Group #16019351 — subgroups

Subgroups — #16019351

Order 3328.

464 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 143 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	#13125495	C₂	1	2	yes	maximal
2	#13117001	C₂	1	4	yes	maximal, complemented
3	#13116944	C₂	1	2	yes	maximal
4	#13116927	C₂	1	2	yes	maximal
5	#13116887	C₂	1	4	yes	maximal, complemented
6	C₁₃×#2151	C₂	1	1	yes	maximal, characteristic
7	#11719223	—	2	2	no	—
8	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
9	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
10	C₁₃×#405	C₂²	1	1	yes	characteristic
11	#11718798	—	2	2	no	—
12	#11718777	C₂²	1	4	yes	—
13	C₂×Dic₁₀₄	C₂²	1	2	yes	—
14	C₂×Dic₁₀₄	C₂²	1	2	yes	—
15	#11718377	—	2	2	no	—
16	#11718364	—	2	2	no	—
17	#11718360	C₂²	1	2	yes	—
18	C₄×Dic₅₂	C₂²	1	4	yes	—
19	#11718344	C₂²	1	2	yes	—
20	C₁₃×C₄⋊C₁₆	C₂²	1	4	yes	—
21	#11718097	C₂²	1	2	yes	—
22	#11718096	C₂²	1	2	yes	—
23	#11718095	C₂²	1	4	yes	—
24	#11718095	C₂²	1	4	yes	—
25	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
27	C₁₃×#117	D₄	1	1	yes	characteristic
28	#88178	—	2	4	no	—
29	C₂×Dic₅₂	—	2	2	no	—
30	C₂×Dic₅₂	C₂³	1	4	yes	—
31	#88143	—	4	4	no	—
32	#88133	—	4	4	no	—
33	#88124	—	2	2	no	—
34	#88123	—	2	2	no	—
35	C₄×Dic₂₆	—	2	4	no	—
36	C₁₃×M₃₂	—	4	1	no	—
37	C₂×C₁₃×C₁₆	—	4	1	no	—
38	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
39	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
40	C₁₃×#97	D₄	1	2	yes	—
41	C₄×C₈×C₁₃	C₂³	1	2	yes	—
42	#88058	C₂³	1	4	yes	—
43	#88057	—	2	2	no	—
44	#88056	—	2	4	no	—
45	#88056	—	2	4	no	—
46	Dic₁₀₄	—	4	2	no	—
47	Dic₁₀₄	—	4	2	no	—
48	#33827	—	13	1	no	maximal, Sylow
49	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
50	C₂×Dic₂₆	—	2	4	no	—
51	C₁₃×M₁₆	—	4	1	no	—
52	C₂×C₈×C₁₃	—	2	1	no	—
53	C₂×C₈×C₁₃	—	2	1	no	—
54	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
55	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
57	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
58	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
59	C₄²×C₁₃	C₂×D₄	1	2	yes	—
60	C₂²⋊Dic₁₃	—	4	4	no	—
61	C₄⋊Dic₁₃	—	2	4	no	—
62	C₄⋊Dic₁₃	—	4	4	no	—
63	#6101	—	4	4	no	—
64	#6101	—	4	4	no	—
65	#6101	—	4	4	no	—
66	C₄×Dic₁₃	—	4	4	no	—
67	Dic₅₂	—	2	4	no	—
68	Dic₅₂	—	2	4	no	—
69	Dic₅₂	—	4	4	no	—
70	C₁₃×C₁₆	—	2	2	no	—
71	C₁₃×C₁₆	—	2	2	no	—
72	#3177	—	13	2	no	—
73	#2229	—	13	4	no	—
74	#2219	—	13	2	no	—
75	#2216	—	13	2	no	—
76	#2209	—	13	4	no	—
77	#2151	D₁₃	1	1	yes	characteristic
78	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₁₃	—	2	1	no	—
80	C₂×C₄×C₁₃	—	2	1	no	—
81	C₂×C₄×C₁₃	C₂×D₈	1	4	yes	—
82	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
84	C₂×Dic₁₃	—	4	4	no	—
85	C₂×Dic₁₃	—	4	4	no	—
86	Dic₂₆	—	4	4	no	—
87	Dic₂₆	—	4	4	no	—
88	C₈×C₁₃	—	2	1	no	—
89	C₈×C₁₃	—	2	2	no	—
90	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
91	C₈×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
92	#541	—	26	2	no	—
93	C₂×Q₃₂	—	13	2	no	—
94	C₂×Q₃₂	—	13	2	no	—
95	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
96	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
97	#499	—	13	2	no	—
98	#494	—	13	2	no	—
99	#491	—	26	2	no	—
100	#470	—	26	2	no	—
101	#467	—	13	4	no	—
102	#441	—	26	2	no	—
103	C₄×Q₁₆	—	13	4	no	—
104	#405	C₂×D₁₃	1	1	yes	characteristic
105	#367	—	13	2	no	—
106	#366	—	13	2	no	—
107	C₄⋊C₁₆	C₂×D₁₃	1	4	yes	—
108	#358	—	13	4	no	—
109	#358	—	13	4	no	—
110	C₂²×C₁₃	—	2	2	no	—
111	C₂²×C₁₃	—	2	1	no	—
112	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
113	C₄×C₁₃	—	2	4	no	—
114	C₄×C₁₃	—	2	1	no	—
115	C₄×C₁₃	#537	1	1	yes	characteristic
116	C₄×C₁₃	#575	1	1	yes	characteristic
117	Dic₁₃	—	8	4	no	—
118	Dic₁₃	—	8	4	no	—
119	C₂×Q₁₆	—	13	4	no	—
120	C₂×Q₁₆	—	26	2	no	—
121	C₂×M₁₆	D₅₂	1	1	yes	characteristic
122	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
123	C₄⋊Q₈	—	26	2	no	—
124	#126	—	52	4	no	—
125	#125	—	26	2	no	—
126	#125	—	52	4	no	—
127	C₂²⋊Q₈	—	26	4	no	—
128	C₄×Q₈	—	26	4	no	—
129	#117	D₅₂	1	1	yes	characteristic
130	Q₃₂	—	52	2	no	—
131	Q₃₂	—	52	2	no	—
132	M₃₂	—	4	1	no	—
133	C₂×C₁₆	—	4	1	no	—
134	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
135	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
136	#107	—	13	4	no	—
137	#106	—	26	2	no	—
138	#103	—	26	4	no	—
139	#103	—	26	4	no	—
140	#97	D₅₂	1	2	yes	—
141	C₄×C₈	C₂²×D₁₃	1	2	yes	—
142	C₂×C₁₃	—	4	1	no	—
143	C₂×C₁₃	#3245	1	1	yes	characteristic
144	C₂×C₁₃	#3417	1	2	yes	—
145	C₂×Q₈	—	26	4	no	—
146	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
147	Q₁₆	—	26	4	no	—
148	Q₁₆	—	26	4	no	—
149	Q₁₆	—	52	4	no	—
150	M₁₆	—	4	1	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	C₂×D₅₂	1	2	yes	—
154	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
155	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
156	C₄⋊C₄	—	26	4	no	—
157	C₄⋊C₄	—	52	4	no	—
158	C₄⋊C₄	—	52	4	no	—
159	C₄⋊C₄	—	52	4	no	—
160	C₄⋊C₄	—	52	4	no	—
161	C₄⋊C₄	D₁₀₄	1	2	yes	—
162	C₂²⋊C₄	—	52	4	no	—
163	C₂²⋊C₄	D₁₀₄	1	2	yes	—
164	C₄²	—	52	4	no	—
165	C₄²	C₂×D₅₂	1	2	yes	—
166	C₁₆	—	2	2	no	—
167	C₁₆	—	2	2	no	—
168	C₁₃	#33827	1	1	yes	characteristic, complemented, Sylow
169	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
170	Q₈	—	52	4	no	—
171	Q₈	—	52	4	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	52	4	no	—
175	C₂×C₄	—	52	4	no	—
176	C₂×C₄	C₂×D₁₀₄	1	4	yes	—
177	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
178	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
179	C₈	—	2	2	no	—
180	C₈	—	2	1	no	—
181	C₈	#88251	1	1	yes	characteristic
182	C₈	#88254	1	1	yes	characteristic
183	C₂²	—	2	1	no	—
184	C₂²	—	2	2	no	—
185	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
186	C₄	—	2	1	no	—
187	C₄	—	2	4	no	—
188	C₄	—	104	4	no	—
189	C₄	—	104	4	no	—
190	C₄	#11719215	1	1	yes	characteristic
191	C₄	#11719453	1	1	yes	characteristic
192	C₂	—	4	1	no	—
193	C₂	#13126146	1	1	yes	characteristic
194	C₂	#13127989	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13125495

C₂

yes

maximal

#13117001

C₂

yes

maximal, complemented

#13116944

C₂

yes

maximal

#13116927

C₂

yes

maximal

#13116887

C₂

yes

maximal, complemented

C₁₃×#2151

C₂

yes

maximal, characteristic

#11719223

—

C₂×C₁₃×M₃₂

C₂²

yes

characteristic

C₂²×C₁₃×C₁₆

C₂²

yes

characteristic

C₁₃×#405

C₂²

yes

characteristic

#11718798

—

#11718777

C₂²

yes

—

C₂×Dic₁₀₄

C₂²

yes

—

C₂×Dic₁₀₄

C₂²

yes

—

#11718377

—

#11718364

—

#11718360

C₂²

yes

—

C₄×Dic₅₂

C₂²

yes

—

#11718344

C₂²

yes

—

C₁₃×C₄⋊C₁₆

C₂²

yes

—

#11718097

C₂²

yes

—

#11718096

C₂²

yes

—

#11718095

C₂²

yes

—

#11718095

C₂²

yes

—

C₂×C₁₃×M₁₆

D₄

yes

characteristic

C₂²×C₈×C₁₃

C₂³

yes

characteristic

C₁₃×#117

D₄

yes

characteristic

#88178

—

C₂×Dic₅₂

—

C₂×Dic₅₂

C₂³

yes

—

—

—

—

—

C₄×Dic₂₆

—

C₁₃×M₃₂

—

C₂×C₁₃×C₁₆

—

C₂×C₁₃×C₁₆

C₂³

yes

—

C₂×C₁₃×C₁₆

C₂³

yes

—

C₁₃×#97

D₄

yes

—

C₄×C₈×C₁₃

C₂³

yes

—

#88058

C₂³

yes

—

#88057

—

#88056

—

#88056

—

Dic₁₀₄

—

Dic₁₀₄

—

#33827

—

maximal, Sylow

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×Dic₂₆

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₂×D₄

yes

—

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃

C₂⁴

yes

characteristic

C₁₃×C₄⋊C₄

D₈

yes

—

C₁₃×C₂²⋊C₄

D₈

yes

—

C₄²×C₁₃

C₂×D₄

yes

—

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#6101

—

#6101

—

#6101

—

C₄×Dic₁₃

—

Dic₅₂

—

Dic₅₂

—

Dic₅₂

—

C₁₃×C₁₆

—

C₁₃×C₁₆

—

—

—

—

—

—

D₁₃

yes

characteristic

C₂³×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₈

yes

—

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

C₂²×D₄

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

H(5,𝔽₂)

yes

characteristic

C₈×C₁₃

2₋¹⁺⁴

yes

characteristic

#541

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×M₃₂

C₂×D₁₃

yes

characteristic

C₂²×C₁₆

C₂×D₁₃

yes

—

—

—

100

#470

—

101

#467

—

102

#441

—

103

C₄×Q₁₆

—

104

#405

C₂×D₁₃

yes

characteristic

105

#367

—

106

#366

—

107

C₄⋊C₁₆

C₂×D₁₃

yes

—

108

#358

—

109

#358

—

110

C₂²×C₁₃

—

111

C₂²×C₁₃

—

112

C₂²×C₁₃

C₂²×D₈

yes

characteristic

113

C₄×C₁₃

—

114

C₄×C₁₃

—

115

C₄×C₁₃

#537

yes

characteristic

116

C₄×C₁₃

#575

yes

characteristic

117

Dic₁₃

—

118

Dic₁₃

—

119

C₂×Q₁₆

—

120

C₂×Q₁₆

—

121

C₂×M₁₆

D₅₂

yes

characteristic

122

C₂²×C₈

C₂²×D₁₃

yes

characteristic

123

C₄⋊Q₈

—

124

#126

—

125

#125

—

126

#125

—

127

C₂²⋊Q₈

—

128

C₄×Q₈

—

129

#117

D₅₂

yes

characteristic

130

Q₃₂

—

131

Q₃₂

—

132

M₃₂

—

133

C₂×C₁₆

—

134

C₂×C₁₆

C₂²×D₁₃

yes

—

135

C₂×C₁₆

C₂²×D₁₃

yes

—

136

#107

—

137

#106

—

138

#103

—

139

#103

—

140

#97

D₅₂

yes

—

141

C₄×C₈

C₂²×D₁₃

yes

—

142

C₂×C₁₃

—

143

C₂×C₁₃

#3245

yes

characteristic

144

C₂×C₁₃

#3417

yes

—

145

C₂×Q₈

—

146

C₂²×C₄

C₂×D₅₂

yes

characteristic

147

Q₁₆

—

148

Q₁₆

—

149

Q₁₆

—

150

M₁₆

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

C₂×D₅₂

yes

—

154

C₂×C₈

C₂×D₅₂

yes

characteristic

155

C₂×C₈

C₂³×D₁₃

yes

characteristic

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

D₁₀₄

yes

—

162

C₂²⋊C₄

—

163

C₂²⋊C₄

D₁₀₄

yes

—

164

C₄²

—

165

C₄²

C₂×D₅₂

yes

—

166

C₁₆

—

167

C₁₆

—

168

C₁₃

#33827

yes

characteristic, complemented, Sylow

169

C₂³

C₂×D₁₀₄

yes

characteristic

170

Q₈

—

171

Q₈

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

C₂×D₁₀₄

yes

—

177

C₂×C₄

C₂×D₁₀₄

yes

characteristic

178

C₂×C₄

C₂²×D₅₂

yes

characteristic

179

C₈

—

180

C₈

—

181

C₈

#88251

yes

characteristic

182

C₈

#88254

yes

characteristic

183

C₂²

—

184

C₂²

—

185

C₂²

C₂²×D₁₀₄

yes

characteristic

186

C₄

—

187

C₄

—

188

C₄

—

104

—

189

C₄

—

104

—

190

C₄

#11719215

yes

characteristic

191

C₄

#11719453

yes

characteristic

192

C₂

—

193

C₂

#13126146

yes

characteristic

194

C₂

#13127989

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₁₀₄
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2151	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂³×C₁₃	C₂×D₈
hypercenter	#2151	D₁₃
hypocenter	C₁₃	#33827
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2151

subgroup

G/H

center

C₂²

C₂²×D₁₀₄

commutator, cosocle

C₂×C₈×C₁₃

C₂⁴

Frattini

C₂×C₈

C₂³×D₁₃

Fitting

C₁₃×#2151

C₂

socle

C₂²×C₁₃

C₂²×D₈

prime radical

C₂³×C₁₃

C₂×D₈

D₁₃

C₁₃

G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2151

N	H ≅ G/N	type
2. #13117001	C₂	semidirect
5. #13116887	C₂	semidirect
168. C₁₃	#33827	semidirect

H ≅ G/N

type

2. #13117001

C₂

semidirect

5. #13116887

C₂

semidirect

168. C₁₃

#33827

semidirect

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

subgroup	index
#13125495	2
#13117001	2
#13116944	2
#13116927	2
#13116887	2
C₁₃×#2151	2
#33827	13

subgroup

index

C₁₃×#2151

Subgroups — #16019351

Conjugacy classes of subgroups

Complements

Maximal subgroups