Group #15763039 — subgroups

Subgroups — #15763039

Order 3328.

430 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 159 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	#13123199	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic
3	#13112720	C₂	1	4	yes	maximal
4	C₁₃×#1831	C₂	1	1	yes	maximal, characteristic
5	C₂×#88124	—	2	1	no	—
6	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
7	C₄³×C₁₃	C₂²	1	1	yes	characteristic
8	#11718597	—	2	1	no	—
9	#11718596	C₂²	1	2	yes	—
10	#11718589	—	2	2	no	—
11	#11718589	C₄	1	4	yes	—
12	C₂×C₄⋊#1055	C₂²	1	2	yes	—
13	C₂×C₄×#1055	—	2	1	no	—
14	C₁₃×#336	—	2	2	no	—
15	#11718116	—	2	1	no	—
16	C₂²×C₈×C₁₃	—	2	2	no	—
17	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
19	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
20	C₂×C₄²×C₁₃	D₄	1	2	yes	—
21	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
22	C₂×C₄⋊Dic₁₃	—	2	2	no	—
23	C₂×C₄⋊Dic₁₃	—	2	2	no	—
24	C₂×C₄⋊Dic₁₃	C₂×C₄	1	4	yes	—
25	C₂×#6101	—	2	4	no	—
26	C₂²×#1055	—	2	2	no	—
27	#88124	—	4	4	no	—
28	C₁₃×C₄⋊C₈	—	2	4	no	—
29	#88071	—	2	4	no	—
30	C₄⋊#1055	—	2	2	no	—
31	C₄⋊#1055	Q₈	1	4	yes	—
32	C₄×#1055	—	4	2	no	—
33	#9897	—	13	1	no	maximal, Sylow
34	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
35	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
36	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
39	C₂²×Dic₁₃	—	2	4	no	—
40	C₂×C₈×C₁₃	—	4	2	no	—
41	C₂×C₈×C₁₃	—	4	4	no	—
42	C₄²×C₁₃	—	2	4	no	—
43	C₄²×C₁₃	—	2	2	no	—
44	C₄²×C₁₃	—	4	2	no	—
45	C₄²×C₁₃	SD₁₆	1	2	yes	—
46	C₄²×C₁₃	Q₁₆	1	2	yes	—
47	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
48	C₄²×C₁₃	#41	1	2	yes	—
49	C₄⋊Dic₁₃	—	4	4	no	—
50	C₄⋊Dic₁₃	—	4	4	no	—
51	C₄⋊Dic₁₃	—	4	4	no	—
52	#6101	—	4	8	no	—
53	C₂×#1055	—	2	4	no	—
54	C₂×#1055	—	4	2	no	—
55	C₂×#1055	—	4	2	no	—
56	#2859	—	13	1	no	—
57	#1831	D₁₃	1	1	yes	characteristic
58	C₄×C₄⋊C₈	—	13	1	no	—
59	#1296	—	13	4	no	—
60	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₃	—	2	2	no	—
62	C₂×C₄×C₁₃	—	2	2	no	—
63	C₂×C₄×C₁₃	—	2	2	no	—
64	C₂×C₄×C₁₃	—	2	2	no	—
65	C₂×C₄×C₁₃	—	2	2	no	—
66	C₂×C₄×C₁₃	—	2	2	no	—
67	C₂×C₄×C₁₃	#103	1	4	yes	—
68	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
69	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
70	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
71	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
73	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
74	C₂×C₄×C₁₃	#135	1	4	yes	—
75	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
76	C₂×Dic₁₃	—	4	4	no	—
77	C₂×Dic₁₃	—	4	4	no	—
78	C₂×Dic₁₃	—	4	4	no	—
79	C₈×C₁₃	—	8	2	no	—
80	#1055	—	4	4	no	—
81	C₂×#125	—	26	1	no	—
82	C₂×C₄⋊C₈	—	13	2	no	—
83	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
84	C₂×C₄×C₈	—	26	1	no	—
85	#389	—	13	4	no	—
86	#389	—	26	2	no	—
87	#384	—	13	2	no	—
88	#382	—	26	1	no	—
89	C₄³	C₂×D₁₃	1	1	yes	characteristic
90	#336	—	2	2	no	—
91	#336	—	26	1	no	—
92	C₂²×C₁₃	#391	1	1	yes	characteristic
93	C₂²×C₁₃	#413	1	2	yes	—
94	C₂²×C₁₃	C₂×#103	1	1	yes	characteristic
95	C₂²×C₁₃	#416	1	2	yes	—
96	C₂²×C₁₃	#418	1	1	yes	characteristic
97	C₄×C₁₃	—	2	4	no	—
98	C₄×C₁₃	—	2	2	no	—
99	C₄×C₁₃	—	4	2	no	—
100	C₄×C₁₃	#476	1	2	yes	—
101	C₄×C₁₃	#477	1	2	yes	—
102	C₄×C₁₃	#488	1	2	yes	—
103	C₄×C₁₃	#499	1	2	yes	—
104	Dic₁₃	—	8	4	no	—
105	C₂²×C₈	—	2	2	no	—
106	C₂²×C₈	—	26	2	no	—
107	#125	—	52	4	no	—
108	C₂×C₄⋊C₄	—	13	4	no	—
109	C₂×C₄⋊C₄	—	26	4	no	—
110	C₂×C₄⋊C₄	—	26	2	no	—
111	C₂×C₄⋊C₄	—	26	2	no	—
112	C₂×C₄²	C₄×D₁₃	1	2	yes	—
113	C₂×C₄²	D₅₂	1	1	yes	characteristic
114	C₂×C₄²	D₅₂	1	1	yes	characteristic
115	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
116	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
117	C₄⋊C₈	—	2	4	no	—
118	C₄⋊C₈	—	13	4	no	—
119	C₄⋊C₈	—	26	2	no	—
120	C₄×C₈	—	52	2	no	—
121	#95	—	26	4	no	—
122	C₂×C₁₃	#1542	1	2	yes	—
123	C₂×C₁₃	#1553	1	2	yes	—
124	C₂×C₁₃	#1987	1	1	yes	characteristic
125	C₂×C₁₃	#1991	1	1	yes	characteristic
126	C₂×C₁₃	#1998	1	1	yes	characteristic
127	C₂²×C₄	—	26	4	no	—
128	C₂²×C₄	#6103	1	2	yes	—
129	C₂²×C₄	#6103	1	2	yes	—
130	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
131	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
132	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
133	C₂×C₈	—	4	2	no	—
134	C₂×C₈	—	4	4	no	—
135	C₂×C₈	—	26	4	no	—
136	C₂×C₈	—	52	2	no	—
137	C₂×C₈	—	52	2	no	—
138	C₄⋊C₄	—	52	4	no	—
139	C₄⋊C₄	—	52	8	no	—
140	C₄⋊C₄	—	52	4	no	—
141	C₄⋊C₄	—	52	4	no	—
142	C₄²	—	2	2	no	—
143	C₄²	—	2	4	no	—
144	C₄²	—	4	2	no	—
145	C₄²	#6095	1	2	yes	—
146	C₄²	Dic₅₂	1	2	yes	—
147	C₄²	#6129	1	2	yes	—
148	C₄²	Q₈×D₁₃	1	2	yes	—
149	C₁₃	#9897	1	1	yes	characteristic, complemented, Sylow
150	C₂³	C₂×#6103	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	52	4	no	—
158	C₂×C₄	—	52	4	no	—
159	C₂×C₄	—	52	4	no	—
160	C₂×C₄	#88056	1	4	yes	—
161	C₂×C₄	#88141	1	2	yes	—
162	C₂×C₄	#88151	1	2	yes	—
163	C₂×C₄	C₂×#6095	1	1	yes	characteristic
164	C₂×C₄	C₂×Dic₅₂	1	1	yes	characteristic
165	C₂×C₄	#88162	1	2	yes	—
166	C₂×C₄	#88190	1	1	yes	characteristic
167	C₂×C₄	#88198	1	1	yes	characteristic
168	C₂×C₄	#88204	1	4	yes	—
169	C₈	—	8	2	no	—
170	C₈	—	52	4	no	—
171	C₂²	#11718635	1	1	yes	characteristic
172	C₂²	#11718663	1	2	yes	—
173	C₂²	C₂×#88056	1	1	yes	characteristic
174	C₂²	#11718792	1	1	yes	characteristic
175	C₂²	#11718793	1	2	yes	—
176	C₄	—	2	4	no	—
177	C₄	—	2	2	no	—
178	C₄	—	4	2	no	—
179	C₄	—	104	4	no	—
180	C₄	#11718503	1	2	yes	—
181	C₄	#11718511	1	2	yes	—
182	C₄	#11718717	1	2	yes	—
183	C₄	#11718725	1	2	yes	—
184	C₂	#13113491	1	2	yes	—
185	C₂	#13113542	1	2	yes	—
186	C₂	#13115783	1	1	yes	characteristic
187	C₂	#13115800	1	1	yes	characteristic
188	C₂	#13115828	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13123199

C₂

yes

maximal, characteristic

C₄×C₄⋊#1055

C₂

yes

maximal, characteristic

#13112720

C₂

yes

maximal

C₁₃×#1831

C₂

yes

maximal, characteristic

C₂×#88124

—

C₂×C₁₃×C₄⋊C₈

C₂²

yes

—

C₄³×C₁₃

C₂²

yes

characteristic

#11718597

—

#11718596

C₂²

yes

—

#11718589

—

#11718589

C₄

yes

—

C₂×C₄⋊#1055

C₂²

yes

—

C₂×C₄×#1055

—

C₁₃×#336

—

#11718116

—

C₂²×C₈×C₁₃

—

C₂×C₄²×C₁₃

C₂×C₄

yes

—

C₂×C₄²×C₁₃

D₄

yes

characteristic

C₂×C₄²×C₁₃

D₄

yes

characteristic

C₂×C₄²×C₁₃

D₄

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

C₂×C₄

yes

—

C₂×#6101

—

C₂²×#1055

—

#88124

—

C₁₃×C₄⋊C₈

—

#88071

—

C₄⋊#1055

—

C₄⋊#1055

Q₈

yes

—

C₄×#1055

—

#9897

—

maximal, Sylow

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×Dic₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

SD₁₆

yes

—

C₄²×C₁₃

Q₁₆

yes

—

C₄²×C₁₃

C₂×Q₈

yes

—

C₄²×C₁₃

#41

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

—

—

—

—

—

D₁₃

yes

characteristic

C₄×C₄⋊C₈

—

#1296

—

C₂³×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#103

yes

—

C₂×C₄×C₁₃

C₄×Q₈

yes

—

C₂×C₄×C₁₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₃

#124

yes

characteristic

C₂×C₄×C₁₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₁₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₁₃

#135

yes

—

C₂×C₄×C₁₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₈×C₁₃

—

#1055

—

C₂×#125

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂×D₁₃

yes

—

C₂×C₄×C₈

—

—

—

—

—

C₄³

C₂×D₁₃

yes

characteristic

#336

—

#336

—

C₂²×C₁₃

#391

yes

characteristic

C₂²×C₁₃

#413

yes

—

C₂²×C₁₃

C₂×#103

yes

characteristic

C₂²×C₁₃

#416

yes

—

C₂²×C₁₃

#418

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

100

C₄×C₁₃

#476

yes

—

101

C₄×C₁₃

#477

yes

—

102

C₄×C₁₃

#488

yes

—

103

C₄×C₁₃

#499

yes

—

104

Dic₁₃

—

105

C₂²×C₈

—

106

C₂²×C₈

—

107

#125

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄²

C₄×D₁₃

yes

—

113

C₂×C₄²

D₅₂

yes

characteristic

114

C₂×C₄²

D₅₂

yes

characteristic

115

C₂×C₄²

C₂²⋊D₁₃

yes

—

116

C₂×C₄²

C₂²×D₁₃

yes

characteristic

117

C₄⋊C₈

—

118

C₄⋊C₈

—

119

C₄⋊C₈

—

120

C₄×C₈

—

121

#95

—

122

C₂×C₁₃

#1542

yes

—

123

C₂×C₁₃

#1553

yes

—

124

C₂×C₁₃

#1987

yes

characteristic

125

C₂×C₁₃

#1991

yes

characteristic

126

C₂×C₁₃

#1998

yes

characteristic

127

C₂²×C₄

—

128

C₂²×C₄

#6103

yes

—

129

C₂²×C₄

#6103

yes

—

130

C₂²×C₄

C₂×C₄×D₁₃

yes

characteristic

131

C₂²×C₄

C₂×D₅₂

yes

characteristic

132

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄²

—

143

C₄²

—

144

C₄²

—

145

C₄²

#6095

yes

—

146

C₄²

Dic₅₂

yes

—

147

C₄²

#6129

yes

—

148

C₄²

Q₈×D₁₃

yes

—

149

C₁₃

#9897

yes

characteristic, complemented, Sylow

150

C₂³

C₂×#6103

yes

characteristic

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

#88056

yes

—

161

C₂×C₄

#88141

yes

—

162

C₂×C₄

#88151

yes

—

163

C₂×C₄

C₂×#6095

yes

characteristic

164

C₂×C₄

C₂×Dic₅₂

yes

characteristic

165

C₂×C₄

#88162

yes

—

166

C₂×C₄

#88190

yes

characteristic

167

C₂×C₄

#88198

yes

characteristic

168

C₂×C₄

#88204

yes

—

169

C₈

—

170

C₈

—

171

C₂²

#11718635

yes

characteristic

172

C₂²

#11718663

yes

—

173

C₂²

C₂×#88056

yes

characteristic

174

C₂²

#11718792

yes

characteristic

175

C₂²

#11718793

yes

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

104

—

180

C₄

#11718503

yes

—

181

C₄

#11718511

yes

—

182

C₄

#11718717

yes

—

183

C₄

#11718725

yes

—

184

C₂

#13113491

yes

—

185

C₂

#13113542

yes

—

186

C₂

#13115783

yes

characteristic

187

C₂

#13115800

yes

characteristic

188

C₂

#13115828

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×#6103
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1831	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1831	D₁₃
hypocenter	C₁₃	#9897
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

subgroup

G/H

center

C₂³

C₂×#6103

commutator

C₂²×C₄×C₁₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1831

C₂

socle, prime radical

C₂³×C₁₃

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂²×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

N	H ≅ G/N	type
149. C₁₃	#9897	semidirect

H ≅ G/N

type

149. C₁₃

#9897

semidirect

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

subgroup	index
#13123199	2
C₄×C₄⋊#1055	2
#13112720	2
C₁₃×#1831	2
#9897	13

subgroup

index

#13123199

C₄×C₄⋊#1055

#13112720

C₁₃×#1831

#9897

Subgroups — #15763039

Conjugacy classes of subgroups

Complements

Maximal subgroups