Group #15767972 — subgroups

Subgroups — #15767972

Order 3328.

442 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 165 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	#13115797	C₂	1	2	yes	maximal
2	#13115638	C₂	1	1	yes	maximal, characteristic
3	#13114996	C₂	1	2	yes	maximal
4	#13114996	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1273	C₂	1	1	yes	maximal, characteristic
6	C₂×#88123	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×#88058	—	2	2	no	—
10	C₂×#88058	—	2	4	no	—
11	C₂×#88056	—	2	2	no	—
12	#11718596	C₂²	1	2	yes	—
13	#11718596	C₂²	1	1	yes	characteristic
14	#11718143	—	2	4	no	—
15	C₂²×Dic₂₆	—	2	1	no	—
16	C₂²×C₈×C₁₃	D₄	1	2	yes	—
17	C₂²×C₈×C₁₃	D₄	1	4	yes	—
18	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
19	C₂×C₄⋊Dic₁₃	—	2	2	no	—
20	C₂×C₄⋊Dic₁₃	—	2	1	no	—
21	C₂×C₄⋊Dic₁₃	—	2	2	no	—
22	C₂×C₄⋊Dic₁₃	—	2	2	no	—
23	C₂×C₄⋊Dic₁₃	—	2	2	no	—
24	C₂×C₄⋊Dic₁₃	—	2	2	no	—
25	#88123	—	2	2	no	—
26	#88123	—	2	1	no	—
27	#88123	Q₈	1	2	yes	—
28	C₄×C₈×C₁₃	—	2	2	no	—
29	C₄×C₈×C₁₃	—	2	2	no	—
30	C₄×C₈×C₁₃	—	2	2	no	—
31	#88058	—	4	8	no	—
32	#88058	—	4	4	no	—
33	#88056	—	4	4	no	—
34	#10674	—	13	1	no	maximal, Sylow
35	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
37	C₂²×Dic₁₃	—	4	1	no	—
38	C₂²×Dic₁₃	—	4	1	no	—
39	C₂²×Dic₁₃	—	4	2	no	—
40	C₂×Dic₂₆	—	2	2	no	—
41	C₂×Dic₂₆	—	4	2	no	—
42	C₂×Dic₂₆	—	4	2	no	—
43	C₂×Dic₂₆	—	4	1	no	—
44	C₂×C₈×C₁₃	—	2	2	no	—
45	C₂×C₈×C₁₃	—	2	4	no	—
46	C₂×C₈×C₁₃	—	2	4	no	—
47	C₂×C₈×C₁₃	—	2	2	no	—
48	C₂×C₈×C₁₃	D₈	1	4	yes	—
49	C₂×C₈×C₁₃	Q₁₆	1	8	yes	—
50	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
51	C₄²×C₁₃	#41	1	1	yes	characteristic
52	C₄²×C₁₃	#41	1	2	yes	—
53	C₄⋊Dic₁₃	—	2	4	no	—
54	C₄⋊Dic₁₃	—	4	2	no	—
55	C₄⋊Dic₁₃	—	4	2	no	—
56	C₄⋊Dic₁₃	—	4	4	no	—
57	C₄⋊Dic₁₃	—	4	4	no	—
58	C₄⋊Dic₁₃	—	4	4	no	—
59	C₄⋊Dic₁₃	—	4	4	no	—
60	#1990	—	13	2	no	—
61	#1956	—	13	1	no	—
62	#1845	—	13	2	no	—
63	#1845	—	13	1	no	—
64	#1273	D₁₃	1	1	yes	characteristic
65	C₂³×C₁₃	#127	1	1	yes	characteristic
66	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
68	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
69	C₂×C₄×C₁₃	#123	1	2	yes	—
70	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
71	C₂×C₄×C₁₃	C₂×Q₁₆	1	4	yes	—
72	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
73	C₂×C₄×C₁₃	#137	1	2	yes	—
74	C₂×Dic₁₃	—	4	2	no	—
75	C₂×Dic₁₃	—	8	1	no	—
76	C₂×Dic₁₃	—	8	1	no	—
77	C₂×Dic₁₃	—	8	1	no	—
78	C₂×Dic₁₃	—	8	2	no	—
79	C₂×Dic₁₃	—	8	2	no	—
80	C₂×Dic₁₃	—	8	2	no	—
81	C₂×Dic₁₃	—	8	2	no	—
82	Dic₂₆	—	4	4	no	—
83	Dic₂₆	—	8	2	no	—
84	C₈×C₁₃	—	2	4	no	—
85	C₈×C₁₃	—	2	8	no	—
86	C₂×C₄⋊Q₈	—	13	1	no	—
87	C₂×#107	—	26	4	no	—
88	C₂×#107	—	26	2	no	—
89	C₂×#103	—	26	2	no	—
90	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
91	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
92	#384	—	13	1	no	—
93	#384	—	13	2	no	—
94	#340	—	26	4	no	—
95	C₂²×C₁₃	#399	1	1	yes	characteristic
96	C₂²×C₁₃	#493	1	1	yes	characteristic
97	C₂²×C₁₃	#494	1	2	yes	—
98	C₂²×C₁₃	#496	1	2	yes	—
99	C₂²×C₁₃	#497	1	1	yes	characteristic
100	C₄×C₁₃	#474	1	2	yes	—
101	C₄×C₁₃	#477	1	4	yes	—
102	C₄×C₁₃	#480	1	2	yes	—
103	C₄×C₁₃	#484	1	4	yes	—
104	Dic₁₃	—	8	2	no	—
105	Dic₁₃	—	16	1	no	—
106	Dic₁₃	—	16	2	no	—
107	C₂²×Q₈	—	26	1	no	—
108	C₂²×C₈	D₅₂	1	2	yes	—
109	C₂²×C₈	D₅₂	1	4	yes	—
110	C₄⋊Q₈	—	13	2	no	—
111	C₄⋊Q₈	—	26	2	no	—
112	C₄⋊Q₈	—	26	1	no	—
113	C₂×C₄⋊C₄	—	26	2	no	—
114	C₂×C₄⋊C₄	—	26	2	no	—
115	C₂×C₄⋊C₄	—	26	1	no	—
116	C₂×C₄⋊C₄	—	26	2	no	—
117	C₂×C₄⋊C₄	—	26	2	no	—
118	C₂×C₄⋊C₄	—	26	2	no	—
119	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
120	#107	—	52	8	no	—
121	#107	—	52	4	no	—
122	#103	—	52	4	no	—
123	C₄×C₈	—	2	2	no	—
124	C₄×C₈	—	2	2	no	—
125	C₄×C₈	—	2	2	no	—
126	C₂×C₁₃	#1713	1	1	yes	characteristic
127	C₂×C₁₃	#1721	1	1	yes	characteristic
128	C₂×C₁₃	#1723	1	2	yes	—
129	C₂×C₁₃	#2093	1	1	yes	characteristic
130	C₂×C₁₃	#2095	1	2	yes	—
131	C₂×Q₈	—	26	2	no	—
132	C₂×Q₈	—	52	1	no	—
133	C₂×Q₈	—	52	2	no	—
134	C₂×Q₈	—	52	2	no	—
135	C₂²×C₄	—	52	1	no	—
136	C₂²×C₄	—	52	2	no	—
137	C₂²×C₄	—	52	1	no	—
138	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
139	C₂²×C₄	C₂×D₅₂	1	2	yes	—
140	C₂×C₈	—	2	4	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	D₁₀₄	1	4	yes	—
145	C₂×C₈	Dic₅₂	1	8	yes	—
146	C₄⋊C₄	—	26	4	no	—
147	C₄⋊C₄	—	52	2	no	—
148	C₄⋊C₄	—	52	2	no	—
149	C₄⋊C₄	—	52	4	no	—
150	C₄⋊C₄	—	52	4	no	—
151	C₄⋊C₄	—	52	4	no	—
152	C₄⋊C₄	—	52	4	no	—
153	C₄²	#6129	1	1	yes	characteristic
154	C₄²	#6129	1	2	yes	—
155	C₄²	Q₈×D₁₃	1	1	yes	characteristic
156	C₁₃	#10674	1	1	yes	characteristic, complemented, Sylow
157	C₂³	#88128	1	1	yes	characteristic
158	Q₈	—	52	4	no	—
159	Q₈	—	104	2	no	—
160	C₂×C₄	—	52	2	no	—
161	C₂×C₄	—	104	1	no	—
162	C₂×C₄	—	104	1	no	—
163	C₂×C₄	—	104	1	no	—
164	C₂×C₄	—	104	2	no	—
165	C₂×C₄	—	104	2	no	—
166	C₂×C₄	—	104	2	no	—
167	C₂×C₄	—	104	2	no	—
168	C₂×C₄	#88140	1	2	yes	—
169	C₂×C₄	#88140	1	1	yes	characteristic
170	C₂×C₄	#88151	1	1	yes	characteristic
171	C₂×C₄	#88151	1	2	yes	—
172	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
173	C₂×C₄	C₂×Dic₅₂	1	4	yes	—
174	C₂×C₄	#88162	1	4	yes	—
175	C₂×C₄	#88163	1	2	yes	—
176	C₈	—	2	8	no	—
177	C₈	—	2	4	no	—
178	C₂²	#11718338	1	1	yes	characteristic
179	C₂²	#11718356	1	1	yes	characteristic
180	C₂²	#11718360	1	2	yes	—
181	C₂²	#11718373	1	2	yes	—
182	C₂²	#11718376	1	1	yes	characteristic
183	C₄	—	104	2	no	—
184	C₄	—	208	1	no	—
185	C₄	—	208	2	no	—
186	C₄	#11718383	1	4	yes	—
187	C₄	#11718397	1	2	yes	—
188	C₄	#11718507	1	2	yes	—
189	C₄	#11718511	1	4	yes	—
190	C₂	#13114454	1	1	yes	characteristic
191	C₂	#13114502	1	1	yes	characteristic
192	C₂	#13114513	1	2	yes	—
193	C₂	#13116363	1	1	yes	characteristic
194	C₂	#13116373	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13115797

C₂

yes

maximal

#13115638

C₂

yes

maximal, characteristic

#13114996

C₂

yes

maximal

#13114996

C₂

yes

maximal, characteristic

C₁₃×#1273

C₂

yes

maximal, characteristic

C₂×#88123

C₂²

yes

characteristic

C₂×C₄×C₈×C₁₃

C₂²

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

—

—

—

C₂²

yes

—

#11718596

C₂²

yes

characteristic

#11718143

—

C₂²×Dic₂₆

—

C₂²×C₈×C₁₃

D₄

yes

—

C₂²×C₈×C₁₃

D₄

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

#88123

—

#88123

—

#88123

Q₈

yes

—

C₄×C₈×C₁₃

—

C₄×C₈×C₁₃

—

C₄×C₈×C₁₃

—

—

—

—

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×D₄

yes

—

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

D₈

yes

—

C₂×C₈×C₁₃

Q₁₆

yes

—

C₄²×C₁₃

C₂×Q₈

yes

C₄²×C₁₃

yes

C₄²×C₁₃

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

—

—

—

—

D₁₃

yes

characteristic

C₂³×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₃

#123

yes

characteristic

C₂×C₄×C₁₃

#123

yes

—

C₂×C₄×C₁₃

C₂×D₈

yes

—

C₂×C₄×C₁₃

C₂×Q₁₆

yes

—

C₂×C₄×C₁₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₁₃

#137

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

—

C₂×C₄⋊Q₈

—

C₂×#107

—

C₂×#107

—

C₂×#103

—

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

C₂×C₄×C₈

C₂×D₁₃

yes

—

#384

—

#384

—

#340

—

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

—

C₂²×C₁₃

#496

yes

—

C₂²×C₁₃

#497

yes

characteristic

100

C₄×C₁₃

#474

yes

—

101

C₄×C₁₃

#477

yes

—

102

C₄×C₁₃

#480

yes

—

103

C₄×C₁₃

#484

yes

—

104

Dic₁₃

—

105

Dic₁₃

—

106

Dic₁₃

—

107

C₂²×Q₈

—

108

C₂²×C₈

D₅₂

yes

—

109

C₂²×C₈

D₅₂

yes

—

110

C₄⋊Q₈

—

111

C₄⋊Q₈

—

112

C₄⋊Q₈

—

113

C₂×C₄⋊C₄

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄⋊C₄

—

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄²

C₂²×D₁₃

yes

characteristic

120

#107

—

121

#107

—

122

#103

—

123

C₄×C₈

—

124

C₄×C₈

—

125

C₄×C₈

—

126

C₂×C₁₃

#1713

yes

characteristic

127

C₂×C₁₃

#1721

yes

characteristic

128

C₂×C₁₃

#1723

yes

—

129

C₂×C₁₃

#2093

yes

characteristic

130

C₂×C₁₃

#2095

yes

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

C₂×D₅₂

yes

characteristic

139

C₂²×C₄

C₂×D₅₂

yes

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

D₁₀₄

yes

—

145

C₂×C₈

Dic₅₂

yes

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄²

#6129

yes

characteristic

154

C₄²

#6129

yes

—

155

C₄²

Q₈×D₁₃

yes

characteristic

156

C₁₃

#10674

yes

characteristic, complemented, Sylow

157

C₂³

#88128

yes

characteristic

158

Q₈

—

159

Q₈

—

104

—

160

C₂×C₄

—

161

C₂×C₄

—

104

—

162

C₂×C₄

—

104

—

163

C₂×C₄

—

104

—

164

C₂×C₄

—

104

—

165

C₂×C₄

—

104

—

166

C₂×C₄

—

104

—

167

C₂×C₄

—

104

—

168

C₂×C₄

#88140

yes

—

169

C₂×C₄

#88140

yes

characteristic

170

C₂×C₄

#88151

yes

characteristic

171

C₂×C₄

#88151

yes

—

172

C₂×C₄

C₂×D₁₀₄

yes

—

173

C₂×C₄

C₂×Dic₅₂

yes

—

174

C₂×C₄

#88162

yes

—

175

C₂×C₄

#88163

yes

—

176

C₈

—

177

C₈

—

178

C₂²

#11718338

yes

characteristic

179

C₂²

#11718356

yes

characteristic

180

C₂²

#11718360

yes

—

181

C₂²

#11718373

yes

—

182

C₂²

#11718376

yes

characteristic

183

C₄

—

104

—

184

C₄

—

208

—

185

C₄

—

208

—

186

C₄

#11718383

yes

—

187

C₄

#11718397

yes

—

188

C₄

#11718507

yes

—

189

C₄

#11718511

yes

—

190

C₂

#13114454

yes

characteristic

191

C₂

#13114502

yes

characteristic

192

C₂

#13114513

yes

—

193

C₂

#13116363

yes

characteristic

194

C₂

#13116373

yes

—

	subgroup	G/H
center	C₂³	#88128
commutator, cosocle	C₂×C₄²×C₁₃	C₂³
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1273	C₂
socle, prime radical	C₂³×C₁₃	#127
hypercenter	#1273	D₁₃
hypocenter	C₁₃	#10674
LCS	G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

subgroup

G/H

center

C₂³

#88128

commutator, cosocle

C₂×C₄²×C₁₃

C₂³

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1273

C₂

C₂³×C₁₃

D₁₃

C₁₃

G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

N	H ≅ G/N	type
156. C₁₃	#10674	semidirect

H ≅ G/N

type

156. C₁₃

#10674

semidirect

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

subgroup	index
#13115797	2
#13115638	2
#13114996	2
C₁₃×#1273	2
#10674	13

subgroup

index

C₁₃×#1273

Subgroups — #15767972

Conjugacy classes of subgroups

Complements

Maximal subgroups