Group #19212178 — subgroups

Subgroups — #19212178

Order 5888.

444 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 159 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	#15673143	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	—	2	8	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	4	no	—
42	C₄⋊C₄×C₂₃	C₂×C₈	1	4	yes	—
43	C₄²×C₂₃	—	2	2	no	—
44	C₄²×C₂₃	—	2	4	no	—
45	C₄²×C₂₃	—	2	4	no	—
46	C₄²×C₂₃	—	4	1	no	—
47	C₄²×C₂₃	—	4	4	no	—
48	C₄²×C₂₃	C₂×C₈	1	2	yes	—
49	C₄²×C₂₃	M₁₆	1	2	yes	—
50	C₂×C₂₃⋊C₈	—	2	8	no	—
51	C₂×C₂₃⋊C₈	—	4	4	no	—
52	C₂×C₂₃⋊C₈	—	4	4	no	—
53	#7568	—	23	1	no	maximal, Sylow
54	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₂₃	—	2	2	no	—
56	C₂×C₄×C₂₃	—	2	1	no	—
57	C₂×C₄×C₂₃	—	2	1	no	—
58	C₂×C₄×C₂₃	—	2	2	no	—
59	C₂×C₄×C₂₃	—	2	2	no	—
60	C₂×C₄×C₂₃	—	2	2	no	—
61	C₂×C₄×C₂₃	—	2	4	no	—
62	C₂×C₄×C₂₃	—	2	1	no	—
63	C₂×C₄×C₂₃	—	2	2	no	—
64	C₂×C₄×C₂₃	—	4	2	no	—
65	C₂×C₄×C₂₃	—	4	2	no	—
66	C₂×C₄×C₂₃	—	4	2	no	—
67	C₂×C₄×C₂₃	—	4	2	no	—
68	C₂×C₄×C₂₃	—	4	2	no	—
69	C₂×C₄×C₂₃	C₂²⋊C₈	1	4	yes	—
70	C₂×C₄×C₂₃	#101	1	2	yes	—
71	C₂×C₄×C₂₃	#104	1	4	yes	—
72	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
74	C₂×C₄×C₂₃	#135	1	4	yes	—
75	C₂₃⋊C₈	—	4	8	no	—
76	C₄×#125	D₂₃	1	1	yes	characteristic
77	#1744	—	23	2	no	—
78	#1271	—	23	4	no	—
79	C₂²×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
80	C₂²×C₂₃	#410	1	1	yes	characteristic
81	C₂²×C₂₃	C₂×#101	1	1	yes	characteristic
82	C₂²×C₂₃	#416	1	2	yes	—
83	C₂²×C₂₃	C₂×#104	1	2	yes	—
84	C₄×C₂₃	—	2	1	no	—
85	C₄×C₂₃	—	2	1	no	—
86	C₄×C₂₃	—	2	4	no	—
87	C₄×C₂₃	—	2	4	no	—
88	C₄×C₂₃	—	4	1	no	—
89	C₄×C₂₃	—	4	4	no	—
90	C₄×C₂₃	—	8	2	no	—
91	C₄×C₂₃	#330	1	4	yes	—
92	C₂×#125	Dic₂₃	1	1	yes	characteristic
93	C₂×C₄×C₈	—	46	2	no	—
94	#389	C₂×D₂₃	1	2	yes	—
95	#387	—	2	2	no	—
96	#382	Dic₂₃	1	1	yes	characteristic
97	C₄×C₄⋊C₄	—	2	2	no	—
98	C₄×C₄⋊C₄	Dic₂₃	1	2	yes	—
99	C₄³	C₂×D₂₃	1	1	yes	characteristic
100	#336	—	23	4	no	—
101	#336	—	46	2	no	—
102	C₂×C₂₃	#1461	1	1	yes	characteristic
103	C₂×C₂₃	#1476	1	2	yes	—
104	C₂×C₂₃	#1500	1	2	yes	—
105	C₂×C₂₃	C₂×#330	1	1	yes	characteristic
106	C₂×C₂₃	#1544	1	1	yes	characteristic
107	C₂²×C₈	—	46	4	no	—
108	#125	—	2	2	no	—
109	#125	C₂₃⋊C₈	1	4	yes	complemented
110	C₂×C₄⋊C₄	—	2	2	no	—
111	C₂×C₄⋊C₄	—	2	2	no	—
112	C₂×C₄⋊C₄	C₂×Dic₂₃	1	2	yes	—
113	C₂×C₄⋊C₄	C₂×Dic₂₃	1	2	yes	—
114	C₂×C₄²	—	2	2	no	—
115	C₂×C₄²	C₂×Dic₂₃	1	1	yes	characteristic
116	C₂×C₄²	C₂×Dic₂₃	1	1	yes	characteristic
117	C₂×C₄²	C₂²⋊D₂₃	1	2	yes	—
118	C₂×C₄²	C₂²⋊D₂₃	1	2	yes	—
119	C₂×C₄²	C₂²×D₂₃	1	1	yes	characteristic
120	C₄×C₈	—	46	8	no	—
121	#95	—	2	2	no	—
122	#95	—	2	2	no	—
123	#95	—	2	2	no	—
124	C₂₃	#7568	1	1	yes	characteristic, complemented, Sylow
125	C₂²×C₄	—	2	2	no	—
126	C₂²×C₄	—	2	2	no	—
127	C₂²×C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
128	C₂²×C₄	C₂²⋊Dic₂₃	1	2	yes	—
129	C₂²×C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
130	C₂²×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
131	C₂²×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
132	C₂×C₈	—	46	8	no	—
133	C₂×C₈	—	92	4	no	—
134	C₂×C₈	—	92	4	no	—
135	C₄⋊C₄	—	2	2	no	—
136	C₄⋊C₄	—	2	8	no	—
137	C₄⋊C₄	—	4	2	no	—
138	C₄⋊C₄	—	4	4	no	—
139	C₄⋊C₄	C₂×C₂₃⋊C₈	1	4	yes	—
140	C₄²	—	2	4	no	—
141	C₄²	—	2	2	no	—
142	C₄²	—	2	4	no	—
143	C₄²	—	4	1	no	—
144	C₄²	—	4	4	no	—
145	C₄²	C₂×C₂₃⋊C₈	1	2	yes	—
146	C₄²	#67267	1	2	yes	—
147	C₂³	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	4	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	4	2	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	C₂²⋊(C₂₃⋊C₈)	1	4	yes	—
163	C₂×C₄	#10625018	1	2	yes	—
164	C₂×C₄	#10625019	1	4	yes	—
165	C₂×C₄	C₂²×C₂₃⋊C₈	1	1	yes	characteristic
166	C₂×C₄	C₂×#67267	1	1	yes	characteristic
167	C₂×C₄	#10625121	1	4	yes	—
168	C₈	—	92	8	no	—
169	C₂²	C₂×C₂²⋊(C₂₃⋊C₈)	1	1	yes	characteristic
170	C₂²	#13092391	1	1	yes	characteristic
171	C₂²	C₂×#10625018	1	1	yes	characteristic
172	C₂²	#13092413	1	2	yes	—
173	C₂²	C₂×#10625019	1	2	yes	—
174	C₄	—	2	4	no	—
175	C₄	—	2	1	no	—
176	C₄	—	2	1	no	—
177	C₄	—	2	4	no	—
178	C₄	—	4	4	no	—
179	C₄	—	4	1	no	—
180	C₄	—	8	2	no	—
181	C₄	#13092159	1	4	yes	—
182	C₂	#15671721	1	1	yes	characteristic
183	C₂	#15671779	1	2	yes	—
184	C₂	#15671883	1	2	yes	—
185	C₂	C₂×#13092159	1	1	yes	characteristic
186	C₂	#15672062	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15673143

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

C₂×C₈

yes

—

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

—

#7568

—

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

C₂²⋊C₈

yes

—

C₂×C₄×C₂₃

#101

yes

—

C₂×C₄×C₂₃

#104

yes

—

C₂×C₄×C₂₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₂₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₂₃

#135

yes

—

C₂₃⋊C₈

—

C₄×#125

D₂₃

yes

characteristic

#1744

—

#1271

—

C₂²×C₂₃

C₂×C₂²⋊C₈

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

—

C₂²×C₂₃

C₂×#104

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#330

yes

—

C₂×#125

Dic₂₃

yes

characteristic

C₂×C₄×C₈

—

#389

C₂×D₂₃

yes

—

#387

—

#382

Dic₂₃

yes

characteristic

C₄×C₄⋊C₄

—

C₄×C₄⋊C₄

Dic₂₃

yes

—

C₄³

C₂×D₂₃

yes

characteristic

100

#336

—

101

#336

—

102

C₂×C₂₃

#1461

yes

characteristic

103

C₂×C₂₃

#1476

yes

—

104

C₂×C₂₃

#1500

yes

—

105

C₂×C₂₃

C₂×#330

yes

characteristic

106

C₂×C₂₃

#1544

yes

characteristic

107

C₂²×C₈

—

108

#125

—

109

#125

C₂₃⋊C₈

yes

complemented

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

C₂×Dic₂₃

yes

—

113

C₂×C₄⋊C₄

C₂×Dic₂₃

yes

—

114

C₂×C₄²

—

115

C₂×C₄²

C₂×Dic₂₃

yes

characteristic

116

C₂×C₄²

C₂×Dic₂₃

yes

characteristic

117

C₂×C₄²

C₂²⋊D₂₃

yes

—

118

C₂×C₄²

C₂²⋊D₂₃

yes

—

119

C₂×C₄²

C₂²×D₂₃

yes

characteristic

120

C₄×C₈

—

121

#95

—

122

#95

—

123

#95

—

124

C₂₃

#7568

yes

characteristic, complemented, Sylow

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

C₂²⋊Dic₂₃

yes

characteristic

128

C₂²×C₄

C₂²⋊Dic₂₃

yes

—

129

C₂²×C₄

C₂²⋊Dic₂₃

yes

characteristic

130

C₂²×C₄

C₂²×Dic₂₃

yes

characteristic

131

C₂²×C₄

C₂×C₂²⋊D₂₃

yes

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄⋊C₄

C₂×C₂₃⋊C₈

yes

—

140

C₄²

—

141

C₄²

—

142

C₄²

—

143

C₄²

—

144

C₄²

—

145

C₄²

C₂×C₂₃⋊C₈

yes

—

146

C₄²

#67267

yes

—

147

C₂³

C₂×C₂²⋊Dic₂₃

yes

characteristic

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

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

C₂²⋊(C₂₃⋊C₈)

yes

—

163

C₂×C₄

#10625018

yes

—

164

C₂×C₄

#10625019

yes

—

165

C₂×C₄

C₂²×C₂₃⋊C₈

yes

characteristic

166

C₂×C₄

C₂×#67267

yes

characteristic

167

C₂×C₄

#10625121

yes

—

168

C₈

—

169

C₂²

C₂×C₂²⋊(C₂₃⋊C₈)

yes

characteristic

170

C₂²

#13092391

yes

characteristic

171

C₂²

C₂×#10625018

yes

characteristic

172

C₂²

#13092413

yes

—

173

C₂²

C₂×#10625019

yes

—

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

#13092159

yes

—

182

C₂

#15671721

yes

characteristic

183

C₂

#15671779

yes

—

184

C₂

#15671883

yes

—

185

C₂

C₂×#13092159

yes

characteristic

186

C₂

#15672062

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	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	#7568
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#125

subgroup

G/H

center

C₂²×C₄

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 ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#125

N	H ≅ G/N	type
15. C₂₃×#125	C₈	semidirect
109. #125	C₂₃⋊C₈	semidirect
124. C₂₃	#7568	semidirect

H ≅ G/N

type

15. C₂₃×#125

C₈

semidirect

109. #125

C₂₃⋊C₈

semidirect

124. C₂₃

#7568

semidirect

Subgroups — #19212178

Conjugacy classes of subgroups

Complements