Group #15753845 — subgroups

Subgroups — #15753845

Order 3328.

444 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 151 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	#13114585	C₂	1	2	yes	maximal
2	C₂×#11718135	C₂	1	2	yes	maximal
3	#13112717	C₂	1	2	yes	maximal
4	C₄×C₁₃×#125	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×#125	C₂²	1	1	yes	characteristic
6	C₁₃×#389	C₄	1	2	yes	—
7	C₁₃×#387	—	2	2	no	—
8	C₁₃×#382	C₂²	1	1	yes	characteristic
9	C₄×C₁₃×C₄⋊C₄	—	2	2	no	—
10	C₄×C₁₃×C₄⋊C₄	C₄	1	2	yes	—
11	C₄³×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×#88043	C₂²	1	4	yes	—
13	#11718135	C₄	1	8	yes	complemented
14	#11718116	—	2	2	no	—
15	#11718116	—	2	2	no	—
16	C₁₃×#125	—	2	2	no	—
17	C₁₃×#125	C₂×C₄	1	4	yes	—
18	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
19	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
20	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
21	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₁₃	—	2	2	no	—
23	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₃	D₄	1	2	yes	—
26	C₂×C₄²×C₁₃	D₄	1	2	yes	—
27	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
28	C₂²×#1055	—	2	4	no	—
29	C₁₃×#95	—	2	2	no	—
30	C₁₃×#95	—	2	2	no	—
31	C₁₃×#95	—	2	2	no	—
32	#88043	—	2	4	no	—
33	#88043	C₂×C₄	1	8	yes	—
34	#8225	—	13	1	no	maximal, Sylow
35	C₂²×C₄×C₁₃	—	2	2	no	—
36	C₂²×C₄×C₁₃	—	2	2	no	—
37	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
39	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
42	C₁₃×C₄⋊C₄	—	2	2	no	—
43	C₁₃×C₄⋊C₄	—	2	8	no	—
44	C₁₃×C₄⋊C₄	—	4	2	no	—
45	C₁₃×C₄⋊C₄	—	4	4	no	—
46	C₁₃×C₄⋊C₄	C₄²	1	4	yes	—
47	C₄²×C₁₃	—	2	2	no	—
48	C₄²×C₁₃	—	4	1	no	—
49	C₄²×C₁₃	—	4	2	no	—
50	C₄²×C₁₃	—	4	2	no	—
51	C₄²×C₁₃	—	4	4	no	—
52	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
53	C₄²×C₁₃	#41	1	2	yes	—
54	C₂×#1055	—	2	8	no	—
55	C₂×#1055	—	4	4	no	—
56	C₂×#1055	—	4	4	no	—
57	C₄×#125	D₁₃	1	1	yes	characteristic
58	#1745	—	13	2	no	—
59	C₂×#330	—	13	2	no	—
60	#1295	—	13	2	no	—
61	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₁₃	—	2	1	no	—
63	C₂×C₄×C₁₃	—	2	1	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	—	2	2	no	—
71	C₂×C₄×C₁₃	—	2	2	no	—
72	C₂×C₄×C₁₃	—	2	2	no	—
73	C₂×C₄×C₁₃	—	4	2	no	—
74	C₂×C₄×C₁₃	—	4	2	no	—
75	C₂×C₄×C₁₃	—	4	2	no	—
76	C₂×C₄×C₁₃	—	4	2	no	—
77	C₂×C₄×C₁₃	—	4	2	no	—
78	C₂×C₄×C₁₃	#101	1	2	yes	—
79	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
80	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₄×D₄	1	4	yes	—
82	#1055	—	4	8	no	—
83	C₂×#125	C₂×D₁₃	1	1	yes	characteristic
84	C₂×#97	—	13	4	no	—
85	#389	Dic₁₃	1	2	yes	—
86	#387	—	2	2	no	—
87	#382	C₂×D₁₃	1	1	yes	characteristic
88	C₄×C₄⋊C₄	—	2	2	no	—
89	C₄×C₄⋊C₄	Dic₁₃	1	2	yes	—
90	C₄³	C₂×D₁₃	1	1	yes	characteristic
91	#336	—	26	2	no	—
92	#336	—	26	2	no	—
93	#330	—	13	8	no	—
94	C₂²×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
95	C₂²×C₁₃	C₂×#101	1	1	yes	characteristic
96	C₂²×C₁₃	#413	1	1	yes	characteristic
97	C₂²×C₁₃	#421	1	2	yes	—
98	C₂²×C₁₃	#421	1	2	yes	—
99	C₄×C₁₃	—	2	1	no	—
100	C₄×C₁₃	—	2	1	no	—
101	C₄×C₁₃	—	2	4	no	—
102	C₄×C₁₃	—	2	2	no	—
103	C₄×C₁₃	—	4	1	no	—
104	C₄×C₁₃	—	4	4	no	—
105	C₄×C₁₃	—	4	2	no	—
106	C₄×C₁₃	—	8	2	no	—
107	C₂²×C₈	—	26	4	no	—
108	#125	—	2	2	no	—
109	#125	C₄×D₁₃	1	4	yes	—
110	C₂×C₄⋊C₄	—	2	2	no	—
111	C₂×C₄⋊C₄	—	2	2	no	—
112	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
113	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
114	C₂×C₄²	—	2	2	no	—
115	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
116	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
117	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
118	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
119	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
120	#97	—	13	8	no	—
121	#97	—	26	4	no	—
122	#95	—	2	2	no	—
123	#95	—	2	2	no	—
124	#95	—	2	2	no	—
125	C₂×C₁₃	#1534	1	1	yes	characteristic
126	C₂×C₁₃	#1535	1	2	yes	—
127	C₂×C₁₃	#1537	1	1	yes	characteristic
128	C₂×C₁₃	C₄×#101	1	1	yes	characteristic
129	C₂×C₁₃	#1759	1	2	yes	—
130	C₂²×C₄	—	2	2	no	—
131	C₂²×C₄	—	2	2	no	—
132	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
133	C₂²×C₄	C₂²⋊Dic₁₃	1	2	yes	—
134	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
135	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
136	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
137	C₂×C₈	—	26	8	no	—
138	C₂×C₈	—	52	4	no	—
139	C₂×C₈	—	52	4	no	—
140	C₄⋊C₄	—	2	2	no	—
141	C₄⋊C₄	—	2	8	no	—
142	C₄⋊C₄	—	4	2	no	—
143	C₄⋊C₄	—	4	4	no	—
144	C₄⋊C₄	C₄×Dic₁₃	1	4	yes	—
145	C₄²	—	2	2	no	—
146	C₄²	—	4	1	no	—
147	C₄²	—	4	2	no	—
148	C₄²	—	4	2	no	—
149	C₄²	—	4	4	no	—
150	C₄²	C₂×C₄×D₁₃	1	2	yes	—
151	C₄²	#6127	1	2	yes	—
152	C₁₃	#8225	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	1	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	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	—	4	2	no	—
170	C₂×C₄	#88076	1	2	yes	—
171	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
172	C₂×C₄	#88180	1	1	yes	characteristic
173	C₂×C₄	C₄×C₂²⋊D₁₃	1	4	yes	—
174	C₈	—	52	8	no	—
175	C₂²	C₄×C₂²⋊Dic₁₃	1	1	yes	characteristic
176	C₂²	C₂×#88076	1	1	yes	characteristic
177	C₂²	#11718897	1	2	yes	—
178	C₂²	#11718897	1	2	yes	—
179	C₂²	#11718898	1	1	yes	characteristic
180	C₄	—	2	1	no	—
181	C₄	—	2	1	no	—
182	C₄	—	2	4	no	—
183	C₄	—	2	2	no	—
184	C₄	—	4	4	no	—
185	C₄	—	4	2	no	—
186	C₄	—	4	1	no	—
187	C₄	—	8	2	no	—
188	C₂	#13113463	1	1	yes	characteristic
189	C₂	#13113467	1	2	yes	—
190	C₂	#13113474	1	1	yes	characteristic
191	C₂	C₄×#88076	1	1	yes	characteristic
192	C₂	#13114646	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13114585

C₂

yes

maximal

C₂×#11718135

C₂

yes

maximal

#13112717

C₂

yes

maximal

C₄×C₁₃×#125

C₂

yes

maximal, characteristic

C₂×C₁₃×#125

C₂²

yes

characteristic

C₁₃×#389

C₄

yes

—

C₁₃×#387

—

C₁₃×#382

C₂²

yes

characteristic

C₄×C₁₃×C₄⋊C₄

—

C₄×C₁₃×C₄⋊C₄

C₄

yes

—

C₄³×C₁₃

C₂²

yes

characteristic

C₂×#88043

C₂²

yes

—

#11718135

C₄

yes

complemented

#11718116

—

#11718116

—

C₁₃×#125

—

C₁₃×#125

C₂×C₄

yes

—

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

—

#88043

—

#88043

C₂×C₄

yes

—

#8225

—

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

C₄²

yes

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

C₂²×C₄

yes

—

C₄²×C₁₃

#41

yes

—

—

—

—

D₁₃

yes

—

—

—

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

—

C₂×C₄×C₁₃

#101

yes

—

C₂×C₄×C₁₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₁₃

#117

yes

characteristic

C₂×C₄×C₁₃

C₄×D₄

yes

—

#1055

—

C₂×#125

C₂×D₁₃

yes

characteristic

C₂×#97

—

#389

Dic₁₃

yes

—

#387

—

#382

C₂×D₁₃

yes

characteristic

C₄×C₄⋊C₄

—

C₄×C₄⋊C₄

Dic₁₃

yes

—

C₄³

C₂×D₁₃

yes

—

—

—

C₂²×C₁₃

C₄×C₂²⋊C₄

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

—

C₂²×C₁₃

#421

yes

—

C₄×C₁₃

—

100

C₄×C₁₃

—

101

C₄×C₁₃

—

102

C₄×C₁₃

—

103

C₄×C₁₃

—

104

C₄×C₁₃

—

105

C₄×C₁₃

—

106

C₄×C₁₃

—

107

C₂²×C₈

—

108

#125

—

109

#125

C₄×D₁₃

yes

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

C₂×Dic₁₃

yes

—

113

C₂×C₄⋊C₄

C₂×Dic₁₃

yes

—

114

C₂×C₄²

—

115

C₂×C₄²

C₂×Dic₁₃

yes

characteristic

116

C₂×C₄²

C₂×Dic₁₃

yes

characteristic

117

C₂×C₄²

C₂²⋊D₁₃

yes

—

118

C₂×C₄²

C₂²⋊D₁₃

yes

—

119

C₂×C₄²

C₂²×D₁₃

yes

characteristic

120

#97

—

121

#97

—

122

#95

—

123

#95

—

124

#95

—

125

C₂×C₁₃

#1534

yes

characteristic

126

C₂×C₁₃

#1535

yes

—

127

C₂×C₁₃

#1537

yes

characteristic

128

C₂×C₁₃

C₄×#101

yes

characteristic

129

C₂×C₁₃

#1759

yes

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

C₂²⋊Dic₁₃

yes

characteristic

133

C₂²×C₄

C₂²⋊Dic₁₃

yes

—

134

C₂²×C₄

C₂²⋊Dic₁₃

yes

characteristic

135

C₂²×C₄

C₂²×Dic₁₃

yes

characteristic

136

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

C₄×Dic₁₃

yes

—

145

C₄²

—

146

C₄²

—

147

C₄²

—

148

C₄²

—

149

C₄²

—

150

C₄²

C₂×C₄×D₁₃

yes

—

151

C₄²

#6127

yes

—

152

C₁₃

#8225

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

—

170

C₂×C₄

#88076

yes

—

171

C₂×C₄

C₂×C₄×Dic₁₃

yes

characteristic

172

C₂×C₄

#88180

yes

characteristic

173

C₂×C₄

C₄×C₂²⋊D₁₃

yes

—

174

C₈

—

175

C₂²

C₄×C₂²⋊Dic₁₃

yes

characteristic

176

C₂²

C₂×#88076

yes

characteristic

177

C₂²

#11718897

yes

—

178

C₂²

#11718897

yes

—

179

C₂²

#11718898

yes

characteristic

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

—

187

C₄

—

188

C₂

#13113463

yes

characteristic

189

C₂

#13113467

yes

—

190

C₂

#13113474

yes

characteristic

191

C₂

C₄×#88076

yes

characteristic

192

C₂

#13114646

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₃
commutator	C₂×C₄×C₁₃	C₂×C₄²
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	#8225
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

C₂×C₄²

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
13. #11718135	C₄	semidirect
152. C₁₃	#8225	semidirect

H ≅ G/N

type

13. #11718135

C₄

semidirect

152. C₁₃

#8225

semidirect

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

subgroup	index
#13114585	2
C₂×#11718135	2
#13112717	2
C₄×C₁₃×#125	2
#8225	13

subgroup

index

C₄×C₁₃×#125

Subgroups — #15753845

Conjugacy classes of subgroups

Complements

Maximal subgroups