Group #15768034 — subgroups

Subgroups — #15768034

Order 3328.

442 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 155 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	#13115798	C₂	1	2	yes	maximal
2	#13114993	C₂	1	1	yes	maximal, characteristic
3	#13114992	C₂	1	2	yes	maximal
4	#13112663	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1956	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×#103	—	2	2	no	—
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₁₃×#384	C₂²	1	1	yes	characteristic
10	C₂×#88057	—	2	2	no	—
11	C₂×#88048	—	2	4	no	—
12	#11718596	C₂²	1	2	yes	—
13	C₂×C₄×#1055	C₂²	1	2	yes	—
14	#11718121	—	2	4	no	—
15	C₂²×Q₈×C₁₃	—	2	1	no	—
16	C₂²×C₈×C₁₃	D₄	1	2	yes	—
17	C₁₃×C₄⋊Q₈	—	2	2	no	—
18	C₁₃×C₄⋊Q₈	—	2	1	no	—
19	C₁₃×C₄⋊Q₈	Q₈	1	2	yes	—
20	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
21	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
22	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
23	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₃	—	2	2	no	—
25	C₂×C₄⋊Dic₁₃	—	2	2	no	—
26	C₂×C₄⋊Dic₁₃	—	2	2	no	—
27	C₂²×#1055	D₄	1	4	yes	—
28	C₁₃×#103	—	4	4	no	—
29	C₄×C₈×C₁₃	—	2	2	no	—
30	#88057	—	4	4	no	—
31	#88048	—	4	8	no	—
32	C₄×#1055	—	2	2	no	—
33	C₄×#1055	—	2	2	no	—
34	#10685	—	13	1	no	maximal, Sylow
35	C₂×Q₈×C₁₃	—	2	2	no	—
36	C₂×Q₈×C₁₃	—	4	1	no	—
37	C₂×Q₈×C₁₃	—	4	2	no	—
38	C₂×Q₈×C₁₃	—	4	2	no	—
39	C₂²×C₄×C₁₃	—	4	1	no	—
40	C₂²×C₄×C₁₃	—	4	1	no	—
41	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
42	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₁₃	—	4	2	no	—
44	C₂×C₈×C₁₃	—	2	2	no	—
45	C₂×C₈×C₁₃	—	2	2	no	—
46	C₂×C₈×C₁₃	SD₁₆	1	4	yes	—
47	C₁₃×C₄⋊C₄	—	2	4	no	—
48	C₁₃×C₄⋊C₄	—	4	2	no	—
49	C₁₃×C₄⋊C₄	—	4	2	no	—
50	C₁₃×C₄⋊C₄	—	4	4	no	—
51	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
52	C₄²×C₁₃	#41	1	1	yes	characteristic
53	C₄²×C₁₃	#41	1	2	yes	—
54	C₄⋊Dic₁₃	—	4	4	no	—
55	C₄⋊Dic₁₃	—	4	4	no	—
56	C₄⋊Dic₁₃	—	4	4	no	—
57	C₂×#1055	—	2	4	no	—
58	C₂×#1055	—	2	4	no	—
59	C₂×#1055	SD₁₆	1	8	yes	—
60	#1990	—	13	2	no	—
61	#1956	D₁₃	1	1	yes	characteristic
62	#1844	—	13	1	no	—
63	#1844	—	13	2	no	—
64	#1273	—	13	1	no	—
65	C₂³×C₁₃	#127	1	1	yes	characteristic
66	Q₈×C₁₃	—	4	4	no	—
67	Q₈×C₁₃	—	8	2	no	—
68	C₂×C₄×C₁₃	—	4	2	no	—
69	C₂×C₄×C₁₃	—	8	1	no	—
70	C₂×C₄×C₁₃	—	8	1	no	—
71	C₂×C₄×C₁₃	—	8	1	no	—
72	C₂×C₄×C₁₃	—	8	2	no	—
73	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
74	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
76	C₂×C₄×C₁₃	#123	1	2	yes	—
77	C₂×C₄×C₁₃	C₂×SD₁₆	1	4	yes	—
78	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
79	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
80	C₂×C₄×C₁₃	#137	1	4	yes	—
81	C₂×Dic₁₃	—	8	2	no	—
82	C₂×Dic₁₃	—	8	2	no	—
83	C₂×Dic₁₃	—	8	2	no	—
84	C₈×C₁₃	—	2	4	no	—
85	#1055	—	2	8	no	—
86	C₂×C₄⋊Q₈	C₂×D₁₃	1	1	yes	characteristic
87	C₂×#106	—	26	4	no	—
88	C₂×#106	—	26	2	no	—
89	C₂×#103	—	2	2	no	—
90	C₂×C₄×C₈	—	13	2	no	—
91	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
92	#384	—	13	2	no	—
93	#384	C₂×D₁₃	1	1	yes	characteristic
94	#340	—	26	4	no	—
95	C₂²×C₁₃	#399	1	1	yes	characteristic
96	C₂²×C₁₃	#492	1	1	yes	characteristic
97	C₂²×C₁₃	#492	1	2	yes	—
98	C₂²×C₁₃	#496	1	1	yes	characteristic
99	C₂²×C₁₃	#497	1	2	yes	—
100	C₄×C₁₃	—	8	2	no	—
101	C₄×C₁₃	—	16	1	no	—
102	C₄×C₁₃	#475	1	4	yes	—
103	C₄×C₁₃	#476	1	2	yes	—
104	C₄×C₁₃	#481	1	2	yes	—
105	C₄×C₁₃	#483	1	4	yes	—
106	Dic₁₃	—	16	2	no	—
107	C₂²×Q₈	—	2	1	no	—
108	C₂²×C₈	—	13	4	no	—
109	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
110	C₄⋊Q₈	—	2	1	no	—
111	C₄⋊Q₈	—	2	2	no	—
112	C₄⋊Q₈	Dic₂₆	1	2	yes	—
113	C₂×C₄⋊C₄	—	2	2	no	—
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	2	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	#106	—	52	8	no	—
121	#106	—	52	4	no	—
122	#103	—	4	4	no	—
123	C₄×C₈	—	2	2	no	—
124	C₄×C₈	—	26	2	no	—
125	C₄×C₈	—	26	2	no	—
126	C₂×C₁₃	#1705	1	1	yes	characteristic
127	C₂×C₁₃	#1714	1	2	yes	—
128	C₂×C₁₃	#1716	1	1	yes	characteristic
129	C₂×C₁₃	#2094	1	1	yes	characteristic
130	C₂×C₁₃	#2096	1	2	yes	—
131	C₂×Q₈	—	2	2	no	—
132	C₂×Q₈	—	4	1	no	—
133	C₂×Q₈	—	4	2	no	—
134	C₂×Q₈	—	4	2	no	—
135	C₂²×C₄	—	4	1	no	—
136	C₂²×C₄	—	4	1	no	—
137	C₂²×C₄	—	52	2	no	—
138	C₂²×C₄	D₄×D₁₃	1	2	yes	—
139	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	13	8	no	—
143	C₂×C₈	—	26	4	no	—
144	C₂×C₈	—	26	4	no	—
145	C₂×C₈	#6105	1	4	yes	—
146	C₄⋊C₄	—	2	4	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄⋊C₄	—	4	4	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄⋊C₄	—	52	4	no	—
151	C₄⋊C₄	—	52	4	no	—
152	C₄⋊C₄	—	52	4	no	—
153	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
154	C₄²	#6127	1	1	yes	characteristic
155	C₄²	#6129	1	2	yes	—
156	C₁₃	#10685	1	1	yes	characteristic, complemented, Sylow
157	C₂³	#88194	1	1	yes	characteristic
158	Q₈	—	4	4	no	—
159	Q₈	—	8	2	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	8	1	no	—
162	C₂×C₄	—	8	1	no	—
163	C₂×C₄	—	8	1	no	—
164	C₂×C₄	—	8	2	no	—
165	C₂×C₄	—	104	2	no	—
166	C₂×C₄	—	104	2	no	—
167	C₂×C₄	—	104	2	no	—
168	C₂×C₄	#88132	1	2	yes	—
169	C₂×C₄	#88137	1	2	yes	—
170	C₂×C₄	SD₁₆×D₁₃	1	4	yes	—
171	C₂×C₄	#88172	1	4	yes	—
172	C₂×C₄	#88178	1	1	yes	characteristic
173	C₂×C₄	C₂×#6105	1	2	yes	—
174	C₂×C₄	#88189	1	1	yes	characteristic
175	C₂×C₄	#88203	1	2	yes	—
176	C₈	—	2	4	no	—
177	C₈	—	26	8	no	—
178	C₂²	#11718610	1	1	yes	characteristic
179	C₂²	#11718719	1	1	yes	characteristic
180	C₂²	#11718737	1	1	yes	characteristic
181	C₂²	#11718836	1	2	yes	—
182	C₂²	#11718851	1	2	yes	—
183	C₄	—	8	2	no	—
184	C₄	—	16	1	no	—
185	C₄	—	208	2	no	—
186	C₄	#11718452	1	4	yes	—
187	C₄	#11718466	1	4	yes	—
188	C₄	#11718669	1	2	yes	—
189	C₄	#11718695	1	2	yes	—
190	C₂	#13114422	1	1	yes	characteristic
191	C₂	#13114460	1	2	yes	—
192	C₂	#13114467	1	1	yes	characteristic
193	C₂	#13116369	1	1	yes	characteristic
194	C₂	#13116377	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13115798

C₂

yes

maximal

#13114993

C₂

yes

maximal, characteristic

#13114992

C₂

yes

maximal

#13112663

C₂

yes

maximal, characteristic

C₁₃×#1956

C₂

yes

maximal, characteristic

C₂×C₁₃×C₄⋊Q₈

C₂²

yes

characteristic

C₂×C₁₃×#103

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₁₃×#384

C₂²

yes

—

—

C₂²

yes

—

C₂×C₄×#1055

C₂²

yes

—

#11718121

—

C₂²×Q₈×C₁₃

—

C₂²×C₈×C₁₃

D₄

yes

—

C₁₃×C₄⋊Q₈

—

C₁₃×C₄⋊Q₈

—

C₁₃×C₄⋊Q₈

Q₈

yes

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₂²×#1055

D₄

yes

—

C₁₃×#103

—

C₄×C₈×C₁₃

—

—

—

—

—

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×Dic₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

SD₁₆

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂×Q₈

yes

C₄²×C₁₃

yes

C₄²×C₁₃

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₂×#1055

—

C₂×#1055

—

C₂×#1055

SD₁₆

yes

—

#1990

—

#1956

D₁₃

yes

—

—

—

C₂³×C₁₃

#127

yes

characteristic

Q₈×C₁₃

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₃

C₂²⋊Q₈

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

—

C₂×C₄×C₁₃

C₂×SD₁₆

yes

—

C₂×C₄×C₁₃

C₂×SD₁₆

yes

—

C₂×C₄×C₁₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₁₃

#137

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₈×C₁₃

—

#1055

—

C₂×C₄⋊Q₈

C₂×D₁₃

yes

—

—

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

#384

—

#384

C₂×D₁₃

yes

characteristic

#340

—

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

—

C₂²×C₁₃

#496

yes

characteristic

C₂²×C₁₃

#497

yes

—

100

C₄×C₁₃

—

101

C₄×C₁₃

—

102

C₄×C₁₃

#475

yes

—

103

C₄×C₁₃

#476

yes

—

104

C₄×C₁₃

#481

yes

—

105

C₄×C₁₃

#483

yes

—

106

Dic₁₃

—

107

C₂²×Q₈

—

108

C₂²×C₈

—

109

C₂²×C₈

C₂²⋊D₁₃

yes

—

110

C₄⋊Q₈

—

111

C₄⋊Q₈

—

112

C₄⋊Q₈

Dic₂₆

yes

—

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

#106

—

121

#106

—

122

#103

—

123

C₄×C₈

—

124

C₄×C₈

—

125

C₄×C₈

—

126

C₂×C₁₃

#1705

yes

characteristic

127

C₂×C₁₃

#1714

yes

—

128

C₂×C₁₃

#1716

yes

characteristic

129

C₂×C₁₃

#2094

yes

characteristic

130

C₂×C₁₃

#2096

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

D₄×D₁₃

yes

—

139

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

#6105

yes

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄²

C₂×Dic₂₆

yes

characteristic

154

C₄²

#6127

yes

characteristic

155

C₄²

#6129

yes

—

156

C₁₃

#10685

yes

characteristic, complemented, Sylow

157

C₂³

#88194

yes

characteristic

158

Q₈

—

159

Q₈

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

104

—

166

C₂×C₄

—

104

—

167

C₂×C₄

—

104

—

168

C₂×C₄

#88132

yes

—

169

C₂×C₄

#88137

yes

—

170

C₂×C₄

SD₁₆×D₁₃

yes

—

171

C₂×C₄

#88172

yes

—

172

C₂×C₄

#88178

yes

characteristic

173

C₂×C₄

C₂×#6105

yes

—

174

C₂×C₄

#88189

yes

characteristic

175

C₂×C₄

#88203

yes

—

176

C₈

—

177

C₈

—

178

C₂²

#11718610

yes

characteristic

179

C₂²

#11718719

yes

characteristic

180

C₂²

#11718737

yes

characteristic

181

C₂²

#11718836

yes

—

182

C₂²

#11718851

yes

—

183

C₄

—

184

C₄

—

185

C₄

—

208

—

186

C₄

#11718452

yes

—

187

C₄

#11718466

yes

—

188

C₄

#11718669

yes

—

189

C₄

#11718695

yes

—

190

C₂

#13114422

yes

characteristic

191

C₂

#13114460

yes

—

192

C₂

#13114467

yes

characteristic

193

C₂

#13116369

yes

characteristic

194

C₂

#13116377

yes

—

	subgroup	G/H
center	C₂³	#88194
commutator, cosocle	C₂×C₄²×C₁₃	C₂³
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1956	C₂
socle, prime radical	C₂³×C₁₃	#127
hypercenter	#1956	D₁₃
hypocenter	C₁₃	#10685
LCS	G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1956

subgroup

G/H

center

C₂³

#88194

commutator, cosocle

C₂×C₄²×C₁₃

C₂³

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1956

C₂

C₂³×C₁₃

D₁₃

C₁₃

G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1956

N	H ≅ G/N	type
156. C₁₃	#10685	semidirect

H ≅ G/N

type

156. C₁₃

#10685

semidirect

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

subgroup	index
#13115798	2
#13114993	2
#13114992	2
#13112663	2
C₁₃×#1956	2
#10685	13

subgroup

index

C₁₃×#1956

Subgroups — #15768034

Conjugacy classes of subgroups

Complements

Maximal subgroups