Group #15756588 — subgroups

Subgroups — #15756588

Order 3328.

362 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 147 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	#13114987	C₂	1	2	yes	maximal
2	#13114581	C₂	1	1	yes	maximal, characteristic
3	#13112671	C₂	1	1	yes	maximal, characteristic
4	#13112670	C₂	1	2	yes	maximal
5	C₁₃×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88054	—	2	2	no	—
9	C₂×#88054	—	2	2	no	—
10	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊#1055	C₂²	1	2	yes	—
12	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
13	#11718142	C₄	1	2	yes	—
14	#11718142	C₄	1	2	yes	—
15	C₂²×C₈×C₁₃	—	2	2	no	—
16	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
17	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
18	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
19	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
20	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
21	C₂×C₄×Dic₁₃	Q₈	1	2	yes	—
22	C₂²×#1055	—	2	2	no	—
23	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
24	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
25	C₁₃×C₄⋊C₈	—	2	2	no	—
26	C₁₃×C₄⋊C₈	—	2	2	no	—
27	C₄×C₈×C₁₃	—	2	2	no	—
28	#88054	—	4	4	no	—
29	#88054	—	4	4	no	—
30	C₄⋊#1055	—	2	2	no	—
31	C₄⋊#1055	—	2	2	no	—
32	C₄×#1055	—	2	2	no	—
33	#8787	—	13	1	no	maximal, Sylow
34	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
37	C₂²×Dic₁₃	C₄⋊C₄	1	2	yes	—
38	C₂²×Dic₁₃	C₄⋊C₄	1	2	yes	—
39	C₂×C₈×C₁₃	—	2	2	no	—
40	C₂×C₈×C₁₃	—	2	2	no	—
41	C₂×C₈×C₁₃	—	2	2	no	—
42	C₂×C₈×C₁₃	—	4	2	no	—
43	C₂×C₈×C₁₃	—	4	2	no	—
44	C₂×C₈×C₁₃	—	4	2	no	—
45	C₄²×C₁₃	#41	1	2	yes	—
46	C₄²×C₁₃	#41	1	2	yes	—
47	C₄×Dic₁₃	—	2	2	no	—
48	C₄×Dic₁₃	—	2	2	no	—
49	C₄×Dic₁₃	—	4	2	no	—
50	C₄×Dic₁₃	D₈	1	2	yes	—
51	C₄×Dic₁₃	SD₁₆	1	4	yes	—
52	C₄×Dic₁₃	Q₁₆	1	2	yes	—
53	C₂×#1055	—	2	1	no	—
54	C₂×#1055	—	2	1	no	—
55	C₂×#1055	—	2	2	no	—
56	C₂×#1055	—	2	2	no	—
57	C₂×#1055	—	4	2	no	—
58	C₂×#1055	—	4	2	no	—
59	C₂×#1055	—	4	2	no	—
60	#1842	—	13	2	no	—
61	#1744	—	13	1	no	—
62	#1277	—	13	2	no	—
63	#1277	—	13	1	no	—
64	#1277	D₁₃	1	1	yes	characteristic
65	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₃	—	2	2	no	—
67	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
68	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
69	C₂×C₄×C₁₃	#124	1	2	yes	—
70	C₂×C₄×C₁₃	#125	1	2	yes	—
71	C₂×C₄×C₁₃	#131	1	2	yes	—
72	C₂×C₄×C₁₃	#131	1	2	yes	—
73	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
74	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
75	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
76	C₂×Dic₁₃	—	2	2	no	—
77	C₂×Dic₁₃	—	2	2	no	—
78	C₂×Dic₁₃	—	2	2	no	—
79	C₂×Dic₁₃	—	2	2	no	—
80	C₂×Dic₁₃	#106	1	4	yes	—
81	C₂×Dic₁₃	#107	1	4	yes	—
82	C₈×C₁₃	—	4	2	no	—
83	C₈×C₁₃	—	8	2	no	—
84	#1055	—	4	1	no	—
85	#1055	—	4	1	no	—
86	#1055	—	8	2	no	—
87	C₂×C₄⋊C₈	—	13	2	no	—
88	C₂×C₄⋊C₈	—	26	2	no	—
89	C₂×C₄⋊C₈	—	26	2	no	—
90	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
91	C₂×C₄×C₈	—	13	1	no	—
92	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
93	C₄³	—	13	1	no	—
94	#336	—	13	2	no	—
95	#336	—	13	2	no	—
96	C₂²×C₁₃	#382	1	1	yes	characteristic
97	C₂²×C₁₃	#424	1	2	yes	—
98	C₂²×C₁₃	C₂×#106	1	1	yes	characteristic
99	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
100	C₂²×C₁₃	#430	1	2	yes	—
101	C₄×C₁₃	—	2	2	no	—
102	C₄×C₁₃	#433	1	1	yes	characteristic
103	C₄×C₁₃	#433	1	1	yes	characteristic
104	C₄×C₁₃	#433	1	2	yes	—
105	C₄×C₁₃	#486	1	2	yes	—
106	C₄×C₁₃	#487	1	2	yes	—
107	Dic₁₃	—	2	4	no	—
108	Dic₁₃	—	2	4	no	—
109	C₂²×C₈	—	2	2	no	—
110	C₂²×C₈	—	13	1	no	—
111	C₂²×C₈	—	13	1	no	—
112	C₂²×C₈	—	26	2	no	—
113	C₂²×C₈	C₄×D₁₃	1	2	yes	—
114	C₂×C₄²	—	13	1	no	—
115	C₂×C₄²	—	13	1	no	—
116	C₂×C₄²	—	13	2	no	—
117	C₂×C₄²	—	13	2	no	—
118	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	2	2	no	—
121	C₄⋊C₈	—	26	2	no	—
122	C₄⋊C₈	—	26	2	no	—
123	C₄⋊C₈	—	52	4	no	—
124	C₄⋊C₈	—	52	4	no	—
125	C₄×C₈	—	2	2	no	—
126	C₄×C₈	—	26	2	no	—
127	C₂×C₁₃	#1570	1	2	yes	—
128	C₂×C₁₃	#1571	1	2	yes	—
129	C₂×C₁₃	#1832	1	1	yes	characteristic
130	C₂×C₁₃	#1834	1	1	yes	characteristic
131	C₂×C₁₃	#1838	1	1	yes	characteristic
132	C₂²×C₄	—	13	2	no	—
133	C₂²×C₄	—	13	2	no	—
134	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
135	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
136	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	—	4	2	no	—
143	C₂×C₈	—	26	1	no	—
144	C₂×C₈	—	26	1	no	—
145	C₂×C₈	—	26	2	no	—
146	C₂×C₈	—	26	2	no	—
147	C₂×C₈	—	52	2	no	—
148	C₂×C₈	—	52	2	no	—
149	C₂×C₈	—	52	2	no	—
150	C₄²	—	13	2	no	—
151	C₄²	—	13	4	no	—
152	C₄²	—	13	2	no	—
153	C₄²	—	26	2	no	—
154	C₄²	—	26	2	no	—
155	C₄²	—	52	2	no	—
156	C₄²	#6127	1	2	yes	—
157	C₄²	#6129	1	2	yes	—
158	C₁₃	#8787	1	1	yes	characteristic, complemented, Sylow
159	C₂³	C₄⋊C₄×D₁₃	1	1	yes	characteristic
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	13	4	no	—
162	C₂×C₄	—	13	4	no	—
163	C₂×C₄	—	26	2	no	—
164	C₂×C₄	—	26	2	no	—
165	C₂×C₄	—	26	2	no	—
166	C₂×C₄	—	26	2	no	—
167	C₂×C₄	#88126	1	1	yes	characteristic
168	C₂×C₄	#88131	1	1	yes	characteristic
169	C₂×C₄	#88139	1	2	yes	—
170	C₂×C₄	#88143	1	2	yes	—
171	C₂×C₄	#88155	1	2	yes	—
172	C₂×C₄	#88161	1	2	yes	—
173	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
174	C₂×C₄	SD₁₆×D₁₃	1	2	yes	—
175	C₂×C₄	Q₁₆×D₁₃	1	1	yes	characteristic
176	C₈	—	4	2	no	—
177	C₈	—	8	2	no	—
178	C₈	—	52	1	no	—
179	C₈	—	52	1	no	—
180	C₈	—	104	2	no	—
181	C₂²	#11718311	1	1	yes	characteristic
182	C₂²	#11718500	1	2	yes	—
183	C₂²	D₁₃×#106	1	1	yes	characteristic
184	C₂²	D₁₃×#107	1	1	yes	characteristic
185	C₂²	#11718554	1	2	yes	—
186	C₄	—	2	2	no	—
187	C₄	—	26	4	no	—
188	C₄	—	26	4	no	—
189	C₄	#11718351	1	1	yes	characteristic
190	C₄	#11718370	1	1	yes	characteristic
191	C₄	#11718381	1	2	yes	—
192	C₄	#11718421	1	2	yes	—
193	C₄	#11718453	1	2	yes	—
194	C₂	#13113605	1	2	yes	—
195	C₂	#13113609	1	2	yes	—
196	C₂	#13114954	1	1	yes	characteristic
197	C₂	#13114962	1	1	yes	characteristic
198	C₂	#13114975	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13114987

C₂

yes

maximal

#13114581

C₂

yes

maximal, characteristic

#13112671

C₂

yes

maximal, characteristic

#13112670

C₂

yes

maximal

C₁₃×#1277

C₂

yes

maximal, characteristic

C₂×C₁₃×C₄⋊C₈

C₂²

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₂×#88054

—

C₂×#88054

—

C₄²×Dic₁₃

C₂²

yes

characteristic

C₂×C₄⋊#1055

C₂²

yes

—

C₂×C₄×#1055

C₂²

yes

characteristic

#11718142

C₄

yes

—

#11718142

C₄

yes

—

C₂²×C₈×C₁₃

—

C₂²×C₈×C₁₃

C₂×C₄

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×C₄×Dic₁₃

C₂×C₄

yes

—

C₂×C₄×Dic₁₃

D₄

yes

characteristic

C₂×C₄×Dic₁₃

D₄

yes

characteristic

C₂×C₄×Dic₁₃

Q₈

yes

—

C₂²×#1055

—

C₂²×#1055

C₂×C₄

yes

characteristic

C₂²×#1055

C₂×C₄

yes

characteristic

C₁₃×C₄⋊C₈

—

C₁₃×C₄⋊C₈

—

C₄×C₈×C₁₃

—

—

—

—

—

—

—

C₂²×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×Q₈

yes

characteristic

C₂²×Dic₁₃

C₄⋊C₄

yes

—

C₂²×Dic₁₃

C₄⋊C₄

yes

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₄²×C₁₃

#41

yes

—

C₄²×C₁₃

#41

yes

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

D₈

yes

—

C₄×Dic₁₃

SD₁₆

yes

—

C₄×Dic₁₃

Q₁₆

yes

—

—

—

—

—

—

—

—

—

—

—

—

D₁₃

yes

characteristic

C₂³×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

—

C₂×C₄×C₁₃

#125

yes

—

C₂×C₄×C₁₃

#131

yes

—

C₂×C₄×C₁₃

#131

yes

—

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

C₂×SD₁₆

yes

—

C₂×C₄×C₁₃

C₂×Q₁₆

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

#106

yes

—

C₂×Dic₁₃

#107

yes

—

C₈×C₁₃

—

C₈×C₁₃

—

#1055

—

#1055

—

#1055

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂×D₁₃

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

C₄³

—

#336

—

#336

—

C₂²×C₁₃

#382

yes

characteristic

C₂²×C₁₃

#424

yes

—

C₂²×C₁₃

yes

C₂²×C₁₃

yes

100

C₂²×C₁₃

#430

yes

—

101

C₄×C₁₃

—

102

C₄×C₁₃

#433

yes

characteristic

103

C₄×C₁₃

#433

yes

characteristic

104

C₄×C₁₃

#433

yes

—

105

C₄×C₁₃

#486

yes

—

106

C₄×C₁₃

#487

yes

—

107

Dic₁₃

—

108

Dic₁₃

—

109

C₂²×C₈

—

110

C₂²×C₈

—

111

C₂²×C₈

—

112

C₂²×C₈

—

113

C₂²×C₈

C₄×D₁₃

yes

—

114

C₂×C₄²

—

115

C₂×C₄²

—

116

C₂×C₄²

—

117

C₂×C₄²

—

118

C₂×C₄²

C₂²×D₁₃

yes

characteristic

119

C₄⋊C₈

—

120

C₄⋊C₈

—

121

C₄⋊C₈

—

122

C₄⋊C₈

—

123

C₄⋊C₈

—

124

C₄⋊C₈

—

125

C₄×C₈

—

126

C₄×C₈

—

127

C₂×C₁₃

#1570

yes

—

128

C₂×C₁₃

#1571

yes

—

129

C₂×C₁₃

#1832

yes

characteristic

130

C₂×C₁₃

#1834

yes

characteristic

131

C₂×C₁₃

#1838

yes

characteristic

132

C₂²×C₄

—

133

C₂²×C₄

—

134

C₂²×C₄

C₂×C₄×D₁₃

yes

characteristic

135

C₂²×C₄

D₄×D₁₃

yes

characteristic

136

C₂²×C₄

Q₈×D₁₃

yes

characteristic

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₄²

—

151

C₄²

—

152

C₄²

—

153

C₄²

—

154

C₄²

—

155

C₄²

—

156

C₄²

#6127

yes

—

157

C₄²

#6129

yes

—

158

C₁₃

#8787

yes

characteristic, complemented, Sylow

159

C₂³

C₄⋊C₄×D₁₃

yes

characteristic

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

#88126

yes

characteristic

168

C₂×C₄

#88131

yes

characteristic

169

C₂×C₄

#88139

yes

—

170

C₂×C₄

#88143

yes

—

171

C₂×C₄

#88155

yes

—

172

C₂×C₄

#88161

yes

—

173

C₂×C₄

D₈×D₁₃

yes

characteristic

174

C₂×C₄

SD₁₆×D₁₃

yes

—

175

C₂×C₄

Q₁₆×D₁₃

yes

characteristic

176

C₈

—

177

C₈

—

178

C₈

—

179

C₈

—

180

C₈

—

104

—

181

C₂²

#11718311

yes

characteristic

182

C₂²

#11718500

yes

—

183

C₂²

D₁₃×#106

yes

characteristic

184

C₂²

D₁₃×#107

yes

characteristic

185

C₂²

#11718554

yes

—

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

#11718351

yes

characteristic

190

C₄

#11718370

yes

characteristic

191

C₄

#11718381

yes

—

192

C₄

#11718421

yes

—

193

C₄

#11718453

yes

—

194

C₂

#13113605

yes

—

195

C₂

#13113609

yes

—

196

C₂

#13114954

yes

characteristic

197

C₂

#13114962

yes

characteristic

198

C₂

#13114975

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₃
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1277	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1277	D₁₃
hypocenter	C₁₃	#8787
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1277

subgroup

G/H

center

C₂²×C₄

D₄×D₁₃

commutator

C₂²×C₄×C₁₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1277

C₂

socle, prime radical

C₂³×C₁₃

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1277

N	H ≅ G/N	type
158. C₁₃	#8787	semidirect

H ≅ G/N

type

158. C₁₃

#8787

semidirect

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

subgroup	index
#13114987	2
#13114581	2
#13112671	2
#13112670	2
C₁₃×#1277	2
#8787	13

subgroup

index

C₁₃×#1277

Subgroups — #15756588

Conjugacy classes of subgroups

Complements

Maximal subgroups