Group #15763049 — subgroups

Subgroups — #15763049

Order 3328.

430 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 167 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	#13114953	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic
3	#13112719	C₂	1	4	yes	maximal
4	C₁₃×#2859	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×#125	—	2	1	no	—
6	C₁₃×#389	—	2	2	no	—
7	C₁₃×#389	C₄	1	4	yes	—
8	C₁₃×#384	C₂²	1	2	yes	—
9	C₁₃×#382	—	2	1	no	—
10	C₄³×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊#1055	C₂²	1	2	yes	—
12	C₂×C₄⋊#1055	C₂²	1	2	yes	—
13	C₂×C₄×#1055	—	2	1	no	—
14	#11718116	—	2	1	no	—
15	#11718116	—	2	2	no	—
16	C₁₃×#125	—	4	2	no	—
17	C₁₃×#125	—	4	2	no	—
18	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
19	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
20	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
21	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
22	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₃	D₄	1	2	yes	—
25	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
27	C₂²×#1055	—	2	2	no	—
28	C₂²×#1055	—	2	2	no	—
29	C₁₃×#95	—	2	4	no	—
30	C₄⋊#1055	—	2	2	no	—
31	C₄⋊#1055	—	2	4	no	—
32	C₄⋊#1055	Q₈	1	4	yes	—
33	C₄×#1055	—	4	2	no	—
34	#9899	—	13	1	no	maximal, Sylow
35	C₂²×C₄×C₁₃	—	2	4	no	—
36	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
37	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
38	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
41	C₁₃×C₄⋊C₄	—	4	4	no	—
42	C₁₃×C₄⋊C₄	—	4	8	no	—
43	C₁₃×C₄⋊C₄	—	4	2	no	—
44	C₁₃×C₄⋊C₄	—	4	2	no	—
45	C₁₃×C₄⋊C₄	—	4	4	no	—
46	C₄²×C₁₃	—	2	4	no	—
47	C₄²×C₁₃	—	2	2	no	—
48	C₄²×C₁₃	—	4	2	no	—
49	C₄²×C₁₃	D₈	1	2	yes	—
50	C₄²×C₁₃	SD₁₆	1	2	yes	—
51	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
52	C₄²×C₁₃	#41	1	2	yes	—
53	C₂×#1055	—	2	4	no	—
54	C₂×#1055	—	4	2	no	—
55	C₂×#1055	—	4	2	no	—
56	C₂×#1055	—	4	2	no	—
57	C₂×#1055	—	4	4	no	—
58	#2859	D₁₃	1	1	yes	characteristic
59	#1831	—	13	1	no	—
60	C₄×C₄⋊C₈	—	13	1	no	—
61	#1296	—	13	4	no	—
62	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	—	2	2	no	—
68	C₂×C₄×C₁₃	—	2	2	no	—
69	C₂×C₄×C₁₃	—	4	4	no	—
70	C₂×C₄×C₁₃	—	4	4	no	—
71	C₂×C₄×C₁₃	—	4	4	no	—
72	C₂×C₄×C₁₃	#102	1	4	yes	—
73	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
74	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
75	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
76	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₁₃	#135	1	4	yes	—
80	C₂×C₄×C₁₃	#137	1	2	yes	—
81	#1055	—	4	4	no	—
82	#1055	—	8	2	no	—
83	C₂×#125	—	2	1	no	—
84	C₂×C₄⋊C₈	—	13	2	no	—
85	C₂×C₄⋊C₈	—	13	2	no	—
86	C₂×C₄×C₈	—	26	1	no	—
87	#389	—	2	2	no	—
88	#389	Dic₁₃	1	4	yes	—
89	#384	C₂×D₁₃	1	2	yes	—
90	#382	—	2	1	no	—
91	C₄³	C₂×D₁₃	1	1	yes	characteristic
92	#336	—	26	2	no	—
93	#336	—	26	1	no	—
94	C₂²×C₁₃	#391	1	1	yes	characteristic
95	C₂²×C₁₃	#413	1	2	yes	—
96	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
97	C₂²×C₁₃	#416	1	2	yes	—
98	C₂²×C₁₃	#419	1	1	yes	characteristic
99	C₄×C₁₃	—	2	4	no	—
100	C₄×C₁₃	—	2	2	no	—
101	C₄×C₁₃	—	4	2	no	—
102	C₄×C₁₃	—	8	4	no	—
103	C₄×C₁₃	#474	1	2	yes	—
104	C₄×C₁₃	#475	1	2	yes	—
105	C₄×C₁₃	#488	1	2	yes	—
106	C₄×C₁₃	#499	1	2	yes	—
107	C₂²×C₈	—	26	2	no	—
108	C₂²×C₈	—	26	2	no	—
109	#125	—	4	2	no	—
110	#125	—	4	2	no	—
111	C₂×C₄⋊C₄	—	2	4	no	—
112	C₂×C₄⋊C₄	—	2	2	no	—
113	C₂×C₄⋊C₄	—	2	2	no	—
114	C₂×C₄⋊C₄	C₂×Dic₁₃	1	4	yes	—
115	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
116	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
117	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
118	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
119	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
120	C₄⋊C₈	—	13	4	no	—
121	C₄⋊C₈	—	26	2	no	—
122	C₄⋊C₈	—	26	4	no	—
123	C₄×C₈	—	52	2	no	—
124	#95	—	2	4	no	—
125	C₂×C₁₃	#1540	1	2	yes	—
126	C₂×C₁₃	#1554	1	2	yes	—
127	C₂×C₁₃	#1988	1	1	yes	characteristic
128	C₂×C₁₃	#1991	1	1	yes	characteristic
129	C₂×C₁₃	#1998	1	1	yes	characteristic
130	C₂²×C₄	—	2	4	no	—
131	C₂²×C₄	C₂²⋊Dic₁₃	1	2	yes	—
132	C₂²×C₄	C₂²⋊Dic₁₃	1	2	yes	—
133	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
134	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
135	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
136	C₂×C₈	—	26	4	no	—
137	C₂×C₈	—	52	2	no	—
138	C₂×C₈	—	52	2	no	—
139	C₂×C₈	—	52	2	no	—
140	C₂×C₈	—	52	4	no	—
141	C₄⋊C₄	—	4	4	no	—
142	C₄⋊C₄	—	4	8	no	—
143	C₄⋊C₄	—	4	2	no	—
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	—	4	4	no	—
146	C₄²	—	2	2	no	—
147	C₄²	—	2	4	no	—
148	C₄²	—	4	2	no	—
149	C₄²	#6104	1	2	yes	—
150	C₄²	#6105	1	2	yes	—
151	C₄²	Q₈×D₁₃	1	2	yes	—
152	C₄²	#6131	1	2	yes	—
153	C₁₃	#9899	1	1	yes	characteristic, complemented, Sylow
154	C₂³	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	—	4	4	no	—
163	C₂×C₄	—	4	4	no	—
164	C₂×C₄	#88072	1	4	yes	—
165	C₂×C₄	C₂×#6104	1	1	yes	characteristic
166	C₂×C₄	C₂×#6105	1	1	yes	characteristic
167	C₂×C₄	#88196	1	2	yes	—
168	C₂×C₄	#88198	1	1	yes	characteristic
169	C₂×C₄	Q₈×Dic₁₃	1	2	yes	—
170	C₂×C₄	#88200	1	2	yes	—
171	C₂×C₄	#88201	1	1	yes	characteristic
172	C₂×C₄	#88204	1	4	yes	—
173	C₈	—	52	4	no	—
174	C₈	—	104	2	no	—
175	C₂²	C₂×#88072	1	1	yes	characteristic
176	C₂²	#11718886	1	1	yes	characteristic
177	C₂²	#11718891	1	1	yes	characteristic
178	C₂²	#11718894	1	2	yes	—
179	C₂²	#11718898	1	2	yes	—
180	C₄	—	2	4	no	—
181	C₄	—	2	2	no	—
182	C₄	—	4	2	no	—
183	C₄	—	8	4	no	—
184	C₄	#11718725	1	2	yes	—
185	C₄	#11718729	1	2	yes	—
186	C₄	#11718748	1	2	yes	—
187	C₄	#11718755	1	2	yes	—
188	C₂	#13113485	1	2	yes	—
189	C₂	#13113546	1	2	yes	—
190	C₂	#13115789	1	1	yes	characteristic
191	C₂	#13115802	1	1	yes	characteristic
192	C₂	#13115830	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13114953

C₂

yes

maximal, characteristic

C₄×C₄⋊#1055

C₂

yes

maximal, characteristic

#13112719

C₂

yes

maximal

C₁₃×#2859

C₂

yes

maximal, characteristic

C₂×C₁₃×#125

—

C₁₃×#389

—

C₁₃×#389

C₄

yes

—

C₁₃×#384

C₂²

yes

—

C₁₃×#382

—

C₄³×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊#1055

C₂²

yes

—

C₂×C₄⋊#1055

C₂²

yes

—

C₂×C₄×#1055

—

#11718116

—

#11718116

—

C₁₃×#125

—

C₁₃×#125

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄²×C₁₃

C₂×C₄

yes

—

C₂×C₄²×C₁₃

D₄

yes

characteristic

C₂×C₄²×C₁₃

D₄

yes

—

C₂×C₄²×C₁₃

D₄

yes

characteristic

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂²×#1055

—

C₂²×#1055

—

C₁₃×#95

—

C₄⋊#1055

—

C₄⋊#1055

—

C₄⋊#1055

Q₈

yes

—

C₄×#1055

—

#9899

—

maximal, Sylow

C₂²×C₄×C₁₃

—

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

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

D₈

yes

—

C₄²×C₁₃

SD₁₆

yes

—

C₄²×C₁₃

C₂×Q₈

yes

—

C₄²×C₁₃

#41

yes

—

—

—

—

—

—

D₁₃

yes

characteristic

#1831

—

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

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#102

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×D₈

yes

characteristic

C₂×C₄×C₁₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₁₃

#135

yes

—

C₂×C₄×C₁₃

#137

yes

—

#1055

—

#1055

—

C₂×#125

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

#389

—

#389

Dic₁₃

yes

—

#384

C₂×D₁₃

yes

—

#382

—

C₄³

C₂×D₁₃

yes

characteristic

#336

—

#336

—

C₂²×C₁₃

#391

yes

characteristic

C₂²×C₁₃

#413

yes

—

C₂²×C₁₃

C₂×#102

yes

characteristic

C₂²×C₁₃

#416

yes

—

C₂²×C₁₃

#419

yes

characteristic

C₄×C₁₃

—

100

C₄×C₁₃

—

101

C₄×C₁₃

—

102

C₄×C₁₃

—

103

C₄×C₁₃

#474

yes

—

104

C₄×C₁₃

#475

yes

—

105

C₄×C₁₃

#488

yes

—

106

C₄×C₁₃

#499

yes

—

107

C₂²×C₈

—

108

C₂²×C₈

—

109

#125

—

110

#125

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

—

114

C₂×C₄⋊C₄

C₂×Dic₁₃

yes

—

115

C₂×C₄²

C₂×Dic₁₃

yes

—

116

C₂×C₄²

C₂²⋊D₁₃

yes

characteristic

117

C₂×C₄²

C₂²⋊D₁₃

yes

—

118

C₂×C₄²

C₂²⋊D₁₃

yes

characteristic

119

C₂×C₄²

C₂²×D₁₃

yes

characteristic

120

C₄⋊C₈

—

121

C₄⋊C₈

—

122

C₄⋊C₈

—

123

C₄×C₈

—

124

#95

—

125

C₂×C₁₃

#1540

yes

—

126

C₂×C₁₃

#1554

yes

—

127

C₂×C₁₃

#1988

yes

characteristic

128

C₂×C₁₃

#1991

yes

characteristic

129

C₂×C₁₃

#1998

yes

characteristic

130

C₂²×C₄

—

131

C₂²×C₄

C₂²⋊Dic₁₃

yes

—

132

C₂²×C₄

C₂²⋊Dic₁₃

yes

—

133

C₂²×C₄

C₂²×Dic₁₃

yes

characteristic

134

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

135

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄²

—

147

C₄²

—

148

C₄²

—

149

C₄²

#6104

yes

—

150

C₄²

#6105

yes

—

151

C₄²

Q₈×D₁₃

yes

—

152

C₄²

#6131

yes

—

153

C₁₃

#9899

yes

characteristic, complemented, Sylow

154

C₂³

C₂×C₂²⋊Dic₁₃

yes

characteristic

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

#88072

yes

—

165

C₂×C₄

C₂×#6104

yes

characteristic

166

C₂×C₄

C₂×#6105

yes

characteristic

167

C₂×C₄

#88196

yes

—

168

C₂×C₄

#88198

yes

characteristic

169

C₂×C₄

Q₈×Dic₁₃

yes

—

170

C₂×C₄

#88200

yes

—

171

C₂×C₄

#88201

yes

characteristic

172

C₂×C₄

#88204

yes

—

173

C₈

—

174

C₈

—

104

—

175

C₂²

C₂×#88072

yes

characteristic

176

C₂²

#11718886

yes

characteristic

177

C₂²

#11718891

yes

characteristic

178

C₂²

#11718894

yes

—

179

C₂²

#11718898

yes

—

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

#11718725

yes

—

185

C₄

#11718729

yes

—

186

C₄

#11718748

yes

—

187

C₄

#11718755

yes

—

188

C₂

#13113485

yes

—

189

C₂

#13113546

yes

—

190

C₂

#13115789

yes

characteristic

191

C₂

#13115802

yes

characteristic

192

C₂

#13115830

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₃
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#2859	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#2859	D₁₃
hypocenter	C₁₃	#9899
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2859

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₁₃

commutator

C₂²×C₄×C₁₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#2859

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 ◁ #2859

N	H ≅ G/N	type
153. C₁₃	#9899	semidirect

H ≅ G/N

type

153. C₁₃

#9899

semidirect

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

subgroup	index
#13114953	2
C₄×C₄⋊#1055	2
#13112719	2
C₁₃×#2859	2
#9899	13

subgroup

index

#13114953

C₄×C₄⋊#1055

#13112719

C₁₃×#2859

#9899

Subgroups — #15763049

Conjugacy classes of subgroups

Complements

Maximal subgroups