Group #16790288 — subgroups

Subgroups — #16790288

Order 3328.

370 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 120 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	#13131803	C₂	1	1	yes	maximal, characteristic
2	#13131177	C₂	1	2	yes	maximal, complemented
3	#13131066	C₂	1	1	yes	maximal, characteristic
4	#13129891	C₂	1	2	yes	maximal, complemented
5	#13114639	C₂	1	1	yes	maximal, characteristic
6	C₂×#88202	C₂²	1	1	yes	characteristic
7	#11719162	—	2	4	no	—
8	C₂×#88106	—	2	2	no	—
9	C₂×#88106	C₂²	1	1	yes	characteristic
10	C₂×#88106	C₂²	1	2	yes	—
11	C₂²×#6092	C₂²	1	1	yes	characteristic
12	C₁₃×#413	C₄	1	1	yes	characteristic
13	#11718901	C₄	1	1	yes	characteristic
14	#11718783	C₂²	1	2	yes	—
15	C₂²⋊#6092	—	2	2	no	—
16	#11718261	—	2	4	no	—
17	#11718151	C₄	1	2	yes	—
18	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
20	#88202	—	2	2	no	—
21	#88202	—	2	2	no	—
22	#88180	C₂×C₄	1	2	yes	—
23	C₂×#6099	C₂×C₄	1	1	yes	characteristic
24	C₂×#6099	Q₈	1	2	yes	—
25	C₂²×#1055	D₄	1	2	yes	—
26	C₂²×#1055	C₂³	1	1	yes	characteristic
27	#88106	—	2	4	no	—
28	#88106	—	2	2	no	—
29	#88106	—	2	2	no	—
30	#88106	D₄	1	4	yes	—
31	C₂×#6092	—	2	4	no	—
32	C₂×#6092	—	2	2	no	—
33	C₂×#6092	—	2	1	no	—
34	C₂×#6092	—	2	1	no	—
35	C₂×#6092	—	2	2	no	—
36	C₁₃×#101	C₈	1	2	yes	—
37	C₁₃×#100	C₈	1	2	yes	—
38	#88074	—	2	2	no	—
39	#88055	—	2	2	no	—
40	C₈×Dic₁₃	—	2	2	no	—
41	#12916	—	13	1	no	maximal, Sylow
42	C₁₃×#41	—	2	2	no	—
43	C₁₃×#41	—	2	2	no	—
44	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
45	C₂×D₄×C₁₃	C₂×C₈	1	1	yes	characteristic
46	C₂×D₄×C₁₃	M₁₆	1	2	yes	—
47	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
49	C₁₃×M₁₆	—	2	2	no	—
50	C₁₃×M₁₆	C₂×C₈	1	4	yes	—
51	C₂×C₈×C₁₃	—	2	2	no	—
52	C₂²⋊Dic₁₃	—	2	2	no	—
53	C₄⋊Dic₁₃	—	2	2	no	—
54	C₄×Dic₁₃	—	2	2	no	—
55	#6099	—	2	2	no	—
56	#6099	—	2	4	no	—
57	C₂×#1055	—	2	2	no	—
58	C₂×#1055	—	2	2	no	—
59	C₂×#1055	C₄⋊C₄	1	2	yes	—
60	C₂×#1055	C₂×D₄	1	2	yes	—
61	C₂×#1055	C₂×D₄	1	1	yes	characteristic
62	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
63	C₂×#1055	#41	1	2	yes	—
64	#6092	—	2	4	no	—
65	#6092	—	4	1	no	—
66	#6092	—	4	1	no	—
67	#3407	—	13	1	no	—
68	#2153	—	13	2	no	—
69	#2113	—	13	1	no	—
70	#1757	—	13	1	no	—
71	#1379	—	13	2	no	—
72	C₂³×C₁₃	C₄⋊C₈	1	2	yes	—
73	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
74	Q₈×C₁₃	—	2	2	no	—
75	D₄×C₁₃	—	2	4	no	—
76	D₄×C₁₃	—	2	2	no	—
77	C₂×C₄×C₁₃	—	2	2	no	—
78	C₂×C₄×C₁₃	—	2	2	no	—
79	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
80	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
82	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
83	C₂×C₄×C₁₃	#131	1	2	yes	—
84	C₂×Dic₁₃	—	4	2	no	—
85	C₈×C₁₃	—	2	4	no	—
86	#1055	—	2	2	no	—
87	#1055	#120	1	1	yes	characteristic
88	#1055	C₂²⋊Q₈	1	2	yes	—
89	#1055	#123	1	1	yes	characteristic
90	C₂×#131	—	13	1	no	—
91	#504	—	26	4	no	—
92	C₂×M₃₂	—	13	2	no	—
93	C₂×M₃₂	—	13	1	no	—
94	C₂×M₃₂	—	26	2	no	—
95	C₂²×C₁₆	—	13	1	no	—
96	#413	#261	1	1	yes	characteristic
97	#410	—	13	1	no	—
98	#405	—	13	2	no	—
99	#364	—	26	4	no	—
100	C₂²⋊C₁₆	—	26	2	no	—
101	#344	—	13	2	no	—
102	C₂²×C₁₃	—	2	2	no	—
103	C₂²×C₁₃	—	4	1	no	—
104	C₂²×C₁₃	—	4	2	no	—
105	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
106	C₂²×C₁₃	#435	1	2	yes	—
107	C₄×C₁₃	—	2	2	no	—
108	C₄×C₁₃	#385	1	1	yes	characteristic
109	C₄×C₁₃	#424	1	1	yes	characteristic
110	C₄×C₁₃	C₈×D₄	1	2	yes	—
111	Dic₁₃	—	8	2	no	—
112	C₂×#41	C₂×#261	1	1	yes	characteristic
113	#131	—	26	2	no	—
114	#131	—	26	2	no	—
115	C₂×M₁₆	—	13	2	no	—
116	C₂×M₁₆	—	13	1	no	—
117	C₂×M₁₆	C₂×#261	1	2	yes	—
118	C₂²×C₈	—	13	1	no	—
119	C₂²×C₈	—	13	2	no	—
120	#117	—	13	2	no	—
121	M₃₂	—	13	4	no	—
122	M₃₂	—	26	2	no	—
123	M₃₂	—	26	4	no	—
124	M₃₂	—	26	2	no	—
125	C₂×C₁₆	—	26	2	no	—
126	C₂×C₁₆	—	26	4	no	—
127	C₂×C₁₆	—	26	1	no	—
128	C₂×C₁₆	—	26	1	no	—
129	C₂×C₁₆	—	26	2	no	—
130	#101	#1057	1	2	yes	—
131	#100	#1057	1	2	yes	—
132	#99	—	26	2	no	—
133	#97	—	26	2	no	—
134	C₄×C₈	—	26	2	no	—
135	C₂×C₁₃	—	2	2	no	—
136	C₂×C₁₃	—	2	1	no	—
137	C₂×C₁₃	—	4	2	no	—
138	C₂×C₁₃	#1850	1	1	yes	characteristic
139	#41	—	2	2	no	—
140	#41	—	2	2	no	—
141	C₂×Q₈	C₂×#1057	1	1	yes	characteristic
142	C₂×D₄	#6118	1	2	yes	—
143	C₂×D₄	C₂×#1057	1	1	yes	characteristic
144	C₂²×C₄	C₄⋊#261	1	2	yes	—
145	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
146	M₁₆	—	2	2	no	—
147	M₁₆	—	26	4	no	—
148	M₁₆	—	26	2	no	—
149	M₁₆	C₂×#1057	1	4	yes	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	13	1	no	—
152	C₂×C₈	—	13	1	no	—
153	C₂×C₈	—	13	2	no	—
154	C₂×C₈	—	13	2	no	—
155	C₂×C₈	—	13	2	no	—
156	C₂×C₈	—	26	2	no	—
157	C₂×C₈	—	26	2	no	—
158	C₄⋊C₄	—	26	2	no	—
159	C₂²⋊C₄	—	26	2	no	—
160	C₄²	—	26	2	no	—
161	C₁₆	—	26	4	no	—
162	C₁₆	—	52	1	no	—
163	C₁₆	—	52	1	no	—
164	C₁₃	#12916	1	1	yes	characteristic, complemented, Sylow
165	C₂³	C₄⋊#1057	1	2	yes	—
166	C₂³	C₂×#6118	1	1	yes	characteristic
167	Q₈	—	2	2	no	—
168	D₄	—	2	4	no	—
169	D₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	52	2	no	—
173	C₂×C₄	C₄⋊#1057	1	2	yes	—
174	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
175	C₂×C₄	#88238	1	2	yes	—
176	C₂×C₄	D₄×#261	1	2	yes	—
177	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
178	C₈	—	2	4	no	—
179	C₈	—	13	1	no	—
180	C₈	—	13	1	no	—
181	C₈	—	13	2	no	—
182	C₈	—	26	2	no	—
183	C₂²	—	2	2	no	—
184	C₂²	—	4	1	no	—
185	C₂²	—	4	2	no	—
186	C₂²	#11719064	1	2	yes	—
187	C₂²	C₂×C₄⋊#1057	1	1	yes	characteristic
188	C₄	—	2	2	no	—
189	C₄	—	104	2	no	—
190	C₄	#11719067	1	1	yes	characteristic
191	C₄	#11719070	1	1	yes	characteristic
192	C₄	D₄×#1057	1	2	yes	—
193	C₂	—	2	2	no	—
194	C₂	—	2	1	no	—
195	C₂	—	4	2	no	—
196	C₂	#13130779	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13131803

C₂

yes

maximal, characteristic

#13131177

C₂

yes

maximal, complemented

#13131066

C₂

yes

maximal, characteristic

#13129891

C₂

yes

maximal, complemented

#13114639

C₂

yes

maximal, characteristic

C₂×#88202

C₂²

yes

—

—

C₂²

yes

characteristic

C₂×#88106

C₂²

yes

—

C₂²×#6092

C₂²

yes

characteristic

C₁₃×#413

C₄

yes

characteristic

#11718901

C₄

yes

characteristic

#11718783

C₂²

yes

—

C₂²⋊#6092

—

#11718261

—

#11718151

C₄

yes

—

C₂×C₁₃×#41

C₂×C₄

yes

characteristic

C₂×C₁₃×M₁₆

C₂×C₄

yes

—

#88202

—

#88202

—

#88180

C₂×C₄

yes

—

C₂×#6099

C₂×C₄

yes

characteristic

C₂×#6099

Q₈

yes

—

C₂²×#1055

D₄

yes

—

C₂²×#1055

C₂³

yes

—

—

—

D₄

yes

—

—

—

—

—

—

C₁₃×#101

C₈

yes

—

C₁₃×#100

C₈

yes

—

#88074

—

#88055

—

C₈×Dic₁₃

—

#12916

—

maximal, Sylow

C₁₃×#41

—

C₁₃×#41

—

C₂×Q₈×C₁₃

C₂×C₈

yes

characteristic

C₂×D₄×C₁₃

C₂×C₈

yes

characteristic

C₂×D₄×C₁₃

M₁₆

yes

—

C₂²×C₄×C₁₃

C₄⋊C₄

yes

—

C₂²×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₁₃×M₁₆

—

C₁₃×M₁₆

C₂×C₈

yes

—

C₂×C₈×C₁₃

—

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄×Dic₁₃

—

—

—

—

—

C₄⋊C₄

yes

—

C₂×#1055

C₂×D₄

yes

—

C₂×#1055

C₂×D₄

yes

characteristic

C₂×#1055

C₂×Q₈

yes

characteristic

C₂×#1055

#41

yes

—

—

—

—

—

—

—

—

—

C₂³×C₁₃

C₄⋊C₈

yes

—

C₂³×C₁₃

C₂×M₁₆

yes

characteristic

Q₈×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₄⋊C₈

yes

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₄×D₄

yes

—

C₂×C₄×C₁₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₃

#131

yes

—

C₂×Dic₁₃

—

C₈×C₁₃

—

—

yes

C₂²⋊Q₈

yes

—

yes

—

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂²×C₁₆

—

yes

—

—

—

100

C₂²⋊C₁₆

—

101

#344

—

102

C₂²×C₁₃

—

103

C₂²×C₁₃

—

104

C₂²×C₁₃

—

105

C₂²×C₁₃

C₂×C₄⋊C₈

yes

characteristic

106

C₂²×C₁₃

#435

yes

—

107

C₄×C₁₃

—

108

C₄×C₁₃

#385

yes

characteristic

109

C₄×C₁₃

#424

yes

characteristic

110

C₄×C₁₃

C₈×D₄

yes

—

111

Dic₁₃

—

112

C₂×#41

C₂×#261

yes

characteristic

113

#131

—

114

#131

—

115

C₂×M₁₆

—

116

C₂×M₁₆

—

117

C₂×M₁₆

C₂×#261

yes

—

118

C₂²×C₈

—

119

C₂²×C₈

—

120

#117

—

121

M₃₂

—

122

M₃₂

—

123

M₃₂

—

124

M₃₂

—

125

C₂×C₁₆

—

126

C₂×C₁₆

—

127

C₂×C₁₆

—

128

C₂×C₁₆

—

129

C₂×C₁₆

—

130

#101

#1057

yes

—

131

#100

#1057

yes

—

132

#99

—

133

#97

—

134

C₄×C₈

—

135

C₂×C₁₃

—

136

C₂×C₁₃

—

137

C₂×C₁₃

—

138

C₂×C₁₃

#1850

yes

characteristic

139

#41

—

140

#41

—

141

C₂×Q₈

C₂×#1057

yes

characteristic

142

C₂×D₄

#6118

yes

—

143

C₂×D₄

C₂×#1057

yes

characteristic

144

C₂²×C₄

C₄⋊#261

yes

—

145

C₂²×C₄

C₂²×#261

yes

characteristic

146

M₁₆

—

147

M₁₆

—

148

M₁₆

—

149

M₁₆

C₂×#1057

yes

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₄⋊C₄

—

159

C₂²⋊C₄

—

160

C₄²

—

161

C₁₆

—

162

C₁₆

—

163

C₁₆

—

164

C₁₃

#12916

yes

characteristic, complemented, Sylow

165

C₂³

C₄⋊#1057

yes

—

166

C₂³

C₂×#6118

yes

characteristic

167

Q₈

—

168

D₄

—

169

D₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

C₄⋊#1057

yes

—

174

C₂×C₄

C₂×C₄⋊#261

yes

characteristic

175

C₂×C₄

#88238

yes

—

176

C₂×C₄

D₄×#261

yes

—

177

C₂×C₄

C₂²×#1057

yes

characteristic

178

C₈

—

179

C₈

—

180

C₈

—

181

C₈

—

182

C₈

—

183

C₂²

—

184

C₂²

—

185

C₂²

—

186

C₂²

#11719064

yes

—

187

C₂²

C₂×C₄⋊#1057

yes

characteristic

188

C₄

—

189

C₄

—

104

—

190

C₄

#11719067

yes

characteristic

191

C₄

#11719070

yes

characteristic

192

C₄

D₄×#1057

yes

—

193

C₂

—

194

C₂

—

195

C₂

—

196

C₂

#13130779

yes

characteristic

	subgroup	G/H
center	C₄	#11719070
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₁₃×#413	C₄
socle	C₂×C₁₃	#1850
prime radical	C₂×C₁₃×#41	C₂×C₄
cosocle	C₂²×#1055	C₂³
hypercenter	#413	#261
hypocenter	C₁₃	#12916
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

subgroup

G/H

center

C₄

#11719070

commutator

C₂×C₄×C₁₃

C₂²×C₈

Frattini

C₂²×C₄

C₂²×#261

Fitting

C₁₃×#413

C₄

socle

C₂×C₁₃

#1850

prime radical

C₂×C₁₃×#41

C₂×C₄

cosocle

C₂²×#1055

C₂³

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

N	H ≅ G/N	type
2. #13131177	C₂	semidirect
4. #13129891	C₂	semidirect
164. C₁₃	#12916	semidirect

H ≅ G/N

type

2. #13131177

C₂

semidirect

4. #13129891

C₂

semidirect

164. C₁₃

#12916

semidirect

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

subgroup	index
#13131803	2
#13131177	2
#13131066	2
#13129891	2
#13114639	2
#12916	13

subgroup

index

Subgroups — #16790288

Conjugacy classes of subgroups

Complements

Maximal subgroups