Group #15756591 — subgroups

Subgroups — #15756591

Order 3328.

322 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 131 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	#13114952	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	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊#1055	C₂²	1	2	yes	—
10	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
11	#11718142	—	2	2	no	—
12	#11718142	—	2	2	no	—
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	C₄⋊#1055	—	2	2	no	—
29	C₄⋊#1055	—	2	2	no	—
30	C₄×#1055	—	2	2	no	—
31	#8788	—	13	1	no	maximal, Sylow
32	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
35	C₂²×Dic₁₃	C₄⋊C₄	1	2	yes	—
36	C₂²×Dic₁₃	C₄⋊C₄	1	2	yes	—
37	C₂×C₈×C₁₃	—	2	2	no	—
38	C₂×C₈×C₁₃	—	2	2	no	—
39	C₂×C₈×C₁₃	—	2	2	no	—
40	C₂×C₈×C₁₃	—	4	2	no	—
41	C₂×C₈×C₁₃	—	4	2	no	—
42	C₂×C₈×C₁₃	—	4	2	no	—
43	C₄²×C₁₃	#41	1	2	yes	—
44	C₄²×C₁₃	#41	1	2	yes	—
45	C₄×Dic₁₃	—	2	2	no	—
46	C₄×Dic₁₃	—	2	2	no	—
47	C₄×Dic₁₃	—	2	1	no	—
48	C₄×Dic₁₃	—	2	2	no	—
49	C₄×Dic₁₃	—	2	1	no	—
50	C₄×Dic₁₃	—	4	2	no	—
51	C₂×#1055	—	2	1	no	—
52	C₂×#1055	—	2	1	no	—
53	C₂×#1055	—	2	2	no	—
54	C₂×#1055	—	2	2	no	—
55	C₂×#1055	—	4	2	no	—
56	C₂×#1055	—	4	2	no	—
57	C₂×#1055	—	4	2	no	—
58	#1831	—	13	2	no	—
59	#1744	—	13	1	no	—
60	#1277	—	13	2	no	—
61	#1277	—	13	1	no	—
62	#1277	D₁₃	1	1	yes	characteristic
63	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₁₃	—	2	2	no	—
65	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
66	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
67	C₂×C₄×C₁₃	#124	1	2	yes	—
68	C₂×C₄×C₁₃	#125	1	2	yes	—
69	C₂×C₄×C₁₃	#131	1	2	yes	—
70	C₂×C₄×C₁₃	#131	1	2	yes	—
71	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
72	C₂×C₄×C₁₃	#135	1	2	yes	—
73	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
74	C₂×Dic₁₃	—	2	2	no	—
75	C₂×Dic₁₃	—	2	2	no	—
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	C₈×C₁₃	—	4	2	no	—
81	C₈×C₁₃	—	8	2	no	—
82	#1055	—	4	1	no	—
83	#1055	—	4	1	no	—
84	#1055	—	8	2	no	—
85	C₂×C₄⋊C₈	—	13	2	no	—
86	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
87	C₂×C₄×C₈	—	13	1	no	—
88	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
89	C₄³	—	13	1	no	—
90	#336	—	13	2	no	—
91	#336	—	13	2	no	—
92	#336	—	26	2	no	—
93	#336	—	26	2	no	—
94	C₂²×C₁₃	#382	1	1	yes	characteristic
95	C₂²×C₁₃	#424	1	2	yes	—
96	C₂²×C₁₃	#427	1	1	yes	characteristic
97	C₂²×C₁₃	#427	1	1	yes	characteristic
98	C₂²×C₁₃	#430	1	2	yes	—
99	C₄×C₁₃	—	2	2	no	—
100	C₄×C₁₃	#433	1	1	yes	characteristic
101	C₄×C₁₃	#433	1	1	yes	characteristic
102	C₄×C₁₃	#433	1	2	yes	—
103	C₄×C₁₃	#488	1	2	yes	—
104	C₄×C₁₃	#488	1	2	yes	—
105	Dic₁₃	—	4	2	no	—
106	Dic₁₃	—	4	2	no	—
107	C₂²×C₈	—	2	2	no	—
108	C₂²×C₈	—	13	1	no	—
109	C₂²×C₈	—	13	1	no	—
110	C₂²×C₈	—	26	2	no	—
111	C₂²×C₈	C₄×D₁₃	1	2	yes	—
112	C₂×C₄²	—	13	2	no	—
113	C₂×C₄²	—	13	1	no	—
114	C₂×C₄²	—	13	1	no	—
115	C₂×C₄²	—	13	2	no	—
116	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
117	C₄⋊C₈	—	2	2	no	—
118	C₄⋊C₈	—	2	2	no	—
119	C₄⋊C₈	—	26	2	no	—
120	C₄⋊C₈	—	26	2	no	—
121	C₄×C₈	—	2	2	no	—
122	C₄×C₈	—	26	2	no	—
123	C₂×C₁₃	#1572	1	2	yes	—
124	C₂×C₁₃	#1572	1	2	yes	—
125	C₂×C₁₃	#1832	1	1	yes	characteristic
126	C₂×C₁₃	#1835	1	1	yes	characteristic
127	C₂×C₁₃	#1838	1	1	yes	characteristic
128	C₂²×C₄	—	13	2	no	—
129	C₂²×C₄	—	13	2	no	—
130	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
131	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
132	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	—	4	2	no	—
137	C₂×C₈	—	4	2	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	26	1	no	—
140	C₂×C₈	—	26	1	no	—
141	C₂×C₈	—	26	2	no	—
142	C₂×C₈	—	26	2	no	—
143	C₂×C₈	—	52	2	no	—
144	C₂×C₈	—	52	2	no	—
145	C₂×C₈	—	52	2	no	—
146	C₄²	—	26	2	no	—
147	C₄²	—	26	2	no	—
148	C₄²	—	26	1	no	—
149	C₄²	—	26	2	no	—
150	C₄²	—	26	1	no	—
151	C₄²	—	52	2	no	—
152	C₄²	#6127	1	2	yes	—
153	C₄²	#6129	1	2	yes	—
154	C₁₃	#8788	1	1	yes	characteristic, complemented, Sylow
155	C₂³	C₄⋊C₄×D₁₃	1	1	yes	characteristic
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	26	2	no	—
158	C₂×C₄	—	26	2	no	—
159	C₂×C₄	—	26	2	no	—
160	C₂×C₄	—	26	2	no	—
161	C₂×C₄	—	26	2	no	—
162	C₂×C₄	—	26	2	no	—
163	C₂×C₄	#88126	1	1	yes	characteristic
164	C₂×C₄	#88131	1	1	yes	characteristic
165	C₂×C₄	#88139	1	2	yes	—
166	C₂×C₄	#88143	1	2	yes	—
167	C₂×C₄	#88155	1	2	yes	—
168	C₂×C₄	#88161	1	2	yes	—
169	C₂×C₄	#88166	1	1	yes	characteristic
170	C₂×C₄	#88170	1	2	yes	—
171	C₂×C₄	#88173	1	1	yes	characteristic
172	C₈	—	4	2	no	—
173	C₈	—	8	2	no	—
174	C₈	—	52	1	no	—
175	C₈	—	52	1	no	—
176	C₈	—	104	2	no	—
177	C₂²	#11718311	1	1	yes	characteristic
178	C₂²	#11718500	1	2	yes	—
179	C₂²	#11718521	1	1	yes	characteristic
180	C₂²	#11718541	1	1	yes	characteristic
181	C₂²	#11718554	1	2	yes	—
182	C₄	—	2	2	no	—
183	C₄	—	52	2	no	—
184	C₄	—	52	2	no	—
185	C₄	#11718351	1	1	yes	characteristic
186	C₄	#11718370	1	1	yes	characteristic
187	C₄	#11718381	1	2	yes	—
188	C₄	#11718429	1	2	yes	—
189	C₄	#11718461	1	2	yes	—
190	C₂	#13113613	1	2	yes	—
191	C₂	#13113615	1	2	yes	—
192	C₂	#13114954	1	1	yes	characteristic
193	C₂	#13114965	1	1	yes	characteristic
194	C₂	#13114975	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13114952

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

C₄²×Dic₁₃

C₂²

yes

characteristic

C₂×C₄⋊#1055

C₂²

yes

—

C₂×C₄×#1055

C₂²

yes

—

—

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

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

—

—

—

—

—

—

—

—

—

—

—

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

#135

yes

characteristic

C₂×C₄×C₁₃

#135

yes

—

C₂×C₄×C₁₃

#135

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

#1055

—

#1055

—

#1055

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂×D₁₃

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

C₄³

—

—

—

—

—

C₂²×C₁₃

#382

yes

characteristic

C₂²×C₁₃

#424

yes

—

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

—

C₄×C₁₃

—

100

C₄×C₁₃

#433

yes

characteristic

101

C₄×C₁₃

#433

yes

characteristic

102

C₄×C₁₃

#433

yes

—

103

C₄×C₁₃

#488

yes

—

104

C₄×C₁₃

#488

yes

—

105

Dic₁₃

—

106

Dic₁₃

—

107

C₂²×C₈

—

108

C₂²×C₈

—

109

C₂²×C₈

—

110

C₂²×C₈

—

111

C₂²×C₈

C₄×D₁₃

yes

—

112

C₂×C₄²

—

113

C₂×C₄²

—

114

C₂×C₄²

—

115

C₂×C₄²

—

116

C₂×C₄²

C₂²×D₁₃

yes

characteristic

117

C₄⋊C₈

—

118

C₄⋊C₈

—

119

C₄⋊C₈

—

120

C₄⋊C₈

—

121

C₄×C₈

—

122

C₄×C₈

—

123

C₂×C₁₃

#1572

yes

—

124

C₂×C₁₃

#1572

yes

—

125

C₂×C₁₃

#1832

yes

characteristic

126

C₂×C₁₃

#1835

yes

characteristic

127

C₂×C₁₃

#1838

yes

characteristic

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

C₂×C₄×D₁₃

yes

characteristic

131

C₂²×C₄

D₄×D₁₃

yes

characteristic

132

C₂²×C₄

Q₈×D₁₃

yes

characteristic

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

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

C₄²

—

147

C₄²

—

148

C₄²

—

149

C₄²

—

150

C₄²

—

151

C₄²

—

152

C₄²

#6127

yes

—

153

C₄²

#6129

yes

—

154

C₁₃

#8788

yes

characteristic, complemented, Sylow

155

C₂³

C₄⋊C₄×D₁₃

yes

characteristic

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

#88126

yes

characteristic

164

C₂×C₄

#88131

yes

characteristic

165

C₂×C₄

#88139

yes

—

166

C₂×C₄

#88143

yes

—

167

C₂×C₄

#88155

yes

—

168

C₂×C₄

#88161

yes

—

169

C₂×C₄

#88166

yes

characteristic

170

C₂×C₄

#88170

yes

—

171

C₂×C₄

#88173

yes

characteristic

172

C₈

—

173

C₈

—

174

C₈

—

175

C₈

—

176

C₈

—

104

—

177

C₂²

#11718311

yes

characteristic

178

C₂²

#11718500

yes

—

179

C₂²

#11718521

yes

characteristic

180

C₂²

#11718541

yes

characteristic

181

C₂²

#11718554

yes

—

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

#11718351

yes

characteristic

186

C₄

#11718370

yes

characteristic

187

C₄

#11718381

yes

—

188

C₄

#11718429

yes

—

189

C₄

#11718461

yes

—

190

C₂

#13113613

yes

—

191

C₂

#13113615

yes

—

192

C₂

#13114954

yes

characteristic

193

C₂

#13114965

yes

characteristic

194

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	#8788
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
154. C₁₃	#8788	semidirect

H ≅ G/N

type

154. C₁₃

#8788

semidirect

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

subgroup	index
#13114952	2
#13114581	2
#13112671	2
#13112670	2
C₁₃×#1277	2
#8788	13

subgroup

index

C₁₃×#1277

Subgroups — #15756591

Conjugacy classes of subgroups

Complements

Maximal subgroups