Group #19212324 — subgroups

Subgroups — #19212324

Order 5888.

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

normal subgroups only

This group has 408 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	#15673514	C₂	1	2	yes	maximal
2	#15671220	C₂	1	4	yes	maximal
3	C₄×C₂₃×#125	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊(C₂₃⋊C₈)	—	2	2	no	—
5	#13092172	—	2	2	no	—
6	#13092172	C₂²	1	4	yes	—
7	C₂×C₂₃×#125	C₄	1	1	yes	characteristic
8	C₂₃×#389	C₂²	1	2	yes	—
9	C₂₃×#387	—	2	2	no	—
10	C₂₃×#382	C₄	1	1	yes	characteristic
11	C₄×C₄⋊C₄×C₂₃	—	2	2	no	—
12	C₄×C₄⋊C₄×C₂₃	C₄	1	2	yes	—
13	C₄³×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×#125	—	2	2	no	—
15	C₂₃×#125	C₈	1	4	yes	complemented
16	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
17	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
18	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
19	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₂₃	—	2	2	no	—
21	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₂₃	D₄	1	2	yes	—
24	C₂×C₄²×C₂₃	D₄	1	2	yes	—
25	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
26	C₂²×C₂₃⋊C₈	—	2	4	no	—
27	C₂₃×#95	—	2	2	no	—
28	C₂₃×#95	—	2	2	no	—
29	C₂₃×#95	—	2	2	no	—
30	C₄⋊(C₂₃⋊C₈)	—	4	4	no	—
31	C₂²×C₄×C₂₃	—	2	2	no	—
32	C₂²×C₄×C₂₃	—	2	2	no	—
33	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
35	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
38	C₄⋊C₄×C₂₃	—	2	8	no	—
39	C₄⋊C₄×C₂₃	—	2	2	no	—
40	C₄⋊C₄×C₂₃	—	4	2	no	—
41	C₄⋊C₄×C₂₃	—	4	2	no	—
42	C₄⋊C₄×C₂₃	—	4	2	no	—
43	C₄⋊C₄×C₂₃	C₂×C₈	1	4	yes	—
44	C₄²×C₂₃	—	2	2	no	—
45	C₄²×C₂₃	—	2	2	no	—
46	C₄²×C₂₃	—	2	2	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	#7602	—	23	1	no	maximal, Sylow
56	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₂₃	—	2	2	no	—
58	C₂×C₄×C₂₃	—	2	1	no	—
59	C₂×C₄×C₂₃	—	2	1	no	—
60	C₂×C₄×C₂₃	—	2	2	no	—
61	C₂×C₄×C₂₃	—	2	2	no	—
62	C₂×C₄×C₂₃	—	2	2	no	—
63	C₂×C₄×C₂₃	—	2	2	no	—
64	C₂×C₄×C₂₃	—	2	4	no	—
65	C₂×C₄×C₂₃	—	2	1	no	—
66	C₂×C₄×C₂₃	—	2	2	no	—
67	C₂×C₄×C₂₃	—	4	2	no	—
68	C₂×C₄×C₂₃	—	4	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	C₂²⋊C₈	1	4	yes	—
73	C₂×C₄×C₂₃	#101	1	2	yes	—
74	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
76	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
77	C₂×C₄×C₂₃	#137	1	2	yes	—
78	C₂₃⋊C₈	—	8	4	no	—
79	C₄×#125	D₂₃	1	1	yes	characteristic
80	#1831	—	23	2	no	—
81	#1271	—	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×#101	1	1	yes	characteristic
85	C₂²×C₂₃	#417	1	2	yes	—
86	C₂²×C₂₃	#421	1	2	yes	—
87	C₄×C₂₃	—	2	1	no	—
88	C₄×C₂₃	—	2	1	no	—
89	C₄×C₂₃	—	2	4	no	—
90	C₄×C₂₃	—	2	2	no	—
91	C₄×C₂₃	—	2	2	no	—
92	C₄×C₂₃	—	2	2	no	—
93	C₄×C₂₃	—	4	1	no	—
94	C₄×C₂₃	—	4	4	no	—
95	C₄×C₂₃	—	8	2	no	—
96	C₂×#125	Dic₂₃	1	1	yes	characteristic
97	C₂×C₄⋊C₈	—	46	2	no	—
98	#389	C₂×D₂₃	1	2	yes	—
99	#387	—	2	2	no	—
100	#382	Dic₂₃	1	1	yes	characteristic
101	C₄×C₄⋊C₄	—	2	2	no	—
102	C₄×C₄⋊C₄	Dic₂₃	1	2	yes	—
103	C₄³	C₂×D₂₃	1	1	yes	characteristic
104	#336	—	23	4	no	—
105	#336	—	46	2	no	—
106	C₂×C₂₃	#1461	1	1	yes	characteristic
107	C₂×C₂₃	#1477	1	2	yes	—
108	C₂×C₂₃	#1512	1	2	yes	—
109	C₂×C₂₃	#1534	1	1	yes	characteristic
110	C₂×C₂₃	#1561	1	1	yes	characteristic
111	C₂²×C₈	—	46	4	no	—
112	#125	—	2	2	no	—
113	#125	C₂₃⋊C₈	1	4	yes	complemented
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	C₂×Dic₂₃	1	2	yes	—
117	C₂×C₄⋊C₄	C₂×Dic₂₃	1	2	yes	—
118	C₂×C₄²	—	2	2	no	—
119	C₂×C₄²	C₂×Dic₂₃	1	1	yes	characteristic
120	C₂×C₄²	C₂×Dic₂₃	1	1	yes	characteristic
121	C₂×C₄²	C₂²⋊D₂₃	1	2	yes	—
122	C₂×C₄²	C₂²⋊D₂₃	1	2	yes	—
123	C₂×C₄²	C₂²×D₂₃	1	1	yes	characteristic
124	C₄⋊C₈	—	92	4	no	—
125	#95	—	2	2	no	—
126	#95	—	2	2	no	—
127	#95	—	2	2	no	—
128	C₂₃	#7602	1	1	yes	characteristic, complemented, Sylow
129	C₂²×C₄	—	2	2	no	—
130	C₂²×C₄	—	2	2	no	—
131	C₂²×C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
132	C₂²×C₄	C₂²⋊Dic₂₃	1	2	yes	—
133	C₂²×C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
134	C₂²×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
135	C₂²×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
136	C₂×C₈	—	92	4	no	—
137	C₂×C₈	—	92	4	no	—
138	C₂×C₈	—	92	4	no	—
139	C₄⋊C₄	—	2	2	no	—
140	C₄⋊C₄	—	2	8	no	—
141	C₄⋊C₄	—	4	2	no	—
142	C₄⋊C₄	—	4	2	no	—
143	C₄⋊C₄	—	4	2	no	—
144	C₄⋊C₄	C₂×C₂₃⋊C₈	1	4	yes	—
145	C₄²	—	2	2	no	—
146	C₄²	—	2	2	no	—
147	C₄²	—	2	2	no	—
148	C₄²	—	4	2	no	—
149	C₄²	—	4	1	no	—
150	C₄²	—	4	4	no	—
151	C₄²	C₂×C₂₃⋊C₈	1	2	yes	—
152	C₄²	#67267	1	2	yes	—
153	C₂³	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	4	2	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	C₂²⋊(C₂₃⋊C₈)	1	4	yes	—
170	C₂×C₄	#10625018	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	#10625120	1	2	yes	—
174	C₂×C₄	#10625122	1	2	yes	—
175	C₈	—	184	4	no	—
176	C₂²	C₂×C₂²⋊(C₂₃⋊C₈)	1	1	yes	characteristic
177	C₂²	#13092391	1	1	yes	characteristic
178	C₂²	C₂×#10625018	1	1	yes	characteristic
179	C₂²	#13092418	1	2	yes	—
180	C₂²	#13092434	1	2	yes	—
181	C₄	—	2	2	no	—
182	C₄	—	2	2	no	—
183	C₄	—	2	1	no	—
184	C₄	—	2	1	no	—
185	C₄	—	2	2	no	—
186	C₄	—	2	4	no	—
187	C₄	—	4	1	no	—
188	C₄	—	4	4	no	—
189	C₄	—	8	2	no	—
190	C₂	#15671721	1	1	yes	characteristic
191	C₂	#15671784	1	2	yes	—
192	C₂	#15671935	1	2	yes	—
193	C₂	#15672024	1	1	yes	characteristic
194	C₂	#15672135	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15673514

C₂

yes

maximal

#15671220

C₂

yes

maximal

C₄×C₂₃×#125

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₂₃⋊C₈)

—

#13092172

—

#13092172

C₂²

yes

—

C₂×C₂₃×#125

C₄

yes

characteristic

C₂₃×#389

C₂²

yes

—

C₂₃×#387

—

C₂₃×#382

C₄

yes

characteristic

C₄×C₄⋊C₄×C₂₃

—

C₄×C₄⋊C₄×C₂₃

C₄

yes

—

C₄³×C₂₃

C₂²

yes

characteristic

C₂₃×#125

—

C₂₃×#125

C₈

yes

complemented

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

C₂×C₄

yes

—

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

—

C₂₃×#95

—

C₂₃×#95

—

C₄⋊(C₂₃⋊C₈)

—

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

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

C₂×C₈

yes

—

C₄²×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

—

#7602

—

maximal, Sylow

C₂³×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

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

—

C₂×C₄×C₂₃

C₂²⋊C₈

yes

—

C₂×C₄×C₂₃

#101

yes

—

C₂×C₄×C₂₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₂₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₂₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₂₃

#137

yes

—

C₂₃⋊C₈

—

C₄×#125

D₂₃

yes

characteristic

#1831

—

#1271

—

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

—

C₄×C₂₃

—

C₂×#125

Dic₂₃

yes

characteristic

C₂×C₄⋊C₈

—

#389

C₂×D₂₃

yes

—

#387

—

100

#382

Dic₂₃

yes

characteristic

101

C₄×C₄⋊C₄

—

102

C₄×C₄⋊C₄

Dic₂₃

yes

—

103

C₄³

C₂×D₂₃

yes

characteristic

104

#336

—

105

#336

—

106

C₂×C₂₃

#1461

yes

characteristic

107

C₂×C₂₃

#1477

yes

—

108

C₂×C₂₃

#1512

yes

—

109

C₂×C₂₃

#1534

yes

characteristic

110

C₂×C₂₃

#1561

yes

characteristic

111

C₂²×C₈

—

112

#125

—

113

#125

C₂₃⋊C₈

yes

complemented

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄⋊C₄

C₂×Dic₂₃

yes

—

117

C₂×C₄⋊C₄

C₂×Dic₂₃

yes

—

118

C₂×C₄²

—

119

C₂×C₄²

C₂×Dic₂₃

yes

characteristic

120

C₂×C₄²

C₂×Dic₂₃

yes

characteristic

121

C₂×C₄²

C₂²⋊D₂₃

yes

—

122

C₂×C₄²

C₂²⋊D₂₃

yes

—

123

C₂×C₄²

C₂²×D₂₃

yes

characteristic

124

C₄⋊C₈

—

125

#95

—

126

#95

—

127

#95

—

128

C₂₃

#7602

yes

characteristic, complemented, Sylow

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

C₂²⋊Dic₂₃

yes

characteristic

132

C₂²×C₄

C₂²⋊Dic₂₃

yes

—

133

C₂²×C₄

C₂²⋊Dic₂₃

yes

characteristic

134

C₂²×C₄

C₂²×Dic₂₃

yes

characteristic

135

C₂²×C₄

C₂×C₂²⋊D₂₃

yes

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

C₂×C₂₃⋊C₈

yes

—

145

C₄²

—

146

C₄²

—

147

C₄²

—

148

C₄²

—

149

C₄²

—

150

C₄²

—

151

C₄²

C₂×C₂₃⋊C₈

yes

—

152

C₄²

#67267

yes

—

153

C₂³

C₂×C₂²⋊Dic₂₃

yes

characteristic

154

C₂×C₄

—

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

#10625018

yes

—

171

C₂×C₄

C₂²×C₂₃⋊C₈

yes

characteristic

172

C₂×C₄

C₂×#67267

yes

characteristic

173

C₂×C₄

#10625120

yes

—

174

C₂×C₄

#10625122

yes

—

175

C₈

—

184

—

176

C₂²

C₂×C₂²⋊(C₂₃⋊C₈)

yes

characteristic

177

C₂²

#13092391

yes

characteristic

178

C₂²

C₂×#10625018

yes

characteristic

179

C₂²

#13092418

yes

—

180

C₂²

#13092434

yes

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

—

190

C₂

#15671721

yes

characteristic

191

C₂

#15671784

yes

—

192

C₂

#15671935

yes

—

193

C₂

#15672024

yes

characteristic

194

C₂

#15672135

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₂₃
commutator	C₂×C₄×C₂₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₂₃
Fitting	C₄×C₂₃×#125	C₂
socle	C₂³×C₂₃	C₂×C₂²⋊C₄
hypercenter	C₄×#125	D₂₃
hypocenter	C₂₃	#7602
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₂₃

commutator

C₂×C₄×C₂₃

C₂²×C₈

Frattini

C₂×C₄²

C₂²×D₂₃

Fitting

C₄×C₂₃×#125

C₂

socle

C₂³×C₂₃

C₂×C₂²⋊C₄

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#125

N	H ≅ G/N	type
15. C₂₃×#125	C₈	semidirect
113. #125	C₂₃⋊C₈	semidirect
128. C₂₃	#7602	semidirect

H ≅ G/N

type

15. C₂₃×#125

C₈

semidirect

113. #125

C₂₃⋊C₈

semidirect

128. C₂₃

#7602

semidirect

Subgroups — #19212324

Conjugacy classes of subgroups

Complements