Group #16784959 — subgroups

Subgroups — #16784959

Order 3328.

542 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 132 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	#13131017	C₂	1	4	yes	maximal
2	#13130771	C₂	1	1	yes	maximal, characteristic
3	#13130030	C₂	1	1	yes	maximal, characteristic
4	D₁₃×#494	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₁₆×D₁₃	—	2	2	no	—
6	D₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₂×#88101	—	2	2	no	—
8	#11719059	C₂²	1	2	yes	—
9	C₄×D₁₃⋊C₈	C₂²	1	2	yes	—
10	C₁₃×#494	C₄	1	1	yes	characteristic, complemented
11	#11718842	—	2	2	no	—
12	#11718753	C₄	1	2	yes	complemented
13	#11718360	C₄	1	1	yes	characteristic, complemented
14	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
15	#11718251	—	2	2	no	—
16	#11718246	—	2	8	no	—
17	C₂×Q₈×D₁₃	—	2	2	no	—
18	C₂×C₄⋊#261	—	2	4	no	—
19	C₂×C₄⋊#261	—	2	2	no	—
20	C₂×D₁₃⋊C₈	D₄	1	4	yes	—
21	C₂×C₁₃×Q₁₆	—	2	2	no	—
22	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
23	#88198	—	2	2	no	—
24	C₂×#6107	—	2	4	no	—
25	Q₁₆×D₁₃	—	4	4	no	—
26	Q₁₆×D₁₃	—	4	4	no	—
27	C₂×Dic₅₂	—	2	2	no	—
28	C₂×C₈×D₁₃	D₄	1	2	yes	—
29	C₄⋊C₄×D₁₃	—	2	4	no	—
30	#88142	—	2	4	no	—
31	C₄²×D₁₃	C₂³	1	1	yes	characteristic
32	#88123	C₂×C₄	1	2	yes	—
33	C₄×#1057	—	2	2	no	—
34	C₄×#1057	—	2	2	no	—
35	#88101	—	4	4	no	—
36	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₈×Dic₁₃	D₄	1	2	yes	—
38	C₄×#1055	C₂×C₄	1	1	yes	characteristic
39	#9839	—	13	1	no	maximal, Sylow
40	C₂²×#261	—	4	2	no	—
41	C₂×Q₈×C₁₃	—	2	2	no	—
42	Q₈×D₁₃	—	4	4	no	—
43	Q₈×D₁₃	—	4	4	no	—
44	C₂×C₄×D₁₃	—	4	2	no	—
45	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×D₁₃	C₂×D₄	1	2	yes	—
47	C₂×Dic₂₆	—	2	2	no	—
48	C₂×Dic₂₆	—	4	2	no	—
49	C₂×#1057	—	2	4	no	—
50	C₄⋊#261	—	4	4	no	—
51	C₄⋊#261	—	4	8	no	—
52	D₁₃⋊C₈	—	2	8	no	—
53	C₁₃×Q₁₆	—	4	4	no	—
54	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
55	C₁₃×C₄⋊C₄	—	2	4	no	—
56	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
57	#6107	—	4	8	no	—
58	C₄⋊Dic₁₃	—	2	4	no	—
59	#6101	—	4	4	no	—
60	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
61	C₄×Dic₁₃	#41	1	2	yes	—
62	C₂×#1055	C₂²⋊C₄	1	2	yes	—
63	Dic₅₂	—	4	4	no	—
64	C₈×D₁₃	D₈	1	4	yes	—
65	C₈×D₁₃	SD₁₆	1	4	yes	—
66	C₂×#494	—	13	1	no	—
67	#1990	—	13	4	no	—
68	#1844	—	13	1	no	—
69	C₂×#322	—	13	1	no	—
70	C₂²×D₁₃	#127	1	1	yes	characteristic
71	C₂×#261	—	8	2	no	—
72	C₂×#261	—	8	4	no	—
73	Q₈×C₁₃	—	4	4	no	—
74	C₂×C₄×C₁₃	—	4	2	no	—
75	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
77	C₂×Dic₁₃	—	4	2	no	—
78	C₂×Dic₁₃	#121	1	2	yes	—
79	C₂×Dic₁₃	#124	1	1	yes	characteristic
80	C₄×D₁₃	—	8	2	no	—
81	C₄×D₁₃	—	8	2	no	—
82	C₄×D₁₃	C₂×D₈	1	2	yes	—
83	C₄×D₁₃	C₂×SD₁₆	1	2	yes	—
84	C₄×D₁₃	#137	1	8	yes	—
85	Dic₂₆	—	4	4	no	—
86	Dic₂₆	—	8	4	no	—
87	#1057	—	4	4	no	—
88	C₈×C₁₃	#102	1	4	yes	—
89	#1055	#102	1	4	yes	—
90	C₂²×Q₁₆	—	26	2	no	—
91	C₂×C₄⋊Q₈	—	13	2	no	—
92	#494	—	13	2	no	—
93	#494	—	13	1	no	—
94	#494	—	26	2	no	—
95	#494	#261	1	1	yes	characteristic, complemented
96	C₂×#106	—	26	2	no	—
97	C₂×C₄×C₈	—	13	1	no	—
98	C₂×C₄×C₈	—	13	2	no	—
99	#384	—	13	2	no	—
100	#340	—	26	8	no	—
101	#322	—	26	2	no	—
102	C₂²×C₁₃	#390	1	1	yes	characteristic
103	C₂×D₁₃	#492	1	1	yes	characteristic
104	C₂×D₁₃	#493	1	1	yes	characteristic
105	C₂×D₁₃	#497	1	4	yes	—
106	#261	—	16	2	no	—
107	C₄×C₁₃	—	8	2	no	—
108	C₄×C₁₃	C₂×#102	1	2	yes	—
109	C₄×C₁₃	#441	1	4	yes	—
110	Dic₁₃	—	8	2	no	—
111	Dic₁₃	#470	1	4	yes	—
112	Dic₁₃	#486	1	2	yes	—
113	C₂²×Q₈	—	26	2	no	—
114	C₂×Q₁₆	—	2	2	no	—
115	C₂×Q₁₆	—	26	4	no	—
116	C₂×Q₁₆	—	26	2	no	—
117	C₂×Q₁₆	—	52	4	no	—
118	C₂×Q₁₆	—	52	4	no	—
119	C₂²×C₈	—	13	2	no	—
120	C₂²×C₈	—	13	4	no	—
121	C₄⋊Q₈	—	13	2	no	—
122	C₄⋊Q₈	—	26	4	no	—
123	C₄⋊Q₈	—	26	2	no	—
124	C₄⋊Q₈	C₂×#261	1	2	yes	—
125	C₂×C₄⋊C₄	—	26	4	no	—
126	C₂×C₄⋊C₄	—	26	4	no	—
127	C₂×C₄⋊C₄	—	26	2	no	—
128	C₂×C₄²	—	13	1	no	—
129	#106	—	52	4	no	—
130	C₄×C₈	—	13	2	no	—
131	C₄×C₈	—	13	1	no	—
132	C₄×C₈	—	26	2	no	—
133	C₄×C₈	—	26	2	no	—
134	C₄×C₈	C₂×#261	1	1	yes	characteristic
135	C₂×C₁₃	#1957	1	1	yes	characteristic
136	C₂×C₁₃	#1971	1	2	yes	—
137	D₁₃	#1716	1	2	yes	—
138	D₁₃	#1722	1	2	yes	—
139	C₂×Q₈	—	2	2	no	—
140	C₂×Q₈	—	26	2	no	—
141	C₂×Q₈	—	52	2	no	—
142	C₂×Q₈	—	52	4	no	—
143	C₂×Q₈	—	52	4	no	—
144	C₂²×C₄	—	13	2	no	—
145	C₂²×C₄	—	13	1	no	—
146	C₂²×C₄	—	52	2	no	—
147	C₂²×C₄	—	52	2	no	—
148	Q₁₆	—	4	4	no	—
149	Q₁₆	—	52	8	no	—
150	Q₁₆	—	52	4	no	—
151	C₂×C₈	—	13	2	no	—
152	C₂×C₈	—	13	4	no	—
153	C₂×C₈	—	13	4	no	—
154	C₂×C₈	—	26	4	no	—
155	C₂×C₈	—	26	8	no	—
156	C₂×C₈	C₂²⋊#261	1	2	yes	—
157	C₄⋊C₄	—	2	4	no	—
158	C₄⋊C₄	—	26	4	no	—
159	C₄⋊C₄	—	52	4	no	—
160	C₄⋊C₄	—	52	4	no	—
161	C₄⋊C₄	—	52	8	no	—
162	C₄²	—	13	2	no	—
163	C₄²	—	13	1	no	—
164	C₄²	C₂²×#261	1	1	yes	characteristic
165	C₁₃	#9839	1	1	yes	characteristic, complemented, Sylow
166	C₂³	—	13	1	no	—
167	Q₈	—	4	4	no	—
168	Q₈	—	52	4	no	—
169	Q₈	—	104	4	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	13	2	no	—
172	C₂×C₄	—	13	1	no	—
173	C₂×C₄	—	13	8	no	—
174	C₂×C₄	—	13	2	no	—
175	C₂×C₄	—	13	2	no	—
176	C₂×C₄	—	52	2	no	—
177	C₂×C₄	—	104	2	no	—
178	C₂×C₄	—	104	2	no	—
179	C₂×C₄	—	104	2	no	—
180	C₂×C₄	—	104	4	no	—
181	C₂×C₄	D₄×#261	1	2	yes	—
182	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
183	C₈	—	13	4	no	—
184	C₈	—	52	4	no	—
185	C₈	#88115	1	4	yes	—
186	C₂²	—	13	1	no	—
187	C₂²	—	13	1	no	—
188	C₂²	—	13	4	no	—
189	C₂²	#11719151	1	1	yes	characteristic
190	C₄	—	8	2	no	—
191	C₄	—	13	2	no	—
192	C₄	—	13	4	no	—
193	C₄	—	104	2	no	—
194	C₄	—	208	2	no	—
195	C₄	#11719110	1	4	yes	—
196	C₄	C₂×#88115	1	2	yes	—
197	C₂	—	13	2	no	—
198	C₂	—	13	2	no	—
199	C₂	#13130964	1	1	yes	characteristic
200	C₂	#13130984	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13131017

C₂

yes

maximal

#13130771

C₂

yes

maximal, characteristic

#13130030

C₂

yes

maximal, characteristic

D₁₃×#494

C₂

yes

maximal, characteristic

C₂×Q₁₆×D₁₃

—

D₁₃×C₄⋊Q₈

C₂²

yes

—

C₂×#88101

—

#11719059

C₂²

yes

—

C₄×D₁₃⋊C₈

C₂²

yes

—

C₁₃×#494

C₄

yes

characteristic, complemented

#11718842

—

#11718753

C₄

yes

complemented

#11718360

C₄

yes

characteristic, complemented

C₄×C₈×D₁₃

C₂²

yes

characteristic

#11718251

—

#11718246

—

C₂×Q₈×D₁₃

—

C₂×C₄⋊#261

—

C₂×C₄⋊#261

—

C₂×D₁₃⋊C₈

D₄

yes

—

C₂×C₁₃×Q₁₆

—

C₁₃×C₄⋊Q₈

C₂×C₄

yes

—

#88198

—

C₂×#6107

—

Q₁₆×D₁₃

—

Q₁₆×D₁₃

—

C₂×Dic₅₂

—

C₂×C₈×D₁₃

D₄

yes

—

C₄⋊C₄×D₁₃

—

#88142

—

C₄²×D₁₃

C₂³

yes

characteristic

#88123

C₂×C₄

yes

—

C₄×#1057

—

C₄×#1057

—

#88101

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₈×Dic₁₃

D₄

yes

—

C₄×#1055

C₂×C₄

yes

characteristic

#9839

—

maximal, Sylow

C₂²×#261

—

C₂×Q₈×C₁₃

—

Q₈×D₁₃

—

Q₈×D₁₃

—

C₂×C₄×D₁₃

—

C₂×C₄×D₁₃

C₂×D₄

yes

characteristic

C₂×C₄×D₁₃

C₂×D₄

yes

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×#1057

—

C₄⋊#261

—

C₄⋊#261

—

D₁₃⋊C₈

—

C₁₃×Q₁₆

—

C₂×C₈×C₁₃

C₂²⋊C₄

yes

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂²×C₄

yes

characteristic

#6107

—

C₄⋊Dic₁₃

—

#6101

—

C₄×Dic₁₃

C₂×D₄

yes

characteristic

C₄×Dic₁₃

#41

yes

—

C₂×#1055

C₂²⋊C₄

yes

—

Dic₅₂

—

C₈×D₁₃

D₈

yes

—

C₈×D₁₃

SD₁₆

yes

—

—

—

—

—

C₂²×D₁₃

yes

—

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₄×D₄

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

#121

yes

—

C₂×Dic₁₃

#124

yes

characteristic

C₄×D₁₃

—

C₄×D₁₃

—

C₄×D₁₃

C₂×D₈

yes

—

C₄×D₁₃

C₂×SD₁₆

yes

—

C₄×D₁₃

#137

yes

—

Dic₂₆

—

Dic₂₆

—

#1057

—

C₈×C₁₃

#102

yes

—

#1055

#102

yes

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

—

—

—

yes

characteristic, complemented

C₂×#106

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

#384

—

100

#340

—

101

#322

—

102

C₂²×C₁₃

#390

yes

characteristic

103

C₂×D₁₃

#492

yes

characteristic

104

C₂×D₁₃

#493

yes

characteristic

105

C₂×D₁₃

#497

yes

—

106

#261

—

107

C₄×C₁₃

—

108

C₄×C₁₃

C₂×#102

yes

—

109

C₄×C₁₃

#441

yes

—

110

Dic₁₃

—

111

Dic₁₃

#470

yes

—

112

Dic₁₃

#486

yes

—

113

C₂²×Q₈

—

114

C₂×Q₁₆

—

115

C₂×Q₁₆

—

116

C₂×Q₁₆

—

117

C₂×Q₁₆

—

118

C₂×Q₁₆

—

119

C₂²×C₈

—

120

C₂²×C₈

—

121

C₄⋊Q₈

—

122

C₄⋊Q₈

—

123

C₄⋊Q₈

—

124

C₄⋊Q₈

C₂×#261

yes

—

125

C₂×C₄⋊C₄

—

126

C₂×C₄⋊C₄

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄²

—

129

#106

—

130

C₄×C₈

—

131

C₄×C₈

—

132

C₄×C₈

—

133

C₄×C₈

—

134

C₄×C₈

C₂×#261

yes

characteristic

135

C₂×C₁₃

#1957

yes

characteristic

136

C₂×C₁₃

#1971

yes

—

137

D₁₃

#1716

yes

—

138

D₁₃

#1722

yes

—

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂×Q₈

—

142

C₂×Q₈

—

143

C₂×Q₈

—

144

C₂²×C₄

—

145

C₂²×C₄

—

146

C₂²×C₄

—

147

C₂²×C₄

—

148

Q₁₆

—

149

Q₁₆

—

150

Q₁₆

—

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

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄²

—

163

C₄²

—

164

C₄²

C₂²×#261

yes

characteristic

165

C₁₃

#9839

yes

characteristic, complemented, Sylow

166

C₂³

—

167

Q₈

—

168

Q₈

—

169

Q₈

—

104

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

104

—

178

C₂×C₄

—

104

—

179

C₂×C₄

—

104

—

180

C₂×C₄

—

104

—

181

C₂×C₄

D₄×#261

yes

—

182

C₂×C₄

C₂×C₂²⋊#261

yes

characteristic

183

C₈

—

184

C₈

—

185

C₈

#88115

yes

—

186

C₂²

—

187

C₂²

—

188

C₂²

—

189

C₂²

#11719151

yes

characteristic

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

—

104

—

194

C₄

—

208

—

195

C₄

#11719110

yes

—

196

C₄

C₂×#88115

yes

—

197

C₂

—

198

C₂

—

199

C₂

#13130964

yes

characteristic

200

C₂

#13130984

yes

—

	subgroup	G/H
center	C₂²	#11719151
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₄²	C₂²×#261
Fitting	C₁₃×#494	C₄
socle	C₂²×C₁₃	#390
prime radical	C₂²×D₁₃	#127
cosocle	C₄²×D₁₃	C₂³
hypercenter	#494	#261
hypocenter	C₁₃	#9839
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

#11719151

commutator

C₄²×C₁₃

C₂²×C₄

Frattini

C₄²

C₂²×#261

Fitting

C₁₃×#494

C₄

C₂²×C₁₃

C₂²×D₁₃

C₄²×D₁₃

C₂³

C₁₃

G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
10. C₁₃×#494	C₄	semidirect
12. #11718753	C₄	semidirect
13. #11718360	C₄	semidirect
95. #494	#261	semidirect
165. C₁₃	#9839	semidirect

H ≅ G/N

type

10. C₁₃×#494

C₄

semidirect

12. #11718753

C₄

semidirect

13. #11718360

C₄

semidirect

95. #494

#261

semidirect

165. C₁₃

#9839

semidirect

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

subgroup	index
#13131017	2
#13130771	2
#13130030	2
D₁₃×#494	2
#9839	13

subgroup

index

D₁₃×#494

Subgroups — #16784959

Conjugacy classes of subgroups

Complements

Maximal subgroups