Group #15779073 — subgroups

Subgroups — #15779073

Order 3328.

358 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 127 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	C₂×#11718801	C₂	1	1	yes	maximal, characteristic
2	#13116608	C₂	1	2	yes	maximal, complemented
3	#13114886	C₂	1	1	yes	maximal, characteristic
4	#13112931	C₂	1	2	yes	maximal, complemented
5	C₁₃×#1812	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
7	C₁₃×#427	C₂²	1	2	yes	—
8	C₁₃×#410	—	2	1	no	—
9	#11718898	—	2	1	no	—
10	#11718801	—	2	4	no	—
11	C₂×#88059	C₂²	1	2	yes	—
12	C₂×#88052	C₄	1	2	yes	complemented
13	C₂×#88052	C₂²	1	1	yes	characteristic
14	C₂²×#6090	C₄	1	2	yes	complemented
15	C₂²×#6090	C₂²	1	1	yes	characteristic
16	C₁₃×H(3,ℤ/4)	—	2	1	no	—
17	#11718152	—	2	1	no	—
18	#11718079	—	2	4	no	—
19	C₂×C₁₃×#41	D₄	1	1	yes	characteristic
20	C₁₃×#131	—	2	2	no	—
21	C₁₃×#131	—	2	2	no	—
22	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
23	C₂×C₁₃×M₁₆	Q₈	1	2	yes	—
24	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
26	C₁₃×#117	—	2	2	no	—
27	C₂×#6099	—	2	2	no	—
28	C₁₃×#107	—	2	2	no	—
29	C₁₃×#106	—	2	2	no	—
30	C₁₃×#99	—	4	2	no	—
31	#88076	—	4	1	no	—
32	#88073	—	4	1	no	—
33	#88059	—	2	2	no	—
34	#88059	—	2	2	no	—
35	#88052	—	2	4	no	—
36	#88052	D₄	1	4	yes	—
37	C₂×#6090	—	2	2	no	—
38	C₂×#6090	—	2	4	no	—
39	C₂×#6090	C₂×C₄	1	1	yes	characteristic
40	C₂×#6090	C₂×C₄	1	1	yes	characteristic
41	C₂×#6090	C₂×C₄	1	2	yes	—
42	#12832	—	13	1	no	maximal, Sylow
43	C₁₃×#41	—	2	2	no	—
44	C₁₃×#41	—	2	2	no	—
45	C₂×Q₈×C₁₃	SD₁₆	1	1	yes	characteristic
46	C₂×D₄×C₁₃	SD₁₆	1	2	yes	—
47	C₂×D₄×C₁₃	SD₁₆	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
50	C₁₃×M₁₆	—	2	2	no	—
51	C₁₃×M₁₆	—	2	4	no	—
52	C₂×C₈×C₁₃	—	2	2	no	—
53	C₂×C₈×C₁₃	—	2	2	no	—
54	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
55	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
57	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₈×C₁₃	#41	1	2	yes	—
59	C₁₃×C₄⋊C₄	—	4	2	no	—
60	C₁₃×C₂²⋊C₄	—	4	2	no	—
61	C₄²×C₁₃	—	4	2	no	—
62	#6099	—	4	2	no	—
63	#6099	—	4	2	no	—
64	C₂×#1055	—	4	2	no	—
65	#6090	—	2	1	no	—
66	#6090	—	2	1	no	—
67	#6090	C₂²⋊C₄	1	4	yes	—
68	C₂×#504	—	13	1	no	—
69	#2155	—	13	2	no	—
70	#1813	—	13	1	no	—
71	#1812	D₁₃	1	1	yes	characteristic
72	#1382	—	13	2	no	—
73	C₂³×C₁₃	#106	1	2	yes	—
74	C₂³×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
75	Q₈×C₁₃	—	2	2	no	—
76	D₄×C₁₃	—	2	4	no	—
77	D₄×C₁₃	—	2	2	no	—
78	C₂×C₄×C₁₃	—	2	2	no	—
79	C₂×C₄×C₁₃	—	2	2	no	—
80	C₂×C₄×C₁₃	—	8	2	no	—
81	C₂×C₄×C₁₃	#106	1	2	yes	—
82	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₃	#121	1	2	yes	—
84	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
85	C₂×C₄×C₁₃	#135	1	2	yes	—
86	C₈×C₁₃	—	2	2	no	—
87	C₈×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₈×C₁₃	#117	1	1	yes	characteristic
89	C₈×C₁₃	C₂²⋊Q₈	1	2	yes	—
90	#1055	—	8	2	no	—
91	C₂×#131	C₂×D₁₃	1	1	yes	characteristic
92	#504	—	26	4	no	—
93	C₂×M₃₂	—	13	1	no	—
94	C₂×M₃₂	—	13	2	no	—
95	C₂²×C₁₆	—	13	2	no	—
96	C₂²×C₁₆	—	13	1	no	—
97	C₂×#108	—	13	2	no	—
98	#427	C₂×D₁₃	1	2	yes	—
99	#413	—	26	1	no	—
100	#410	—	2	1	no	—
101	#365	—	26	4	no	—
102	#338	—	26	1	no	—
103	H(3,ℤ/4)	—	2	1	no	—
104	C₂²×C₁₃	—	2	2	no	—
105	C₂²×C₁₃	—	4	1	no	—
106	C₂²×C₁₃	—	4	2	no	—
107	C₂²×C₁₃	C₂×#106	1	1	yes	characteristic
108	C₂²×C₁₃	#465	1	2	yes	—
109	C₄×C₁₃	—	2	2	no	—
110	C₄×C₁₃	—	16	2	no	—
111	C₄×C₁₃	#380	1	1	yes	characteristic
112	C₄×C₁₃	#427	1	1	yes	characteristic
113	C₄×C₁₃	#465	1	2	yes	—
114	C₂×#41	C₂²⋊D₁₃	1	1	yes	characteristic
115	#131	—	2	2	no	—
116	#131	—	2	2	no	—
117	C₂×M₁₆	—	26	2	no	—
118	C₂×M₁₆	Dic₂₆	1	2	yes	—
119	C₂×M₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
120	C₂²×C₈	C₄×D₁₃	1	2	yes	—
121	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
122	#117	—	2	2	no	—
123	M₃₂	—	13	4	no	—
124	M₃₂	—	26	4	no	—
125	C₂×C₁₆	—	13	2	no	—
126	C₂×C₁₆	—	13	1	no	—
127	C₂×C₁₆	—	13	1	no	—
128	C₂×C₁₆	—	26	4	no	—
129	C₂×C₁₆	—	26	2	no	—
130	#108	—	26	2	no	—
131	#108	—	26	2	no	—
132	#107	—	2	2	no	—
133	#106	—	2	2	no	—
134	#101	—	52	1	no	—
135	#100	—	52	1	no	—
136	#99	—	4	2	no	—
137	C₂×C₁₃	—	2	2	no	—
138	C₂×C₁₃	—	2	1	no	—
139	C₂×C₁₃	—	4	2	no	—
140	C₂×C₁₃	#1807	1	1	yes	characteristic
141	#41	—	2	2	no	—
142	#41	—	2	2	no	—
143	C₂×Q₈	#6106	1	1	yes	characteristic
144	C₂×D₄	#6105	1	2	yes	—
145	C₂×D₄	#6106	1	1	yes	characteristic
146	C₂²×C₄	#6101	1	2	yes	—
147	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
148	M₁₆	—	2	2	no	—
149	M₁₆	—	2	4	no	—
150	M₁₆	—	52	2	no	—
151	M₁₆	—	52	2	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	52	2	no	—
155	C₂×C₈	#6101	1	2	yes	—
156	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
157	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
158	C₂×C₈	D₄×D₁₃	1	2	yes	—
159	C₂×C₈	#6129	1	2	yes	—
160	C₄⋊C₄	—	4	2	no	—
161	C₂²⋊C₄	—	4	2	no	—
162	C₄²	—	4	2	no	—
163	C₁₆	—	13	4	no	—
164	C₁₆	—	26	1	no	—
165	C₁₆	—	26	1	no	—
166	C₁₃	#12832	1	1	yes	characteristic, complemented, Sylow
167	C₂³	#88048	1	2	yes	—
168	C₂³	C₂×#6105	1	1	yes	characteristic
169	Q₈	—	2	2	no	—
170	D₄	—	2	4	no	—
171	D₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	8	2	no	—
175	C₂×C₄	#88048	1	2	yes	—
176	C₂×C₄	C₂×#6101	1	1	yes	characteristic
177	C₂×C₄	#88193	1	2	yes	—
178	C₂×C₄	C₂×#6106	1	1	yes	characteristic
179	C₂×C₄	#88204	1	2	yes	—
180	C₈	—	2	2	no	—
181	C₈	—	104	2	no	—
182	C₈	#88131	1	1	yes	characteristic
183	C₈	#88132	1	2	yes	—
184	C₈	C₂²⋊C₄×D₁₃	1	1	yes	characteristic
185	C₂²	—	2	2	no	—
186	C₂²	—	4	1	no	—
187	C₂²	—	4	2	no	—
188	C₂²	C₂×#88048	1	1	yes	characteristic
189	C₂²	#11718702	1	2	yes	—
190	C₄	—	2	2	no	—
191	C₄	—	16	2	no	—
192	C₄	#11718603	1	1	yes	characteristic
193	C₄	#11718659	1	1	yes	characteristic
194	C₄	#11718709	1	2	yes	—
195	C₂	—	2	2	no	—
196	C₂	—	2	1	no	—
197	C₂	—	4	2	no	—
198	C₂	#13114859	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11718801

C₂

yes

maximal, characteristic

#13116608

C₂

yes

maximal, complemented

#13114886

C₂

yes

maximal, characteristic

#13112931

C₂

yes

maximal, complemented

C₁₃×#1812

C₂

yes

maximal, characteristic

C₂×C₁₃×#131

C₂²

yes

characteristic

C₁₃×#427

C₂²

yes

—

C₁₃×#410

—

#11718898

—

#11718801

—

C₂×#88059

C₂²

yes

—

C₂×#88052

C₄

yes

complemented

C₂×#88052

C₂²

yes

characteristic

C₂²×#6090

C₄

yes

complemented

C₂²×#6090

C₂²

yes

characteristic

C₁₃×H(3,ℤ/4)

—

#11718152

—

#11718079

—

C₂×C₁₃×#41

D₄

yes

characteristic

C₁₃×#131

—

C₁₃×#131

—

C₂×C₁₃×M₁₆

D₄

yes

characteristic

C₂×C₁₃×M₁₆

Q₈

yes

—

C₂²×C₈×C₁₃

C₂×C₄

yes

—

C₂²×C₈×C₁₃

C₂³

yes

characteristic

C₁₃×#117

—

C₂×#6099

—

C₁₃×#107

—

C₁₃×#106

—

C₁₃×#99

—

—

—

—

—

—

D₄

yes

—

C₂×#6090

—

C₂×#6090

—

C₂×#6090

C₂×C₄

yes

characteristic

C₂×#6090

C₂×C₄

yes

characteristic

C₂×#6090

C₂×C₄

yes

—

#12832

—

maximal, Sylow

C₁₃×#41

—

C₁₃×#41

—

C₂×Q₈×C₁₃

SD₁₆

yes

characteristic

C₂×D₄×C₁₃

SD₁₆

yes

—

C₂×D₄×C₁₃

SD₁₆

yes

characteristic

C₂²×C₄×C₁₃

C₄⋊C₄

yes

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₁₃×M₁₆

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₂×C₈×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₃

C₂×D₄

yes

—

C₂×C₈×C₁₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₃

#41

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

—

—

—

—

—

—

C₂²⋊C₄

yes

—

—

—

—

D₁₃

yes

characteristic

#1382

—

C₂³×C₁₃

#106

yes

—

C₂³×C₁₃

C₂×SD₁₆

yes

characteristic

Q₈×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#106

yes

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

#121

yes

—

C₂×C₄×C₁₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₁₃

#135

yes

—

C₈×C₁₃

—

C₈×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₁₃

#117

yes

characteristic

C₈×C₁₃

C₂²⋊Q₈

yes

—

#1055

—

C₂×#131

C₂×D₁₃

yes

characteristic

#504

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂×#108

—

#427

C₂×D₁₃

yes

—

#413

—

100

#410

—

101

#365

—

102

#338

—

103

H(3,ℤ/4)

—

104

C₂²×C₁₃

—

105

C₂²×C₁₃

—

106

C₂²×C₁₃

—

107

C₂²×C₁₃

C₂×#106

yes

characteristic

108

C₂²×C₁₃

#465

yes

—

109

C₄×C₁₃

—

110

C₄×C₁₃

—

111

C₄×C₁₃

#380

yes

characteristic

112

C₄×C₁₃

#427

yes

characteristic

113

C₄×C₁₃

#465

yes

—

114

C₂×#41

C₂²⋊D₁₃

yes

characteristic

115

#131

—

116

#131

—

117

C₂×M₁₆

—

118

C₂×M₁₆

Dic₂₆

yes

—

119

C₂×M₁₆

C₂²⋊D₁₃

yes

characteristic

120

C₂²×C₈

C₄×D₁₃

yes

—

121

C₂²×C₈

C₂²×D₁₃

yes

characteristic

122

#117

—

123

M₃₂

—

124

M₃₂

—

125

C₂×C₁₆

—

126

C₂×C₁₆

—

127

C₂×C₁₆

—

128

C₂×C₁₆

—

129

C₂×C₁₆

—

130

#108

—

131

#108

—

132

#107

—

133

#106

—

134

#101

—

135

#100

—

136

#99

—

137

C₂×C₁₃

—

138

C₂×C₁₃

—

139

C₂×C₁₃

—

140

C₂×C₁₃

#1807

yes

characteristic

141

#41

—

142

#41

—

143

C₂×Q₈

#6106

yes

characteristic

144

C₂×D₄

#6105

yes

—

145

C₂×D₄

#6106

yes

characteristic

146

C₂²×C₄

#6101

yes

—

147

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

148

M₁₆

—

149

M₁₆

—

150

M₁₆

—

151

M₁₆

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

#6101

yes

—

156

C₂×C₈

C₂×Dic₂₆

yes

characteristic

157

C₂×C₈

C₂×C₄×D₁₃

yes

characteristic

158

C₂×C₈

D₄×D₁₃

yes

—

159

C₂×C₈

#6129

yes

—

160

C₄⋊C₄

—

161

C₂²⋊C₄

—

162

C₄²

—

163

C₁₆

—

164

C₁₆

—

165

C₁₆

—

166

C₁₃

#12832

yes

characteristic, complemented, Sylow

167

C₂³

#88048

yes

—

168

C₂³

C₂×#6105

yes

characteristic

169

Q₈

—

170

D₄

—

171

D₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

#88048

yes

—

176

C₂×C₄

C₂×#6101

yes

characteristic

177

C₂×C₄

#88193

yes

—

178

C₂×C₄

C₂×#6106

yes

characteristic

179

C₂×C₄

#88204

yes

—

180

C₈

—

181

C₈

—

104

—

182

C₈

#88131

yes

characteristic

183

C₈

#88132

yes

—

184

C₈

C₂²⋊C₄×D₁₃

yes

characteristic

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

C₂×#88048

yes

characteristic

189

C₂²

#11718702

yes

—

190

C₄

—

191

C₄

—

192

C₄

#11718603

yes

characteristic

193

C₄

#11718659

yes

characteristic

194

C₄

#11718709

yes

—

195

C₂

—

196

C₂

—

197

C₂

—

198

C₂

#13114859

yes

characteristic

	subgroup	G/H
center	C₄	#11718659
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₁₃×#1812	C₂
socle	C₂×C₁₃	#1807
prime radical	C₂×C₁₃×#41	D₄
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#1812	D₁₃
hypocenter	C₁₃	#12832
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131 ◁ #1812

subgroup

G/H

center

C₄

#11718659

commutator

C₂×C₈×C₁₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₁₃

Fitting

C₁₃×#1812

C₂

socle

C₂×C₁₃

#1807

prime radical

C₂×C₁₃×#41

D₄

cosocle

C₂²×C₈×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131 ◁ #1812

N	H ≅ G/N	type
2. #13116608	C₂	semidirect
4. #13112931	C₂	semidirect
12. C₂×#88052	C₄	semidirect
14. C₂²×#6090	C₄	semidirect
166. C₁₃	#12832	semidirect

H ≅ G/N

type

2. #13116608

C₂

semidirect

4. #13112931

C₂

semidirect

12. C₂×#88052

C₄

semidirect

14. C₂²×#6090

C₄

semidirect

166. C₁₃

#12832

semidirect

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

subgroup	index
C₂×#11718801	2
#13116608	2
#13114886	2
#13112931	2
C₁₃×#1812	2
#12832	13

subgroup

index

C₁₃×#1812

Subgroups — #15779073

Conjugacy classes of subgroups

Complements

Maximal subgroups