Group #16790859 — subgroups

Subgroups — #16790859

Order 3328.

522 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 136 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	C₂×C₄⋊#6092	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11718259	C₂	1	4	yes	maximal, complemented
3	C₂×#11718231	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11718495	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₈×D₁₃	C₂²	1	2	yes	—
6	C₂×C₄×D₅₂	C₄	1	1	yes	characteristic
7	C₂²×#6092	C₂²	1	2	yes	—
8	C₂²×#6092	C₂²	1	2	yes	—
9	C₂×C₁₃×C₄⋊C₈	C₄	1	1	yes	characteristic
10	C₂×#88060	C₄	1	2	yes	—
11	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
12	#11718495	—	2	2	no	—
13	#11718495	—	2	2	no	—
14	#11718259	—	2	8	no	—
15	#11718231	—	2	2	no	—
16	C₄⋊#6092	—	2	2	no	—
17	C₂²×D₅₂	C₈	1	1	yes	characteristic
18	C₂²×C₄×D₁₃	C₂×C₄	1	2	yes	—
19	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂×#6103	C₈	1	2	yes	—
22	C₂×C₄⋊Dic₁₃	C₈	1	1	yes	characteristic
23	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
24	C₂²×#1055	C₂³	1	1	yes	characteristic
25	C₂×C₈×D₁₃	—	2	2	no	—
26	C₂×C₈×D₁₃	—	2	2	no	—
27	C₂×C₈×D₁₃	—	2	4	no	—
28	C₂×C₈×D₁₃	—	2	4	no	—
29	C₄×D₅₂	—	2	2	no	—
30	C₄×D₅₂	—	2	2	no	—
31	C₂×#6092	—	2	2	no	—
32	C₂×#6092	—	2	2	no	—
33	C₂×#6092	—	2	2	no	—
34	C₂×#6092	—	2	4	no	—
35	C₂×#6092	D₄	1	4	yes	complemented
36	C₁₃×C₄⋊C₈	—	2	2	no	—
37	#88060	—	2	4	no	—
38	C₄×#1055	D₄	1	2	yes	—
39	C₄×#1055	D₄	1	2	yes	—
40	#13190	—	13	1	no	maximal, Sylow
41	C₂³×D₁₃	C₂×C₈	1	2	yes	—
42	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
45	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
46	C₂×D₅₂	—	2	4	no	—
47	C₂×D₅₂	—	2	2	no	—
48	C₂×C₄×D₁₃	—	2	2	no	—
49	C₂×C₄×D₁₃	—	2	4	no	—
50	C₂×C₄×D₁₃	—	2	2	no	—
51	C₂×C₄×D₁₃	—	2	4	no	—
52	C₂×C₈×C₁₃	—	2	2	no	—
53	C₂×C₈×C₁₃	—	2	4	no	—
54	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
55	C₄²×C₁₃	M₁₆	1	2	yes	—
56	#6103	—	2	4	no	—
57	C₄⋊Dic₁₃	—	2	2	no	—
58	C₂×#1055	—	2	2	no	—
59	C₂×#1055	C₂²⋊C₄	1	2	yes	—
60	C₂×#1055	C₂×D₄	1	1	yes	characteristic
61	C₂×#1055	C₂×D₄	1	2	yes	—
62	C₂×#1055	C₂×D₄	1	1	yes	characteristic
63	C₂×#1055	#41	1	2	yes	—
64	C₈×D₁₃	—	4	4	no	—
65	C₈×D₁₃	—	4	2	no	—
66	C₈×D₁₃	—	4	2	no	—
67	#6092	—	2	4	no	—
68	#6092	—	4	2	no	—
69	C₂×C₈×D₄	—	13	1	no	—
70	C₂×C₄⋊C₁₆	—	13	1	no	—
71	C₂×C₂²⋊C₁₆	—	13	4	no	—
72	C₂×#346	—	13	1	no	—
73	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
74	C₂²×D₁₃	—	2	2	no	—
75	C₂²×D₁₃	—	2	4	no	—
76	C₂²×D₁₃	—	2	2	no	—
77	C₂²×D₁₃	—	2	4	no	—
78	C₂²×D₁₃	M₃₂	1	4	yes	—
79	C₂×C₄×C₁₃	—	2	2	no	—
80	C₂×C₄×C₁₃	—	2	2	no	—
81	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
82	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
83	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
85	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
86	C₂×C₄×C₁₃	#131	1	2	yes	—
87	C₂×Dic₁₃	—	2	2	no	—
88	C₂×Dic₁₃	—	2	4	no	—
89	D₅₂	—	2	8	no	—
90	C₄×D₁₃	—	4	2	no	—
91	C₄×D₁₃	—	4	2	no	—
92	C₄×D₁₃	—	4	4	no	—
93	C₈×C₁₃	—	4	2	no	—
94	#1055	—	2	2	no	—
95	#1055	#121	1	2	yes	—
96	#1055	#124	1	1	yes	characteristic
97	#1055	#127	1	1	yes	characteristic
98	C₂³×C₈	—	13	2	no	—
99	C₂×C₄×D₄	—	13	1	no	—
100	C₂²×C₁₆	—	13	2	no	—
101	C₂²×C₁₆	—	13	2	no	—
102	C₈×D₄	—	26	2	no	—
103	C₈×D₄	—	26	2	no	—
104	C₂×C₄⋊C₈	#261	1	1	yes	characteristic
105	C₂×C₂²⋊C₈	—	13	2	no	—
106	C₂×C₄×C₈	—	13	1	no	—
107	C₄⋊C₁₆	—	26	2	no	—
108	C₂²⋊C₁₆	—	26	8	no	—
109	#346	—	26	2	no	—
110	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
111	C₂²×C₁₃	C₈×D₄	1	2	yes	—
112	C₂²×C₁₃	C₂×M₃₂	1	2	yes	—
113	C₂²×C₁₃	#504	1	2	yes	—
114	C₂×D₁₃	—	2	4	no	—
115	C₂×D₁₃	—	2	8	no	—
116	C₂×D₁₃	—	4	2	no	—
117	C₂×D₁₃	—	4	4	no	—
118	C₂×D₁₃	—	4	2	no	—
119	C₄×C₁₃	—	2	2	no	—
120	C₄×C₁₃	—	2	2	no	—
121	C₄×C₁₃	#390	1	1	yes	characteristic
122	C₄×C₁₃	#408	1	1	yes	characteristic
123	C₄×C₁₃	#436	1	2	yes	—
124	Dic₁₃	—	4	2	no	—
125	C₂²×D₄	—	13	1	no	—
126	C₂³×C₄	—	13	2	no	—
127	C₂²×C₈	—	13	1	no	—
128	C₂²×C₈	—	13	1	no	—
129	C₂²×C₈	—	26	4	no	—
130	C₂²×C₈	—	26	4	no	—
131	C₂²×C₈	—	26	2	no	—
132	C₂²×C₈	—	26	2	no	—
133	C₂²×C₈	C₂×#261	1	2	yes	—
134	C₄×D₄	—	26	2	no	—
135	C₄×D₄	—	26	2	no	—
136	C₂×C₄⋊C₄	—	13	1	no	—
137	C₂×C₂²⋊C₄	—	13	2	no	—
138	C₂×C₄²	C₂×#261	1	1	yes	characteristic
139	C₂×C₁₆	—	13	4	no	—
140	C₂×C₁₆	—	26	2	no	—
141	C₂×C₁₆	—	26	2	no	—
142	C₂×C₁₆	—	26	4	no	—
143	C₂×C₁₆	—	26	2	no	—
144	C₄⋊C₈	—	2	2	no	—
145	C₂²⋊C₈	—	26	4	no	—
146	C₄×C₈	—	13	2	no	—
147	C₄×C₈	—	13	2	no	—
148	C₂×C₁₃	#1954	1	1	yes	characteristic
149	C₂×C₁₃	#2112	1	1	yes	characteristic
150	C₂×C₁₃	#2113	1	1	yes	characteristic
151	C₂×C₁₃	#2168	1	4	yes	—
152	D₁₃	—	4	4	no	—
153	C₂⁴	—	13	2	no	—
154	C₂×D₄	—	26	4	no	—
155	C₂×D₄	—	26	2	no	—
156	C₂²×C₄	—	13	2	no	—
157	C₂²×C₄	—	26	2	no	—
158	C₂²×C₄	—	26	4	no	—
159	C₂²×C₄	—	26	4	no	—
160	C₂²×C₄	—	26	2	no	—
161	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
162	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
163	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
164	C₂×C₈	—	2	4	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	13	2	no	—
167	C₂×C₈	—	13	2	no	—
168	C₂×C₈	—	13	1	no	—
169	C₂×C₈	—	13	1	no	—
170	C₂×C₈	—	13	2	no	—
171	C₂×C₈	—	26	2	no	—
172	C₂×C₈	—	52	4	no	—
173	C₂×C₈	—	52	2	no	—
174	C₂×C₈	—	52	2	no	—
175	C₄⋊C₄	—	26	2	no	—
176	C₂²⋊C₄	—	26	4	no	—
177	C₄²	#6118	1	2	yes	—
178	C₄²	C₂²⋊#261	1	2	yes	—
179	C₁₆	—	26	4	no	—
180	C₁₆	—	52	2	no	—
181	C₁₃	#13190	1	1	yes	characteristic, complemented, Sylow
182	C₂³	—	13	4	no	—
183	C₂³	—	26	4	no	—
184	C₂³	—	26	2	no	—
185	C₂³	—	26	4	no	—
186	C₂³	—	26	2	no	—
187	C₂³	C₂×D₁₃⋊C₈	1	1	yes	characteristic
188	D₄	—	26	8	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	26	4	no	—
192	C₂×C₄	—	26	2	no	—
193	C₂×C₄	—	52	4	no	—
194	C₂×C₄	—	52	2	no	—
195	C₂×C₄	—	52	2	no	—
196	C₂×C₄	#88111	1	2	yes	—
197	C₂×C₄	#88111	1	2	yes	—
198	C₂×C₄	C₂×#6118	1	1	yes	characteristic
199	C₂×C₄	D₄×#261	1	2	yes	—
200	C₂×C₄	#88240	1	2	yes	—
201	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
202	C₈	—	4	2	no	—
203	C₈	—	13	2	no	—
204	C₈	—	13	1	no	—
205	C₈	—	13	1	no	—
206	C₈	—	26	2	no	—
207	C₂²	—	26	4	no	—
208	C₂²	—	26	8	no	—
209	C₂²	—	52	2	no	—
210	C₂²	—	52	2	no	—
211	C₂²	—	52	4	no	—
212	C₂²	#11719073	1	2	yes	—
213	C₂²	D₁₃⋊M₃₂	1	2	yes	—
214	C₂²	#11719096	1	2	yes	—
215	C₂²	C₂×#88111	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	2	2	no	—
218	C₄	—	52	2	no	—
219	C₄	#11719076	1	2	yes	—
220	C₄	#11719151	1	1	yes	characteristic
221	C₄	#11719159	1	1	yes	characteristic
222	C₂	—	52	4	no	—
223	C₂	#13130958	1	1	yes	characteristic
224	C₂	#13131065	1	1	yes	characteristic
225	C₂	#13131068	1	1	yes	characteristic
226	C₂	#13131219	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#6092

C₂

yes

maximal, characteristic, complemented

C₂×#11718259

C₂

yes

maximal, complemented

C₂×#11718231

C₂

yes

maximal, characteristic, complemented

C₂×#11718495

C₂

yes

maximal, characteristic

C₂²×C₈×D₁₃

C₂²

yes

—

C₂×C₄×D₅₂

C₄

yes

characteristic

C₂²×#6092

C₂²

yes

—

C₂²×#6092

C₂²

yes

—

C₂×C₁₃×C₄⋊C₈

C₄

yes

characteristic

C₂×#88060

C₄

yes

—

C₂×C₄×#1055

C₂²

yes

—

—

—

—

—

C₂²×D₅₂

C₈

yes

characteristic

C₂²×C₄×D₁₃

C₂×C₄

yes

—

C₂²×C₈×C₁₃

C₂×C₄

yes

—

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

C₂×#6103

C₈

yes

—

C₂×C₄⋊Dic₁₃

C₈

yes

characteristic

C₂²×#1055

C₂×C₄

yes

characteristic

C₂²×#1055

C₂³

yes

characteristic

C₂×C₈×D₁₃

—

C₂×C₈×D₁₃

—

C₂×C₈×D₁₃

—

C₂×C₈×D₁₃

—

C₄×D₅₂

—

C₄×D₅₂

—

—

—

—

—

D₄

yes

complemented

C₁₃×C₄⋊C₈

—

#88060

—

C₄×#1055

D₄

yes

—

C₄×#1055

D₄

yes

—

#13190

—

maximal, Sylow

C₂³×D₁₃

C₂×C₈

yes

—

C₂²×C₄×C₁₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₁₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂²×Dic₁₃

C₂×C₈

yes

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×C₄×D₁₃

—

C₂×C₄×D₁₃

—

C₂×C₄×D₁₃

—

C₂×C₄×D₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₄²×C₁₃

C₂²⋊C₄

yes

—

C₄²×C₁₃

M₁₆

yes

—

#6103

—

C₄⋊Dic₁₃

—

C₂×#1055

—

C₂×#1055

C₂²⋊C₄

yes

—

C₂×#1055

C₂×D₄

yes

characteristic

C₂×#1055

C₂×D₄

yes

—

C₂×#1055

C₂×D₄

yes

characteristic

C₂×#1055

#41

yes

—

C₈×D₁₃

—

C₈×D₁₃

—

C₈×D₁₃

—

#6092

—

#6092

—

C₂×C₈×D₄

—

C₂×C₄⋊C₁₆

—

C₂×C₂²⋊C₁₆

—

C₂×#346

—

C₂³×C₁₃

C₂²×C₈

yes

characteristic

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

M₃₂

yes

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊C₈

yes

—

C₂×C₄×C₁₃

C₂²⋊C₈

yes

—

C₂×C₄×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₄×D₄

yes

—

C₂×C₄×C₁₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₁₃

#131

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

D₅₂

—

C₄×D₁₃

—

C₄×D₁₃

—

C₄×D₁₃

—

C₈×C₁₃

—

#1055

—

#1055

#121

yes

—

yes

yes

C₂³×C₈

—

C₂×C₄×D₄

—

100

C₂²×C₁₆

—

101

C₂²×C₁₆

—

102

C₈×D₄

—

103

C₈×D₄

—

104

C₂×C₄⋊C₈

#261

yes

characteristic

105

C₂×C₂²⋊C₈

—

106

C₂×C₄×C₈

—

107

C₄⋊C₁₆

—

108

C₂²⋊C₁₆

—

109

#346

—

110

C₂²×C₁₃

C₂×C₂²⋊C₈

yes

characteristic

111

C₂²×C₁₃

C₈×D₄

yes

—

112

C₂²×C₁₃

C₂×M₃₂

yes

—

113

C₂²×C₁₃

#504

yes

—

114

C₂×D₁₃

—

115

C₂×D₁₃

—

116

C₂×D₁₃

—

117

C₂×D₁₃

—

118

C₂×D₁₃

—

119

C₄×C₁₃

—

120

C₄×C₁₃

—

121

C₄×C₁₃

#390

yes

characteristic

122

C₄×C₁₃

#408

yes

characteristic

123

C₄×C₁₃

#436

yes

—

124

Dic₁₃

—

125

C₂²×D₄

—

126

C₂³×C₄

—

127

C₂²×C₈

—

128

C₂²×C₈

—

129

C₂²×C₈

—

130

C₂²×C₈

—

131

C₂²×C₈

—

132

C₂²×C₈

—

133

C₂²×C₈

C₂×#261

yes

—

134

C₄×D₄

—

135

C₄×D₄

—

136

C₂×C₄⋊C₄

—

137

C₂×C₂²⋊C₄

—

138

C₂×C₄²

C₂×#261

yes

characteristic

139

C₂×C₁₆

—

140

C₂×C₁₆

—

141

C₂×C₁₆

—

142

C₂×C₁₆

—

143

C₂×C₁₆

—

144

C₄⋊C₈

—

145

C₂²⋊C₈

—

146

C₄×C₈

—

147

C₄×C₈

—

148

C₂×C₁₃

#1954

yes

characteristic

149

C₂×C₁₃

#2112

yes

characteristic

150

C₂×C₁₃

#2113

yes

characteristic

151

C₂×C₁₃

#2168

yes

—

152

D₁₃

—

153

C₂⁴

—

154

C₂×D₄

—

155

C₂×D₄

—

156

C₂²×C₄

—

157

C₂²×C₄

—

158

C₂²×C₄

—

159

C₂²×C₄

—

160

C₂²×C₄

—

161

C₂²×C₄

D₁₃⋊C₈

yes

characteristic

162

C₂²×C₄

D₁₃⋊C₈

yes

characteristic

163

C₂²×C₄

C₂²×#261

yes

characteristic

164

C₂×C₈

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

—

168

C₂×C₈

—

169

C₂×C₈

—

170

C₂×C₈

—

171

C₂×C₈

—

172

C₂×C₈

—

173

C₂×C₈

—

174

C₂×C₈

—

175

C₄⋊C₄

—

176

C₂²⋊C₄

—

177

C₄²

#6118

yes

—

178

C₄²

C₂²⋊#261

yes

—

179

C₁₆

—

180

C₁₆

—

181

C₁₃

#13190

yes

characteristic, complemented, Sylow

182

C₂³

—

183

C₂³

—

184

C₂³

—

185

C₂³

—

186

C₂³

—

187

C₂³

C₂×D₁₃⋊C₈

yes

characteristic

188

D₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

#88111

yes

—

197

C₂×C₄

#88111

yes

—

198

C₂×C₄

C₂×#6118

yes

characteristic

199

C₂×C₄

D₄×#261

yes

—

200

C₂×C₄

#88240

yes

—

201

C₂×C₄

C₂×C₂²⋊#261

yes

characteristic

202

C₈

—

203

C₈

—

204

C₈

—

205

C₈

—

206

C₈

—

207

C₂²

—

208

C₂²

—

209

C₂²

—

210

C₂²

—

211

C₂²

—

212

C₂²

#11719073

yes

—

213

C₂²

D₁₃⋊M₃₂

yes

—

214

C₂²

#11719096

yes

—

215

C₂²

C₂×#88111

yes

characteristic

216

C₄

—

217

C₄

—

218

C₄

—

219

C₄

#11719076

yes

—

220

C₄

#11719151

yes

characteristic

221

C₄

#11719159

yes

characteristic

222

C₂

—

223

C₂

#13130958

yes

characteristic

224

C₂

#13131065

yes

characteristic

225

C₂

#13131068

yes

characteristic

226

C₂

#13131219

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#261
commutator, socle	C₂³×C₁₃	C₂²×C₈
Fitting	C₂×C₁₃×C₄⋊C₈	C₄
prime radical	C₂²×D₅₂	C₈
cosocle	C₂²×#1055	C₂³
hypercenter	C₂×C₄⋊C₈	#261
hypocenter	C₁₃	#13190
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×#261

commutator, socle

C₂³×C₁₃

C₂²×C₈

Fitting

C₂×C₁₃×C₄⋊C₈

C₄

prime radical

C₂²×D₅₂

C₈

cosocle

C₂²×#1055

C₂³

C₂×C₄⋊C₈

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
1. C₂×C₄⋊#6092	C₂	semidirect
2. C₂×#11718259	C₂	semidirect
3. C₂×#11718231	C₂	semidirect
35. C₂×#6092	D₄	semidirect
181. C₁₃	#13190	semidirect

H ≅ G/N

type

1. C₂×C₄⋊#6092

C₂

semidirect

2. C₂×#11718259

C₂

semidirect

3. C₂×#11718231

C₂

semidirect

35. C₂×#6092

D₄

semidirect

181. C₁₃

#13190

semidirect

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

subgroup	index
C₂×C₄⋊#6092	2
C₂×#11718259	2
C₂×#11718231	2
C₂×#11718495	2
#13190	13

subgroup

index

C₂×C₄⋊#6092

Subgroups — #16790859

Conjugacy classes of subgroups

Complements

Maximal subgroups