Group #19156355 — subgroups

Subgroups — C₂₃×#8729 #19156355

Order 5888.

514 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 166 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×#2285	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1797	C₂	1	2	yes	maximal
3	C₂₃×#1296	C₂	1	4	yes	maximal, complemented
4	C₂×C₂₃×#117	C₄	1	2	yes	—
5	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
6	C₂×C₂₃×C₄⋊C₈	C₂²	1	4	yes	—
7	C₂×C₂₃×C₂²⋊C₈	—	2	2	no	—
8	C₂₃×#384	C₂²	1	2	yes	—
9	C₂₃×#382	C₄	1	2	yes	—
10	C₂₃×#380	—	2	2	no	—
11	C₄×C₄⋊C₄×C₂₃	—	2	2	no	—
12	C₂₃×#336	—	2	2	no	—
13	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
14	C₂³×C₄×C₂₃	D₄	1	2	yes	—
15	C₂²×C₈×C₂₃	—	2	4	no	—
16	C₂₃×#117	—	4	4	no	—
17	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
18	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
19	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
20	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
21	C₂×C₄²×C₂₃	—	2	4	no	—
22	C₂×C₄²×C₂₃	—	2	2	no	—
23	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₂₃	D₄	1	2	yes	—
25	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
26	C₂₃×C₄⋊C₈	—	2	8	no	—
27	C₂₃×C₂²⋊C₈	—	4	4	no	—
28	C₂₃×#95	—	2	4	no	—
29	C₂⁴×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₂₃	—	2	2	no	—
31	C₂²×C₄×C₂₃	—	2	2	no	—
32	C₂²×C₄×C₂₃	—	2	2	no	—
33	C₂²×C₄×C₂₃	—	2	4	no	—
34	C₂²×C₄×C₂₃	—	2	4	no	—
35	C₂²×C₄×C₂₃	—	2	2	no	—
36	C₂²×C₄×C₂₃	—	2	2	no	—
37	C₂²×C₄×C₂₃	—	2	4	no	—
38	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
39	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
42	C₂×C₈×C₂₃	—	4	4	no	—
43	C₂×C₈×C₂₃	—	4	4	no	—
44	C₂×C₈×C₂₃	—	4	4	no	—
45	C₄⋊C₄×C₂₃	—	4	4	no	—
46	C₄⋊C₄×C₂₃	—	4	4	no	—
47	C₄⋊C₄×C₂₃	—	4	2	no	—
48	C₄⋊C₄×C₂₃	—	4	2	no	—
49	C₂²⋊C₄×C₂₃	—	4	4	no	—
50	C₄²×C₂₃	—	2	2	no	—
51	C₄²×C₂₃	—	2	4	no	—
52	C₄²×C₂₃	—	4	4	no	—
53	C₄²×C₂₃	#41	1	4	yes	—
54	#8729	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂³×C₂₃	—	2	1	no	—
56	C₂³×C₂₃	—	2	1	no	—
57	C₂³×C₂₃	—	2	2	no	—
58	C₂³×C₂₃	—	2	2	no	—
59	C₂³×C₂₃	#100	1	2	yes	—
60	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₂₃	—	2	8	no	—
62	C₂×C₄×C₂₃	—	2	1	no	—
63	C₂×C₄×C₂₃	—	2	1	no	—
64	C₂×C₄×C₂₃	—	2	1	no	—
65	C₂×C₄×C₂₃	—	2	1	no	—
66	C₂×C₄×C₂₃	—	2	2	no	—
67	C₂×C₄×C₂₃	—	2	2	no	—
68	C₂×C₄×C₂₃	—	2	2	no	—
69	C₂×C₄×C₂₃	—	4	4	no	—
70	C₂×C₄×C₂₃	—	4	4	no	—
71	C₂×C₄×C₂₃	—	4	4	no	—
72	C₂×C₄×C₂₃	—	4	4	no	—
73	C₂×C₄×C₂₃	—	4	4	no	—
74	C₂×C₄×C₂₃	#101	1	2	yes	—
75	C₂×C₄×C₂₃	#117	1	2	yes	—
76	C₂×C₄×C₂₃	#123	1	4	yes	—
77	C₂×C₄×C₂₃	#135	1	4	yes	—
78	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
79	C₂×C₄×C₂₃	#137	1	2	yes	—
80	C₈×C₂₃	—	8	4	no	—
81	#2285	C₂×C₂₃	1	1	yes	characteristic
82	#1797	C₂×C₂₃	1	2	yes	—
83	#1296	C₂×C₂₃	1	4	yes	complemented
84	C₂²×C₂₃	—	2	4	no	—
85	C₂²×C₂₃	—	2	2	no	—
86	C₂²×C₂₃	—	4	1	no	—
87	C₂²×C₂₃	—	4	1	no	—
88	C₂²×C₂₃	—	4	2	no	—
89	C₂²×C₂₃	#381	1	1	yes	characteristic
90	C₂²×C₂₃	C₂×#100	1	1	yes	characteristic
91	C₂²×C₂₃	C₂×#101	1	1	yes	characteristic
92	C₂²×C₂₃	#416	1	2	yes	—
93	C₂²×C₂₃	#417	1	2	yes	—
94	C₄×C₂₃	—	2	2	no	—
95	C₄×C₂₃	—	2	2	no	—
96	C₄×C₂₃	—	4	2	no	—
97	C₄×C₂₃	—	8	4	no	—
98	C₄×C₂₃	#482	1	4	yes	—
99	C₄×C₂₃	#485	1	4	yes	—
100	C₂×#117	C₄×C₂₃	1	2	yes	—
101	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
102	C₂×C₄⋊C₈	C₂²×C₂₃	1	4	yes	—
103	C₂×C₂²⋊C₈	—	2	2	no	—
104	#384	C₂²×C₂₃	1	2	yes	—
105	#382	C₄×C₂₃	1	2	yes	—
106	#380	—	2	2	no	—
107	C₄×C₄⋊C₄	—	2	2	no	—
108	#336	—	2	2	no	—
109	C₂×C₂₃	—	4	2	no	—
110	C₂×C₂₃	#1544	1	1	yes	characteristic
111	C₂×C₂₃	#1551	1	2	yes	—
112	C₂×C₂₃	#1561	1	1	yes	characteristic
113	C₂×C₂₃	#1821	1	1	yes	characteristic
114	C₂×C₂₃	#1825	1	2	yes	—
115	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
116	C₂³×C₄	D₄×C₂₃	1	2	yes	—
117	C₂²×C₈	—	2	4	no	—
118	#117	—	4	4	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
122	C₂×C₂²⋊C₄	—	2	2	no	—
123	C₂×C₄²	—	2	4	no	—
124	C₂×C₄²	—	2	2	no	—
125	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
126	C₂×C₄²	D₄×C₂₃	1	2	yes	—
127	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
128	C₄⋊C₈	—	2	8	no	—
129	C₂²⋊C₈	—	4	4	no	—
130	#95	—	2	4	no	—
131	C₂₃	#8729	1	1	yes	characteristic, direct product factor, Sylow
132	C₂⁴	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
133	C₂²×C₄	—	2	2	no	—
134	C₂²×C₄	—	2	2	no	—
135	C₂²×C₄	—	2	2	no	—
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	—	2	4	no	—
138	C₂²×C₄	—	2	2	no	—
139	C₂²×C₄	—	2	2	no	—
140	C₂²×C₄	—	2	4	no	—
141	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
142	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
143	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
144	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
145	C₂×C₈	—	4	4	no	—
146	C₂×C₈	—	4	4	no	—
147	C₂×C₈	—	4	4	no	—
148	C₄⋊C₄	—	4	4	no	—
149	C₄⋊C₄	—	4	4	no	—
150	C₄⋊C₄	—	4	2	no	—
151	C₄⋊C₄	—	4	2	no	—
152	C₂²⋊C₄	—	4	4	no	—
153	C₄²	—	2	2	no	—
154	C₄²	—	2	4	no	—
155	C₄²	—	4	4	no	—
156	C₄²	#41×C₂₃	1	4	yes	—
157	C₂³	—	2	1	no	—
158	C₂³	—	2	1	no	—
159	C₂³	—	2	2	no	—
160	C₂³	—	2	2	no	—
161	C₂³	C₂₃×#100	1	2	yes	—
162	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
163	C₂×C₄	—	2	8	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	—	4	4	no	—
173	C₂×C₄	—	4	4	no	—
174	C₂×C₄	—	4	4	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	C₂₃×#101	1	2	yes	—
177	C₂×C₄	C₂₃×#117	1	2	yes	—
178	C₂×C₄	C₂₃×#123	1	4	yes	—
179	C₂×C₄	C₂₃×#135	1	4	yes	—
180	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
181	C₂×C₄	C₂₃×#137	1	2	yes	—
182	C₈	—	8	4	no	—
183	C₂²	—	2	4	no	—
184	C₂²	—	2	2	no	—
185	C₂²	—	4	1	no	—
186	C₂²	—	4	1	no	—
187	C₂²	—	4	2	no	—
188	C₂²	C₂₃×#381	1	1	yes	characteristic
189	C₂²	C₂×C₂₃×#100	1	1	yes	characteristic
190	C₂²	C₂×C₂₃×#101	1	1	yes	characteristic
191	C₂²	C₂₃×#416	1	2	yes	—
192	C₂²	C₂₃×#417	1	2	yes	—
193	C₄	—	2	2	no	—
194	C₄	—	2	2	no	—
195	C₄	—	4	2	no	—
196	C₄	—	8	4	no	—
197	C₄	C₂₃×#482	1	4	yes	—
198	C₄	C₂₃×#485	1	4	yes	—
199	C₂	—	4	2	no	—
200	C₂	C₂₃×#1544	1	1	yes	characteristic
201	C₂	C₂₃×#1551	1	2	yes	—
202	C₂	C₂₃×#1561	1	1	yes	characteristic
203	C₂	C₂₃×#1821	1	1	yes	characteristic
204	C₂	C₂₃×#1825	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#2285

C₂

yes

maximal, characteristic

C₂₃×#1797

C₂

yes

maximal

C₂₃×#1296

C₂

yes

maximal, complemented

C₂×C₂₃×#117

C₄

yes

—

C₂²×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₂₃×C₄⋊C₈

C₂²

yes

—

C₂×C₂₃×C₂²⋊C₈

—

C₂₃×#384

C₂²

yes

—

C₂₃×#382

C₄

yes

—

C₂₃×#380

—

C₄×C₄⋊C₄×C₂₃

—

C₂₃×#336

—

C₂³×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂³×C₄×C₂₃

D₄

yes

—

C₂²×C₈×C₂₃

—

C₂₃×#117

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

C₂×C₄

yes

—

C₂×C₂²⋊C₄×C₂₃

—

C₂×C₄²×C₂₃

—

C₂×C₄²×C₂₃

—

C₂×C₄²×C₂₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₂₃

D₄

yes

—

C₂×C₄²×C₂₃

C₂³

yes

characteristic

C₂₃×C₄⋊C₈

—

C₂₃×C₂²⋊C₈

—

C₂₃×#95

—

C₂⁴×C₂₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

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

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

#41

yes

—

#8729

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

#100

yes

—

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

#101

yes

—

C₂×C₄×C₂₃

#117

yes

—

C₂×C₄×C₂₃

#123

yes

—

C₂×C₄×C₂₃

#135

yes

—

C₂×C₄×C₂₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₂₃

#137

yes

—

C₈×C₂₃

—

#2285

C₂×C₂₃

yes

characteristic

#1797

C₂×C₂₃

yes

—

#1296

C₂×C₂₃

yes

complemented

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

—

C₂²×C₂₃

#417

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#482

yes

—

C₄×C₂₃

#485

yes

—

100

C₂×#117

C₄×C₂₃

yes

—

101

C₂²×C₄²

C₂²×C₂₃

yes

characteristic

102

C₂×C₄⋊C₈

C₂²×C₂₃

yes

—

103

C₂×C₂²⋊C₈

—

104

#384

C₂²×C₂₃

yes

—

105

#382

C₄×C₂₃

yes

—

106

#380

—

107

C₄×C₄⋊C₄

—

108

#336

—

109

C₂×C₂₃

—

110

C₂×C₂₃

#1544

yes

characteristic

111

C₂×C₂₃

#1551

yes

—

112

C₂×C₂₃

#1561

yes

characteristic

113

C₂×C₂₃

#1821

yes

characteristic

114

C₂×C₂₃

#1825

yes

—

115

C₂³×C₄

C₂×C₄×C₂₃

yes

characteristic

116

C₂³×C₄

D₄×C₂₃

yes

—

117

C₂²×C₈

—

118

#117

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

C₂×C₄×C₂₃

yes

—

122

C₂×C₂²⋊C₄

—

123

C₂×C₄²

—

124

C₂×C₄²

—

125

C₂×C₄²

C₂×C₄×C₂₃

yes

characteristic

126

C₂×C₄²

D₄×C₂₃

yes

—

127

C₂×C₄²

C₂³×C₂₃

yes

characteristic

128

C₄⋊C₈

—

129

C₂²⋊C₈

—

130

#95

—

131

C₂₃

#8729

yes

characteristic, direct product factor, Sylow

132

C₂⁴

C₂²⋊C₄×C₂₃

yes

characteristic

133

C₂²×C₄

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

—

140

C₂²×C₄

—

141

C₂²×C₄

C₂²⋊C₄×C₂₃

yes

—

142

C₂²×C₄

C₂²⋊C₄×C₂₃

yes

characteristic

143

C₂²×C₄

C₂²×C₄×C₂₃

yes

characteristic

144

C₂²×C₄

C₂×D₄×C₂₃

yes

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₂²⋊C₄

—

153

C₄²

—

154

C₄²

—

155

C₄²

—

156

C₄²

#41×C₂₃

yes

—

157

C₂³

—

158

C₂³

—

159

C₂³

—

160

C₂³

—

161

C₂³

C₂₃×#100

yes

—

162

C₂³

C₂×C₂²⋊C₄×C₂₃

yes

characteristic

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

C₂₃×#101

yes

—

177

C₂×C₄

C₂₃×#117

yes

—

178

C₂×C₄

C₂₃×#123

yes

—

179

C₂×C₄

C₂₃×#135

yes

—

180

C₂×C₄

C₂₃×AGL(1,ℤ/8)

yes

—

181

C₂×C₄

C₂₃×#137

yes

—

182

C₈

—

183

C₂²

—

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

C₂₃×#381

yes

characteristic

189

C₂²

C₂×C₂₃×#100

yes

characteristic

190

C₂²

C₂×C₂₃×#101

yes

characteristic

191

C₂²

C₂₃×#416

yes

—

192

C₂²

C₂₃×#417

yes

—

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

C₂₃×#482

yes

—

198

C₄

C₂₃×#485

yes

—

199

C₂

—

200

C₂

C₂₃×#1544

yes

characteristic

201

C₂

C₂₃×#1551

yes

—

202

C₂

C₂₃×#1561

yes

characteristic

203

C₂

C₂₃×#1821

yes

characteristic

204

C₂

C₂₃×#1825

yes

—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
prime radical	C₂⁴×C₂₃	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

subgroup

G/H

center, socle

C₂³×C₂₃

C₂×C₂²⋊C₄

commutator

C₂²×C₄

C₂²×C₄×C₂₃

Frattini, cosocle

C₂×C₄²

C₂³×C₂₃

prime radical

C₂⁴×C₂₃

C₂²⋊C₄

LCS

G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

N	H ≅ G/N	type
3. C₂₃×#1296	C₂	semidirect
54. #8729	C₂₃	direct (N × H)
83. #1296	C₂×C₂₃	semidirect
131. C₂₃	#8729	direct (N × H)

H ≅ G/N

type

3. C₂₃×#1296

C₂

semidirect

54. #8729

C₂₃

direct (N × H)

83. #1296

C₂×C₂₃

semidirect

131. C₂₃

#8729

direct (N × H)

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

subgroup	index
C₂₃×#2285	2
C₂₃×#1797	2
C₂₃×#1296	2
#8729	23

subgroup

index

C₂₃×#2285

C₂₃×#1797

C₂₃×#1296

#8729

Subgroups — C₂₃×#8729 #19156355

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#8729 #19156355

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8729 #19156355