Group #16807355 — subgroups

Subgroups — #16807355

Order 3328.

556 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 210 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	#13131219	C₂	1	8	yes	maximal, complemented
2	#13131197	C₂	1	2	yes	maximal, complemented
3	#13131067	C₂	1	4	yes	maximal, complemented
4	#13123547	C₂	1	1	yes	maximal, characteristic
5	C₂×#88161	C₂²	1	2	yes	—
6	#11719182	C₄	1	1	yes	characteristic
7	C₂×#88098	C₂²	1	4	yes	—
8	C₂×D₁₃⋊C₁₆	C₂²	1	4	yes	—
9	C₁₃×#423	C₄	1	1	yes	characteristic
10	#11718790	C₄	1	2	yes	—
11	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
12	#11718499	C₄	1	4	yes	—
13	#11718495	C₂²	1	4	yes	—
14	C₂²⋊#6092	C₂²	1	4	yes	—
15	#11718259	C₂²	1	8	yes	—
16	#11718231	C₂²	1	4	yes	—
17	C₄⋊#6092	C₂²	1	4	yes	—
18	C₂×#6127	C₂×C₄	1	2	yes	—
19	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
21	#88184	C₈	1	4	yes	—
22	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
23	C₂²×#1055	C₂³	1	1	yes	characteristic
24	#88161	—	4	2	no	—
25	#88161	—	4	2	no	—
26	C₂×#6094	C₂×C₄	1	4	yes	—
27	C₂×C₈×D₁₃	C₂³	1	4	yes	—
28	#88129	C₈	1	2	yes	—
29	#88128	C₈	1	1	yes	characteristic
30	C₄×D₅₂	C₂×C₄	1	4	yes	—
31	#88123	C₈	1	1	yes	characteristic
32	C₂×#6092	C₂³	1	8	yes	—
33	#88098	—	4	4	no	—
34	D₁₃⋊C₁₆	—	4	4	no	—
35	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
36	#88060	C₂×C₄	1	8	yes	—
37	C₄×#1055	C₂×C₄	1	2	yes	—
38	C₄×#1055	C₂³	1	2	yes	—
39	#33494	—	13	1	no	maximal, Sylow
40	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
41	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₂²⋊D₁₃	C₂×C₈	1	4	yes	—
43	#6127	—	4	2	no	—
44	#6127	—	4	2	no	—
45	C₂×D₅₂	C₂×C₈	1	4	yes	—
46	C₂×D₅₂	C₂×C₈	1	2	yes	—
47	C₂×C₄×D₁₃	C₂²×C₄	1	4	yes	—
48	C₂×Dic₂₆	C₂×C₈	1	2	yes	—
49	C₁₃×M₁₆	—	4	2	no	—
50	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
51	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
52	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
53	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
54	#6103	C₂×C₈	1	8	yes	—
55	C₄⋊Dic₁₃	C₂×C₈	1	4	yes	—
56	C₂×#1055	—	2	2	no	—
57	C₂×#1055	—	4	1	no	—
58	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
59	C₂×#1055	C₂²×C₄	1	2	yes	—
60	C₂×#1055	C₂⁴	1	1	yes	characteristic
61	#6094	—	4	4	no	—
62	C₈×D₁₃	—	2	4	no	—
63	C₈×D₁₃	—	2	4	no	—
64	#6092	—	2	8	no	—
65	#2931	—	13	1	no	—
66	#2168	—	13	8	no	—
67	#2163	—	13	2	no	—
68	#2113	—	13	4	no	—
69	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
70	C₂²×D₁₃	C₂²×C₈	1	4	yes	—
71	C₂×C₄×C₁₃	—	2	1	no	—
72	C₂×C₄×C₁₃	—	2	1	no	—
73	C₂×C₄×C₁₃	—	4	2	no	—
74	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
75	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
76	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
78	C₂²⋊D₁₃	—	4	4	no	—
79	C₂×Dic₁₃	C₂²×C₈	1	4	yes	—
80	D₅₂	—	4	4	no	—
81	D₅₂	—	4	2	no	—
82	C₄×D₁₃	—	2	4	no	—
83	C₄×D₁₃	—	2	4	no	—
84	Dic₂₆	—	4	2	no	—
85	C₈×C₁₃	—	2	4	no	—
86	#1055	—	2	1	no	—
87	#1055	—	2	2	no	—
88	#1055	H(5,𝔽₂)	1	2	yes	—
89	C₂×#131	—	13	2	no	—
90	#532	—	13	1	no	—
91	C₂×M₃₂	—	13	4	no	—
92	C₂²×C₁₆	—	13	4	no	—
93	#436	—	13	4	no	—
94	C₈×D₄	—	13	4	no	—
95	#423	#261	1	1	yes	characteristic
96	#408	—	13	2	no	—
97	C₂×C₄×C₈	—	13	1	no	—
98	C₄⋊C₁₆	—	13	4	no	—
99	C₂²⋊C₁₆	—	13	4	no	—
100	C₂²⋊C₁₆	—	13	8	no	—
101	#346	—	13	4	no	—
102	C₂²×C₁₃	—	2	2	no	—
103	C₂²×C₁₃	—	2	1	no	—
104	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
105	C₂×D₁₃	—	2	8	no	—
106	C₂×D₁₃	—	2	4	no	—
107	C₄×C₁₃	—	2	2	no	—
108	C₄×C₁₃	—	2	2	no	—
109	C₄×C₁₃	—	2	1	no	—
110	C₄×C₁₃	#518	1	1	yes	characteristic
111	C₄×C₁₃	#568	1	1	yes	characteristic
112	Dic₁₃	—	2	4	no	—
113	C₂×#41	—	13	2	no	—
114	#131	—	52	2	no	—
115	#131	—	52	2	no	—
116	C₂×M₁₆	—	13	4	no	—
117	C₂×M₁₆	C₂×#261	1	2	yes	—
118	C₂²×C₈	—	13	1	no	—
119	C₂²×C₈	—	13	1	no	—
120	C₂²×C₈	—	13	4	no	—
121	C₄⋊Q₈	—	13	1	no	—
122	#127	—	13	1	no	—
123	#124	—	13	2	no	—
124	#121	—	13	4	no	—
125	C₄×D₄	—	13	4	no	—
126	C₂×C₄²	C₂×#261	1	1	yes	characteristic
127	M₃₂	—	52	4	no	—
128	C₂×C₁₆	—	13	8	no	—
129	C₂×C₁₆	—	52	4	no	—
130	C₄⋊C₈	C₂×#261	1	4	yes	—
131	C₂²⋊C₈	—	13	8	no	—
132	C₄×C₈	—	13	2	no	—
133	C₄×C₈	—	13	2	no	—
134	C₂×C₁₃	—	4	1	no	—
135	C₂×C₁₃	#2972	1	1	yes	characteristic
136	C₂×C₁₃	#3407	1	2	yes	—
137	D₁₃	—	4	4	no	—
138	#41	—	52	2	no	—
139	#41	—	52	2	no	—
140	C₂×Q₈	—	13	2	no	—
141	C₂×D₄	—	13	4	no	—
142	C₂×D₄	—	13	4	no	—
143	C₂×D₄	—	13	2	no	—
144	C₂²×C₄	—	13	4	no	—
145	C₂²×C₄	D₁₃⋊C₈	1	2	yes	—
146	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
147	M₁₆	—	4	2	no	—
148	M₁₆	—	52	4	no	—
149	C₂×C₈	—	13	1	no	—
150	C₂×C₈	—	13	2	no	—
151	C₂×C₈	—	13	1	no	—
152	C₂×C₈	—	26	4	no	—
153	C₂×C₈	—	26	4	no	—
154	C₂×C₈	—	26	2	no	—
155	C₂×C₈	—	52	1	no	—
156	C₂×C₈	C₂²×#261	1	4	yes	—
157	C₄⋊C₄	—	13	4	no	—
158	C₂²⋊C₄	—	13	8	no	—
159	C₄²	D₁₃⋊C₈	1	1	yes	characteristic
160	C₄²	D₁₃⋊C₈	1	1	yes	characteristic
161	C₄²	C₂²×#261	1	2	yes	—
162	C₁₆	—	26	8	no	—
163	C₁₃	#33494	1	1	yes	characteristic, complemented, Sylow
164	C₂³	—	13	4	no	—
165	C₂³	C₂×D₁₃⋊C₈	1	1	yes	characteristic
166	Q₈	—	52	2	no	—
167	D₄	—	52	4	no	—
168	D₄	—	52	2	no	—
169	D₄	—	52	4	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	—	13	4	no	—
174	C₂×C₄	—	26	4	no	—
175	C₂×C₄	—	26	4	no	—
176	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
177	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
178	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
179	C₂×C₄	C₂³×#261	1	1	yes	characteristic
180	C₈	—	2	4	no	—
181	C₈	—	13	2	no	—
182	C₈	—	26	1	no	—
183	C₈	—	26	2	no	—
184	C₂²	—	2	2	no	—
185	C₂²	—	2	1	no	—
186	C₂²	—	26	8	no	—
187	C₂²	—	26	4	no	—
188	C₂²	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
189	C₄	—	2	2	no	—
190	C₄	—	2	2	no	—
191	C₄	—	2	1	no	—
192	C₄	—	26	4	no	—
193	C₄	#11719623	1	1	yes	characteristic
194	C₄	#11719625	1	1	yes	characteristic
195	C₂	—	4	1	no	—
196	C₂	—	52	4	no	—
197	C₂	#13131686	1	1	yes	characteristic
198	C₂	#13131805	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13131219

C₂

yes

maximal, complemented

#13131197

C₂

yes

maximal, complemented

#13131067

C₂

yes

maximal, complemented

#13123547

C₂

yes

maximal, characteristic

C₂×#88161

C₂²

yes

—

#11719182

C₄

yes

characteristic

C₂×#88098

C₂²

yes

—

C₂×D₁₃⋊C₁₆

C₂²

yes

—

C₁₃×#423

C₄

yes

characteristic

#11718790

C₄

yes

—

C₂×C₄×#1055

C₂²

yes

characteristic

#11718499

C₄

yes

—

#11718495

C₂²

yes

—

C₂²⋊#6092

C₂²

yes

—

#11718259

C₂²

yes

—

#11718231

C₂²

yes

—

C₄⋊#6092

C₂²

yes

—

C₂×#6127

C₂×C₄

yes

—

C₂×C₁₃×M₁₆

C₂×C₄

yes

—

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

#88184

C₈

yes

—

C₂²×#1055

C₂×C₄

yes

characteristic

C₂²×#1055

C₂³

yes

—

—

C₂×C₄

yes

—

C₂×C₈×D₁₃

C₂³

yes

—

#88129

C₈

yes

—

#88128

C₈

yes

characteristic

C₄×D₅₂

C₂×C₄

yes

—

#88123

C₈

yes

characteristic

C₂×#6092

C₂³

yes

—

#88098

—

D₁₃⋊C₁₆

—

C₁₃×C₄⋊C₈

C₂×C₄

yes

—

#88060

C₂×C₄

yes

—

C₄×#1055

C₂×C₄

yes

—

C₄×#1055

C₂³

yes

—

#33494

—

maximal, Sylow

C₂²×C₄×C₁₃

C₂×C₈

yes

—

C₂²×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₂²⋊D₁₃

C₂×C₈

yes

—

#6127

—

#6127

—

C₂×D₅₂

C₂×C₈

yes

—

C₂×D₅₂

C₂×C₈

yes

—

C₂×C₄×D₁₃

C₂²×C₄

yes

—

C₂×Dic₂₆

C₂×C₈

yes

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

C₂²×C₄

yes

—

C₄²×C₁₃

C₂×C₈

yes

characteristic

C₄²×C₁₃

C₂×C₈

yes

characteristic

C₄²×C₁₃

C₂²×C₄

yes

—

#6103

C₂×C₈

yes

—

C₄⋊Dic₁₃

C₂×C₈

yes

—

C₂×#1055

—

C₂×#1055

—

C₂×#1055

C₂²×C₄

yes

characteristic

C₂×#1055

C₂²×C₄

yes

—

C₂×#1055

C₂⁴

yes

characteristic

#6094

—

C₈×D₁₃

—

C₈×D₁₃

—

—

—

—

—

—

C₂³×C₁₃

C₂²×C₈

yes

characteristic

C₂²×D₁₃

C₂²×C₈

yes

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²×C₈

yes

—

C₂×C₄×C₁₃

C₂²×C₈

yes

—

C₂×C₄×C₁₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₃

C₂³×C₄

yes

characteristic

C₂²⋊D₁₃

—

C₂×Dic₁₃

C₂²×C₈

yes

—

D₅₂

—

D₅₂

—

C₄×D₁₃

—

C₄×D₁₃

—

Dic₂₆

—

C₈×C₁₃

—

—

—

yes

—

C₂×#131

—

#532

—

C₂×M₃₂

—

C₂²×C₁₆

—

#436

—

C₈×D₄

—

yes

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

100

C₂²⋊C₁₆

—

101

#346

—

102

C₂²×C₁₃

—

103

C₂²×C₁₃

—

104

C₂²×C₁₃

C₂³×C₈

yes

characteristic

105

C₂×D₁₃

—

106

C₂×D₁₃

—

107

C₄×C₁₃

—

108

C₄×C₁₃

—

109

C₄×C₁₃

—

110

C₄×C₁₃

#518

yes

characteristic

111

C₄×C₁₃

#568

yes

characteristic

112

Dic₁₃

—

113

C₂×#41

—

114

#131

—

115

#131

—

116

C₂×M₁₆

—

117

C₂×M₁₆

C₂×#261

yes

—

118

C₂²×C₈

—

119

C₂²×C₈

—

120

C₂²×C₈

—

121

C₄⋊Q₈

—

122

#127

—

123

#124

—

124

#121

—

125

C₄×D₄

—

126

C₂×C₄²

C₂×#261

yes

characteristic

127

M₃₂

—

128

C₂×C₁₆

—

129

C₂×C₁₆

—

130

C₄⋊C₈

C₂×#261

yes

—

131

C₂²⋊C₈

—

132

C₄×C₈

—

133

C₄×C₈

—

134

C₂×C₁₃

—

135

C₂×C₁₃

#2972

yes

characteristic

136

C₂×C₁₃

#3407

yes

—

137

D₁₃

—

138

#41

—

139

#41

—

140

C₂×Q₈

—

141

C₂×D₄

—

142

C₂×D₄

—

143

C₂×D₄

—

144

C₂²×C₄

—

145

C₂²×C₄

D₁₃⋊C₈

yes

—

146

C₂²×C₄

C₂²×#261

yes

characteristic

147

M₁₆

—

148

M₁₆

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

C₂²×#261

yes

—

157

C₄⋊C₄

—

158

C₂²⋊C₄

—

159

C₄²

D₁₃⋊C₈

yes

characteristic

160

C₄²

D₁₃⋊C₈

yes

characteristic

161

C₄²

C₂²×#261

yes

—

162

C₁₆

—

163

C₁₃

#33494

yes

characteristic, complemented, Sylow

164

C₂³

—

165

C₂³

C₂×D₁₃⋊C₈

yes

characteristic

166

Q₈

—

167

D₄

—

168

D₄

—

169

D₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

C₂×D₁₃⋊C₈

yes

characteristic

177

C₂×C₄

C₂×D₁₃⋊C₈

yes

—

178

C₂×C₄

C₂×D₁₃⋊C₈

yes

—

179

C₂×C₄

C₂³×#261

yes

characteristic

180

C₈

—

181

C₈

—

182

C₈

—

183

C₈

—

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

C₂²×D₁₃⋊C₈

yes

characteristic

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

#11719623

yes

characteristic

194

C₄

#11719625

yes

characteristic

195

C₂

—

196

C₂

—

197

C₂

#13131686

yes

characteristic

198

C₂

#13131805

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₁₃×#423	C₄
prime radical	#11719182	C₄
cosocle	C₂×#1055	C₂⁴
hypercenter	#423	#261
hypocenter	C₁₃	#33494
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #423

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×#261

commutator, socle

C₂²×C₁₃

C₂³×C₈

Fitting

C₁₃×#423

C₄

C₄

C₂⁴

C₁₃

G ▷ C₂²×C₁₃ ▷ C₁₃

C₁ ◁ C₂×C₄ ◁ #423

N	H ≅ G/N	type
1. #13131219	C₂	semidirect
2. #13131197	C₂	semidirect
3. #13131067	C₂	semidirect
163. C₁₃	#33494	semidirect

H ≅ G/N

type

1. #13131219

C₂

semidirect

2. #13131197

C₂

semidirect

3. #13131067

C₂

semidirect

163. C₁₃

#33494

semidirect

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

subgroup	index
#13131219	2
#13131197	2
#13131067	2
#13123547	2
#33494	13

subgroup

index

Subgroups — #16807355

Conjugacy classes of subgroups

Complements

Maximal subgroups