Group #19158249 — subgroups

Subgroups — C₂₃×#10623 #19158249

Order 5888.

350 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 134 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×#1984	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1834	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1803	C₂	1	2	yes	maximal
4	C₂₃×#1296	C₂	1	2	yes	maximal
5	C₂₃×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₂₃×#103	—	2	2	no	—
10	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂₃×#384	C₂²	1	2	yes	—
12	C₂₃×#340	—	2	2	no	—
13	C₂₃×#340	—	2	2	no	—
14	C₂₃×#336	—	2	2	no	—
15	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₂₃	—	2	1	no	—
17	C₂²×C₈×C₂₃	—	2	2	no	—
18	C₂²×C₈×C₂₃	D₄	1	2	yes	—
19	C₄×Q₈×C₂₃	—	2	2	no	—
20	C₄×Q₈×C₂₃	—	2	2	no	—
21	C₂×C₄⋊C₄×C₂₃	—	2	1	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	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₂₃	—	2	1	no	—
27	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
28	C₂₃×C₄⋊C₈	—	2	2	no	—
29	C₂₃×C₄⋊C₈	—	2	2	no	—
30	C₂₃×C₄⋊C₈	—	2	2	no	—
31	C₂₃×#103	—	4	4	no	—
32	C₄×C₈×C₂₃	—	2	2	no	—
33	C₂×Q₈×C₂₃	—	2	1	no	—
34	C₂×Q₈×C₂₃	—	2	2	no	—
35	C₂×Q₈×C₂₃	—	2	2	no	—
36	C₂×Q₈×C₂₃	D₈	1	2	yes	—
37	C₂²×C₄×C₂₃	—	2	1	no	—
38	C₂²×C₄×C₂₃	—	2	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	1	yes	characteristic
42	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
43	C₂×C₈×C₂₃	—	2	2	no	—
44	C₂×C₈×C₂₃	—	2	2	no	—
45	C₂×C₈×C₂₃	—	2	2	no	—
46	C₂×C₈×C₂₃	—	4	2	no	—
47	C₂×C₈×C₂₃	—	4	1	no	—
48	C₂×C₈×C₂₃	—	4	2	no	—
49	C₂×C₈×C₂₃	—	4	2	no	—
50	C₂×C₈×C₂₃	—	4	2	no	—
51	C₄⋊C₄×C₂₃	—	2	2	no	—
52	C₄⋊C₄×C₂₃	—	4	2	no	—
53	C₄⋊C₄×C₂₃	—	4	2	no	—
54	C₄⋊C₄×C₂₃	—	4	2	no	—
55	C₄⋊C₄×C₂₃	—	4	2	no	—
56	C₄⋊C₄×C₂₃	—	4	2	no	—
57	C₄⋊C₄×C₂₃	—	8	2	no	—
58	C₄²×C₂₃	—	4	1	no	—
59	C₄²×C₂₃	—	4	1	no	—
60	C₄²×C₂₃	#41	1	2	yes	—
61	C₄²×C₂₃	#41	1	2	yes	—
62	#10623	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂³×C₂₃	#121	1	1	yes	characteristic
64	Q₈×C₂₃	—	2	4	no	—
65	Q₈×C₂₃	—	4	2	no	—
66	C₂×C₄×C₂₃	—	2	2	no	—
67	C₂×C₄×C₂₃	—	2	2	no	—
68	C₂×C₄×C₂₃	—	4	1	no	—
69	C₂×C₄×C₂₃	—	4	1	no	—
70	C₂×C₄×C₂₃	—	4	1	no	—
71	C₂×C₄×C₂₃	—	4	2	no	—
72	C₂×C₄×C₂₃	—	8	2	no	—
73	C₂×C₄×C₂₃	—	8	2	no	—
74	C₂×C₄×C₂₃	—	8	2	no	—
75	C₂×C₄×C₂₃	#123	1	2	yes	—
76	C₂×C₄×C₂₃	#123	1	1	yes	characteristic
77	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
78	C₂×C₄×C₂₃	#126	1	2	yes	—
79	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₂₃	#135	1	1	yes	characteristic
81	C₂×C₄×C₂₃	#135	1	2	yes	—
82	C₂×C₄×C₂₃	#137	1	2	yes	—
83	C₂×C₄×C₂₃	#137	1	2	yes	—
84	C₈×C₂₃	—	4	2	no	—
85	C₈×C₂₃	—	8	2	no	—
86	C₈×C₂₃	—	8	1	no	—
87	#1984	C₂×C₂₃	1	1	yes	characteristic
88	#1834	C₂×C₂₃	1	1	yes	characteristic
89	#1803	C₂×C₂₃	1	2	yes	—
90	#1296	C₂×C₂₃	1	2	yes	—
91	#1277	C₂×C₂₃	1	1	yes	characteristic
92	C₂²×C₂₃	#397	1	1	yes	characteristic
93	C₂²×C₂₃	#464	1	2	yes	—
94	C₂²×C₂₃	#466	1	1	yes	characteristic
95	C₂²×C₂₃	#470	1	1	yes	characteristic
96	C₂²×C₂₃	#471	1	2	yes	—
97	C₄×C₂₃	—	2	2	no	—
98	C₄×C₂₃	—	4	2	no	—
99	C₄×C₂₃	—	8	1	no	—
100	C₄×C₂₃	—	16	2	no	—
101	C₄×C₂₃	#480	1	2	yes	—
102	C₄×C₂₃	#485	1	2	yes	—
103	C₄×C₂₃	#485	1	2	yes	—
104	C₄×C₂₃	#488	1	1	yes	characteristic
105	C₄×C₂₃	#491	1	1	yes	characteristic
106	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
107	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
108	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
109	C₂×#103	—	2	2	no	—
110	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
111	#384	C₂²×C₂₃	1	2	yes	—
112	#340	—	2	2	no	—
113	#340	—	2	2	no	—
114	#336	—	2	2	no	—
115	C₂×C₂₃	#1682	1	2	yes	—
116	C₂×C₂₃	#1704	1	2	yes	—
117	C₂×C₂₃	#2074	1	1	yes	characteristic
118	C₂×C₂₃	#2075	1	1	yes	characteristic
119	C₂×C₂₃	#2079	1	1	yes	characteristic
120	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
121	C₂²×C₈	—	2	1	no	—
122	C₂²×C₈	—	2	2	no	—
123	C₂²×C₈	D₄×C₂₃	1	2	yes	—
124	C₄×Q₈	—	2	2	no	—
125	C₄×Q₈	—	2	2	no	—
126	C₂×C₄⋊C₄	—	2	1	no	—
127	C₂×C₄⋊C₄	—	2	2	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
131	C₂×C₄²	—	2	1	no	—
132	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
133	C₄⋊C₈	—	2	2	no	—
134	C₄⋊C₈	—	2	2	no	—
135	C₄⋊C₈	—	2	2	no	—
136	#103	—	4	4	no	—
137	C₄×C₈	—	2	2	no	—
138	C₂₃	#10623	1	1	yes	characteristic, direct product factor, Sylow
139	C₂×Q₈	—	2	1	no	—
140	C₂×Q₈	—	2	2	no	—
141	C₂×Q₈	—	2	2	no	—
142	C₂×Q₈	D₈×C₂₃	1	2	yes	—
143	C₂²×C₄	—	2	1	no	—
144	C₂²×C₄	—	2	1	no	—
145	C₂²×C₄	—	4	2	no	—
146	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
147	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
148	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	—	4	1	no	—
154	C₂×C₈	—	4	2	no	—
155	C₂×C₈	—	4	2	no	—
156	C₂×C₈	—	4	2	no	—
157	C₄⋊C₄	—	2	2	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	4	2	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	—	8	2	no	—
164	C₄²	—	4	1	no	—
165	C₄²	—	4	1	no	—
166	C₄²	#41×C₂₃	1	2	yes	—
167	C₄²	#41×C₂₃	1	2	yes	—
168	C₂³	C₂₃×#121	1	1	yes	characteristic
169	Q₈	—	2	4	no	—
170	Q₈	—	4	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	4	1	no	—
174	C₂×C₄	—	4	1	no	—
175	C₂×C₄	—	4	1	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	8	2	no	—
178	C₂×C₄	—	8	2	no	—
179	C₂×C₄	—	8	2	no	—
180	C₂×C₄	C₂₃×#123	1	2	yes	—
181	C₂×C₄	C₂₃×#123	1	1	yes	characteristic
182	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
183	C₂×C₄	C₂₃×#126	1	2	yes	—
184	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
185	C₂×C₄	C₂₃×#135	1	1	yes	characteristic
186	C₂×C₄	C₂₃×#135	1	2	yes	—
187	C₂×C₄	C₂₃×#137	1	2	yes	—
188	C₂×C₄	C₂₃×#137	1	2	yes	—
189	C₈	—	4	2	no	—
190	C₈	—	8	2	no	—
191	C₈	—	8	1	no	—
192	C₂²	C₂₃×#397	1	1	yes	characteristic
193	C₂²	C₂₃×#464	1	2	yes	—
194	C₂²	C₂₃×#466	1	1	yes	characteristic
195	C₂²	C₂₃×#470	1	1	yes	characteristic
196	C₂²	C₂₃×#471	1	2	yes	—
197	C₄	—	2	2	no	—
198	C₄	—	4	2	no	—
199	C₄	—	8	1	no	—
200	C₄	—	16	2	no	—
201	C₄	C₂₃×#480	1	2	yes	—
202	C₄	C₂₃×#485	1	2	yes	—
203	C₄	C₂₃×#485	1	2	yes	—
204	C₄	C₂₃×#488	1	1	yes	characteristic
205	C₄	C₂₃×#491	1	1	yes	characteristic
206	C₂	C₂₃×#1682	1	2	yes	—
207	C₂	C₂₃×#1704	1	2	yes	—
208	C₂	C₂₃×#2074	1	1	yes	characteristic
209	C₂	C₂₃×#2075	1	1	yes	characteristic
210	C₂	C₂₃×#2079	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#1984

C₂

yes

maximal, characteristic

C₂₃×#1834

C₂

yes

maximal, characteristic

C₂₃×#1803

C₂

yes

maximal

C₂₃×#1296

C₂

yes

maximal

C₂₃×#1277

C₂

yes

maximal, characteristic

C₂×C₄×Q₈×C₂₃

C₂²

yes

characteristic

C₂×C₂₃×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₂₃×C₄⋊C₈

C₂²

yes

—

C₂×C₂₃×#103

—

C₂×C₄×C₈×C₂₃

C₂²

yes

characteristic

C₂₃×#384

C₂²

yes

—

C₂₃×#340

—

C₂₃×#340

—

C₂₃×#336

—

C₂²×Q₈×C₂₃

D₄

yes

characteristic

C₂²×C₈×C₂₃

—

C₂²×C₈×C₂₃

—

C₂²×C₈×C₂₃

D₄

yes

—

C₄×Q₈×C₂₃

—

C₄×Q₈×C₂₃

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

D₄

yes

characteristic

C₂×C₄²×C₂₃

—

C₂×C₄²×C₂₃

C₂³

yes

characteristic

C₂₃×C₄⋊C₈

—

C₂₃×C₄⋊C₈

—

C₂₃×C₄⋊C₈

—

C₂₃×#103

—

C₄×C₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

D₈

yes

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₂₃

#41

yes

characteristic

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×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

—

C₄²×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

#41

yes

—

C₄²×C₂₃

#41

yes

—

#10623

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₃

#121

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

—

C₂×C₄×C₂₃

#123

yes

—

C₂×C₄×C₂₃

yes

C₂×C₄×C₂₃

yes

C₂×C₄×C₂₃

yes

—

C₂×C₄×C₂₃

C₂×D₈

yes

C₂×C₄×C₂₃

yes

C₂×C₄×C₂₃

yes

—

C₂×C₄×C₂₃

#137

yes

—

C₂×C₄×C₂₃

#137

yes

—

C₈×C₂₃

—

C₈×C₂₃

—

C₈×C₂₃

—

#1984

C₂×C₂₃

yes

characteristic

#1834

C₂×C₂₃

yes

characteristic

#1803

C₂×C₂₃

yes

—

#1296

C₂×C₂₃

yes

—

#1277

C₂×C₂₃

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

—

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

100

C₄×C₂₃

—

101

C₄×C₂₃

#480

yes

—

102

C₄×C₂₃

#485

yes

—

103

C₄×C₂₃

#485

yes

—

104

C₄×C₂₃

#488

yes

characteristic

105

C₄×C₂₃

#491

yes

characteristic

106

C₂×C₄×Q₈

C₂²×C₂₃

yes

characteristic

107

C₂×C₄⋊C₈

C₂²×C₂₃

yes

—

108

C₂×C₄⋊C₈

C₂²×C₂₃

yes

characteristic

109

C₂×#103

—

110

C₂×C₄×C₈

C₂²×C₂₃

yes

characteristic

111

#384

C₂²×C₂₃

yes

—

112

#340

—

113

#340

—

114

#336

—

115

C₂×C₂₃

#1682

yes

—

116

C₂×C₂₃

#1704

yes

—

117

C₂×C₂₃

#2074

yes

characteristic

118

C₂×C₂₃

#2075

yes

characteristic

119

C₂×C₂₃

#2079

yes

characteristic

120

C₂²×Q₈

D₄×C₂₃

yes

characteristic

121

C₂²×C₈

—

122

C₂²×C₈

—

123

C₂²×C₈

D₄×C₂₃

yes

—

124

C₄×Q₈

—

125

C₄×Q₈

—

126

C₂×C₄⋊C₄

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

D₄×C₂₃

yes

characteristic

131

C₂×C₄²

—

132

C₂×C₄²

C₂³×C₂₃

yes

characteristic

133

C₄⋊C₈

—

134

C₄⋊C₈

—

135

C₄⋊C₈

—

136

#103

—

137

C₄×C₈

—

138

C₂₃

#10623

yes

characteristic, direct product factor, Sylow

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂×Q₈

—

142

C₂×Q₈

D₈×C₂₃

yes

—

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

—

146

C₂²×C₄

C₂×D₄×C₂₃

yes

characteristic

147

C₂²×C₄

C₂×D₄×C₂₃

yes

characteristic

148

C₂²×C₄

#41×C₂₃

yes

characteristic

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄²

—

165

C₄²

—

166

C₄²

#41×C₂₃

yes

—

167

C₄²

#41×C₂₃

yes

—

168

C₂³

C₂₃×#121

yes

characteristic

169

Q₈

—

170

Q₈

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

C₂₃×#123

yes

—

181

C₂×C₄

C₂₃×#123

yes

characteristic

182

C₂×C₄

C₂₃×#124

yes

characteristic

183

C₂×C₄

C₂₃×#126

yes

—

184

C₂×C₄

C₂×D₈×C₂₃

yes

characteristic

185

C₂×C₄

C₂₃×#135

yes

characteristic

186

C₂×C₄

C₂₃×#135

yes

—

187

C₂×C₄

C₂₃×#137

yes

—

188

C₂×C₄

C₂₃×#137

yes

—

189

C₈

—

190

C₈

—

191

C₈

—

192

C₂²

C₂₃×#397

yes

characteristic

193

C₂²

C₂₃×#464

yes

—

194

C₂²

C₂₃×#466

yes

characteristic

195

C₂²

C₂₃×#470

yes

characteristic

196

C₂²

C₂₃×#471

yes

—

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

—

201

C₄

C₂₃×#480

yes

—

202

C₄

C₂₃×#485

yes

—

203

C₄

C₂₃×#485

yes

—

204

C₄

C₂₃×#488

yes

characteristic

205

C₄

C₂₃×#491

yes

characteristic

206

C₂

C₂₃×#1682

yes

—

207

C₂

C₂₃×#1704

yes

—

208

C₂

C₂₃×#2074

yes

characteristic

209

C₂

C₂₃×#2075

yes

characteristic

210

C₂

C₂₃×#2079

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	#121
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

#121

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
62. #10623	C₂₃	direct (N × H)
138. C₂₃	#10623	direct (N × H)

H ≅ G/N

type

62. #10623

C₂₃

direct (N × H)

138. C₂₃

#10623

direct (N × H)

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

subgroup	index
C₂₃×#1984	2
C₂₃×#1834	2
C₂₃×#1803	2
C₂₃×#1296	2
C₂₃×#1277	2
#10623	23

subgroup

index

C₂₃×#1984

C₂₃×#1834

C₂₃×#1803

C₂₃×#1296

C₂₃×#1277

#10623

Subgroups — C₂₃×#10623 #19158249

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#10623 #19158249

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#10623 #19158249