Group #15781156 — subgroups

Subgroups — #15781156

Order 3328.

486 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 167 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×#11718156	C₂	1	4	yes	maximal
2	C₂×#11718053	C₂	1	1	yes	maximal, characteristic
3	#13115000	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃×#494	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₃×Q₁₆	—	2	1	no	—
6	C₂²×C₁₃×Q₁₆	C₂²	1	2	yes	—
7	C₂×C₁₃×C₄⋊Q₈	—	2	1	no	—
8	C₁₃×#494	—	2	2	no	—
9	C₁₃×#494	C₄	1	4	yes	complemented
10	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×#88058	C₂²	1	2	yes	—
12	C₂×#88057	—	2	1	no	—
13	C₂²×#6090	C₂²	1	2	yes	—
14	#11718597	—	2	1	no	—
15	#11718156	—	2	8	no	—
16	#11718053	—	2	2	no	—
17	C₂²×Q₈×C₁₃	—	2	2	no	—
18	C₂×C₁₃×Q₁₆	—	2	4	no	—
19	C₂×C₁₃×Q₁₆	—	2	4	no	—
20	C₂×C₁₃×Q₁₆	—	2	2	no	—
21	C₂×C₁₃×Q₁₆	—	2	4	no	—
22	C₂×C₁₃×Q₁₆	—	4	2	no	—
23	C₂×C₁₃×Q₁₆	—	4	2	no	—
24	C₂×C₁₃×Q₁₆	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
27	C₁₃×C₄⋊Q₈	—	2	4	no	—
28	C₁₃×C₄⋊Q₈	—	4	2	no	—
29	C₂×C₁₃×C₄⋊C₄	—	4	2	no	—
30	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
31	C₂×C₄⋊Dic₁₃	—	2	2	no	—
32	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₁₃	D₄	1	2	yes	—
34	#88071	—	4	2	no	—
35	#88058	—	2	4	no	—
36	#88057	—	4	2	no	—
37	C₂×#6090	—	2	2	no	—
38	C₂×#6090	—	2	2	no	—
39	C₂×#6090	D₄	1	4	yes	—
40	#13207	—	13	1	no	maximal, Sylow
41	C₂×Q₈×C₁₃	—	2	4	no	—
42	C₂×Q₈×C₁₃	—	4	2	no	—
43	C₂×Q₈×C₁₃	—	4	2	no	—
44	C₂×Q₈×C₁₃	—	4	2	no	—
45	C₂×Q₈×C₁₃	—	4	2	no	—
46	C₂×Q₈×C₁₃	—	4	2	no	—
47	C₂²×C₄×C₁₃	—	4	2	no	—
48	C₂²×C₄×C₁₃	D₈	1	2	yes	—
49	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×Dic₁₃	—	4	2	no	—
51	C₁₃×Q₁₆	—	2	8	no	—
52	C₁₃×Q₁₆	—	4	8	no	—
53	C₁₃×Q₁₆	—	4	4	no	—
54	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
55	C₂×C₈×C₁₃	D₈	1	2	yes	—
56	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
57	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
59	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₁₃	#41	1	2	yes	—
61	C₁₃×C₄⋊C₄	—	4	8	no	—
62	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
63	C₄²×C₁₃	SD₁₆	1	2	yes	—
64	C₄⋊Dic₁₃	—	4	2	no	—
65	C₄⋊Dic₁₃	—	4	2	no	—
66	#6090	—	2	4	no	—
67	C₂×#494	D₁₃	1	1	yes	characteristic
68	C₂×#358	—	13	4	no	—
69	C₂×#346	—	13	1	no	—
70	#1846	—	13	1	no	—
71	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
72	Q₈×C₁₃	—	4	4	no	—
73	Q₈×C₁₃	—	4	4	no	—
74	Q₈×C₁₃	—	8	2	no	—
75	Q₈×C₁₃	—	8	2	no	—
76	C₂×C₄×C₁₃	—	2	4	no	—
77	C₂×C₄×C₁₃	—	4	4	no	—
78	C₂×C₄×C₁₃	—	8	2	no	—
79	C₂×C₄×C₁₃	—	8	2	no	—
80	C₂×C₄×C₁₃	#102	1	4	yes	—
81	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
83	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
87	C₂×Dic₁₃	—	8	2	no	—
88	C₂×Dic₁₃	—	8	4	no	—
89	C₈×C₁₃	#102	1	4	yes	—
90	C₈×C₁₃	C₄×D₄	1	2	yes	—
91	C₈×C₁₃	#121	1	2	yes	—
92	C₂²×Q₁₆	—	2	1	no	—
93	C₂²×Q₁₆	C₂×D₁₃	1	2	yes	—
94	C₂×C₄⋊Q₈	—	2	1	no	—
95	C₂²×C₁₆	—	13	2	no	—
96	#494	—	2	2	no	—
97	#494	Dic₁₃	1	4	yes	complemented
98	C₂×#107	—	13	2	no	—
99	C₂×#106	—	26	1	no	—
100	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
101	#382	—	26	1	no	—
102	#358	—	26	8	no	—
103	#346	—	26	2	no	—
104	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
105	C₂²×C₁₃	#486	1	1	yes	characteristic
106	C₂²×C₁₃	#493	1	1	yes	characteristic
107	C₂²×C₁₃	#510	1	4	yes	—
108	C₄×C₁₃	—	2	4	no	—
109	C₄×C₁₃	—	8	4	no	—
110	C₄×C₁₃	#390	1	1	yes	characteristic
111	C₄×C₁₃	C₂×#102	1	1	yes	characteristic
112	C₄×C₁₃	#486	1	1	yes	characteristic
113	C₄×C₁₃	#492	1	1	yes	characteristic
114	Dic₁₃	—	16	2	no	—
115	C₂²×Q₈	—	2	2	no	—
116	C₂×Q₁₆	—	2	4	no	—
117	C₂×Q₁₆	—	2	4	no	—
118	C₂×Q₁₆	—	2	4	no	—
119	C₂×Q₁₆	—	2	2	no	—
120	C₂×Q₁₆	—	4	2	no	—
121	C₂×Q₁₆	—	4	2	no	—
122	C₂×Q₁₆	C₂×Dic₁₃	1	4	yes	—
123	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
124	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
125	C₄⋊Q₈	—	2	4	no	—
126	C₄⋊Q₈	—	4	2	no	—
127	C₂×C₄⋊C₄	—	4	2	no	—
128	C₂×C₄⋊C₄	—	26	2	no	—
129	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
130	C₂×C₁₆	—	13	4	no	—
131	C₂×C₁₆	—	26	2	no	—
132	C₂×C₁₆	—	26	2	no	—
133	#107	—	26	4	no	—
134	#106	—	52	2	no	—
135	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
136	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
137	#95	—	52	2	no	—
138	C₂×C₁₃	#1957	1	1	yes	characteristic
139	C₂×C₁₃	#2141	1	2	yes	—
140	C₂×C₁₃	#2245	1	2	yes	—
141	C₂×C₁₃	#2251	1	2	yes	—
142	C₂×Q₈	—	2	4	no	—
143	C₂×Q₈	—	4	2	no	—
144	C₂×Q₈	—	4	2	no	—
145	C₂×Q₈	—	4	2	no	—
146	C₂×Q₈	—	4	2	no	—
147	C₂×Q₈	—	4	2	no	—
148	C₂²×C₄	—	4	2	no	—
149	C₂²×C₄	—	52	2	no	—
150	C₂²×C₄	#6104	1	2	yes	—
151	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
152	Q₁₆	—	2	8	no	—
153	Q₁₆	—	4	8	no	—
154	Q₁₆	—	4	4	no	—
155	C₂×C₈	#6104	1	2	yes	—
156	C₂×C₈	#6105	1	2	yes	—
157	C₂×C₈	C₂²⋊Dic₁₃	1	2	yes	—
158	C₂×C₈	D₄×D₁₃	1	2	yes	—
159	C₂×C₈	#6129	1	2	yes	—
160	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
161	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
162	C₄⋊C₄	—	4	8	no	—
163	C₄⋊C₄	—	52	2	no	—
164	C₄⋊C₄	—	52	2	no	—
165	C₄²	#6105	1	2	yes	—
166	C₄²	C₂²⋊Dic₁₃	1	2	yes	—
167	C₁₆	—	26	4	no	—
168	C₁₃	#13207	1	1	yes	characteristic, complemented, Sylow
169	C₂³	C₂×#6104	1	1	yes	characteristic
170	Q₈	—	4	4	no	—
171	Q₈	—	4	4	no	—
172	Q₈	—	8	2	no	—
173	Q₈	—	8	2	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	—	8	2	no	—
177	C₂×C₄	—	8	2	no	—
178	C₂×C₄	—	104	2	no	—
179	C₂×C₄	—	104	4	no	—
180	C₂×C₄	#88072	1	4	yes	—
181	C₂×C₄	C₂×#6104	1	1	yes	characteristic
182	C₂×C₄	C₂×#6105	1	1	yes	characteristic
183	C₂×C₄	C₂×#6105	1	1	yes	characteristic
184	C₂×C₄	#88190	1	1	yes	characteristic
185	C₂×C₄	#88194	1	1	yes	characteristic
186	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
187	C₈	#88072	1	4	yes	—
188	C₈	D₄×Dic₁₃	1	2	yes	—
189	C₈	#88192	1	2	yes	—
190	C₂²	#11718731	1	1	yes	characteristic
191	C₂²	#11718734	1	1	yes	characteristic
192	C₂²	#11718840	1	4	yes	—
193	C₂²	C₂×#88072	1	1	yes	characteristic
194	C₄	—	2	4	no	—
195	C₄	—	8	4	no	—
196	C₄	—	208	2	no	—
197	C₄	#11718731	1	1	yes	characteristic
198	C₄	#11718737	1	1	yes	characteristic
199	C₄	C₂×#88072	1	1	yes	characteristic
200	C₄	#11718882	1	1	yes	characteristic
201	C₂	#13115643	1	1	yes	characteristic
202	C₂	#13116556	1	2	yes	—
203	C₂	#13117081	1	2	yes	—
204	C₂	#13117105	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11718156

C₂

yes

maximal

C₂×#11718053

C₂

yes

maximal, characteristic

#13115000

C₂

yes

maximal, characteristic

C₂×C₁₃×#494

C₂

yes

maximal, characteristic

C₂²×C₁₃×Q₁₆

—

C₂²×C₁₃×Q₁₆

C₂²

yes

—

C₂×C₁₃×C₄⋊Q₈

—

C₁₃×#494

—

C₁₃×#494

C₄

yes

complemented

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₂×#88058

C₂²

yes

—

C₂×#88057

—

C₂²×#6090

C₂²

yes

—

#11718597

—

#11718156

—

#11718053

—

C₂²×Q₈×C₁₃

—

C₂×C₁₃×Q₁₆

—

C₂×C₁₃×Q₁₆

—

C₂×C₁₃×Q₁₆

—

C₂×C₁₃×Q₁₆

—

C₂×C₁₃×Q₁₆

—

C₂×C₁₃×Q₁₆

—

C₂×C₁₃×Q₁₆

C₂×C₄

yes

—

C₂²×C₈×C₁₃

D₄

yes

characteristic

C₂²×C₈×C₁₃

C₂³

yes

characteristic

C₁₃×C₄⋊Q₈

—

C₁₃×C₄⋊Q₈

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₄²×C₁₃

D₄

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₄×C₈×C₁₃

D₄

yes

—

—

—

—

—

—

D₄

yes

—

#13207

—

maximal, Sylow

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

D₈

yes

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×Dic₁₃

—

C₁₃×Q₁₆

—

C₁₃×Q₁₆

—

C₁₃×Q₁₆

—

C₂×C₈×C₁₃

C₂²⋊C₄

yes

—

C₂×C₈×C₁₃

D₈

yes

—

C₂×C₈×C₁₃

SD₁₆

yes

—

C₂×C₈×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₃

C₂×D₄

yes

—

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃

#41

yes

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂²⋊C₄

yes

—

C₄²×C₁₃

SD₁₆

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#6090

—

C₂×#494

D₁₃

yes

—

—

—

C₂³×C₁₃

C₂×D₈

yes

characteristic

Q₈×C₁₃

—

Q₈×C₁₃

—

Q₈×C₁₃

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#102

yes

—

C₂×C₄×C₁₃

C₂×C₂²⋊C₄

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₁₃

C₂×SD₁₆

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₈×C₁₃

#102

yes

—

C₈×C₁₃

C₄×D₄

yes

—

C₈×C₁₃

#121

yes

—

C₂²×Q₁₆

—

C₂²×Q₁₆

C₂×D₁₃

yes

—

C₂×C₄⋊Q₈

—

C₂²×C₁₆

—

#494

—

#494

Dic₁₃

yes

complemented

C₂×#107

—

C₂×#106

—

100

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

101

#382

—

102

#358

—

103

#346

—

104

C₂²×C₁₃

C₂×#102

yes

characteristic

105

C₂²×C₁₃

#486

yes

characteristic

106

C₂²×C₁₃

#493

yes

characteristic

107

C₂²×C₁₃

#510

yes

—

108

C₄×C₁₃

—

109

C₄×C₁₃

—

110

C₄×C₁₃

#390

yes

characteristic

111

C₄×C₁₃

C₂×#102

yes

characteristic

112

C₄×C₁₃

#486

yes

characteristic

113

C₄×C₁₃

#492

yes

characteristic

114

Dic₁₃

—

115

C₂²×Q₈

—

116

C₂×Q₁₆

—

117

C₂×Q₁₆

—

118

C₂×Q₁₆

—

119

C₂×Q₁₆

—

120

C₂×Q₁₆

—

121

C₂×Q₁₆

—

122

C₂×Q₁₆

C₂×Dic₁₃

yes

—

123

C₂²×C₈

C₂²⋊D₁₃

yes

characteristic

124

C₂²×C₈

C₂²×D₁₃

yes

characteristic

125

C₄⋊Q₈

—

126

C₄⋊Q₈

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄²

C₂²⋊D₁₃

yes

characteristic

130

C₂×C₁₆

—

131

C₂×C₁₆

—

132

C₂×C₁₆

—

133

#107

—

134

#106

—

135

C₄×C₈

C₂×Dic₁₃

yes

—

136

C₄×C₈

C₂²⋊D₁₃

yes

—

137

#95

—

138

C₂×C₁₃

#1957

yes

characteristic

139

C₂×C₁₃

#2141

yes

—

140

C₂×C₁₃

#2245

yes

—

141

C₂×C₁₃

#2251

yes

—

142

C₂×Q₈

—

143

C₂×Q₈

—

144

C₂×Q₈

—

145

C₂×Q₈

—

146

C₂×Q₈

—

147

C₂×Q₈

—

148

C₂²×C₄

—

149

C₂²×C₄

—

150

C₂²×C₄

#6104

yes

—

151

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

152

Q₁₆

—

153

Q₁₆

—

154

Q₁₆

—

155

C₂×C₈

#6104

yes

—

156

C₂×C₈

#6105

yes

—

157

C₂×C₈

C₂²⋊Dic₁₃

yes

—

158

C₂×C₈

D₄×D₁₃

yes

—

159

C₂×C₈

#6129

yes

—

160

C₂×C₈

C₂²×Dic₁₃

yes

characteristic

161

C₂×C₈

C₂×C₂²⋊D₁₃

yes

characteristic

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄²

#6105

yes

—

166

C₄²

C₂²⋊Dic₁₃

yes

—

167

C₁₆

—

168

C₁₃

#13207

yes

characteristic, complemented, Sylow

169

C₂³

C₂×#6104

yes

characteristic

170

Q₈

—

171

Q₈

—

172

Q₈

—

173

Q₈

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

104

—

179

C₂×C₄

—

104

—

180

C₂×C₄

#88072

yes

—

181

C₂×C₄

C₂×#6104

yes

characteristic

182

C₂×C₄

C₂×#6105

yes

characteristic

183

C₂×C₄

C₂×#6105

yes

characteristic

184

C₂×C₄

#88190

yes

characteristic

185

C₂×C₄

#88194

yes

characteristic

186

C₂×C₄

C₂×C₂²⋊Dic₁₃

yes

characteristic

187

C₈

#88072

yes

—

188

C₈

D₄×Dic₁₃

yes

—

189

C₈

#88192

yes

—

190

C₂²

#11718731

yes

characteristic

191

C₂²

#11718734

yes

characteristic

192

C₂²

#11718840

yes

—

193

C₂²

C₂×#88072

yes

characteristic

194

C₄

—

195

C₄

—

196

C₄

—

208

—

197

C₄

#11718731

yes

characteristic

198

C₄

#11718737

yes

characteristic

199

C₄

C₂×#88072

yes

characteristic

200

C₄

#11718882

yes

characteristic

201

C₂

#13115643

yes

characteristic

202

C₂

#13116556

yes

—

203

C₂

#13117081

yes

—

204

C₂

#13117105

yes

—

	subgroup	G/H
center	C₂³	C₂×#6104
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂×C₁₃×#494	C₂
socle, prime radical	C₂³×C₁₃	C₂×D₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×#494	D₁₃
hypocenter	C₁₃	#13207
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

subgroup

G/H

center

C₂³

C₂×#6104

commutator

C₂×C₈×C₁₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₁₃

Fitting

C₂×C₁₃×#494

C₂

socle, prime radical

C₂³×C₁₃

C₂×D₈

cosocle

C₂²×C₈×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

N	H ≅ G/N	type
9. C₁₃×#494	C₄	semidirect
97. #494	Dic₁₃	semidirect
168. C₁₃	#13207	semidirect

H ≅ G/N

type

9. C₁₃×#494

C₄

semidirect

97. #494

Dic₁₃

semidirect

168. C₁₃

#13207

semidirect

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

subgroup	index
C₂×#11718156	2
C₂×#11718053	2
#13115000	2
C₂×C₁₃×#494	2
#13207	13

subgroup

index

C₂×C₁₃×#494

Subgroups — #15781156

Conjugacy classes of subgroups

Complements

Maximal subgroups