Group #16019291 — subgroups

Subgroups — #16019291

Order 3328.

484 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 151 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	#13125422	C₂	1	2	yes	maximal
2	#13116979	C₂	1	4	yes	maximal, complemented
3	#13116944	C₂	1	4	yes	maximal
4	#13116887	C₂	1	4	yes	maximal, complemented
5	C₁₃×#2150	C₂	1	1	yes	maximal, characteristic
6	#11719175	—	2	2	no	—
7	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	#11718777	—	2	2	no	—
10	#11718777	C₂²	1	4	yes	—
11	C₂×Dic₁₀₄	C₂²	1	4	yes	—
12	#11718360	C₂²	1	2	yes	—
13	C₄×Dic₅₂	—	2	2	no	—
14	C₄×Dic₅₂	C₂²	1	4	yes	—
15	#11718343	C₂²	1	2	yes	—
16	#11718342	—	2	2	no	—
17	C₁₃×C₄⋊C₁₆	C₂²	1	4	yes	—
18	#11718097	C₂²	1	4	yes	—
19	#11718095	C₂²	1	8	yes	—
20	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
22	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
23	#88180	—	4	2	no	—
24	#88178	—	2	4	no	—
25	#88178	—	4	2	no	—
26	C₂×Dic₅₂	—	2	2	no	—
27	C₂×Dic₅₂	C₂³	1	4	yes	—
28	#88124	—	4	2	no	—
29	#88123	—	2	2	no	—
30	#88123	—	2	2	no	—
31	C₄×Dic₂₆	—	2	4	no	—
32	C₄×Dic₂₆	—	4	2	no	—
33	C₁₃×M₃₂	—	4	2	no	—
34	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
35	C₄×C₈×C₁₃	D₄	1	2	yes	—
36	C₄×C₈×C₁₃	C₂³	1	2	yes	—
37	#88058	—	2	2	no	—
38	#88058	C₂³	1	4	yes	—
39	#88056	—	2	4	no	—
40	#88056	—	2	4	no	—
41	Dic₁₀₄	—	4	4	no	—
42	#33821	—	13	1	no	maximal, Sylow
43	C₂²×C₄×C₁₃	D₈	1	2	yes	—
44	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
45	C₂×Dic₂₆	—	2	4	no	—
46	C₂×Dic₂₆	—	4	2	no	—
47	C₂×C₈×C₁₃	—	2	1	no	—
48	C₂×C₈×C₁₃	—	2	1	no	—
49	C₂×C₈×C₁₃	—	4	1	no	—
50	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
52	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
53	C₄²×C₁₃	D₈	1	1	yes	characteristic
54	C₄²×C₁₃	D₈	1	1	yes	characteristic
55	C₄²×C₁₃	C₂×D₄	1	2	yes	—
56	C₂²⋊Dic₁₃	—	4	4	no	—
57	C₄⋊Dic₁₃	—	2	4	no	—
58	C₄⋊Dic₁₃	—	4	2	no	—
59	C₄⋊Dic₁₃	—	4	2	no	—
60	C₄⋊Dic₁₃	—	4	2	no	—
61	#6101	—	4	4	no	—
62	#6101	—	4	4	no	—
63	C₄×Dic₁₃	—	4	4	no	—
64	Dic₅₂	—	2	4	no	—
65	Dic₅₂	—	2	4	no	—
66	Dic₅₂	—	4	4	no	—
67	C₁₃×C₁₆	—	2	4	no	—
68	#3170	—	13	2	no	—
69	#2225	—	13	4	no	—
70	#2219	—	13	4	no	—
71	#2209	—	13	4	no	—
72	#2150	D₁₃	1	1	yes	characteristic
73	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
74	C₂×C₄×C₁₃	—	2	1	no	—
75	C₂×C₄×C₁₃	—	2	1	no	—
76	C₂×C₄×C₁₃	—	2	4	no	—
77	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
79	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
80	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
81	C₂×Dic₁₃	—	4	4	no	—
82	C₂×Dic₁₃	—	4	4	no	—
83	Dic₂₆	—	4	4	no	—
84	Dic₂₆	—	4	4	no	—
85	Dic₂₆	—	8	4	no	—
86	C₈×C₁₃	—	2	1	no	—
87	C₈×C₁₃	—	2	2	no	—
88	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
89	C₈×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
90	#533	—	26	2	no	—
91	C₂×Q₃₂	—	13	4	no	—
92	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
93	#500	—	13	2	no	—
94	#494	—	13	2	no	—
95	#487	—	26	2	no	—
96	#467	—	13	4	no	—
97	#467	—	26	2	no	—
98	C₄×Q₁₆	—	13	4	no	—
99	C₄×Q₁₆	—	26	2	no	—
100	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
101	#367	—	13	4	no	—
102	C₄⋊C₁₆	C₂×D₁₃	1	4	yes	—
103	#358	—	13	8	no	—
104	C₂²×C₁₃	—	2	2	no	—
105	C₂²×C₁₃	—	2	1	no	—
106	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
107	C₄×C₁₃	—	2	2	no	—
108	C₄×C₁₃	—	2	1	no	—
109	C₄×C₁₃	#510	1	4	yes	—
110	C₄×C₁₃	#537	1	1	yes	characteristic
111	C₄×C₁₃	#575	1	1	yes	characteristic
112	Dic₁₃	—	8	4	no	—
113	Dic₁₃	—	8	4	no	—
114	C₂×Q₁₆	—	13	4	no	—
115	C₂×Q₁₆	—	26	2	no	—
116	C₂²×C₈	D₅₂	1	1	yes	characteristic
117	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
118	C₄⋊Q₈	—	26	2	no	—
119	C₄⋊Q₈	—	26	2	no	—
120	#125	—	52	2	no	—
121	C₂²⋊Q₈	—	26	4	no	—
122	C₂²⋊Q₈	—	52	2	no	—
123	C₄×Q₈	—	26	4	no	—
124	C₄×Q₈	—	52	2	no	—
125	#117	—	52	2	no	—
126	C₂×C₄²	D₅₂	1	1	yes	characteristic
127	Q₃₂	—	52	4	no	—
128	M₃₂	—	4	2	no	—
129	C₂×C₁₆	C₂²×D₁₃	1	4	yes	—
130	#107	—	13	4	no	—
131	#107	—	26	2	no	—
132	#103	—	26	4	no	—
133	#103	—	26	4	no	—
134	C₄×C₈	D₅₂	1	2	yes	—
135	C₄×C₈	C₂²×D₁₃	1	2	yes	—
136	C₂×C₁₃	—	4	1	no	—
137	C₂×C₁₃	#3245	1	1	yes	characteristic
138	C₂×C₁₃	C₂×#510	1	2	yes	—
139	C₂×Q₈	—	26	4	no	—
140	C₂×Q₈	—	52	2	no	—
141	C₂²×C₄	D₁₀₄	1	2	yes	—
142	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
143	Q₁₆	—	26	4	no	—
144	Q₁₆	—	26	4	no	—
145	Q₁₆	—	52	4	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	2	1	no	—
148	C₂×C₈	—	4	1	no	—
149	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
150	C₂×C₈	C₂×D₅₂	1	2	yes	—
151	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
152	C₄⋊C₄	—	26	4	no	—
153	C₄⋊C₄	—	52	2	no	—
154	C₄⋊C₄	—	52	2	no	—
155	C₄⋊C₄	—	52	4	no	—
156	C₄⋊C₄	—	52	4	no	—
157	C₄⋊C₄	—	52	2	no	—
158	C₂²⋊C₄	—	52	4	no	—
159	C₄²	—	52	4	no	—
160	C₄²	D₁₀₄	1	1	yes	characteristic
161	C₄²	D₁₀₄	1	1	yes	characteristic
162	C₄²	C₂×D₅₂	1	2	yes	—
163	C₁₆	—	2	4	no	—
164	C₁₃	#33821	1	1	yes	characteristic, complemented, Sylow
165	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
166	Q₈	—	52	4	no	—
167	Q₈	—	52	4	no	—
168	Q₈	—	104	4	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	52	4	no	—
173	C₂×C₄	—	52	4	no	—
174	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
175	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
176	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
177	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
178	C₈	—	2	2	no	—
179	C₈	—	2	1	no	—
180	C₈	#88251	1	1	yes	characteristic
181	C₈	#88254	1	1	yes	characteristic
182	C₂²	—	2	1	no	—
183	C₂²	—	2	2	no	—
184	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
185	C₄	—	2	1	no	—
186	C₄	—	2	2	no	—
187	C₄	—	104	4	no	—
188	C₄	—	104	4	no	—
189	C₄	#11718570	1	4	yes	—
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	C₂×#11718570	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13125422

C₂

yes

maximal

#13116979

C₂

yes

maximal, complemented

#13116944

C₂

yes

maximal

#13116887

C₂

yes

maximal, complemented

C₁₃×#2150

C₂

yes

maximal, characteristic

#11719175

—

C₂×C₁₃×M₃₂

C₂²

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

#11718777

—

#11718777

C₂²

yes

—

C₂×Dic₁₀₄

C₂²

yes

—

#11718360

C₂²

yes

—

C₄×Dic₅₂

—

C₄×Dic₅₂

C₂²

yes

—

#11718343

C₂²

yes

—

#11718342

—

C₁₃×C₄⋊C₁₆

C₂²

yes

—

#11718097

C₂²

yes

—

#11718095

C₂²

yes

—

C₂²×C₈×C₁₃

D₄

yes

characteristic

C₂²×C₈×C₁₃

C₂³

yes

characteristic

C₂×C₄²×C₁₃

D₄

yes

—

—

—

C₂×Dic₅₂

—

C₂×Dic₅₂

C₂³

yes

—

#88124

—

#88123

—

#88123

—

C₄×Dic₂₆

—

C₄×Dic₂₆

—

C₁₃×M₃₂

—

C₂×C₁₃×C₁₆

C₂³

yes

—

C₄×C₈×C₁₃

D₄

yes

—

C₄×C₈×C₁₃

C₂³

yes

—

#88058

—

#88058

C₂³

yes

—

#88056

—

#88056

—

Dic₁₀₄

—

#33821

—

maximal, Sylow

C₂²×C₄×C₁₃

D₈

yes

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃

C₂×D₄

yes

—

C₂×C₈×C₁₃

C₂⁴

yes

characteristic

C₄²×C₁₃

D₈

yes

characteristic

C₄²×C₁₃

D₈

yes

characteristic

C₄²×C₁₃

C₂×D₄

yes

—

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#6101

—

#6101

—

C₄×Dic₁₃

—

Dic₅₂

—

Dic₅₂

—

Dic₅₂

—

C₁₃×C₁₆

—

—

—

—

—

D₁₃

yes

characteristic

C₂³×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

—

C₂×C₄×C₁₃

C₂×D₈

yes

—

C₂×C₄×C₁₃

C₂²×D₄

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

H(5,𝔽₂)

yes

characteristic

C₈×C₁₃

2₋¹⁺⁴

yes

characteristic

#533

—

C₂×Q₃₂

—

C₂×M₃₂

C₂×D₁₃

yes

—

—

—

—

—

—

C₄×Q₁₆

—

C₄×Q₁₆

—

100

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

101

#367

—

102

C₄⋊C₁₆

C₂×D₁₃

yes

—

103

#358

—

104

C₂²×C₁₃

—

105

C₂²×C₁₃

—

106

C₂²×C₁₃

C₂²×D₈

yes

characteristic

107

C₄×C₁₃

—

108

C₄×C₁₃

—

109

C₄×C₁₃

#510

yes

—

110

C₄×C₁₃

#537

yes

characteristic

111

C₄×C₁₃

#575

yes

characteristic

112

Dic₁₃

—

113

Dic₁₃

—

114

C₂×Q₁₆

—

115

C₂×Q₁₆

—

116

C₂²×C₈

D₅₂

yes

characteristic

117

C₂²×C₈

C₂²×D₁₃

yes

characteristic

118

C₄⋊Q₈

—

119

C₄⋊Q₈

—

120

#125

—

121

C₂²⋊Q₈

—

122

C₂²⋊Q₈

—

123

C₄×Q₈

—

124

C₄×Q₈

—

125

#117

—

126

C₂×C₄²

D₅₂

yes

characteristic

127

Q₃₂

—

128

M₃₂

—

129

C₂×C₁₆

C₂²×D₁₃

yes

—

130

#107

—

131

#107

—

132

#103

—

133

#103

—

134

C₄×C₈

D₅₂

yes

—

135

C₄×C₈

C₂²×D₁₃

yes

—

136

C₂×C₁₃

—

137

C₂×C₁₃

#3245

yes

characteristic

138

C₂×C₁₃

C₂×#510

yes

—

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂²×C₄

D₁₀₄

yes

—

142

C₂²×C₄

C₂×D₅₂

yes

characteristic

143

Q₁₆

—

144

Q₁₆

—

145

Q₁₆

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

C₂×D₅₂

yes

characteristic

150

C₂×C₈

C₂×D₅₂

yes

—

151

C₂×C₈

C₂³×D₁₃

yes

characteristic

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₂²⋊C₄

—

159

C₄²

—

160

C₄²

D₁₀₄

yes

characteristic

161

C₄²

D₁₀₄

yes

characteristic

162

C₄²

C₂×D₅₂

yes

—

163

C₁₆

—

164

C₁₃

#33821

yes

characteristic, complemented, Sylow

165

C₂³

C₂×D₁₀₄

yes

characteristic

166

Q₈

—

167

Q₈

—

168

Q₈

—

104

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

C₂×D₁₀₄

yes

characteristic

175

C₂×C₄

C₂×D₁₀₄

yes

—

176

C₂×C₄

C₂×D₁₀₄

yes

—

177

C₂×C₄

C₂²×D₅₂

yes

characteristic

178

C₈

—

179

C₈

—

180

C₈

#88251

yes

characteristic

181

C₈

#88254

yes

characteristic

182

C₂²

—

183

C₂²

—

184

C₂²

C₂²×D₁₀₄

yes

characteristic

185

C₄

—

186

C₄

—

187

C₄

—

104

—

188

C₄

—

104

—

189

C₄

#11718570

yes

—

190

C₄

#11719215

yes

characteristic

191

C₄

#11719453

yes

characteristic

192

C₂

—

193

C₂

#13126146

yes

characteristic

194

C₂

C₂×#11718570

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₁₀₄
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2150	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂³×C₁₃	C₂×D₈
hypercenter	#2150	D₁₃
hypocenter	C₁₃	#33821
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

subgroup

G/H

center

C₂²

C₂²×D₁₀₄

commutator, cosocle

C₂×C₈×C₁₃

C₂⁴

Frattini

C₂×C₈

C₂³×D₁₃

Fitting

C₁₃×#2150

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 ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

N	H ≅ G/N	type
2. #13116979	C₂	semidirect
4. #13116887	C₂	semidirect
164. C₁₃	#33821	semidirect

H ≅ G/N

type

2. #13116979

C₂

semidirect

4. #13116887

C₂

semidirect

164. C₁₃

#33821

semidirect

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

subgroup	index
#13125422	2
#13116979	2
#13116944	2
#13116887	2
C₁₃×#2150	2
#33821	13

subgroup

index

C₁₃×#2150

Subgroups — #16019291

Conjugacy classes of subgroups

Complements

Maximal subgroups