Group #15673830 — subgroups

Subgroups — #15673830

Order 2944.

360 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 109 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	C₂×#10625095	C₂	1	1	yes	maximal, characteristic
2	C₂×#10624998	C₂	1	2	yes	maximal, complemented
3	#13092277	C₂	1	1	yes	maximal, characteristic
4	#13092180	C₂	1	2	yes	maximal, complemented
5	C₂²×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
7	C₂²×Dic₄₆	C₂²	1	1	yes	characteristic
8	C₂×M₁₆×C₂₃	—	2	2	no	—
9	C₂×M₁₆×C₂₃	—	2	4	no	—
10	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
11	C₂×C₂²⋊Dic₂₃	—	2	1	no	—
12	C₂×C₂²⋊Dic₂₃	—	2	1	no	—
13	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
14	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
15	#10625095	—	2	2	no	—
16	#10625095	C₄	1	4	yes	complemented
17	C₂×Dic₂₃⋊C₄	—	2	1	no	—
18	C₂₃⋊#95	—	2	1	no	—
19	#10624998	—	2	4	no	—
20	C₂⁴×C₂₃	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₂₃	—	2	2	no	—
22	C₂²×C₄×C₂₃	—	2	2	no	—
23	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₂₃	D₄	1	2	yes	—
26	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
27	C₂²×Dic₂₃	—	2	1	no	—
28	C₂²×Dic₂₃	—	2	1	no	—
29	C₂²×Dic₂₃	—	2	2	no	—
30	C₂×Dic₄₆	—	2	2	no	—
31	C₂×Dic₄₆	—	2	2	no	—
32	C₂×Dic₄₆	—	2	1	no	—
33	C₂×Dic₄₆	C₂×C₄	1	2	yes	—
34	M₁₆×C₂₃	—	2	8	no	—
35	C₂×C₈×C₂₃	—	2	2	no	—
36	C₂×C₈×C₂₃	—	2	2	no	—
37	C₂×C₈×C₂₃	D₄	1	4	yes	—
38	C₂²⋊Dic₂₃	—	2	4	no	—
39	C₂²⋊Dic₂₃	—	4	2	no	—
40	C₄⋊Dic₂₃	—	2	1	no	—
41	C₄⋊Dic₂₃	—	2	2	no	—
42	C₄⋊Dic₂₃	—	2	2	no	—
43	C₄⋊Dic₂₃	C₂×C₄	1	2	yes	—
44	Dic₂₃⋊C₄	—	2	4	no	—
45	C₂³×C₂₃	—	2	2	no	—
46	C₂³×C₂₃	—	2	1	no	—
47	C₂³×C₂₃	—	2	1	no	—
48	C₂³×C₂₃	—	2	2	no	—
49	C₂³×C₂₃	C₂²⋊C₄	1	2	yes	—
50	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₂₃	—	2	2	no	—
52	C₂×C₄×C₂₃	—	2	4	no	—
53	C₂×C₄×C₂₃	—	2	2	no	—
54	C₂×C₄×C₂₃	—	2	2	no	—
55	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
56	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₂₃	#41	1	2	yes	—
60	C₂×Dic₂₃	—	2	2	no	—
61	C₂×Dic₂₃	—	2	2	no	—
62	C₂×Dic₂₃	—	4	2	no	—
63	C₂×Dic₂₃	—	4	1	no	—
64	C₂×Dic₂₃	—	4	1	no	—
65	C₂×Dic₂₃	—	4	2	no	—
66	C₂×Dic₂₃	—	4	2	no	—
67	C₂×Dic₂₃	—	4	2	no	—
68	Dic₄₆	—	2	4	no	—
69	Dic₄₆	—	4	1	no	—
70	Dic₄₆	—	4	1	no	—
71	C₈×C₂₃	—	2	4	no	—
72	#1894	—	23	1	no	maximal, Sylow
73	C₂²×C₂₃	—	2	2	no	—
74	C₂²×C₂₃	—	2	2	no	—
75	C₂²×C₂₃	—	2	2	no	—
76	C₂²×C₂₃	—	4	2	no	—
77	C₂²×C₂₃	—	4	2	no	—
78	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₂₃	#121	1	2	yes	—
80	C₂²×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
81	C₂²×C₂₃	#137	1	1	yes	characteristic
82	C₂²×C₂₃	#137	1	2	yes	—
83	C₄×C₂₃	—	2	2	no	—
84	C₄×C₂₃	C₄×D₄	1	2	yes	—
85	C₄×C₂₃	#120	1	1	yes	characteristic
86	C₄×C₂₃	#123	1	1	yes	characteristic
87	Dic₂₃	—	4	2	no	—
88	Dic₂₃	—	4	2	no	—
89	Dic₂₃	—	8	2	no	—
90	C₂²×M₁₆	D₂₃	1	1	yes	characteristic
91	C₂×C₂²⋊Q₈	—	23	1	no	—
92	C₂×#103	—	23	2	no	—
93	#380	—	23	1	no	—
94	#340	—	23	2	no	—
95	C₂×C₂₃	—	4	2	no	—
96	C₂×C₂₃	#386	1	1	yes	characteristic
97	C₂×C₂₃	#417	1	1	yes	characteristic
98	C₂×C₂₃	#419	1	1	yes	characteristic
99	C₂×C₂₃	#468	1	2	yes	—
100	C₂×C₂₃	#470	1	2	yes	—
101	C₂²×Q₈	—	23	1	no	—
102	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
103	C₂×M₁₆	—	2	2	no	—
104	C₂×M₁₆	—	2	4	no	—
105	C₂²×C₈	C₂×D₂₃	1	2	yes	—
106	C₂²⋊Q₈	—	23	4	no	—
107	C₂²⋊Q₈	—	46	2	no	—
108	C₂×C₄⋊C₄	—	23	1	no	—
109	C₂×C₄⋊C₄	—	23	2	no	—
110	C₂×C₄⋊C₄	—	46	1	no	—
111	C₂×C₂²⋊C₄	—	46	1	no	—
112	C₂×C₂²⋊C₄	—	46	1	no	—
113	#103	—	46	4	no	—
114	#95	—	46	1	no	—
115	C₂₃	#1894	1	1	yes	characteristic, complemented, Sylow
116	C₂⁴	D₉₂	1	1	yes	characteristic
117	C₂×Q₈	—	23	2	no	—
118	C₂×Q₈	—	46	1	no	—
119	C₂×Q₈	—	46	2	no	—
120	C₂×Q₈	—	46	2	no	—
121	C₂²×C₄	—	2	2	no	—
122	C₂²×C₄	—	2	2	no	—
123	C₂²×C₄	—	46	1	no	—
124	C₂²×C₄	—	46	1	no	—
125	C₂²×C₄	—	46	2	no	—
126	C₂²×C₄	C₄×D₂₃	1	2	yes	—
127	C₂²×C₄	D₉₂	1	1	yes	characteristic
128	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
129	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
130	M₁₆	—	2	8	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	C₂²⋊D₂₃	1	4	yes	—
134	C₄⋊C₄	—	23	2	no	—
135	C₄⋊C₄	—	46	4	no	—
136	C₄⋊C₄	—	46	1	no	—
137	C₄⋊C₄	—	46	2	no	—
138	C₄⋊C₄	—	46	2	no	—
139	C₂²⋊C₄	—	46	4	no	—
140	C₂²⋊C₄	—	92	2	no	—
141	C₂³	—	2	1	no	—
142	C₂³	—	2	1	no	—
143	C₂³	—	2	2	no	—
144	C₂³	—	2	2	no	—
145	C₂³	(C₂×D₂₃)⋊C₄	1	2	yes	—
146	C₂³	C₂×D₉₂	1	1	yes	characteristic
147	Q₈	—	46	4	no	—
148	Q₈	—	92	1	no	—
149	Q₈	—	92	1	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	46	2	no	—
155	C₂×C₄	—	46	2	no	—
156	C₂×C₄	—	92	1	no	—
157	C₂×C₄	—	92	1	no	—
158	C₂×C₄	—	92	2	no	—
159	C₂×C₄	—	92	2	no	—
160	C₂×C₄	—	92	2	no	—
161	C₂×C₄	—	92	2	no	—
162	C₂×C₄	(C₂×D₂₃)⋊C₄	1	2	yes	—
163	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
164	C₂×C₄	#67288	1	2	yes	—
165	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
166	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
167	C₈	—	2	4	no	—
168	C₂²	—	2	2	no	—
169	C₂²	—	2	2	no	—
170	C₂²	—	2	2	no	—
171	C₂²	—	4	2	no	—
172	C₂²	—	4	2	no	—
173	C₂²	#10625079	1	1	yes	characteristic
174	C₂²	#10625080	1	2	yes	—
175	C₂²	#10625080	1	1	yes	characteristic
176	C₂²	C₂×(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
177	C₂²	#10625101	1	2	yes	—
178	C₄	—	2	2	no	—
179	C₄	—	92	2	no	—
180	C₄	—	92	2	no	—
181	C₄	—	184	2	no	—
182	C₄	C₄×C₂²⋊D₂₃	1	2	yes	—
183	C₄	#10625100	1	1	yes	characteristic
184	C₄	#10625124	1	1	yes	characteristic
185	C₂	—	4	2	no	—
186	C₂	#13092296	1	1	yes	characteristic
187	C₂	#13092419	1	1	yes	characteristic
188	C₂	#13092427	1	1	yes	characteristic
189	C₂	#13092663	1	2	yes	—
190	C₂	#13092675	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625095

C₂

yes

maximal, characteristic

C₂×#10624998

C₂

yes

maximal, complemented

#13092277

C₂

yes

maximal, characteristic

#13092180

C₂

yes

maximal, complemented

C₂²×M₁₆×C₂₃

C₂

yes

maximal, characteristic

C₂³×C₄×C₂₃

C₂²

yes

characteristic

C₂²×Dic₄₆

C₂²

yes

characteristic

C₂×M₁₆×C₂₃

—

C₂×M₁₆×C₂₃

—

C₂²×C₈×C₂₃

C₂²

yes

—

C₂×C₂²⋊Dic₂₃

—

C₂×C₂²⋊Dic₂₃

—

C₂×C₄⋊Dic₂₃

C₂²

yes

—

C₂×C₄⋊Dic₂₃

C₂²

yes

characteristic

#10625095

—

#10625095

C₄

yes

complemented

C₂×Dic₂₃⋊C₄

—

C₂₃⋊#95

—

#10624998

—

C₂⁴×C₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₂²×Dic₂₃

—

C₂²×Dic₂₃

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₂×Dic₄₆

C₂×C₄

yes

—

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

D₄

yes

—

C₂²⋊Dic₂₃

—

C₂²⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

C₂×C₄

yes

—

Dic₂₃⋊C₄

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

C₂²⋊C₄

yes

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

Dic₄₆

—

Dic₄₆

—

Dic₄₆

—

C₈×C₂₃

—

#1894

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₃

#121

yes

—

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

—

C₄×C₂₃

—

C₄×C₂₃

C₄×D₄

yes

—

C₄×C₂₃

yes

C₄×C₂₃

yes

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

C₂²×M₁₆

D₂₃

yes

characteristic

C₂×C₂²⋊Q₈

—

C₂×#103

—

#380

—

#340

—

C₂×C₂₃

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

—

100

C₂×C₂₃

#470

yes

—

101

C₂²×Q₈

—

102

C₂³×C₄

C₂×D₂₃

yes

characteristic

103

C₂×M₁₆

—

104

C₂×M₁₆

—

105

C₂²×C₈

C₂×D₂₃

yes

—

106

C₂²⋊Q₈

—

107

C₂²⋊Q₈

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₂²⋊C₄

—

112

C₂×C₂²⋊C₄

—

113

#103

—

114

#95

—

115

C₂₃

#1894

yes

characteristic, complemented, Sylow

116

C₂⁴

D₉₂

yes

characteristic

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂×Q₈

—

120

C₂×Q₈

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

C₄×D₂₃

yes

—

127

C₂²×C₄

D₉₂

yes

characteristic

128

C₂²×C₄

C₂²⋊D₂₃

yes

—

129

C₂²×C₄

C₂²×D₂₃

yes

characteristic

130

M₁₆

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

C₂²⋊D₂₃

yes

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₂²⋊C₄

—

140

C₂²⋊C₄

—

141

C₂³

—

142

C₂³

—

143

C₂³

—

144

C₂³

—

145

C₂³

(C₂×D₂₃)⋊C₄

yes

—

146

C₂³

C₂×D₉₂

yes

characteristic

147

Q₈

—

148

Q₈

—

149

Q₈

—

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

(C₂×D₂₃)⋊C₄

yes

—

163

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

164

C₂×C₄

#67288

yes

—

165

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

166

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

167

C₈

—

168

C₂²

—

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

—

173

C₂²

#10625079

yes

characteristic

174

C₂²

#10625080

yes

—

175

C₂²

#10625080

yes

characteristic

176

C₂²

C₂×(C₂×D₂₃)⋊C₄

yes

characteristic

177

C₂²

#10625101

yes

—

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

—

184

—

182

C₄

C₄×C₂²⋊D₂₃

yes

—

183

C₄

#10625100

yes

characteristic

184

C₄

#10625124

yes

characteristic

185

C₂

—

186

C₂

#13092296

yes

characteristic

187

C₂

#13092419

yes

characteristic

188

C₂

#13092427

yes

characteristic

189

C₂

#13092663

yes

—

190

C₂

#13092675

yes

—

	subgroup	G/H
center	C₂³	C₂×D₉₂
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂²×M₁₆×C₂₃	C₂
socle	C₂³×C₂₃	C₂×D₄
prime radical	C₂⁴×C₂₃	D₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂²×M₁₆	D₂₃
hypocenter	C₂₃	#1894
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×M₁₆

subgroup

G/H

center

C₂³

C₂×D₉₂

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂²×M₁₆×C₂₃

C₂

socle

C₂³×C₂₃

C₂×D₄

prime radical

C₂⁴×C₂₃

D₄

cosocle

C₂²×C₄×C₂₃

C₂³

hypercenter

C₂²×M₁₆

D₂₃

hypocenter

C₂₃

#1894

LCS

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×M₁₆

N	H ≅ G/N	type
2. C₂×#10624998	C₂	semidirect
4. #13092180	C₂	semidirect
16. #10625095	C₄	semidirect
115. C₂₃	#1894	semidirect

H ≅ G/N

type

2. C₂×#10624998

C₂

semidirect

4. #13092180

C₂

semidirect

16. #10625095

C₄

semidirect

115. C₂₃

#1894

semidirect

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

subgroup	index
C₂×#10625095	2
C₂×#10624998	2
#13092277	2
#13092180	2
C₂²×M₁₆×C₂₃	2
#1894	23

subgroup

index

C₂²×M₁₆×C₂₃

Subgroups — #15673830

Conjugacy classes of subgroups

Complements

Maximal subgroups