Group #15779257 — subgroups

Subgroups — #15779257

Order 3328.

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

normal subgroups only

This group has 366 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	#13116450	C₂	1	1	yes	maximal, characteristic
2	#13114882	C₂	1	1	yes	maximal, characteristic
3	#13112950	C₂	1	2	yes	maximal, complemented
4	#13112933	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2916	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#568	—	2	1	no	—
7	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
8	C₁₃×#430	—	2	1	no	—
9	C₁₃×#430	C₄	1	4	yes	complemented
10	C₂×C₁₃×#108	—	2	2	no	—
11	C₂×C₁₃×#108	C₂²	1	2	yes	—
12	#11718901	—	2	1	no	—
13	#11718767	C₂²	1	2	yes	—
14	C₂×#88052	C₂²	1	1	yes	characteristic
15	C₂²×#6090	C₂²	1	1	yes	characteristic
16	#11718149	—	2	1	no	—
17	C₂²⋊#6090	—	2	2	no	—
18	C₂×C₁₃×#41	D₄	1	1	yes	characteristic
19	C₁₃×#131	—	2	2	no	—
20	C₁₃×#131	—	2	2	no	—
21	C₁₃×#131	—	4	4	no	—
22	C₂×C₁₃×M₁₆	—	2	4	no	—
23	C₂×C₁₃×M₁₆	—	2	2	no	—
24	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
25	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	D₄	1	2	yes	—
27	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
28	#88180	—	2	2	no	—
29	C₁₃×#108	—	2	4	no	—
30	C₁₃×#108	—	2	2	no	—
31	C₁₃×#108	C₂×C₄	1	4	yes	—
32	#88074	—	4	2	no	—
33	#88058	—	2	2	no	—
34	#88057	—	2	2	no	—
35	#88052	—	2	2	no	—
36	C₂×#6090	—	2	1	no	—
37	C₂×#6090	—	2	1	no	—
38	C₂×#6090	—	2	2	no	—
39	#12877	—	13	1	no	maximal, Sylow
40	C₁₃×#41	—	2	2	no	—
41	C₁₃×#41	—	2	2	no	—
42	C₂×Q₈×C₁₃	Q₁₆	1	1	yes	characteristic
43	C₂×D₄×C₁₃	SD₁₆	1	2	yes	—
44	C₂×D₄×C₁₃	Q₁₆	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
47	C₁₃×M₁₆	—	2	2	no	—
48	C₁₃×M₁₆	—	2	4	no	—
49	C₁₃×M₁₆	—	2	4	no	—
50	C₁₃×M₁₆	—	4	4	no	—
51	C₁₃×M₁₆	—	4	2	no	—
52	C₂×C₈×C₁₃	—	2	2	no	—
53	C₂×C₈×C₁₃	—	2	2	no	—
54	C₂×C₈×C₁₃	—	4	4	no	—
55	C₂×C₈×C₁₃	—	4	2	no	—
56	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
57	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₁₃	#41	1	2	yes	—
60	C₂×C₈×C₁₃	#41	1	2	yes	—
61	C₂²⋊Dic₁₃	—	4	2	no	—
62	C₄⋊Dic₁₃	—	4	2	no	—
63	C₄×Dic₁₃	—	4	2	no	—
64	#6090	—	4	1	no	—
65	#6090	—	4	1	no	—
66	#2916	D₁₃	1	1	yes	characteristic
67	#2113	—	13	1	no	—
68	#1812	—	13	1	no	—
69	#1389	—	13	2	no	—
70	#1382	—	13	2	no	—
71	C₂³×C₁₃	#103	1	2	yes	—
72	C₂³×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
73	Q₈×C₁₃	—	2	2	no	—
74	D₄×C₁₃	—	2	4	no	—
75	D₄×C₁₃	—	2	2	no	—
76	C₂×C₄×C₁₃	—	2	2	no	—
77	C₂×C₄×C₁₃	—	2	2	no	—
78	C₂×C₄×C₁₃	#103	1	2	yes	—
79	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₃	#123	1	2	yes	—
81	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
82	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
83	C₂×Dic₁₃	—	8	2	no	—
84	C₈×C₁₃	—	2	2	no	—
85	C₈×C₁₃	—	4	4	no	—
86	C₈×C₁₃	#117	1	2	yes	—
87	C₈×C₁₃	#123	1	1	yes	characteristic
88	C₈×C₁₃	#123	1	1	yes	characteristic
89	#568	—	2	1	no	—
90	C₂×#131	C₂×D₁₃	1	1	yes	characteristic
91	C₂×M₃₂	—	13	1	no	—
92	C₂²×C₁₆	—	13	1	no	—
93	#430	—	2	1	no	—
94	#430	Dic₁₃	1	4	yes	complemented
95	C₂×#108	—	2	2	no	—
96	C₂×#108	C₂×D₁₃	1	2	yes	—
97	#427	—	13	2	no	—
98	#410	—	26	1	no	—
99	C₂²⋊C₁₆	—	26	2	no	—
100	H(3,ℤ/4)	—	26	1	no	—
101	C₂²×C₁₃	—	2	2	no	—
102	C₂²×C₁₃	—	4	1	no	—
103	C₂²×C₁₃	—	4	2	no	—
104	C₂²×C₁₃	C₂×#103	1	1	yes	characteristic
105	C₂²×C₁₃	#481	1	2	yes	—
106	C₄×C₁₃	—	2	2	no	—
107	C₄×C₁₃	#381	1	1	yes	characteristic
108	C₄×C₁₃	#418	1	1	yes	characteristic
109	C₄×C₁₃	#484	1	2	yes	—
110	Dic₁₃	—	16	2	no	—
111	C₂×#41	C₂²⋊D₁₃	1	1	yes	characteristic
112	#131	—	2	2	no	—
113	#131	—	2	2	no	—
114	#131	—	4	4	no	—
115	C₂×M₁₆	—	2	4	no	—
116	C₂×M₁₆	—	2	2	no	—
117	C₂×M₁₆	C₂×Dic₁₃	1	2	yes	—
118	C₂×M₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
119	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
120	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
121	#117	—	26	2	no	—
122	M₃₂	—	26	2	no	—
123	C₂×C₁₆	—	26	2	no	—
124	C₂×C₁₆	—	26	1	no	—
125	C₂×C₁₆	—	26	1	no	—
126	#108	—	2	4	no	—
127	#108	—	2	2	no	—
128	#108	C₂×Dic₁₃	1	4	yes	—
129	#107	—	26	2	no	—
130	#106	—	26	2	no	—
131	#99	—	52	2	no	—
132	C₂×C₁₃	—	2	2	no	—
133	C₂×C₁₃	—	2	1	no	—
134	C₂×C₁₃	—	4	2	no	—
135	C₂×C₁₃	#1824	1	1	yes	characteristic
136	#41	—	2	2	no	—
137	#41	—	2	2	no	—
138	C₂×Q₈	#6107	1	1	yes	characteristic
139	C₂×D₄	#6106	1	2	yes	—
140	C₂×D₄	#6107	1	1	yes	characteristic
141	C₂²×C₄	C₂²⋊Dic₁₃	1	2	yes	—
142	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
143	M₁₆	—	2	2	no	—
144	M₁₆	—	2	4	no	—
145	M₁₆	—	2	4	no	—
146	M₁₆	—	4	4	no	—
147	M₁₆	—	4	2	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	4	4	no	—
151	C₂×C₈	—	4	2	no	—
152	C₂×C₈	C₂²⋊Dic₁₃	1	2	yes	—
153	C₂×C₈	#6127	1	2	yes	—
154	C₂×C₈	#6127	1	2	yes	—
155	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
156	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
157	C₄⋊C₄	—	52	2	no	—
158	C₂²⋊C₄	—	52	2	no	—
159	C₄²	—	52	2	no	—
160	C₁₆	—	52	1	no	—
161	C₁₆	—	52	1	no	—
162	C₁₃	#12877	1	1	yes	characteristic, complemented, Sylow
163	C₂³	#88075	1	2	yes	—
164	C₂³	C₂×#6106	1	1	yes	characteristic
165	Q₈	—	2	2	no	—
166	D₄	—	2	4	no	—
167	D₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	104	2	no	—
171	C₂×C₄	#88075	1	2	yes	—
172	C₂×C₄	#88183	1	2	yes	—
173	C₂×C₄	#88186	1	2	yes	—
174	C₂×C₄	C₂×#6107	1	1	yes	characteristic
175	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
176	C₈	—	2	2	no	—
177	C₈	—	4	4	no	—
178	C₈	#88180	1	2	yes	—
179	C₈	#88183	1	1	yes	characteristic
180	C₈	#88183	1	1	yes	characteristic
181	C₂²	—	2	2	no	—
182	C₂²	—	4	1	no	—
183	C₂²	—	4	2	no	—
184	C₂²	#11718629	1	2	yes	—
185	C₂²	C₂×#88075	1	1	yes	characteristic
186	C₄	—	2	2	no	—
187	C₄	—	208	2	no	—
188	C₄	#11718631	1	2	yes	—
189	C₄	#11718872	1	1	yes	characteristic
190	C₄	#11718876	1	1	yes	characteristic
191	C₂	—	2	2	no	—
192	C₂	—	2	1	no	—
193	C₂	—	4	2	no	—
194	C₂	#13114926	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13116450

C₂

yes

maximal, characteristic

#13114882

C₂

yes

maximal, characteristic

#13112950

C₂

yes

maximal, complemented

#13112933

C₂

yes

maximal, complemented

C₁₃×#2916

C₂

yes

maximal, characteristic

C₁₃×#568

—

C₂×C₁₃×#131

C₂²

yes

characteristic

C₁₃×#430

—

C₁₃×#430

C₄

yes

complemented

C₂×C₁₃×#108

—

C₂×C₁₃×#108

C₂²

yes

—

#11718901

—

#11718767

C₂²

yes

—

C₂×#88052

C₂²

yes

characteristic

C₂²×#6090

C₂²

yes

characteristic

#11718149

—

C₂²⋊#6090

—

C₂×C₁₃×#41

D₄

yes

characteristic

C₁₃×#131

—

C₁₃×#131

—

C₁₃×#131

—

C₂×C₁₃×M₁₆

—

C₂×C₁₃×M₁₆

—

C₂×C₁₃×M₁₆

C₂×C₄

yes

—

C₂×C₁₃×M₁₆

D₄

yes

characteristic

C₂²×C₈×C₁₃

D₄

yes

—

C₂²×C₈×C₁₃

C₂³

yes

characteristic

#88180

—

C₁₃×#108

—

C₁₃×#108

—

C₁₃×#108

C₂×C₄

yes

—

—

—

—

—

—

—

—

—

C₁₃×#41

—

C₁₃×#41

—

C₂×Q₈×C₁₃

Q₁₆

yes

characteristic

C₂×D₄×C₁₃

SD₁₆

yes

—

C₂×D₄×C₁₃

Q₁₆

yes

characteristic

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₁₃×M₁₆

—

C₁₃×M₁₆

—

C₁₃×M₁₆

—

C₁₃×M₁₆

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₂²⋊C₄

yes

—

C₂×C₈×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃

#41

yes

—

C₂×C₈×C₁₃

#41

yes

—

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄×Dic₁₃

—

#6090

—

#6090

—

#2916

D₁₃

yes

—

—

—

—

C₂³×C₁₃

#103

yes

—

C₂³×C₁₃

C₂×SD₁₆

yes

characteristic

Q₈×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#103

yes

—

C₂×C₄×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

#123

yes

—

C₂×C₄×C₁₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₁₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

#117

yes

—

C₈×C₁₃

yes

C₈×C₁₃

yes

—

C₂×D₁₃

yes

characteristic

C₂×M₃₂

—

C₂²×C₁₆

—

#430

—

#430

Dic₁₃

yes

complemented

C₂×#108

—

C₂×#108

C₂×D₁₃

yes

—

#427

—

#410

—

C₂²⋊C₁₆

—

100

H(3,ℤ/4)

—

101

C₂²×C₁₃

—

102

C₂²×C₁₃

—

103

C₂²×C₁₃

—

104

C₂²×C₁₃

C₂×#103

yes

characteristic

105

C₂²×C₁₃

#481

yes

—

106

C₄×C₁₃

—

107

C₄×C₁₃

#381

yes

characteristic

108

C₄×C₁₃

#418

yes

characteristic

109

C₄×C₁₃

#484

yes

—

110

Dic₁₃

—

111

C₂×#41

C₂²⋊D₁₃

yes

characteristic

112

#131

—

113

#131

—

114

#131

—

115

C₂×M₁₆

—

116

C₂×M₁₆

—

117

C₂×M₁₆

C₂×Dic₁₃

yes

—

118

C₂×M₁₆

C₂²⋊D₁₃

yes

characteristic

119

C₂²×C₈

C₂²⋊D₁₃

yes

—

120

C₂²×C₈

C₂²×D₁₃

yes

characteristic

121

#117

—

122

M₃₂

—

123

C₂×C₁₆

—

124

C₂×C₁₆

—

125

C₂×C₁₆

—

126

#108

—

127

#108

—

128

#108

C₂×Dic₁₃

yes

—

129

#107

—

130

#106

—

131

#99

—

132

C₂×C₁₃

—

133

C₂×C₁₃

—

134

C₂×C₁₃

—

135

C₂×C₁₃

#1824

yes

characteristic

136

#41

—

137

#41

—

138

C₂×Q₈

#6107

yes

characteristic

139

C₂×D₄

#6106

yes

—

140

C₂×D₄

#6107

yes

characteristic

141

C₂²×C₄

C₂²⋊Dic₁₃

yes

—

142

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

143

M₁₆

—

144

M₁₆

—

145

M₁₆

—

146

M₁₆

—

147

M₁₆

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

C₂²⋊Dic₁₃

yes

—

153

C₂×C₈

#6127

yes

—

154

C₂×C₈

#6127

yes

—

155

C₂×C₈

C₂²×Dic₁₃

yes

characteristic

156

C₂×C₈

C₂×C₂²⋊D₁₃

yes

characteristic

157

C₄⋊C₄

—

158

C₂²⋊C₄

—

159

C₄²

—

160

C₁₆

—

161

C₁₆

—

162

C₁₃

#12877

yes

characteristic, complemented, Sylow

163

C₂³

#88075

yes

—

164

C₂³

C₂×#6106

yes

characteristic

165

Q₈

—

166

D₄

—

167

D₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

104

—

171

C₂×C₄

#88075

yes

—

172

C₂×C₄

#88183

yes

—

173

C₂×C₄

#88186

yes

—

174

C₂×C₄

C₂×#6107

yes

characteristic

175

C₂×C₄

C₂×C₂²⋊Dic₁₃

yes

characteristic

176

C₈

—

177

C₈

—

178

C₈

#88180

yes

—

179

C₈

#88183

yes

characteristic

180

C₈

#88183

yes

characteristic

181

C₂²

—

182

C₂²

—

183

C₂²

—

184

C₂²

#11718629

yes

—

185

C₂²

C₂×#88075

yes

characteristic

186

C₄

—

187

C₄

—

208

—

188

C₄

#11718631

yes

—

189

C₄

#11718872

yes

characteristic

190

C₄

#11718876

yes

characteristic

191

C₂

—

192

C₂

—

193

C₂

—

194

C₂

#13114926

yes

characteristic

	subgroup	G/H
center	C₄	#11718876
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₁₃×#2916	C₂
socle	C₂×C₁₃	#1824
prime radical	C₂×C₁₃×#41	D₄
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#2916	D₁₃
hypocenter	C₁₃	#12877
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #2916

subgroup

G/H

center

C₄

#11718876

commutator

C₂×C₈×C₁₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₁₃

Fitting

C₁₃×#2916

C₂

socle

C₂×C₁₃

#1824

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 ◁ C₂²×C₄ ◁ C₂×#131 ◁ #2916

N	H ≅ G/N	type
3. #13112950	C₂	semidirect
4. #13112933	C₂	semidirect
9. C₁₃×#430	C₄	semidirect
94. #430	Dic₁₃	semidirect
162. C₁₃	#12877	semidirect

H ≅ G/N

type

3. #13112950

C₂

semidirect

4. #13112933

C₂

semidirect

9. C₁₃×#430

C₄

semidirect

94. #430

Dic₁₃

semidirect

162. C₁₃

#12877

semidirect

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

subgroup	index
#13116450	2
#13114882	2
#13112950	2
#13112933	2
C₁₃×#2916	2
#12877	13

subgroup

index

C₁₃×#2916

Subgroups — #15779257

Conjugacy classes of subgroups

Complements

Maximal subgroups