Group #15781152 — subgroups

Subgroups — #15781152

Order 3328.

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

normal subgroups only

This group has 590 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×#11718154	C₂	1	4	yes	maximal, complemented
2	C₂×#11718053	C₂	1	1	yes	maximal, characteristic, complemented
3	#13115000	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₃×#493	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₃×D₈	—	2	1	no	—
6	C₂²×C₁₃×D₈	C₂²	1	2	yes	—
7	C₂×C₁₃×#127	—	2	1	no	—
8	C₁₃×#493	—	2	2	no	—
9	C₁₃×#493	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	#11718154	—	2	8	no	—
16	#11718053	—	2	2	no	—
17	C₂²×D₄×C₁₃	—	2	2	no	—
18	C₂²×D₄×C₁₃	—	4	2	no	—
19	C₂×C₁₃×D₈	—	2	4	no	—
20	C₂×C₁₃×D₈	—	2	4	no	—
21	C₂×C₁₃×D₈	—	2	2	no	—
22	C₂×C₁₃×D₈	—	2	4	no	—
23	C₂×C₁₃×D₈	—	4	2	no	—
24	C₂×C₁₃×D₈	—	4	2	no	—
25	C₂×C₁₃×D₈	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
28	C₁₃×#127	—	2	4	no	—
29	C₁₃×#127	—	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	complemented
40	#13206	—	13	1	no	maximal, Sylow
41	C₂⁴×C₁₃	—	4	2	no	—
42	C₂×D₄×C₁₃	—	2	4	no	—
43	C₂×D₄×C₁₃	—	4	8	no	—
44	C₂×D₄×C₁₃	—	4	2	no	—
45	C₂×D₄×C₁₃	—	4	2	no	—
46	C₂×D₄×C₁₃	—	4	2	no	—
47	C₂×D₄×C₁₃	—	4	2	no	—
48	C₂×D₄×C₁₃	—	4	2	no	—
49	C₂×D₄×C₁₃	—	8	8	no	—
50	C₂²×C₄×C₁₃	D₈	1	2	yes	—
51	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×Dic₁₃	—	4	2	no	—
53	C₁₃×D₈	—	2	8	no	—
54	C₁₃×D₈	—	4	8	no	—
55	C₁₃×D₈	—	4	4	no	—
56	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
57	C₂×C₈×C₁₃	D₈	1	2	yes	—
58	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
59	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₁₃	#41	1	2	yes	—
63	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
64	C₄²×C₁₃	SD₁₆	1	2	yes	—
65	C₄⋊Dic₁₃	—	4	2	no	—
66	C₄⋊Dic₁₃	—	4	2	no	—
67	#6090	—	2	4	no	—
68	C₂×#493	D₁₃	1	1	yes	characteristic
69	C₂×#357	—	13	4	no	—
70	C₂×#346	—	13	1	no	—
71	#1846	—	13	1	no	—
72	C₂³×C₁₃	—	4	4	no	—
73	C₂³×C₁₃	—	8	2	no	—
74	C₂³×C₁₃	—	8	2	no	—
75	C₂³×C₁₃	—	8	8	no	—
76	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
77	D₄×C₁₃	—	4	4	no	—
78	D₄×C₁₃	—	4	4	no	—
79	D₄×C₁₃	—	8	16	no	—
80	D₄×C₁₃	—	8	2	no	—
81	D₄×C₁₃	—	8	2	no	—
82	C₂×C₄×C₁₃	—	2	4	no	—
83	C₂×C₄×C₁₃	#102	1	4	yes	—
84	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
86	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
87	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
88	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
89	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
90	C₂×Dic₁₃	—	8	2	no	—
91	C₂×Dic₁₃	—	8	4	no	—
92	C₈×C₁₃	#102	1	4	yes	—
93	C₈×C₁₃	C₄×D₄	1	2	yes	—
94	C₈×C₁₃	#121	1	2	yes	—
95	C₂²×D₈	—	2	1	no	—
96	C₂²×D₈	C₂×D₁₃	1	2	yes	—
97	C₂×#127	—	2	1	no	—
98	C₂²×C₁₆	—	13	2	no	—
99	#493	—	2	2	no	—
100	#493	Dic₁₃	1	4	yes	complemented
101	C₂×#107	—	13	2	no	—
102	C₂×#106	—	26	1	no	—
103	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
104	#382	—	26	1	no	—
105	#357	—	26	8	no	—
106	#346	—	26	2	no	—
107	C₂²×C₁₃	—	8	8	no	—
108	C₂²×C₁₃	—	8	4	no	—
109	C₂²×C₁₃	—	16	4	no	—
110	C₂²×C₁₃	—	16	4	no	—
111	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
112	C₂²×C₁₃	#486	1	1	yes	characteristic
113	C₂²×C₁₃	#493	1	1	yes	characteristic
114	C₂²×C₁₃	#509	1	4	yes	—
115	C₄×C₁₃	—	2	4	no	—
116	C₄×C₁₃	#390	1	1	yes	characteristic
117	C₄×C₁₃	C₂×#102	1	1	yes	characteristic
118	C₄×C₁₃	#486	1	1	yes	characteristic
119	C₄×C₁₃	#492	1	1	yes	characteristic
120	Dic₁₃	—	16	2	no	—
121	C₂²×D₄	—	2	2	no	—
122	C₂²×D₄	—	4	2	no	—
123	C₂×D₈	—	2	4	no	—
124	C₂×D₈	—	2	4	no	—
125	C₂×D₈	—	2	4	no	—
126	C₂×D₈	—	2	2	no	—
127	C₂×D₈	—	4	2	no	—
128	C₂×D₈	—	4	2	no	—
129	C₂×D₈	C₂×Dic₁₃	1	4	yes	—
130	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
131	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
132	#127	—	2	4	no	—
133	#127	—	4	2	no	—
134	C₂×C₄⋊C₄	—	26	2	no	—
135	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
136	C₂×C₁₆	—	13	4	no	—
137	C₂×C₁₆	—	26	2	no	—
138	C₂×C₁₆	—	26	2	no	—
139	#107	—	26	4	no	—
140	#106	—	52	2	no	—
141	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
142	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
143	#95	—	52	2	no	—
144	C₂×C₁₃	—	16	4	no	—
145	C₂×C₁₃	#1957	1	1	yes	characteristic
146	C₂×C₁₃	#2140	1	2	yes	—
147	C₂×C₁₃	#2244	1	2	yes	—
148	C₂×C₁₃	#2250	1	2	yes	—
149	C₂⁴	—	4	2	no	—
150	C₂×D₄	—	2	4	no	—
151	C₂×D₄	—	4	8	no	—
152	C₂×D₄	—	4	2	no	—
153	C₂×D₄	—	4	2	no	—
154	C₂×D₄	—	4	2	no	—
155	C₂×D₄	—	4	2	no	—
156	C₂×D₄	—	4	2	no	—
157	C₂×D₄	—	8	8	no	—
158	C₂²×C₄	—	52	2	no	—
159	C₂²×C₄	#6104	1	2	yes	—
160	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
161	D₈	—	2	8	no	—
162	D₈	—	4	8	no	—
163	D₈	—	4	4	no	—
164	C₂×C₈	#6104	1	2	yes	—
165	C₂×C₈	#6105	1	2	yes	—
166	C₂×C₈	C₂²⋊Dic₁₃	1	2	yes	—
167	C₂×C₈	D₄×D₁₃	1	2	yes	—
168	C₂×C₈	#6129	1	2	yes	—
169	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
170	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
171	C₄⋊C₄	—	52	2	no	—
172	C₄⋊C₄	—	52	2	no	—
173	C₄²	#6105	1	2	yes	—
174	C₄²	C₂²⋊Dic₁₃	1	2	yes	—
175	C₁₆	—	26	4	no	—
176	C₁₃	#13206	1	1	yes	characteristic, complemented, Sylow
177	C₂³	—	4	4	no	—
178	C₂³	—	8	2	no	—
179	C₂³	—	8	2	no	—
180	C₂³	—	8	8	no	—
181	C₂³	C₂×#6104	1	1	yes	characteristic
182	D₄	—	4	4	no	—
183	D₄	—	4	4	no	—
184	D₄	—	8	16	no	—
185	D₄	—	8	2	no	—
186	D₄	—	8	2	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	104	2	no	—
189	C₂×C₄	—	104	4	no	—
190	C₂×C₄	#88072	1	4	yes	—
191	C₂×C₄	C₂×#6104	1	1	yes	characteristic
192	C₂×C₄	C₂×#6105	1	1	yes	characteristic
193	C₂×C₄	C₂×#6105	1	1	yes	characteristic
194	C₂×C₄	#88190	1	1	yes	characteristic
195	C₂×C₄	#88194	1	1	yes	characteristic
196	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
197	C₈	#88072	1	4	yes	—
198	C₈	D₄×Dic₁₃	1	2	yes	—
199	C₈	#88192	1	2	yes	—
200	C₂²	—	8	8	no	—
201	C₂²	—	8	4	no	—
202	C₂²	—	16	4	no	—
203	C₂²	—	16	4	no	—
204	C₂²	#11718731	1	1	yes	characteristic
205	C₂²	#11718734	1	1	yes	characteristic
206	C₂²	#11718808	1	4	yes	—
207	C₂²	C₂×#88072	1	1	yes	characteristic
208	C₄	—	2	4	no	—
209	C₄	—	208	2	no	—
210	C₄	#11718731	1	1	yes	characteristic
211	C₄	#11718737	1	1	yes	characteristic
212	C₄	C₂×#88072	1	1	yes	characteristic
213	C₄	#11718882	1	1	yes	characteristic
214	C₂	—	16	4	no	—
215	C₂	#13115643	1	1	yes	characteristic
216	C₂	#13116552	1	2	yes	—
217	C₂	#13117077	1	2	yes	—
218	C₂	#13117100	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11718154

C₂

yes

maximal, complemented

C₂×#11718053

C₂

yes

maximal, characteristic, complemented

#13115000

C₂

yes

maximal, characteristic, complemented

C₂×C₁₃×#493

C₂

yes

maximal, characteristic

C₂²×C₁₃×D₈

—

C₂²×C₁₃×D₈

C₂²

yes

—

C₂×C₁₃×#127

—

C₁₃×#493

—

C₁₃×#493

C₄

yes

complemented

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₂×#88058

C₂²

yes

—

C₂×#88057

—

C₂²×#6090

C₂²

yes

—

#11718597

—

#11718154

—

#11718053

—

C₂²×D₄×C₁₃

—

C₂²×D₄×C₁₃

—

C₂×C₁₃×D₈

—

C₂×C₁₃×D₈

—

C₂×C₁₃×D₈

—

C₂×C₁₃×D₈

—

C₂×C₁₃×D₈

—

C₂×C₁₃×D₈

—

C₂×C₁₃×D₈

C₂×C₄

yes

—

C₂²×C₈×C₁₃

D₄

yes

characteristic

C₂²×C₈×C₁₃

C₂³

yes

characteristic

C₁₃×#127

—

C₁₃×#127

—

C₂×C₄²×C₁₃

D₄

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₄×C₈×C₁₃

D₄

yes

—

—

—

—

—

—

D₄

yes

complemented

#13206

—

maximal, Sylow

C₂⁴×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂²×C₄×C₁₃

D₈

yes

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×Dic₁₃

—

C₁₃×D₈

—

C₁₃×D₈

—

C₁₃×D₈

—

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

—

C₄²×C₁₃

C₂²⋊C₄

yes

—

C₄²×C₁₃

SD₁₆

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#6090

—

C₂×#493

D₁₃

yes

—

—

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

C₂×D₈

yes

characteristic

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

D₄×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×D₈

—

C₂²×D₈

C₂×D₁₃

yes

—

C₂×#127

—

C₂²×C₁₆

—

#493

—

100

#493

Dic₁₃

yes

complemented

101

C₂×#107

—

102

C₂×#106

—

103

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

104

#382

—

105

#357

—

106

#346

—

107

C₂²×C₁₃

—

108

C₂²×C₁₃

—

109

C₂²×C₁₃

—

110

C₂²×C₁₃

—

111

C₂²×C₁₃

C₂×#102

yes

characteristic

112

C₂²×C₁₃

#486

yes

characteristic

113

C₂²×C₁₃

#493

yes

characteristic

114

C₂²×C₁₃

#509

yes

—

115

C₄×C₁₃

—

116

C₄×C₁₃

#390

yes

characteristic

117

C₄×C₁₃

C₂×#102

yes

characteristic

118

C₄×C₁₃

#486

yes

characteristic

119

C₄×C₁₃

#492

yes

characteristic

120

Dic₁₃

—

121

C₂²×D₄

—

122

C₂²×D₄

—

123

C₂×D₈

—

124

C₂×D₈

—

125

C₂×D₈

—

126

C₂×D₈

—

127

C₂×D₈

—

128

C₂×D₈

—

129

C₂×D₈

C₂×Dic₁₃

yes

—

130

C₂²×C₈

C₂²⋊D₁₃

yes

characteristic

131

C₂²×C₈

C₂²×D₁₃

yes

characteristic

132

#127

—

133

#127

—

134

C₂×C₄⋊C₄

—

135

C₂×C₄²

C₂²⋊D₁₃

yes

characteristic

136

C₂×C₁₆

—

137

C₂×C₁₆

—

138

C₂×C₁₆

—

139

#107

—

140

#106

—

141

C₄×C₈

C₂×Dic₁₃

yes

—

142

C₄×C₈

C₂²⋊D₁₃

yes

—

143

#95

—

144

C₂×C₁₃

—

145

C₂×C₁₃

#1957

yes

characteristic

146

C₂×C₁₃

#2140

yes

—

147

C₂×C₁₃

#2244

yes

—

148

C₂×C₁₃

#2250

yes

—

149

C₂⁴

—

150

C₂×D₄

—

151

C₂×D₄

—

152

C₂×D₄

—

153

C₂×D₄

—

154

C₂×D₄

—

155

C₂×D₄

—

156

C₂×D₄

—

157

C₂×D₄

—

158

C₂²×C₄

—

159

C₂²×C₄

#6104

yes

—

160

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

161

D₈

—

162

D₈

—

163

D₈

—

164

C₂×C₈

#6104

yes

—

165

C₂×C₈

#6105

yes

—

166

C₂×C₈

C₂²⋊Dic₁₃

yes

—

167

C₂×C₈

D₄×D₁₃

yes

—

168

C₂×C₈

#6129

yes

—

169

C₂×C₈

C₂²×Dic₁₃

yes

characteristic

170

C₂×C₈

C₂×C₂²⋊D₁₃

yes

characteristic

171

C₄⋊C₄

—

172

C₄⋊C₄

—

173

C₄²

#6105

yes

—

174

C₄²

C₂²⋊Dic₁₃

yes

—

175

C₁₆

—

176

C₁₃

#13206

yes

characteristic, complemented, Sylow

177

C₂³

—

178

C₂³

—

179

C₂³

—

180

C₂³

—

181

C₂³

C₂×#6104

yes

characteristic

182

D₄

—

183

D₄

—

184

D₄

—

185

D₄

—

186

D₄

—

187

C₂×C₄

—

188

C₂×C₄

—

104

—

189

C₂×C₄

—

104

—

190

C₂×C₄

#88072

yes

—

191

C₂×C₄

C₂×#6104

yes

characteristic

192

C₂×C₄

C₂×#6105

yes

characteristic

193

C₂×C₄

C₂×#6105

yes

characteristic

194

C₂×C₄

#88190

yes

characteristic

195

C₂×C₄

#88194

yes

characteristic

196

C₂×C₄

C₂×C₂²⋊Dic₁₃

yes

characteristic

197

C₈

#88072

yes

—

198

C₈

D₄×Dic₁₃

yes

—

199

C₈

#88192

yes

—

200

C₂²

—

201

C₂²

—

202

C₂²

—

203

C₂²

—

204

C₂²

#11718731

yes

characteristic

205

C₂²

#11718734

yes

characteristic

206

C₂²

#11718808

yes

—

207

C₂²

C₂×#88072

yes

characteristic

208

C₄

—

209

C₄

—

208

—

210

C₄

#11718731

yes

characteristic

211

C₄

#11718737

yes

characteristic

212

C₄

C₂×#88072

yes

characteristic

213

C₄

#11718882

yes

characteristic

214

C₂

—

215

C₂

#13115643

yes

characteristic

216

C₂

#13116552

yes

—

217

C₂

#13117077

yes

—

218

C₂

#13117100

yes

—

	subgroup	G/H
center	C₂³	C₂×#6104
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting, prime radical	C₂×C₁₃×#493	C₂
socle	C₂³×C₁₃	C₂×D₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×#493	D₁₃
hypocenter	C₁₃	#13206
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#493

subgroup

G/H

center

C₂³

C₂×#6104

commutator

C₂×C₈×C₁₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₁₃

Fitting, prime radical

C₂×C₁₃×#493

C₂

socle

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×#493

N	H ≅ G/N	type
1. C₂×#11718154	C₂	semidirect
2. C₂×#11718053	C₂	semidirect
3. #13115000	C₂	semidirect
9. C₁₃×#493	C₄	semidirect
39. C₂×#6090	D₄	semidirect
100. #493	Dic₁₃	semidirect
176. C₁₃	#13206	semidirect

H ≅ G/N

type

1. C₂×#11718154

C₂

semidirect

2. C₂×#11718053

C₂

semidirect

3. #13115000

C₂

semidirect

9. C₁₃×#493

C₄

semidirect

39. C₂×#6090

D₄

semidirect

100. #493

Dic₁₃

semidirect

176. C₁₃

#13206

semidirect

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

subgroup	index
C₂×#11718154	2
C₂×#11718053	2
#13115000	2
C₂×C₁₃×#493	2
#13206	13

subgroup

index

C₂×C₁₃×#493

Subgroups — #15781152

Conjugacy classes of subgroups

Complements

Maximal subgroups