂	C₂₃×#1708	1	1	yes	characteristic
189	C₂	C₂₃×#1716	1	1	yes	characteristic
190	C₂	C₂₃×#1719	1	2	yes	—
191	C₂	C₂₃×#2094	1	1	yes	characteristic
192	C₂	C₂₃×#2097	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#1990

C₂

yes

maximal

C₂₃×#1956

C₂

yes

maximal, characteristic

C₂₃×#1844

C₂

yes

maximal, characteristic

C₂₃×#1834

C₂

yes

maximal

C₂₃×#1273

C₂

yes

maximal, characteristic

C₂×C₂₃×C₄⋊Q₈

C₂²

yes

characteristic

C₂×C₂₃×#106

—

C₂×C₂₃×#103

—

C₂×C₄×C₈×C₂₃

C₂²

yes

characteristic

C₂×C₄×C₈×C₂₃

C₂²

yes

—

C₂₃×#384

C₂²

yes

characteristic

C₂₃×#384

C₂²

yes

—

C₂₃×#340

—

C₂₃×#340

—

C₂²×Q₈×C₂₃

—

C₂²×C₈×C₂₃

D₄

yes

—

C₂²×C₈×C₂₃

D₄

yes

—

C₂₃×C₄⋊Q₈

—

C₂₃×C₄⋊Q₈

—

C₂₃×C₄⋊Q₈

Q₈

yes

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄²×C₂₃

C₂³

yes

characteristic

C₂₃×#106

—

C₂₃×#103

—

C₄×C₈×C₂₃

—

C₄×C₈×C₂₃

—

C₄×C₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

SD₁₆

yes

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

C₂×Q₈

yes

C₄²×C₂₃

yes

C₄²×C₂₃

yes

—

#10686

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₃

#127

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

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₂₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₂₃

#123

yes

characteristic

C₂×C₄×C₂₃

#123

yes

—

C₂×C₄×C₂₃

C₂×SD₁₆

yes

—

C₂×C₄×C₂₃

#135

yes

—

C₂×C₄×C₂₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₂₃

#137

yes

—

C₈×C₂₃

—

C₈×C₂₃

—

#1990

C₂×C₂₃

yes

—

#1956

C₂×C₂₃

yes

characteristic

#1844

C₂×C₂₃

yes

characteristic

#1834

C₂×C₂₃

yes

—

#1273

C₂×C₂₃

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

—

C₂²×C₂₃

#496

yes

characteristic

C₂²×C₂₃

#497

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#476

yes

—

C₄×C₂₃

#479

yes

—

C₄×C₂₃

#481

yes

—

C₄×C₂₃

#485

yes

—

C₂×C₄⋊Q₈

C₂²×C₂₃

yes

characteristic

C₂×#106

—

C₂×#103

—

100

C₂×C₄×C₈

C₂²×C₂₃

yes

—

101

C₂×C₄×C₈

C₂²×C₂₃

yes

characteristic

102

#384

C₂²×C₂₃

yes

characteristic

103

#384

C₂²×C₂₃

yes

—

104

#340

—

105

#340

—

106

C₂×C₂₃

#1708

yes

characteristic

107

C₂×C₂₃

#1716

yes

characteristic

108

C₂×C₂₃

#1719

yes

—

109

C₂×C₂₃

#2094

yes

characteristic

110

C₂×C₂₃

#2097

yes

—

111

C₂²×Q₈

—

112

C₂²×C₈

D₄×C₂₃

yes

—

113

C₂²×C₈

D₄×C₂₃

yes

—

114

C₄⋊Q₈

—

115

C₄⋊Q₈

—

116

C₄⋊Q₈

Q₈×C₂₃

yes

—

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄²

C₂³×C₂₃

yes

characteristic

124

#106

—

125

#103

—

126

C₄×C₈

—

127

C₄×C₈

—

128

C₄×C₈

—

129

C₂₃

#10686

yes

characteristic, direct product factor, Sylow

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

C₂×D₄×C₂₃

yes

characteristic

138

C₂²×C₄

C₂×D₄×C₂₃

yes

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

SD₁₆×C₂₃

yes

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄²

C₂×Q₈×C₂₃

yes

characteristic

153

C₄²

#41×C₂₃

yes

characteristic

154

C₄²

#41×C₂₃

yes

—

155

C₂³

C₂₃×#127

yes

characteristic

156

Q₈

—

157

Q₈

—

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

C₂₃×C₂²⋊Q₈

yes

characteristic

167

C₂×C₄

C₂₃×C₂²⋊Q₈

yes

—

168

C₂×C₄

C₂₃×#123

yes

characteristic

169

C₂×C₄

C₂₃×#123

yes

—

170

C₂×C₄

C₂×SD₁₆×C₂₃

yes

—

171

C₂×C₄

C₂₃×#135

yes

—

172

C₂×C₄

C₂₃×AGL(1,ℤ/8)

yes

—

173

C₂×C₄

C₂₃×#137

yes

—

174

C₈

—

175

C₈

—

176

C₂²

C₂₃×#399

yes

characteristic

177

C₂²

C₂₃×#492

yes

characteristic

178

C₂²

C₂₃×#495

yes

—

179

C₂²

C₂₃×#496

yes

characteristic

180

C₂²

C₂₃×#497

yes

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

C₂₃×#476

yes

—

185

C₄

C₂₃×#479

yes

—

186

C₄

C₂₃×#481

yes

—

187

C₄

C₂₃×#485

yes

—

188

C₂

C₂₃×#1708

yes

characteristic

189

C₂

C₂₃×#1716

yes

characteristic

190

C₂

C₂₃×#1719

yes

—

191

C₂

C₂₃×#2094

yes

characteristic

192

C₂

C₂₃×#2097

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₂₃

#127

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₂₃

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

N	H ≅ G/N	type
58. #10686	C₂₃	direct (N × H)
129. C₂₃	#10686	direct (N × H)

H ≅ G/N

type

58. #10686

C₂₃

direct (N × H)

129. C₂₃

#10686

direct (N × H)

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

subgroup	index
C₂₃×#1990	2
C₂₃×#1956	2
C₂₃×#1844	2
C₂₃×#1834	2
C₂₃×#1273	2
#10686	23

subgroup

index

C₂₃×#1990

C₂₃×#1956

C₂₃×#1844

C₂₃×#1834

C₂₃×#1273

#10686

Subgroups — C₂₃×#10686 #19158312

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#10686 #19158312

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#10686 #19158312