Group #15767896 — subgroups

Subgroups — #15767896

Order 3328.

372 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 101 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	#13115634	C₂	1	1	yes	maximal, characteristic
2	#13115011	C₂	1	1	yes	maximal, characteristic, complemented
3	#13113611	C₂	1	2	yes	maximal, complemented
4	#13113262	C₂	1	2	yes	maximal
5	C₁₃×#1275	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₅₂	C₂²	1	1	yes	characteristic
7	C₁₃×#431	C₂²	1	2	yes	—
8	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
9	C₂×#88060	—	2	2	no	—
10	C₂×#88059	—	2	2	no	—
11	#11718585	C₂²	1	2	yes	—
12	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
13	#11718051	—	2	4	no	—
14	#11718050	—	2	4	no	—
15	C₂²×D₅₂	Q₈	1	1	yes	characteristic
16	C₂²×C₄×D₁₃	—	2	1	no	—
17	C₂²×C₈×C₁₃	D₄	1	2	yes	—
18	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
19	C₂×#6103	—	2	1	no	—
20	C₂×C₄⋊Dic₁₃	Q₈	1	1	yes	characteristic
21	C₂×#6099	—	2	2	no	—
22	C₂²×#1055	—	2	1	no	—
23	C₄×D₅₂	—	2	2	no	—
24	C₄×D₅₂	—	2	2	no	—
25	C₁₃×C₄⋊C₈	—	2	2	no	—
26	C₁₃×C₂²⋊C₈	—	2	2	no	—
27	C₁₃×#97	—	2	2	no	—
28	C₄×C₈×C₁₃	D₄	1	4	yes	—
29	#88060	—	4	4	no	—
30	#88059	—	4	4	no	—
31	C₄⋊#1055	—	2	2	no	—
32	C₄⋊#1055	—	2	2	no	—
33	C₄⋊#1055	—	2	2	no	—
34	#10660	—	13	1	no	maximal, Sylow
35	C₂³×D₁₃	—	2	1	no	—
36	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
39	C₂²×Dic₁₃	—	2	1	no	—
40	C₂×D₅₂	—	2	2	no	—
41	C₂×D₅₂	—	4	2	no	—
42	C₂×C₄×D₁₃	—	4	1	no	—
43	C₂×C₄×D₁₃	—	4	1	no	—
44	C₂×C₄×D₁₃	—	4	2	no	—
45	C₂×C₄×D₁₃	—	4	2	no	—
46	C₂×C₈×C₁₃	—	2	2	no	—
47	C₂×C₈×C₁₃	—	2	2	no	—
48	C₂×C₈×C₁₃	—	2	4	no	—
49	C₂×C₈×C₁₃	D₈	1	4	yes	—
50	C₄²×C₁₃	C₂×D₄	1	2	yes	—
51	C₄²×C₁₃	#41	1	2	yes	—
52	#6103	—	4	2	no	—
53	C₄⋊Dic₁₃	—	2	2	no	—
54	#6099	—	4	4	no	—
55	C₂×#1055	—	4	1	no	—
56	C₂×#1055	—	4	2	no	—
57	C₂×#1055	—	4	2	no	—
58	#1955	—	13	1	no	—
59	#1849	—	13	1	no	—
60	#1571	—	13	2	no	—
61	#1486	—	13	2	no	—
62	#1275	D₁₃	1	1	yes	characteristic, complemented
63	C₂³×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
64	C₂²×D₁₃	—	2	2	no	—
65	C₂²×D₁₃	—	4	1	no	—
66	C₂²×D₁₃	—	4	2	no	—
67	C₂²×D₁₃	—	4	1	no	—
68	C₂²×D₁₃	—	4	2	no	—
69	C₂×C₄×C₁₃	—	4	1	no	—
70	C₂×C₄×C₁₃	—	4	1	no	—
71	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
72	C₂×C₄×C₁₃	#123	1	2	yes	—
73	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
74	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
75	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
76	C₂×C₄×C₁₃	#137	1	2	yes	—
77	C₂×Dic₁₃	—	4	1	no	—
78	C₂×Dic₁₃	—	4	2	no	—
79	D₅₂	—	2	4	no	—
80	D₅₂	—	4	2	no	—
81	C₄×D₁₃	—	8	2	no	—
82	C₄×D₁₃	—	8	1	no	—
83	C₄×D₁₃	—	8	1	no	—
84	C₈×C₁₃	—	2	4	no	—
85	C₈×C₁₃	—	4	4	no	—
86	#1055	—	8	1	no	—
87	#1055	—	8	2	no	—
88	C₂×C₄×D₄	—	13	1	no	—
89	#431	C₂×D₁₃	1	2	yes	—
90	C₂×#108	—	26	2	no	—
91	#423	—	13	2	no	—
92	C₂×C₄⋊C₈	—	13	1	no	—
93	C₂×C₂²⋊C₈	—	26	2	no	—
94	C₂×#97	C₂×D₁₃	1	1	yes	characteristic
95	#335	—	26	4	no	—
96	D₄⋊C₈	—	26	4	no	—
97	C₂²×C₁₃	—	2	1	no	—
98	C₂²×C₁₃	—	2	1	no	—
99	C₂²×C₁₃	#399	1	1	yes	characteristic
100	C₂²×C₁₃	#474	1	2	yes	—
101	C₂×D₁₃	—	4	4	no	—
102	C₂×D₁₃	—	4	2	no	—
103	C₂×D₁₃	—	8	2	no	—
104	C₂×D₁₃	—	8	1	no	—
105	C₂×D₁₃	—	8	1	no	—
106	C₄×C₁₃	—	2	2	no	—
107	C₄×C₁₃	—	2	2	no	—
108	C₄×C₁₃	#480	1	2	yes	—
109	C₄×C₁₃	#493	1	1	yes	characteristic
110	C₄×C₁₃	#497	1	1	yes	characteristic
111	Dic₁₃	—	8	1	no	—
112	C₂²×D₄	—	13	1	no	—
113	C₂³×C₄	—	26	1	no	—
114	C₂×M₁₆	—	26	2	no	—
115	C₂²×C₈	—	26	1	no	—
116	C₂²×C₈	D₅₂	1	2	yes	—
117	C₄×D₄	—	26	2	no	—
118	C₄×D₄	—	26	2	no	—
119	C₂×C₄⋊C₄	—	13	1	no	—
120	C₂×C₂²⋊C₄	—	26	1	no	—
121	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
122	#108	—	52	4	no	—
123	C₄⋊C₈	—	2	2	no	—
124	C₄⋊C₈	—	26	2	no	—
125	C₄⋊C₈	—	26	2	no	—
126	C₄⋊C₈	—	26	2	no	—
127	C₂²⋊C₈	—	2	2	no	—
128	C₂²⋊C₈	—	52	4	no	—
129	#97	—	2	2	no	—
130	C₄×C₈	D₅₂	1	4	yes	—
131	C₂×C₁₃	—	2	2	no	—
132	C₂×C₁₃	#2091	1	1	yes	characteristic
133	C₂×C₁₃	#2093	1	1	yes	characteristic
134	C₂×C₁₃	#2098	1	1	yes	characteristic
135	D₁₃	—	8	2	no	—
136	C₂⁴	—	26	1	no	—
137	C₂×D₄	—	26	2	no	—
138	C₂×D₄	—	52	2	no	—
139	C₂²×C₄	—	26	1	no	—
140	C₂²×C₄	—	52	1	no	—
141	C₂²×C₄	—	52	1	no	—
142	C₂²×C₄	—	52	2	no	—
143	C₂²×C₄	—	52	2	no	—
144	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
145	C₂²×C₄	#6129	1	1	yes	characteristic
146	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
147	M₁₆	—	52	4	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	4	no	—
151	C₂×C₈	—	52	1	no	—
152	C₂×C₈	—	52	2	no	—
153	C₂×C₈	—	52	2	no	—
154	C₂×C₈	D₁₀₄	1	4	yes	—
155	C₄⋊C₄	—	26	2	no	—
156	C₂²⋊C₄	—	52	2	no	—
157	C₄²	C₂×D₅₂	1	2	yes	—
158	C₄²	#6129	1	2	yes	—
159	C₁₃	#10660	1	1	yes	characteristic, complemented, Sylow
160	C₂³	—	26	2	no	—
161	C₂³	—	52	1	no	—
162	C₂³	—	52	2	no	—
163	C₂³	—	52	2	no	—
164	C₂³	—	52	1	no	—
165	C₂³	#88151	1	1	yes	characteristic
166	D₄	—	26	4	no	—
167	D₄	—	52	2	no	—
168	C₂×C₄	—	4	1	no	—
169	C₂×C₄	—	4	1	no	—
170	C₂×C₄	—	52	2	no	—
171	C₂×C₄	—	52	1	no	—
172	C₂×C₄	—	104	1	no	—
173	C₂×C₄	—	104	1	no	—
174	C₂×C₄	—	104	2	no	—
175	C₂×C₄	#88128	1	1	yes	characteristic
176	C₂×C₄	#88140	1	2	yes	—
177	C₂×C₄	#88140	1	1	yes	characteristic
178	C₂×C₄	#88151	1	2	yes	—
179	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
180	C₂×C₄	#88163	1	2	yes	—
181	C₈	—	2	4	no	—
182	C₈	—	4	4	no	—
183	C₈	—	104	1	no	—
184	C₈	—	104	2	no	—
185	C₂²	—	2	1	no	—
186	C₂²	—	2	1	no	—
187	C₂²	—	52	4	no	—
188	C₂²	—	52	2	no	—
189	C₂²	—	104	2	no	—
190	C₂²	—	104	1	no	—
191	C₂²	—	104	1	no	—
192	C₂²	#11718338	1	1	yes	characteristic
193	C₂²	#11718507	1	2	yes	—
194	C₄	—	2	2	no	—
195	C₄	—	2	2	no	—
196	C₄	—	104	1	no	—
197	C₄	#11718356	1	1	yes	characteristic
198	C₄	#11718376	1	1	yes	characteristic
199	C₄	#11718397	1	2	yes	—
200	C₂	—	2	2	no	—
201	C₂	—	104	2	no	—
202	C₂	#13116353	1	1	yes	characteristic
203	C₂	#13116363	1	1	yes	characteristic
204	C₂	#13116390	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13115634

C₂

yes

maximal, characteristic

#13115011

C₂

yes

maximal, characteristic, complemented

#13113611

C₂

yes

maximal, complemented

#13113262

C₂

yes

maximal

C₁₃×#1275

C₂

yes

maximal, characteristic, complemented

C₂×C₄×D₅₂

C₂²

yes

characteristic

C₁₃×#431

C₂²

yes

—

C₂×C₁₃×#97

C₂²

yes

—

—

C₂²

yes

—

C₂×C₄⋊#1055

C₂²

yes

characteristic

#11718051

—

#11718050

—

C₂²×D₅₂

Q₈

yes

characteristic

C₂²×C₄×D₁₃

—

C₂²×C₈×C₁₃

D₄

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×#6103

—

C₂×C₄⋊Dic₁₃

Q₈

yes

characteristic

C₂×#6099

—

C₂²×#1055

—

C₄×D₅₂

—

C₄×D₅₂

—

C₁₃×C₄⋊C₈

—

C₁₃×C₂²⋊C₈

—

C₁₃×#97

—

C₄×C₈×C₁₃

D₄

yes

—

—

—

—

—

—

—

C₂³×D₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×Q₈

yes

characteristic

C₂²×C₄×C₁₃

#41

yes

characteristic

C₂²×Dic₁₃

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×C₄×D₁₃

—

C₂×C₄×D₁₃

—

C₂×C₄×D₁₃

—

C₂×C₄×D₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

D₈

yes

—

C₄²×C₁₃

C₂×D₄

yes

—

C₄²×C₁₃

#41

yes

—

#6103

—

C₄⋊Dic₁₃

—

—

—

—

—

—

—

—

—

D₁₃

yes

characteristic, complemented

C₂³×C₁₃

C₂²⋊Q₈

yes

characteristic

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₃

#123

yes

—

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

C₂×D₈

yes

—

C₂×C₄×C₁₃

#137

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

D₅₂

—

D₅₂

—

C₄×D₁₃

—

C₄×D₁₃

—

C₄×D₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

#1055

—

#1055

—

C₂×C₄×D₄

—

#431

C₂×D₁₃

yes

—

C₂×#108

—

#423

—

C₂×C₄⋊C₈

—

C₂×C₂²⋊C₈

—

C₂×#97

C₂×D₁₃

yes

characteristic

#335

—

D₄⋊C₈

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#399

yes

characteristic

100

C₂²×C₁₃

#474

yes

—

101

C₂×D₁₃

—

102

C₂×D₁₃

—

103

C₂×D₁₃

—

104

C₂×D₁₃

—

105

C₂×D₁₃

—

106

C₄×C₁₃

—

107

C₄×C₁₃

—

108

C₄×C₁₃

#480

yes

—

109

C₄×C₁₃

#493

yes

characteristic

110

C₄×C₁₃

#497

yes

characteristic

111

Dic₁₃

—

112

C₂²×D₄

—

113

C₂³×C₄

—

114

C₂×M₁₆

—

115

C₂²×C₈

—

116

C₂²×C₈

D₅₂

yes

—

117

C₄×D₄

—

118

C₄×D₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₂²⋊C₄

—

121

C₂×C₄²

C₂²×D₁₃

yes

characteristic

122

#108

—

123

C₄⋊C₈

—

124

C₄⋊C₈

—

125

C₄⋊C₈

—

126

C₄⋊C₈

—

127

C₂²⋊C₈

—

128

C₂²⋊C₈

—

129

#97

—

130

C₄×C₈

D₅₂

yes

—

131

C₂×C₁₃

—

132

C₂×C₁₃

#2091

yes

characteristic

133

C₂×C₁₃

#2093

yes

characteristic

134

C₂×C₁₃

#2098

yes

characteristic

135

D₁₃

—

136

C₂⁴

—

137

C₂×D₄

—

138

C₂×D₄

—

139

C₂²×C₄

—

140

C₂²×C₄

—

141

C₂²×C₄

—

142

C₂²×C₄

—

143

C₂²×C₄

—

144

C₂²×C₄

C₂×D₅₂

yes

characteristic

145

C₂²×C₄

#6129

yes

characteristic

146

C₂²×C₄

Q₈×D₁₃

yes

characteristic

147

M₁₆

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

D₁₀₄

yes

—

155

C₄⋊C₄

—

156

C₂²⋊C₄

—

157

C₄²

C₂×D₅₂

yes

—

158

C₄²

#6129

yes

—

159

C₁₃

#10660

yes

characteristic, complemented, Sylow

160

C₂³

—

161

C₂³

—

162

C₂³

—

163

C₂³

—

164

C₂³

—

165

C₂³

#88151

yes

characteristic

166

D₄

—

167

D₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

104

—

173

C₂×C₄

—

104

—

174

C₂×C₄

—

104

—

175

C₂×C₄

#88128

yes

characteristic

176

C₂×C₄

#88140

yes

—

177

C₂×C₄

#88140

yes

characteristic

178

C₂×C₄

#88151

yes

—

179

C₂×C₄

C₂×D₁₀₄

yes

—

180

C₂×C₄

#88163

yes

—

181

C₈

—

182

C₈

—

183

C₈

—

104

—

184

C₈

—

104

—

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

—

189

C₂²

—

104

—

190

C₂²

—

104

—

191

C₂²

—

104

—

192

C₂²

#11718338

yes

characteristic

193

C₂²

#11718507

yes

—

194

C₄

—

195

C₄

—

196

C₄

—

104

—

197

C₄

#11718356

yes

characteristic

198

C₄

#11718376

yes

characteristic

199

C₄

#11718397

yes

—

200

C₂

—

201

C₂

—

104

—

202

C₂

#13116353

yes

characteristic

203

C₂

#13116363

yes

characteristic

204

C₂

#13116390

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#88128
commutator, cosocle	C₂×C₄²×C₁₃	C₂³
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1275	C₂
socle	C₂²×C₁₃	#399
prime radical	C₂²×D₅₂	Q₈
hypercenter	#1275	D₁₃
hypocenter	C₁₃	#10660
LCS	G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1275

subgroup

G/H

center

C₂×C₄

#88128

commutator, cosocle

C₂×C₄²×C₁₃

C₂³

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1275

C₂

socle

C₂²×C₁₃

#399

prime radical

C₂²×D₅₂

Q₈

D₁₃

C₁₃

G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1275

N	H ≅ G/N	type
2. #13115011	C₂	semidirect
3. #13113611	C₂	semidirect
5. C₁₃×#1275	C₂	semidirect
62. #1275	D₁₃	semidirect
159. C₁₃	#10660	semidirect

H ≅ G/N

type

2. #13115011

C₂

semidirect

3. #13113611

C₂

semidirect

5. C₁₃×#1275

C₂

semidirect

62. #1275

D₁₃

semidirect

159. C₁₃

#10660

semidirect

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

subgroup	index
#13115634	2
#13115011	2
#13113611	2
#13113262	2
C₁₃×#1275	2
#10660	13

subgroup

index

C₁₃×#1275

Subgroups — #15767896

Conjugacy classes of subgroups

Complements

Maximal subgroups