Group #15682028 — subgroups

Subgroups — #15682028

Order 2944.

484 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 197 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×C₄²×D₂₃	C₂	1	1	yes	maximal, characteristic
2	#13092472	C₂	1	4	yes	maximal, complemented
3	#13092468	C₂	1	4	yes	maximal, complemented
4	#13092440	C₂	1	2	yes	maximal, complemented
5	#13092380	C₂	1	2	yes	maximal
6	C₄×#67267	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₄×D₂₃	C₄	1	2	yes	—
9	C₂²×C₄×D₂₃	C₂²	1	1	yes	characteristic
10	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
11	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₂₃	C₄	1	2	yes	—
13	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
14	C₂×#67267	C₂²	1	2	yes	—
15	C₄²×D₂₃	C₄	1	2	yes	—
16	C₄²×D₂₃	C₄	1	2	yes	—
17	C₄²×D₂₃	C₂²	1	2	yes	—
18	C₄²×D₂₃	C₂²	1	2	yes	—
19	C₂₃×#97	C₂²	1	2	yes	complemented
20	C₄×C₈×C₂₃	C₂²	1	2	yes	complemented
21	(C₂×D₂₃)⋊C₈	C₂²	1	8	yes	—
22	Dic₂₃⋊C₈	C₂²	1	8	yes	complemented
23	#10624985	C₂²	1	2	yes	complemented
24	C₄×C₂₃⋊C₈	C₂²	1	2	yes	complemented
25	C₂³×D₂₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
28	C₂²×Dic₂₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×Dic₂₃	C₂×C₄	1	2	yes	—
30	C₂×C₄×D₂₃	—	2	4	no	—
31	C₂×C₄×D₂₃	—	2	8	no	—
32	C₂×C₄×D₂₃	C₂×C₄	1	2	yes	—
33	C₂×C₄×D₂₃	C₂×C₄	1	2	yes	—
34	C₂×C₄×D₂₃	C₂×C₄	1	2	yes	—
35	C₂×C₄×D₂₃	C₂×C₄	1	2	yes	—
36	C₂×C₄×D₂₃	C₂×C₄	1	2	yes	—
37	C₂×C₄×D₂₃	C₂³	1	2	yes	—
38	M₁₆×C₂₃	—	2	4	no	—
39	C₂×C₈×C₂₃	C₂³	1	4	yes	—
40	C₄²×C₂₃	C₂×C₄	1	2	yes	—
41	C₄²×C₂₃	C₂³	1	1	yes	characteristic
42	C₄²×C₂₃	C₂³	1	1	yes	characteristic
43	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
44	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
45	C₄×Dic₂₃	C₂×C₄	1	2	yes	—
46	C₄×Dic₂₃	C₂×C₄	1	2	yes	—
47	C₄×Dic₂₃	C₂×C₄	1	2	yes	—
48	C₄×Dic₂₃	C₂×C₄	1	2	yes	—
49	C₄×Dic₂₃	C₂³	1	2	yes	—
50	#67267	—	2	4	no	—
51	C₂×C₂₃⋊C₈	C₂³	1	4	yes	—
52	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×D₂₃	—	2	2	no	—
54	C₂²×D₂₃	—	2	4	no	—
55	C₂²×D₂₃	C₂²×C₄	1	2	yes	—
56	C₂×C₄×C₂₃	—	2	4	no	—
57	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
60	C₂×C₄×C₂₃	#41	1	4	yes	—
61	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
62	C₂×Dic₂₃	—	2	2	no	—
63	C₂×Dic₂₃	—	2	4	no	—
64	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
65	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
66	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
67	C₄×D₂₃	—	2	4	no	—
68	C₄×D₂₃	—	2	4	no	—
69	C₄×D₂₃	—	2	4	no	—
70	C₄×D₂₃	—	2	4	no	—
71	C₄×D₂₃	—	2	4	no	—
72	C₄×D₂₃	M₁₆	1	8	yes	complemented
73	C₈×C₂₃	—	2	4	no	—
74	C₂₃⋊C₈	—	2	4	no	—
75	#2920	—	23	1	no	maximal, Sylow
76	C₂²×C₂₃	—	2	2	no	—
77	C₂²×C₂₃	#117	1	2	yes	—
78	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
79	C₂×D₂₃	—	2	8	no	—
80	C₂×D₂₃	—	2	2	no	—
81	C₂×D₂₃	—	2	2	no	—
82	C₂×D₂₃	C₂×M₁₆	1	4	yes	—
83	C₄×C₂₃	—	2	2	no	—
84	C₄×C₂₃	#117	1	2	yes	—
85	C₄×C₂₃	C₂×M₁₆	1	4	yes	—
86	C₄×C₂₃	C₂×#41	1	2	yes	—
87	Dic₂₃	—	2	2	no	—
88	Dic₂₃	—	2	2	no	—
89	Dic₂₃	C₂×M₁₆	1	4	yes	—
90	C₂²×C₄²	—	23	1	no	—
91	#432	—	23	4	no	—
92	#431	—	23	4	no	—
93	#423	—	23	2	no	—
94	#407	—	23	2	no	—
95	C₄×M₁₆	—	23	1	no	—
96	C₄×M₁₆	D₂₃	1	1	yes	characteristic, complemented
97	C₂×C₂₃	—	2	2	no	—
98	C₂×C₂₃	C₂×#117	1	1	yes	characteristic
99	C₂×C₂₃	C₂²×M₁₆	1	2	yes	—
100	D₂₃	—	2	4	no	—
101	C₂³×C₄	—	23	2	no	—
102	C₂³×C₄	—	23	1	no	—
103	C₂×M₁₆	—	23	2	no	—
104	C₂×M₁₆	C₂×D₂₃	1	2	yes	—
105	C₂×C₄²	—	23	2	no	—
106	C₂×C₄²	—	23	1	no	—
107	C₂×C₄²	—	23	2	no	—
108	C₂×C₄²	—	23	2	no	—
109	C₂×C₄²	—	23	2	no	—
110	C₂×C₄²	—	23	2	no	—
111	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
112	C₄⋊C₈	—	23	8	no	—
113	C₂²⋊C₈	—	23	8	no	—
114	#97	—	23	2	no	—
115	#97	C₂×D₂₃	1	2	yes	complemented
116	C₄×C₈	—	23	2	no	—
117	C₄×C₈	C₂×D₂₃	1	2	yes	complemented
118	C₂₃	#2920	1	1	yes	characteristic, complemented, Sylow
119	C₂⁴	—	23	1	no	—
120	C₂²×C₄	—	23	2	no	—
121	C₂²×C₄	—	23	2	no	—
122	C₂²×C₄	—	23	1	no	—
123	C₂²×C₄	—	23	2	no	—
124	C₂²×C₄	—	23	2	no	—
125	C₂²×C₄	—	23	2	no	—
126	C₂²×C₄	—	23	2	no	—
127	C₂²×C₄	—	23	2	no	—
128	C₂²×C₄	—	46	8	no	—
129	C₂²×C₄	—	46	4	no	—
130	C₂²×C₄	C₄×D₂₃	1	2	yes	—
131	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
132	M₁₆	—	2	4	no	—
133	M₁₆	—	46	4	no	—
134	C₂×C₈	—	23	4	no	—
135	C₂×C₈	C₂²×D₂₃	1	4	yes	—
136	C₄²	—	23	1	no	—
137	C₄²	—	23	2	no	—
138	C₄²	—	23	2	no	—
139	C₄²	—	23	2	no	—
140	C₄²	—	23	1	no	—
141	C₄²	—	23	2	no	—
142	C₄²	—	23	2	no	—
143	C₄²	C₄×D₂₃	1	2	yes	—
144	C₄²	C₂²×D₂₃	1	1	yes	characteristic
145	C₄²	C₂²×D₂₃	1	1	yes	characteristic
146	C₂³	—	23	2	no	—
147	C₂³	—	46	4	no	—
148	C₂³	—	46	2	no	—
149	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	23	2	no	—
152	C₂×C₄	—	23	8	no	—
153	C₂×C₄	—	23	2	no	—
154	C₂×C₄	—	23	2	no	—
155	C₂×C₄	—	46	4	no	—
156	C₂×C₄	—	46	4	no	—
157	C₂×C₄	—	46	4	no	—
158	C₂×C₄	—	46	2	no	—
159	C₂×C₄	—	46	4	no	—
160	C₂×C₄	—	46	4	no	—
161	C₂×C₄	—	46	4	no	—
162	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
163	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
164	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
165	C₂×C₄	#67288	1	4	yes	—
166	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
167	C₈	—	2	4	no	—
168	C₈	—	46	4	no	—
169	C₂²	—	2	2	no	—
170	C₂²	—	23	4	no	—
171	C₂²	—	46	8	no	—
172	C₂²	—	46	2	no	—
173	C₂²	—	46	2	no	—
174	C₂²	#10625043	1	2	yes	—
175	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
176	C₄	—	2	2	no	—
177	C₄	—	23	4	no	—
178	C₄	—	46	2	no	—
179	C₄	—	46	2	no	—
180	C₄	#10625043	1	2	yes	—
181	C₄	M₁₆×D₂₃	1	4	yes	—
182	C₄	C₂×#67288	1	2	yes	—
183	C₂	—	2	2	no	—
184	C₂	—	46	4	no	—
185	C₂	C₂×#10625043	1	1	yes	characteristic
186	C₂	C₂×M₁₆×D₂₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄²×D₂₃

C₂

yes

maximal, characteristic

#13092472

C₂

yes

maximal, complemented

#13092468

C₂

yes

maximal, complemented

#13092440

C₂

yes

maximal, complemented

#13092380

C₂

yes

maximal

C₄×#67267

C₂

yes

maximal, characteristic, complemented

C₄×M₁₆×C₂₃

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₂₃

C₄

yes

—

C₂²×C₄×D₂₃

C₂²

yes

characteristic

C₂×M₁₆×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄×Dic₂₃

C₄

yes

—

C₂×C₄×Dic₂₃

C₂²

yes

characteristic

C₂×#67267

C₂²

yes

—

C₄²×D₂₃

C₄

yes

—

C₄²×D₂₃

C₄

yes

—

C₄²×D₂₃

C₂²

yes

—

C₄²×D₂₃

C₂²

yes

—

C₂₃×#97

C₂²

yes

complemented

C₄×C₈×C₂₃

C₂²

yes

complemented

(C₂×D₂₃)⋊C₈

C₂²

yes

—

Dic₂₃⋊C₈

C₂²

yes

complemented

#10624985

C₂²

yes

complemented

C₄×C₂₃⋊C₈

C₂²

yes

complemented

C₂³×D₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

C₂×C₄

yes

characteristic

C₂²×Dic₂₃

C₂×C₄

yes

—

C₂×C₄×D₂₃

—

C₂×C₄×D₂₃

—

C₂×C₄×D₂₃

C₂×C₄

yes

—

C₂×C₄×D₂₃

C₂×C₄

yes

—

C₂×C₄×D₂₃

C₂×C₄

yes

—

C₂×C₄×D₂₃

C₂×C₄

yes

—

C₂×C₄×D₂₃

C₂×C₄

yes

—

C₂×C₄×D₂₃

C₂³

yes

—

M₁₆×C₂₃

—

C₂×C₈×C₂₃

C₂³

yes

—

C₄²×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

C₂³

yes

characteristic

C₄²×C₂₃

C₂³

yes

characteristic

C₄×Dic₂₃

C₂×C₄

yes

characteristic

C₄×Dic₂₃

C₂×C₄

yes

characteristic

C₄×Dic₂₃

C₂×C₄

yes

—

C₄×Dic₂₃

C₂×C₄

yes

—

C₄×Dic₂₃

C₂×C₄

yes

—

C₄×Dic₂₃

C₂×C₄

yes

—

C₄×Dic₂₃

C₂³

yes

—

#67267

—

C₂×C₂₃⋊C₈

C₂³

yes

—

C₂³×C₂₃

C₂²×C₄

yes

characteristic

C₂²×D₂₃

—

C₂²×D₂₃

—

C₂²×D₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₃

#41

yes

—

C₂×C₄×C₂₃

C₂⁴

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

C₂²×C₄

yes

—

C₂×Dic₂₃

C₂²×C₄

yes

—

C₂×Dic₂₃

C₂²×C₄

yes

—

C₄×D₂₃

—

C₄×D₂₃

—

C₄×D₂₃

—

C₄×D₂₃

—

C₄×D₂₃

—

C₄×D₂₃

M₁₆

yes

complemented

C₈×C₂₃

—

C₂₃⋊C₈

—

#2920

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

#117

yes

—

C₂²×C₂₃

C₂³×C₄

yes

characteristic

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

C₂×M₁₆

yes

—

C₄×C₂₃

—

C₄×C₂₃

#117

yes

—

C₄×C₂₃

C₂×M₁₆

yes

—

C₄×C₂₃

C₂×#41

yes

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

C₂×M₁₆

yes

—

C₂²×C₄²

—

—

—

—

—

C₄×M₁₆

—

C₄×M₁₆

D₂₃

yes

characteristic, complemented

C₂×C₂₃

—

C₂×C₂₃

C₂×#117

yes

characteristic

C₂×C₂₃

C₂²×M₁₆

yes

—

100

D₂₃

—

101

C₂³×C₄

—

102

C₂³×C₄

—

103

C₂×M₁₆

—

104

C₂×M₁₆

C₂×D₂₃

yes

—

105

C₂×C₄²

—

106

C₂×C₄²

—

107

C₂×C₄²

—

108

C₂×C₄²

—

109

C₂×C₄²

—

110

C₂×C₄²

—

111

C₂×C₄²

C₂×D₂₃

yes

characteristic

112

C₄⋊C₈

—

113

C₂²⋊C₈

—

114

#97

—

115

#97

C₂×D₂₃

yes

complemented

116

C₄×C₈

—

117

C₄×C₈

C₂×D₂₃

yes

complemented

118

C₂₃

#2920

yes

characteristic, complemented, Sylow

119

C₂⁴

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

C₄×D₂₃

yes

—

131

C₂²×C₄

C₂²×D₂₃

yes

characteristic

132

M₁₆

—

133

M₁₆

—

134

C₂×C₈

—

135

C₂×C₈

C₂²×D₂₃

yes

—

136

C₄²

—

137

C₄²

—

138

C₄²

—

139

C₄²

—

140

C₄²

—

141

C₄²

—

142

C₄²

—

143

C₄²

C₄×D₂₃

yes

—

144

C₄²

C₂²×D₂₃

yes

characteristic

145

C₄²

C₂²×D₂₃

yes

characteristic

146

C₂³

—

147

C₂³

—

148

C₂³

—

149

C₂³

C₂×C₄×D₂₃

yes

characteristic

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×C₄×D₂₃

yes

—

163

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

164

C₂×C₄

C₂×C₄×D₂₃

yes

—

165

C₂×C₄

#67288

yes

—

166

C₂×C₄

C₂³×D₂₃

yes

characteristic

167

C₈

—

168

C₈

—

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

—

173

C₂²

—

174

C₂²

#10625043

yes

—

175

C₂²

C₂²×C₄×D₂₃

yes

characteristic

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

#10625043

yes

—

181

C₄

M₁₆×D₂₃

yes

—

182

C₄

C₂×#67288

yes

—

183

C₂

—

184

C₂

—

185

C₂

C₂×#10625043

yes

characteristic

186

C₂

C₂×M₁₆×D₂₃

yes

—

	subgroup	G/H
center	C₄²	C₂²×D₂₃
commutator, socle	C₂²×C₂₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₄×M₁₆×C₂₃	C₂
prime radical	C₂³×D₂₃	C₂×C₄
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	C₄×M₁₆	D₂₃
hypocenter	C₂₃	#2920
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₄²

C₂²×D₂₃

commutator, socle

C₂²×C₂₃

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₂₃

Fitting

C₄×M₁₆×C₂₃

C₂

prime radical

C₂³×D₂₃

C₂×C₄

cosocle

C₂×C₄×C₂₃

C₂⁴

hypercenter

C₄×M₁₆

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂₃

C₁ ◁ C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
2. #13092472	C₂	semidirect
3. #13092468	C₂	semidirect
4. #13092440	C₂	semidirect
6. C₄×#67267	C₂	semidirect
7. C₄×M₁₆×C₂₃	C₂	semidirect
19. C₂₃×#97	C₂²	semidirect
20. C₄×C₈×C₂₃	C₂²	semidirect
22. Dic₂₃⋊C₈	C₂²	semidirect
23. #10624985	C₂²	semidirect
24. C₄×C₂₃⋊C₈	C₂²	semidirect
72. C₄×D₂₃	M₁₆	semidirect
96. C₄×M₁₆	D₂₃	semidirect
115. #97	C₂×D₂₃	semidirect
117. C₄×C₈	C₂×D₂₃	semidirect
118. C₂₃	#2920	semidirect

H ≅ G/N

type

2. #13092472

C₂

semidirect

3. #13092468

C₂

semidirect

4. #13092440

C₂

semidirect

6. C₄×#67267

C₂

semidirect

7. C₄×M₁₆×C₂₃

C₂

semidirect

19. C₂₃×#97

C₂²

semidirect

20. C₄×C₈×C₂₃

C₂²

semidirect

22. Dic₂₃⋊C₈

C₂²

semidirect

23. #10624985

C₂²

semidirect

24. C₄×C₂₃⋊C₈

C₂²

semidirect

72. C₄×D₂₃

M₁₆

semidirect

96. C₄×M₁₆

D₂₃

semidirect

115. #97

C₂×D₂₃

semidirect

117. C₄×C₈

C₂×D₂₃

semidirect

118. C₂₃

#2920

semidirect

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

subgroup	index
C₂×C₄²×D₂₃	2
#13092472	2
#13092468	2
#13092440	2
#13092380	2
C₄×#67267	2
C₄×M₁₆×C₂₃	2
#2920	23

subgroup

index

C₂×C₄²×D₂₃

C₄×M₁₆×C₂₃

Subgroups — #15682028

Conjugacy classes of subgroups

Complements

Maximal subgroups