Group #15751037 — subgroups

Subgroups — #15751037

Order 3328.

408 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 143 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	#13114953	C₂	1	2	yes	maximal
2	#13112659	C₂	1	4	yes	maximal
3	C₄×C₁₃×#125	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃×#125	C₄	1	1	yes	characteristic
5	C₁₃×#389	C₂²	1	2	yes	—
6	C₁₃×#387	—	2	2	no	—
7	C₁₃×#382	C₄	1	1	yes	characteristic
8	C₄×C₁₃×C₄⋊C₄	—	2	2	no	—
9	C₄×C₁₃×C₄⋊C₄	C₄	1	2	yes	—
10	C₄³×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊#1055	—	2	2	no	—
12	#11718116	—	2	2	no	—
13	#11718116	C₂²	1	4	yes	—
14	C₁₃×#125	—	2	2	no	—
15	C₁₃×#125	C₈	1	4	yes	complemented
16	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
17	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
18	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
19	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₁₃	—	2	2	no	—
21	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₃	D₄	1	2	yes	—
24	C₂×C₄²×C₁₃	D₄	1	2	yes	—
25	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
26	C₂²×#1055	—	2	4	no	—
27	C₁₃×#95	—	2	2	no	—
28	C₁₃×#95	—	2	2	no	—
29	C₁₃×#95	—	2	2	no	—
30	C₄⋊#1055	—	4	4	no	—
31	#7602	—	13	1	no	maximal, Sylow
32	C₂²×C₄×C₁₃	—	2	2	no	—
33	C₂²×C₄×C₁₃	—	2	2	no	—
34	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
36	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
39	C₁₃×C₄⋊C₄	—	2	8	no	—
40	C₁₃×C₄⋊C₄	—	2	2	no	—
41	C₁₃×C₄⋊C₄	—	4	2	no	—
42	C₁₃×C₄⋊C₄	—	4	2	no	—
43	C₁₃×C₄⋊C₄	—	4	2	no	—
44	C₁₃×C₄⋊C₄	C₂×C₈	1	4	yes	—
45	C₄²×C₁₃	—	2	2	no	—
46	C₄²×C₁₃	—	2	2	no	—
47	C₄²×C₁₃	—	2	2	no	—
48	C₄²×C₁₃	—	4	2	no	—
49	C₄²×C₁₃	—	4	1	no	—
50	C₄²×C₁₃	—	4	4	no	—
51	C₄²×C₁₃	C₂×C₈	1	2	yes	—
52	C₄²×C₁₃	M₁₆	1	2	yes	—
53	C₂×#1055	—	4	4	no	—
54	C₂×#1055	—	4	4	no	—
55	C₂×#1055	—	4	4	no	—
56	C₄×#125	D₁₃	1	1	yes	characteristic
57	#1831	—	13	2	no	—
58	#1271	—	13	4	no	—
59	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₁₃	—	2	2	no	—
61	C₂×C₄×C₁₃	—	2	1	no	—
62	C₂×C₄×C₁₃	—	2	1	no	—
63	C₂×C₄×C₁₃	—	2	2	no	—
64	C₂×C₄×C₁₃	—	2	2	no	—
65	C₂×C₄×C₁₃	—	2	2	no	—
66	C₂×C₄×C₁₃	—	2	2	no	—
67	C₂×C₄×C₁₃	—	2	4	no	—
68	C₂×C₄×C₁₃	—	2	1	no	—
69	C₂×C₄×C₁₃	—	2	2	no	—
70	C₂×C₄×C₁₃	—	4	2	no	—
71	C₂×C₄×C₁₃	—	4	2	no	—
72	C₂×C₄×C₁₃	—	4	2	no	—
73	C₂×C₄×C₁₃	—	4	2	no	—
74	C₂×C₄×C₁₃	—	4	2	no	—
75	C₂×C₄×C₁₃	C₂²⋊C₈	1	4	yes	—
76	C₂×C₄×C₁₃	#101	1	2	yes	—
77	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
79	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
80	C₂×C₄×C₁₃	#137	1	2	yes	—
81	#1055	—	8	4	no	—
82	C₂×#125	Dic₁₃	1	1	yes	characteristic
83	C₂×C₄⋊C₈	—	26	2	no	—
84	#389	C₂×D₁₃	1	2	yes	—
85	#387	—	2	2	no	—
86	#382	Dic₁₃	1	1	yes	characteristic
87	C₄×C₄⋊C₄	—	2	2	no	—
88	C₄×C₄⋊C₄	Dic₁₃	1	2	yes	—
89	C₄³	C₂×D₁₃	1	1	yes	characteristic
90	#336	—	13	4	no	—
91	#336	—	26	2	no	—
92	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
93	C₂²×C₁₃	#410	1	1	yes	characteristic
94	C₂²×C₁₃	C₂×#101	1	1	yes	characteristic
95	C₂²×C₁₃	#417	1	2	yes	—
96	C₂²×C₁₃	#421	1	2	yes	—
97	C₄×C₁₃	—	2	1	no	—
98	C₄×C₁₃	—	2	1	no	—
99	C₄×C₁₃	—	2	4	no	—
100	C₄×C₁₃	—	2	2	no	—
101	C₄×C₁₃	—	2	2	no	—
102	C₄×C₁₃	—	2	2	no	—
103	C₄×C₁₃	—	4	1	no	—
104	C₄×C₁₃	—	4	4	no	—
105	C₄×C₁₃	—	8	2	no	—
106	C₂²×C₈	—	26	4	no	—
107	#125	—	2	2	no	—
108	#125	#1055	1	4	yes	complemented
109	C₂×C₄⋊C₄	—	2	2	no	—
110	C₂×C₄⋊C₄	—	2	2	no	—
111	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
112	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
113	C₂×C₄²	—	2	2	no	—
114	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
115	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
116	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
117	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
118	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
119	C₄⋊C₈	—	52	4	no	—
120	#95	—	2	2	no	—
121	#95	—	2	2	no	—
122	#95	—	2	2	no	—
123	C₂×C₁₃	#1461	1	1	yes	characteristic
124	C₂×C₁₃	#1477	1	2	yes	—
125	C₂×C₁₃	#1512	1	2	yes	—
126	C₂×C₁₃	#1534	1	1	yes	characteristic
127	C₂×C₁₃	#1561	1	1	yes	characteristic
128	C₂²×C₄	—	2	2	no	—
129	C₂²×C₄	—	2	2	no	—
130	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
131	C₂²×C₄	C₂²⋊Dic₁₃	1	2	yes	—
132	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
133	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
134	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
135	C₂×C₈	—	52	4	no	—
136	C₂×C₈	—	52	4	no	—
137	C₂×C₈	—	52	4	no	—
138	C₄⋊C₄	—	2	8	no	—
139	C₄⋊C₄	—	2	2	no	—
140	C₄⋊C₄	—	4	2	no	—
141	C₄⋊C₄	—	4	2	no	—
142	C₄⋊C₄	—	4	2	no	—
143	C₄⋊C₄	C₂×#1055	1	4	yes	—
144	C₄²	—	2	2	no	—
145	C₄²	—	2	2	no	—
146	C₄²	—	2	2	no	—
147	C₄²	—	4	2	no	—
148	C₄²	—	4	1	no	—
149	C₄²	—	4	4	no	—
150	C₄²	C₂×#1055	1	2	yes	—
151	C₄²	#6099	1	2	yes	—
152	C₁₃	#7602	1	1	yes	characteristic, complemented, Sylow
153	C₂³	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	4	2	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	C₂²⋊#1055	1	4	yes	—
170	C₂×C₄	#88076	1	2	yes	—
171	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
172	C₂×C₄	C₂×#6099	1	1	yes	characteristic
173	C₂×C₄	#88203	1	2	yes	—
174	C₂×C₄	#88205	1	2	yes	—
175	C₈	—	104	4	no	—
176	C₂²	C₂×C₂²⋊#1055	1	1	yes	characteristic
177	C₂²	C₂×#88076	1	1	yes	characteristic
178	C₂²	#11718893	1	2	yes	—
179	C₂²	#11718897	1	2	yes	—
180	C₂²	#11718901	1	1	yes	characteristic
181	C₄	—	2	2	no	—
182	C₄	—	2	2	no	—
183	C₄	—	2	1	no	—
184	C₄	—	2	1	no	—
185	C₄	—	2	2	no	—
186	C₄	—	2	4	no	—
187	C₄	—	4	4	no	—
188	C₄	—	4	1	no	—
189	C₄	—	8	2	no	—
190	C₂	#13113160	1	1	yes	characteristic
191	C₂	#13113223	1	2	yes	—
192	C₂	#13113374	1	2	yes	—
193	C₂	#13113463	1	1	yes	characteristic
194	C₂	#13113574	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13114953

C₂

yes

maximal

#13112659

C₂

yes

maximal

C₄×C₁₃×#125

C₂

yes

maximal, characteristic

C₂×C₁₃×#125

C₄

yes

characteristic

C₁₃×#389

C₂²

yes

—

C₁₃×#387

—

C₁₃×#382

C₄

yes

characteristic

C₄×C₁₃×C₄⋊C₄

—

C₄×C₁₃×C₄⋊C₄

C₄

yes

—

C₄³×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊#1055

—

#11718116

—

#11718116

C₂²

yes

—

C₁₃×#125

—

C₁₃×#125

C₈

yes

complemented

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₁₃×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄²×C₁₃

—

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₃

D₄

yes

—

C₂×C₄²×C₁₃

D₄

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂²×#1055

—

C₁₃×#95

—

C₁₃×#95

—

C₁₃×#95

—

C₄⋊#1055

—

#7602

—

maximal, Sylow

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×D₄

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

C₂×C₈

yes

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

C₂×C₈

yes

—

C₄²×C₁₃

M₁₆

yes

—

—

—

—

D₁₃

yes

characteristic

#1831

—

#1271

—

C₂³×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊C₈

yes

—

C₂×C₄×C₁₃

#101

yes

—

C₂×C₄×C₁₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₁₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₁₃

#137

yes

—

#1055

—

C₂×#125

Dic₁₃

yes

characteristic

C₂×C₄⋊C₈

—

#389

C₂×D₁₃

yes

—

#387

—

#382

Dic₁₃

yes

characteristic

C₄×C₄⋊C₄

—

C₄×C₄⋊C₄

Dic₁₃

yes

—

C₄³

C₂×D₁₃

yes

characteristic

#336

—

#336

—

C₂²×C₁₃

C₂×C₂²⋊C₈

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

—

C₂²×C₁₃

#421

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

100

C₄×C₁₃

—

101

C₄×C₁₃

—

102

C₄×C₁₃

—

103

C₄×C₁₃

—

104

C₄×C₁₃

—

105

C₄×C₁₃

—

106

C₂²×C₈

—

107

#125

—

108

#125

#1055

yes

complemented

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

C₂×Dic₁₃

yes

—

112

C₂×C₄⋊C₄

C₂×Dic₁₃

yes

—

113

C₂×C₄²

—

114

C₂×C₄²

C₂×Dic₁₃

yes

characteristic

115

C₂×C₄²

C₂×Dic₁₃

yes

characteristic

116

C₂×C₄²

C₂²⋊D₁₃

yes

—

117

C₂×C₄²

C₂²⋊D₁₃

yes

—

118

C₂×C₄²

C₂²×D₁₃

yes

characteristic

119

C₄⋊C₈

—

120

#95

—

121

#95

—

122

#95

—

123

C₂×C₁₃

#1461

yes

characteristic

124

C₂×C₁₃

#1477

yes

—

125

C₂×C₁₃

#1512

yes

—

126

C₂×C₁₃

#1534

yes

characteristic

127

C₂×C₁₃

#1561

yes

characteristic

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

C₂²⋊Dic₁₃

yes

characteristic

131

C₂²×C₄

C₂²⋊Dic₁₃

yes

—

132

C₂²×C₄

C₂²⋊Dic₁₃

yes

characteristic

133

C₂²×C₄

C₂²×Dic₁₃

yes

characteristic

134

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

C₂×#1055

yes

—

144

C₄²

—

145

C₄²

—

146

C₄²

—

147

C₄²

—

148

C₄²

—

149

C₄²

—

150

C₄²

C₂×#1055

yes

—

151

C₄²

#6099

yes

—

152

C₁₃

#7602

yes

characteristic, complemented, Sylow

153

C₂³

C₂×C₂²⋊Dic₁₃

yes

characteristic

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

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

C₂²⋊#1055

yes

—

170

C₂×C₄

#88076

yes

—

171

C₂×C₄

C₂²×#1055

yes

characteristic

172

C₂×C₄

C₂×#6099

yes

characteristic

173

C₂×C₄

#88203

yes

—

174

C₂×C₄

#88205

yes

—

175

C₈

—

104

—

176

C₂²

C₂×C₂²⋊#1055

yes

characteristic

177

C₂²

C₂×#88076

yes

characteristic

178

C₂²

#11718893

yes

—

179

C₂²

#11718897

yes

—

180

C₂²

#11718901

yes

characteristic

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

—

190

C₂

#13113160

yes

characteristic

191

C₂

#13113223

yes

—

192

C₂

#13113374

yes

—

193

C₂

#13113463

yes

characteristic

194

C₂

#13113574

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₃
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₄×C₁₃×#125	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₄×#125	D₁₃
hypocenter	C₁₃	#7602
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₁₃

commutator

C₂×C₄×C₁₃

C₂²×C₈

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₄×C₁₃×#125

C₂

socle, prime radical

C₂³×C₁₃

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#125

N	H ≅ G/N	type
15. C₁₃×#125	C₈	semidirect
108. #125	#1055	semidirect
152. C₁₃	#7602	semidirect

H ≅ G/N

type

15. C₁₃×#125

C₈

semidirect

108. #125

#1055

semidirect

152. C₁₃

#7602

semidirect

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

subgroup	index
#13114953	2
#13112659	2
C₄×C₁₃×#125	2
#7602	13

subgroup

index

#13114953

#13112659

C₄×C₁₃×#125

#7602

Subgroups — #15751037

Conjugacy classes of subgroups

Complements

Maximal subgroups