Group #16777533 — subgroups

Subgroups — #16777533

Order 3328.

246 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 84 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	D₁₃⋊(C₄⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	#13130773	C₂	1	1	yes	maximal, characteristic
3	D₁₃×#334	C₂	1	1	yes	maximal, characteristic
4	C₂×#88102	C₄	1	2	yes	—
5	C₂×#88101	—	2	1	no	—
6	C₂×D₁₃⋊C₁₆	—	2	1	no	—
7	#11719060	—	2	1	no	—
8	D₁₃×C₄⋊C₈	—	2	1	no	—
9	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
10	#11718260	C₄	1	2	yes	complemented
11	C₄⋊#6092	—	2	1	no	—
12	C₁₃×#334	C₄	1	1	yes	characteristic, complemented
13	#11718072	—	2	1	no	—
14	#11718049	C₄	1	1	yes	characteristic, complemented
15	C₂×C₄⋊#261	—	2	2	no	—
16	C₂×C₈×D₁₃	—	4	1	no	—
17	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
19	C₄²×D₁₃	D₄	1	1	yes	characteristic
20	(C₂×#261)⋊C₄	—	4	1	no	—
21	(C₂×#261)⋊C₄	—	4	1	no	—
22	C₂×#6092	—	2	2	no	—
23	#88102	—	4	2	no	—
24	#88101	—	4	1	no	—
25	#88101	—	4	1	no	—
26	D₁₃⋊C₁₆	—	4	1	no	—
27	D₁₃⋊C₁₆	—	4	1	no	—
28	C₁₃×C₄⋊C₈	—	2	1	no	—
29	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
30	#88054	—	4	1	no	—
31	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
32	C₈×Dic₁₃	Q₈	1	1	yes	characteristic
33	C₄⋊#1055	—	2	1	no	—
34	C₄×#1055	D₄	1	1	yes	characteristic
35	#7275	—	13	1	no	maximal, Sylow
36	C₂²×#261	—	4	2	no	—
37	C₂×C₄×D₁₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₄×D₁₃	D₈	1	1	yes	characteristic
39	C₂×C₄×D₁₃	SD₁₆	1	1	yes	characteristic
40	C₄⋊#261	—	4	2	no	—
41	C₄⋊#261	—	4	2	no	—
42	C₂×C₈×C₁₃	—	4	1	no	—
43	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
44	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
45	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₄×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
47	C₄×Dic₁₃	SD₁₆	1	1	yes	characteristic
48	C₄×Dic₁₃	Q₁₆	1	1	yes	characteristic
49	C₂×#1055	—	4	1	no	—
50	C₂×#1055	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
52	C₈×D₁₃	—	2	1	no	—
53	C₈×D₁₃	—	2	1	no	—
54	C₈×D₁₃	—	8	2	no	—
55	C₈×D₁₃	D₈	1	1	yes	characteristic
56	C₈×D₁₃	SD₁₆	1	1	yes	characteristic
57	C₈×D₁₃	SD₁₆	1	1	yes	characteristic
58	C₈×D₁₃	Q₁₆	1	1	yes	characteristic
59	#6092	—	4	2	no	—
60	C₂×C₄⋊C₁₆	—	13	1	no	—
61	#1846	—	13	1	no	—
62	C₂×#334	—	13	1	no	—
63	C₂²×D₁₃	#102	1	1	yes	characteristic
64	C₂×#261	—	8	2	no	—
65	C₂×#261	—	8	2	no	—
66	C₂×#261	—	8	2	no	—
67	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
68	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
69	C₂×C₄×C₁₃	#103	1	1	yes	characteristic
70	C₂×Dic₁₃	#103	1	1	yes	characteristic
71	C₂×Dic₁₃	#106	1	1	yes	characteristic
72	C₂×Dic₁₃	#107	1	1	yes	characteristic
73	C₄×D₁₃	—	2	2	no	—
74	C₄×D₁₃	—	2	2	no	—
75	C₄×D₁₃	#100	1	1	yes	characteristic
76	C₄×D₁₃	#101	1	1	yes	characteristic
77	C₄×D₁₃	#102	1	1	yes	characteristic
78	C₄×D₁₃	#103	1	1	yes	characteristic
79	C₈×C₁₃	—	2	1	no	—
80	C₈×C₁₃	—	8	1	no	—
81	C₈×C₁₃	#106	1	1	yes	characteristic
82	C₈×C₁₃	#107	1	1	yes	characteristic
83	#1055	—	2	1	no	—
84	#1055	—	8	1	no	—
85	#1055	#102	1	1	yes	characteristic
86	#1055	#103	1	1	yes	characteristic
87	C₂²×C₁₆	—	26	1	no	—
88	C₂×#107	—	13	2	no	—
89	C₂×#106	—	26	1	no	—
90	C₂×C₄⋊C₈	—	26	1	no	—
91	C₂×C₄×C₈	—	13	1	no	—
92	#382	—	26	1	no	—
93	C₄⋊C₁₆	—	13	2	no	—
94	C₄⋊C₁₆	—	26	1	no	—
95	#334	—	13	1	no	—
96	#334	—	26	1	no	—
97	#334	#261	1	1	yes	characteristic, complemented
98	C₂²×C₁₃	#340	1	1	yes	characteristic
99	C₂×D₁₃	#331	1	1	yes	characteristic
100	C₂×D₁₃	#332	1	1	yes	characteristic
101	C₂×D₁₃	#359	1	2	yes	—
102	C₂×D₁₃	#360	1	2	yes	—
103	#261	—	16	2	no	—
104	C₄×C₁₃	—	2	1	no	—
105	C₄×C₁₃	—	2	1	no	—
106	C₄×C₁₃	#340	1	1	yes	characteristic
107	C₄×C₁₃	#343	1	1	yes	characteristic
108	Dic₁₃	—	2	1	no	—
109	Dic₁₃	—	2	1	no	—
110	Dic₁₃	#332	1	1	yes	characteristic
111	Dic₁₃	#333	1	1	yes	characteristic
112	C₂²×C₈	—	13	1	no	—
113	C₂²×C₈	—	13	1	no	—
114	C₂²×C₈	—	52	1	no	—
115	C₂×C₄⋊C₄	—	26	2	no	—
116	C₂×C₄²	—	13	1	no	—
117	C₂×C₁₆	—	26	2	no	—
118	C₂×C₁₆	—	52	1	no	—
119	C₂×C₁₆	—	52	1	no	—
120	#107	—	52	2	no	—
121	#106	—	52	1	no	—
122	#106	—	52	1	no	—
123	C₄⋊C₈	—	2	1	no	—
124	C₄⋊C₈	—	26	1	no	—
125	C₄⋊C₈	—	52	1	no	—
126	C₄×C₈	—	13	1	no	—
127	C₄×C₈	—	13	1	no	—
128	C₄×C₈	—	13	1	no	—
129	C₄×C₈	C₂×#261	1	1	yes	characteristic
130	#95	—	52	1	no	—
131	#95	—	52	1	no	—
132	C₂×C₁₃	#1296	1	1	yes	characteristic
133	C₂×C₁₃	#1382	1	1	yes	characteristic
134	C₂×C₁₃	#1386	1	1	yes	characteristic
135	D₁₃	#1360	1	2	yes	—
136	D₁₃	#1363	1	2	yes	—
137	C₂²×C₄	—	13	1	no	—
138	C₂²×C₄	—	13	1	no	—
139	C₂²×C₄	—	13	1	no	—
140	C₂²×C₄	—	52	2	no	—
141	C₂×C₈	—	4	1	no	—
142	C₂×C₈	—	13	1	no	—
143	C₂×C₈	—	13	1	no	—
144	C₂×C₈	—	13	1	no	—
145	C₂×C₈	—	13	1	no	—
146	C₂×C₈	—	13	1	no	—
147	C₂×C₈	—	13	1	no	—
148	C₂×C₈	—	26	1	no	—
149	C₂×C₈	—	26	1	no	—
150	C₂×C₈	—	52	1	no	—
151	C₂×C₈	—	104	2	no	—
152	C₂×C₈	C₄×#261	1	1	yes	characteristic
153	C₂×C₈	C₄⋊#261	1	1	yes	characteristic
154	C₄⋊C₄	—	52	2	no	—
155	C₄⋊C₄	—	52	2	no	—
156	C₄²	—	13	1	no	—
157	C₄²	—	13	1	no	—
158	C₄²	—	13	1	no	—
159	C₄²	C₂²⋊#261	1	1	yes	characteristic
160	C₁₆	—	52	2	no	—
161	C₁₃	#7275	1	1	yes	characteristic, complemented, Sylow
162	C₂³	—	13	1	no	—
163	C₂×C₄	—	13	1	no	—
164	C₂×C₄	—	13	1	no	—
165	C₂×C₄	—	13	1	no	—
166	C₂×C₄	—	13	1	no	—
167	C₂×C₄	—	13	1	no	—
168	C₂×C₄	—	13	1	no	—
169	C₂×C₄	—	13	1	no	—
170	C₂×C₄	—	26	2	no	—
171	C₂×C₄	—	26	2	no	—
172	C₂×C₄	—	104	2	no	—
173	C₂×C₄	—	104	2	no	—
174	C₂×C₄	—	104	2	no	—
175	C₂×C₄	(C₂×#261)⋊C₄	1	1	yes	characteristic
176	C₂×C₄	#88115	1	1	yes	characteristic
177	C₂×C₄	#88117	1	1	yes	characteristic
178	C₈	—	2	1	no	—
179	C₈	—	8	1	no	—
180	C₈	—	13	1	no	—
181	C₈	—	13	1	no	—
182	C₈	—	26	1	no	—
183	C₈	—	104	1	no	—
184	C₈	#88101	1	1	yes	characteristic
185	C₈	#88102	1	1	yes	characteristic
186	C₂²	—	13	2	no	—
187	C₂²	—	13	2	no	—
188	C₂²	—	13	1	no	—
189	C₂²	—	13	1	no	—
190	C₂²	#11718246	1	1	yes	characteristic
191	C₄	—	2	1	no	—
192	C₄	—	2	1	no	—
193	C₄	—	13	1	no	—
194	C₄	—	13	1	no	—
195	C₄	—	26	1	no	—
196	C₄	—	26	1	no	—
197	C₄	—	208	2	no	—
198	C₄	#11718265	1	1	yes	characteristic
199	C₄	#11718271	1	1	yes	characteristic
200	C₂	—	13	2	no	—
201	C₂	—	13	2	no	—
202	C₂	#13129715	1	1	yes	characteristic
203	C₂	#13129899	1	1	yes	characteristic
204	C₂	#13129909	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₁₃⋊(C₄⋊C₁₆)

C₂

yes

maximal, characteristic

#13130773

C₂

yes

maximal, characteristic

D₁₃×#334

C₂

yes

maximal, characteristic

C₂×#88102

C₄

yes

—

C₂×#88101

—

C₂×D₁₃⋊C₁₆

—

#11719060

—

D₁₃×C₄⋊C₈

—

C₄×C₈×D₁₃

C₂²

yes

characteristic

#11718260

C₄

yes

complemented

C₄⋊#6092

—

C₁₃×#334

C₄

yes

characteristic, complemented

#11718072

—

#11718049

C₄

yes

characteristic, complemented

C₂×C₄⋊#261

—

C₂×C₈×D₁₃

—

C₂×C₈×D₁₃

C₂×C₄

yes

characteristic

C₂×C₈×D₁₃

D₄

yes

characteristic

C₄²×D₁₃

D₄

yes

characteristic

(C₂×#261)⋊C₄

—

(C₂×#261)⋊C₄

—

—

—

—

—

D₁₃⋊C₁₆

—

D₁₃⋊C₁₆

—

C₁₃×C₄⋊C₈

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

#88054

—

C₈×Dic₁₃

C₂×C₄

yes

characteristic

C₈×Dic₁₃

Q₈

yes

characteristic

C₄⋊#1055

—

C₄×#1055

D₄

yes

characteristic

#7275

—

maximal, Sylow

C₂²×#261

—

C₂×C₄×D₁₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×D₁₃

D₈

yes

characteristic

C₂×C₄×D₁₃

SD₁₆

yes

characteristic

C₄⋊#261

—

C₄⋊#261

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₄²

yes

characteristic

C₂×C₈×C₁₃

C₄⋊C₄

yes

characteristic

C₄²×C₁₃

C₂²⋊C₄

yes

characteristic

C₄×Dic₁₃

C₄⋊C₄

yes

characteristic

C₄×Dic₁₃

SD₁₆

yes

characteristic

C₄×Dic₁₃

Q₁₆

yes

characteristic

C₂×#1055

—

C₂×#1055

C₂²⋊C₄

yes

characteristic

C₂×#1055

C₄⋊C₄

yes

characteristic

C₈×D₁₃

—

C₈×D₁₃

—

C₈×D₁₃

—

C₈×D₁₃

D₈

yes

characteristic

C₈×D₁₃

SD₁₆

yes

characteristic

C₈×D₁₃

SD₁₆

yes

characteristic

C₈×D₁₃

Q₁₆

yes

characteristic

#6092

—

C₂×C₄⋊C₁₆

—

#1846

—

C₂×#334

—

C₂²×D₁₃

yes

—

—

—

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×Dic₁₃

yes

C₂×Dic₁₃

yes

C₂×Dic₁₃

yes

C₄×D₁₃

—

C₄×D₁₃

—

C₄×D₁₃

yes

C₄×D₁₃

yes

C₄×D₁₃

yes

C₄×D₁₃

yes

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

yes

C₈×C₁₃

yes

—

—

yes

yes

C₂²×C₁₆

—

C₂×#107

—

C₂×#106

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

#382

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

—

—

yes

characteristic, complemented

C₂²×C₁₃

yes

C₂×D₁₃

yes

100

C₂×D₁₃

#332

yes

characteristic

101

C₂×D₁₃

#359

yes

—

102

C₂×D₁₃

#360

yes

—

103

#261

—

104

C₄×C₁₃

—

105

C₄×C₁₃

—

106

C₄×C₁₃

#340

yes

characteristic

107

C₄×C₁₃

#343

yes

characteristic

108

Dic₁₃

—

109

Dic₁₃

—

110

Dic₁₃

#332

yes

characteristic

111

Dic₁₃

#333

yes

characteristic

112

C₂²×C₈

—

113

C₂²×C₈

—

114

C₂²×C₈

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄²

—

117

C₂×C₁₆

—

118

C₂×C₁₆

—

119

C₂×C₁₆

—

120

#107

—

121

#106

—

122

#106

—

123

C₄⋊C₈

—

124

C₄⋊C₈

—

125

C₄⋊C₈

—

126

C₄×C₈

—

127

C₄×C₈

—

128

C₄×C₈

—

129

C₄×C₈

C₂×#261

yes

characteristic

130

#95

—

131

#95

—

132

C₂×C₁₃

#1296

yes

characteristic

133

C₂×C₁₃

#1382

yes

characteristic

134

C₂×C₁₃

#1386

yes

characteristic

135

D₁₃

#1360

yes

—

136

D₁₃

#1363

yes

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

—

140

C₂²×C₄

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

104

—

152

C₂×C₈

C₄×#261

yes

characteristic

153

C₂×C₈

C₄⋊#261

yes

characteristic

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄²

—

157

C₄²

—

158

C₄²

—

159

C₄²

C₂²⋊#261

yes

characteristic

160

C₁₆

—

161

C₁₃

#7275

yes

characteristic, complemented, Sylow

162

C₂³

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

104

—

173

C₂×C₄

—

104

—

174

C₂×C₄

—

104

—

175

C₂×C₄

(C₂×#261)⋊C₄

yes

characteristic

176

C₂×C₄

#88115

yes

characteristic

177

C₂×C₄

#88117

yes

characteristic

178

C₈

—

179

C₈

—

180

C₈

—

181

C₈

—

182

C₈

—

183

C₈

—

104

—

184

C₈

#88101

yes

characteristic

185

C₈

#88102

yes

characteristic

186

C₂²

—

187

C₂²

—

188

C₂²

—

189

C₂²

—

190

C₂²

#11718246

yes

characteristic

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

—

208

—

198

C₄

#11718265

yes

characteristic

199

C₄

#11718271

yes

characteristic

200

C₂

—

201

C₂

—

202

C₂

#13129715

yes

characteristic

203

C₂

#13129899

yes

characteristic

204

C₂

#13129909

yes

characteristic

	subgroup	G/H
center	C₂²	#11718246
commutator	C₂×C₈×C₁₃	C₄²
Frattini	C₄×C₈	C₂×#261
Fitting	C₁₃×#334	C₄
socle	C₂²×C₁₃	#340
prime radical	C₂²×D₁₃	#102
cosocle	C₄×C₈×D₁₃	C₂²
hypercenter	#334	#261
hypocenter	C₁₃	#7275
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #334

subgroup

G/H

center

C₂²

#11718246

commutator

C₂×C₈×C₁₃

C₄²

Frattini

C₄×C₈

C₂×#261

Fitting

C₁₃×#334

C₄

C₂²×C₁₃

C₂²×D₁₃

C₄×C₈×D₁₃

C₂²

C₁₃

G ▷ C₂×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #334

N	H ≅ G/N	type
10. #11718260	C₄	semidirect
12. C₁₃×#334	C₄	semidirect
14. #11718049	C₄	semidirect
97. #334	#261	semidirect
161. C₁₃	#7275	semidirect

H ≅ G/N

type

10. #11718260

C₄

semidirect

12. C₁₃×#334

C₄

semidirect

14. #11718049

C₄

semidirect

97. #334

#261

semidirect

161. C₁₃

#7275

semidirect

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

subgroup	index
D₁₃⋊(C₄⋊C₁₆)	2
#13130773	2
D₁₃×#334	2
#7275	13

subgroup

index

D₁₃⋊(C₄⋊C₁₆)

#13130773

D₁₃×#334

#7275

Subgroups — #16777533

Conjugacy classes of subgroups

Complements

Maximal subgroups