Group #16785022 — subgroups

Subgroups — #16785022

Order 3328.

514 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 112 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	#13130775	C₂	1	1	yes	maximal, characteristic, complemented
2	#13130182	C₂	1	4	yes	maximal, complemented
3	#13130030	C₂	1	1	yes	maximal, characteristic, complemented
4	#13125263	C₂	1	1	yes	maximal, characteristic
5	C₂×#88166	—	2	2	no	—
6	#11719421	C₂²	1	2	yes	—
7	C₂×#88103	—	2	2	no	—
8	#11719052	C₂²	1	2	yes	—
9	C₄×D₁₃⋊C₈	C₂²	1	2	yes	—
10	C₁₃×#493	C₄	1	1	yes	characteristic
11	#11718816	—	2	2	no	—
12	#11718737	C₄	1	2	yes	—
13	#11718360	C₄	1	1	yes	characteristic
14	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
15	D₄⋊#1057	—	2	8	no	—
16	#11718251	—	2	2	no	—
17	C₂×#6129	—	2	2	no	—
18	C₂×#6118	—	2	2	no	—
19	C₂×D₁₃⋊C₈	—	2	2	no	—
20	C₂×C₁₃×D₈	—	2	2	no	—
21	C₁₃×#127	C₂×C₄	1	2	yes	—
22	#88192	—	2	4	no	—
23	#88190	—	2	2	no	—
24	D₄×Dic₁₃	—	2	4	no	—
25	C₂×#6105	—	2	4	no	—
26	#88166	—	4	4	no	—
27	#88166	—	4	4	no	—
28	C₂×Dic₅₂	—	2	2	no	—
29	C₂×C₈×D₁₃	D₄	1	2	yes	—
30	C₄²×D₁₃	C₂³	1	1	yes	characteristic
31	#88123	C₂×C₄	1	2	yes	—
32	C₄⋊#1057	—	2	4	no	—
33	C₄×#1057	—	2	2	no	—
34	C₄×#1057	D₄	1	4	yes	—
35	#88103	—	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	#9862	—	13	1	no	maximal, Sylow
40	C₂×D₄×C₁₃	—	2	2	no	—
41	C₂×D₄×C₁₃	—	2	4	no	—
42	C₂×C₂²⋊D₁₃	—	4	2	no	—
43	C₂²×Dic₁₃	—	4	2	no	—
44	#6129	—	4	4	no	—
45	#6129	—	4	4	no	—
46	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×D₁₃	#41	1	2	yes	—
48	C₂×Dic₂₆	—	2	2	no	—
49	C₂×#1057	—	2	4	no	—
50	C₂×#1057	—	4	2	no	—
51	#6118	—	4	4	no	—
52	D₁₃⋊C₈	—	4	4	no	—
53	C₁₃×D₈	—	4	4	no	—
54	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
55	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²⋊Dic₁₃	—	4	4	no	—
57	#6105	—	4	8	no	—
58	C₄⋊Dic₁₃	—	2	4	no	—
59	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
60	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
61	C₂×#1055	C₂²⋊C₄	1	2	yes	—
62	Dic₅₂	—	4	4	no	—
63	C₈×D₁₃	D₈	1	4	yes	—
64	C₈×D₁₃	SD₁₆	1	4	yes	—
65	#3147	—	13	1	no	—
66	#1848	—	13	1	no	—
67	#1492	—	13	4	no	—
68	C₂×#322	—	13	1	no	—
69	C₂³×C₁₃	—	4	2	no	—
70	C₂²×D₁₃	#124	1	1	yes	characteristic
71	D₄×C₁₃	—	4	4	no	—
72	D₄×C₁₃	—	4	8	no	—
73	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
75	C₂²⋊D₁₃	—	8	4	no	—
76	C₂×Dic₁₃	—	4	2	no	—
77	C₂×Dic₁₃	—	8	2	no	—
78	C₂×Dic₁₃	—	8	2	no	—
79	C₂×Dic₁₃	#121	1	2	yes	—
80	C₂×Dic₁₃	#127	1	1	yes	characteristic
81	C₄×D₁₃	—	2	4	no	—
82	C₄×D₁₃	C₂×D₈	1	2	yes	—
83	C₄×D₁₃	C₂×SD₁₆	1	2	yes	—
84	Dic₂₆	—	4	4	no	—
85	#1057	—	4	4	no	—
86	#1057	—	8	2	no	—
87	C₈×C₁₃	#102	1	4	yes	—
88	#1055	#102	1	4	yes	—
89	C₂×#135	—	26	2	no	—
90	#532	—	13	2	no	—
91	#495	—	26	2	no	—
92	#494	—	13	1	no	—
93	#493	#261	1	1	yes	characteristic
94	#492	—	13	2	no	—
95	C₂×#108	—	26	2	no	—
96	#423	—	13	2	no	—
97	C₂×C₄×C₈	—	13	1	no	—
98	C₂×C₄×C₈	—	13	2	no	—
99	D₄⋊C₈	—	26	8	no	—
100	#322	—	26	2	no	—
101	C₂²×C₁₃	—	8	2	no	—
102	C₂²×C₁₃	—	8	4	no	—
103	C₂²×C₁₃	#390	1	1	yes	characteristic
104	C₂×D₁₃	—	2	2	no	—
105	C₂×D₁₃	#486	1	2	yes	—
106	C₄×C₁₃	C₂×#102	1	2	yes	—
107	C₄×C₁₃	#444	1	4	yes	—
108	Dic₁₃	—	2	2	no	—
109	Dic₁₃	—	8	2	no	—
110	Dic₁₃	#492	1	1	yes	characteristic
111	Dic₁₃	#493	1	1	yes	characteristic
112	C₂×#41	—	26	2	no	—
113	#135	—	52	4	no	—
114	#135	—	52	4	no	—
115	C₂×Q₁₆	—	26	2	no	—
116	C₂×SD₁₆	—	26	4	no	—
117	C₂×D₈	—	2	2	no	—
118	C₂×M₁₆	—	26	2	no	—
119	C₂²×C₈	—	13	2	no	—
120	C₂²×C₈	—	26	2	no	—
121	C₄⋊Q₈	—	13	2	no	—
122	#127	C₂×#261	1	2	yes	—
123	#124	—	26	2	no	—
124	#121	—	26	4	no	—
125	C₄×D₄	—	26	4	no	—
126	C₂×C₄²	—	13	1	no	—
127	#108	—	52	4	no	—
128	C₄⋊C₈	—	26	4	no	—
129	C₄×C₈	—	13	2	no	—
130	C₄×C₈	—	13	4	no	—
131	C₄×C₈	—	13	1	no	—
132	C₄×C₈	—	26	2	no	—
133	C₄×C₈	C₂×#261	1	1	yes	characteristic
134	C₂×C₁₃	—	16	2	no	—
135	C₂×C₁₃	#1957	1	1	yes	characteristic
136	C₂×C₁₃	#1975	1	2	yes	—
137	D₁₃	—	2	2	no	—
138	#41	—	52	4	no	—
139	#41	—	52	4	no	—
140	C₂×Q₈	—	26	2	no	—
141	C₂×D₄	—	2	4	no	—
142	C₂×D₄	—	2	2	no	—
143	C₂×D₄	—	52	2	no	—
144	C₂²×C₄	—	13	1	no	—
145	C₂²×C₄	—	13	2	no	—
146	C₂²×C₄	—	52	2	no	—
147	Q₁₆	—	52	4	no	—
148	SD₁₆	—	52	8	no	—
149	D₈	—	4	4	no	—
150	M₁₆	—	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	—	52	4	no	—
156	C₂×C₈	—	52	2	no	—
157	C₂×C₈	C₂²⋊#261	1	2	yes	—
158	C₄⋊C₄	—	26	4	no	—
159	C₂²⋊C₄	—	52	4	no	—
160	C₄²	—	13	2	no	—
161	C₄²	—	13	1	no	—
162	C₄²	C₂²×#261	1	1	yes	characteristic
163	C₁₃	#9862	1	1	yes	characteristic, complemented, Sylow
164	C₂³	—	4	2	no	—
165	C₂³	—	13	1	no	—
166	Q₈	—	52	4	no	—
167	D₄	—	4	8	no	—
168	D₄	—	4	4	no	—
169	D₄	—	104	4	no	—
170	C₂×C₄	—	13	1	no	—
171	C₂×C₄	—	13	2	no	—
172	C₂×C₄	—	13	2	no	—
173	C₂×C₄	—	13	2	no	—
174	C₂×C₄	—	26	4	no	—
175	C₂×C₄	—	52	2	no	—
176	C₂×C₄	—	104	2	no	—
177	C₂×C₄	—	104	2	no	—
178	C₂×C₄	D₄×#261	1	2	yes	—
179	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
180	C₈	—	13	4	no	—
181	C₈	—	52	4	no	—
182	C₈	—	104	2	no	—
183	C₈	#88115	1	4	yes	—
184	C₂²	—	8	2	no	—
185	C₂²	—	8	4	no	—
186	C₂²	—	13	2	no	—
187	C₂²	—	26	2	no	—
188	C₂²	#11719151	1	1	yes	characteristic
189	C₄	—	13	1	no	—
190	C₄	—	13	1	no	—
191	C₄	—	26	2	no	—
192	C₄	—	104	2	no	—
193	C₄	#11719104	1	4	yes	—
194	C₄	C₂×#88115	1	2	yes	—
195	C₂	—	16	2	no	—
196	C₂	—	26	2	no	—
197	C₂	#13130964	1	1	yes	characteristic
198	C₂	#13130991	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13130775

C₂

yes

maximal, characteristic, complemented

#13130182

C₂

yes

maximal, complemented

#13130030

C₂

yes

maximal, characteristic, complemented

#13125263

C₂

yes

maximal, characteristic

C₂×#88166

—

#11719421

C₂²

yes

—

C₂×#88103

—

#11719052

C₂²

yes

—

C₄×D₁₃⋊C₈

C₂²

yes

—

C₁₃×#493

C₄

yes

characteristic

#11718816

—

#11718737

C₄

yes

—

#11718360

C₄

yes

characteristic

C₄×C₈×D₁₃

C₂²

yes

—

—

—

—

C₂×D₁₃⋊C₈

—

C₂×C₁₃×D₈

—

C₁₃×#127

C₂×C₄

yes

—

#88192

—

#88190

—

D₄×Dic₁₃

—

C₂×#6105

—

#88166

—

#88166

—

C₂×Dic₅₂

—

C₂×C₈×D₁₃

D₄

yes

—

C₄²×D₁₃

C₂³

yes

characteristic

#88123

C₂×C₄

yes

—

C₄⋊#1057

—

C₄×#1057

—

C₄×#1057

D₄

yes

—

#88103

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₈×Dic₁₃

D₄

yes

—

C₄×#1055

C₂×C₄

yes

characteristic

#9862

—

maximal, Sylow

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×C₂²⋊D₁₃

—

C₂²×Dic₁₃

—

#6129

—

#6129

—

C₂×C₄×D₁₃

C₂×D₄

yes

characteristic

C₂×C₄×D₁₃

#41

yes

—

C₂×Dic₂₆

—

C₂×#1057

—

C₂×#1057

—

#6118

—

D₁₃⋊C₈

—

C₁₃×D₈

—

C₂×C₈×C₁₃

C₂²⋊C₄

yes

—

C₄²×C₁₃

C₂²×C₄

yes

characteristic

C₂²⋊Dic₁₃

—

#6105

—

C₄⋊Dic₁₃

—

C₄×Dic₁₃

C₂×D₄

yes

characteristic

C₄×Dic₁₃

C₂×D₄

yes

—

C₂×#1055

C₂²⋊C₄

yes

—

Dic₅₂

—

C₈×D₁₃

D₈

yes

—

C₈×D₁₃

SD₁₆

yes

—

—

—

—

—

C₂³×C₁₃

—

C₂²×D₁₃

#124

yes

characteristic

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₄×D₄

yes

—

C₂²⋊D₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

#121

yes

—

C₂×Dic₁₃

#127

yes

characteristic

C₄×D₁₃

—

C₄×D₁₃

C₂×D₈

yes

—

C₄×D₁₃

C₂×SD₁₆

yes

—

Dic₂₆

—

#1057

—

#1057

—

C₈×C₁₃

#102

yes

—

#1055

#102

yes

—

—

—

—

—

yes

—

—

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

D₄⋊C₈

—

100

#322

—

101

C₂²×C₁₃

—

102

C₂²×C₁₃

—

103

C₂²×C₁₃

#390

yes

characteristic

104

C₂×D₁₃

—

105

C₂×D₁₃

#486

yes

—

106

C₄×C₁₃

C₂×#102

yes

—

107

C₄×C₁₃

#444

yes

—

108

Dic₁₃

—

109

Dic₁₃

—

110

Dic₁₃

#492

yes

characteristic

111

Dic₁₃

#493

yes

characteristic

112

C₂×#41

—

113

#135

—

114

#135

—

115

C₂×Q₁₆

—

116

C₂×SD₁₆

—

117

C₂×D₈

—

118

C₂×M₁₆

—

119

C₂²×C₈

—

120

C₂²×C₈

—

121

C₄⋊Q₈

—

122

#127

C₂×#261

yes

—

123

#124

—

124

#121

—

125

C₄×D₄

—

126

C₂×C₄²

—

127

#108

—

128

C₄⋊C₈

—

129

C₄×C₈

—

130

C₄×C₈

—

131

C₄×C₈

—

132

C₄×C₈

—

133

C₄×C₈

C₂×#261

yes

characteristic

134

C₂×C₁₃

—

135

C₂×C₁₃

#1957

yes

characteristic

136

C₂×C₁₃

#1975

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

—

146

C₂²×C₄

—

147

Q₁₆

—

148

SD₁₆

—

149

D₈

—

150

M₁₆

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

C₂²⋊#261

yes

—

158

C₄⋊C₄

—

159

C₂²⋊C₄

—

160

C₄²

—

161

C₄²

—

162

C₄²

C₂²×#261

yes

characteristic

163

C₁₃

#9862

yes

characteristic, complemented, Sylow

164

C₂³

—

165

C₂³

—

166

Q₈

—

167

D₄

—

168

D₄

—

169

D₄

—

104

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

104

—

177

C₂×C₄

—

104

—

178

C₂×C₄

D₄×#261

yes

—

179

C₂×C₄

C₂×C₂²⋊#261

yes

characteristic

180

C₈

—

181

C₈

—

182

C₈

—

104

—

183

C₈

#88115

yes

—

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

#11719151

yes

characteristic

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

104

—

193

C₄

#11719104

yes

—

194

C₄

C₂×#88115

yes

—

195

C₂

—

196

C₂

—

197

C₂

#13130964

yes

characteristic

198

C₂

#13130991

yes

—

	subgroup	G/H
center	C₂²	#11719151
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₄²	C₂²×#261
Fitting	C₁₃×#493	C₄
socle	C₂²×C₁₃	#390
prime radical	#13125263	C₂
cosocle	C₄²×D₁₃	C₂³
hypercenter	#493	#261
hypocenter	C₁₃	#9862
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#11719151

commutator

C₄²×C₁₃

C₂²×C₄

Frattini

C₄²

C₂²×#261

Fitting

C₁₃×#493

C₄

C₂²×C₁₃

C₂

C₄²×D₁₃

C₂³

C₁₃

G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #13130775	C₂	semidirect
2. #13130182	C₂	semidirect
3. #13130030	C₂	semidirect
163. C₁₃	#9862	semidirect

H ≅ G/N

type

1. #13130775

C₂

semidirect

2. #13130182

C₂

semidirect

3. #13130030

C₂

semidirect

163. C₁₃

#9862

semidirect

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

subgroup	index
#13130775	2
#13130182	2
#13130030	2
#13125263	2
#9862	13

subgroup

index

Subgroups — #16785022

Conjugacy classes of subgroups

Complements

Maximal subgroups