Group #15765545 — subgroups

Subgroups — #15765545

Order 3328.

510 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 139 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×#11718343	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718511	C₂	1	2	yes	maximal
3	#13115772	C₂	1	1	yes	maximal, characteristic
4	C₂×#11718073	C₂	1	2	yes	maximal
5	C₁₃×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂×#88123	C₂²	1	2	yes	—
7	C₂×C₄×Dic₂₆	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×#88058	—	2	2	no	—
11	C₂×#88058	—	2	2	no	—
12	C₂×#88056	—	2	2	no	—
13	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
14	#11718511	—	2	4	no	—
15	#11718511	—	2	4	no	—
16	#11718343	—	2	4	no	—
17	#11718142	—	2	2	no	—
18	#11718073	—	2	4	no	—
19	C₂²×Dic₂₆	—	2	2	no	—
20	C₂²×Dic₂₆	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₃	—	2	2	no	—
22	C₂²×C₈×C₁₃	D₄	1	2	yes	—
23	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₃	—	2	2	no	—
25	C₂×C₄⋊Dic₁₃	—	2	2	no	—
26	C₂×C₄⋊Dic₁₃	D₄	1	1	yes	characteristic
27	C₂×#6101	—	2	1	no	—
28	C₂×C₄×Dic₁₃	—	2	1	no	—
29	C₂²×#1055	—	2	1	no	—
30	#88123	—	2	2	no	—
31	#88123	—	2	2	no	—
32	#88123	—	2	2	no	—
33	#88123	D₄	1	4	yes	—
34	C₄×Dic₂₆	—	2	2	no	—
35	C₄×Dic₂₆	—	2	2	no	—
36	C₁₃×C₄⋊C₈	—	2	2	no	—
37	C₁₃×C₄⋊C₈	D₄	1	4	yes	—
38	C₄×C₈×C₁₃	—	2	2	no	—
39	#88058	—	2	8	no	—
40	#88058	—	2	8	no	—
41	#88056	—	2	8	no	—
42	C₄⋊#1055	—	2	2	no	—
43	#10279	—	13	1	no	maximal, Sylow
44	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
47	C₂²×Dic₁₃	—	2	1	no	—
48	C₂²×Dic₁₃	—	2	1	no	—
49	C₂²×Dic₁₃	—	4	2	no	—
50	C₂×Dic₂₆	—	2	2	no	—
51	C₂×Dic₂₆	—	2	2	no	—
52	C₂×Dic₂₆	—	2	1	no	—
53	C₂×Dic₂₆	—	2	4	no	—
54	C₂×Dic₂₆	—	4	2	no	—
55	C₂×Dic₂₆	—	8	4	no	—
56	C₂×Dic₂₆	Q₁₆	1	2	yes	—
57	C₂×C₈×C₁₃	—	2	2	no	—
58	C₂×C₈×C₁₃	—	2	2	no	—
59	C₂×C₈×C₁₃	—	2	4	no	—
60	C₂×C₈×C₁₃	—	4	2	no	—
61	C₂×C₈×C₁₃	—	4	2	no	—
62	C₂×C₈×C₁₃	Q₁₆	1	4	yes	—
63	C₄²×C₁₃	C₂×D₄	1	2	yes	—
64	C₄²×C₁₃	C₂×D₄	1	2	yes	—
65	C₄⋊Dic₁₃	—	2	2	no	—
66	C₄⋊Dic₁₃	—	2	4	no	—
67	C₄⋊Dic₁₃	—	2	4	no	—
68	C₄⋊Dic₁₃	—	4	2	no	—
69	C₄⋊Dic₁₃	—	4	2	no	—
70	#6101	—	4	2	no	—
71	C₄×Dic₁₃	—	4	1	no	—
72	C₄×Dic₁₃	—	4	1	no	—
73	C₂×#1055	—	4	1	no	—
74	C₂×#1055	—	4	2	no	—
75	C₂×#500	—	13	1	no	—
76	C₂×#477	—	13	2	no	—
77	#1984	—	13	1	no	—
78	C₂×#333	—	13	2	no	—
79	#1277	D₁₃	1	1	yes	characteristic
80	C₂³×C₁₃	#121	1	1	yes	characteristic
81	C₂×C₄×C₁₃	—	2	2	no	—
82	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
83	C₂×C₄×C₁₃	#120	1	2	yes	—
84	C₂×C₄×C₁₃	#121	1	2	yes	—
85	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
87	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
88	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
89	C₂×C₄×C₁₃	#137	1	2	yes	—
90	C₂×C₄×C₁₃	#137	1	2	yes	—
91	C₂×Dic₁₃	—	2	2	no	—
92	C₂×Dic₁₃	—	4	1	no	—
93	C₂×Dic₁₃	—	4	1	no	—
94	C₂×Dic₁₃	—	4	1	no	—
95	C₂×Dic₁₃	—	4	4	no	—
96	C₂×Dic₁₃	—	4	2	no	—
97	C₂×Dic₁₃	—	8	4	no	—
98	Dic₂₆	—	2	4	no	—
99	Dic₂₆	—	4	2	no	—
100	Dic₂₆	—	8	4	no	—
101	Dic₂₆	—	8	4	no	—
102	C₈×C₁₃	—	2	4	no	—
103	C₈×C₁₃	—	4	4	no	—
104	#1055	—	8	1	no	—
105	C₂×C₄⋊Q₈	—	13	2	no	—
106	C₂×C₄×Q₈	—	13	1	no	—
107	#500	—	26	4	no	—
108	#477	—	26	4	no	—
109	#477	—	26	4	no	—
110	C₂×#107	—	26	2	no	—
111	C₂×#107	—	26	2	no	—
112	C₂×C₄⋊C₈	—	13	1	no	—
113	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
114	C₂×#103	—	26	2	no	—
115	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
116	#336	—	26	2	no	—
117	#333	—	26	4	no	—
118	C₂²×C₁₃	#392	1	1	yes	characteristic
119	C₂²×C₁₃	#462	1	2	yes	—
120	C₂²×C₁₃	#467	1	1	yes	characteristic
121	C₂²×C₁₃	#470	1	1	yes	characteristic
122	C₂²×C₁₃	#472	1	2	yes	—
123	C₄×C₁₃	—	2	2	no	—
124	C₄×C₁₃	#451	1	2	yes	—
125	C₄×C₁₃	#451	1	2	yes	—
126	C₄×C₁₃	#452	1	2	yes	—
127	C₄×C₁₃	#494	1	1	yes	characteristic
128	C₄×C₁₃	#497	1	1	yes	characteristic
129	Dic₁₃	—	4	2	no	—
130	Dic₁₃	—	8	4	no	—
131	Dic₁₃	—	8	1	no	—
132	C₂²×Q₈	—	13	1	no	—
133	C₂²×Q₈	—	26	2	no	—
134	C₂²×C₈	—	2	2	no	—
135	C₂²×C₈	—	26	1	no	—
136	C₂²×C₈	D₅₂	1	2	yes	—
137	C₄⋊Q₈	—	13	4	no	—
138	C₄⋊Q₈	—	26	2	no	—
139	C₄⋊Q₈	—	26	2	no	—
140	C₄⋊Q₈	—	26	2	no	—
141	C₄×Q₈	—	26	2	no	—
142	C₄×Q₈	—	26	2	no	—
143	C₂×C₄⋊C₄	—	13	1	no	—
144	C₂×C₄⋊C₄	—	26	2	no	—
145	C₂×C₄⋊C₄	—	26	2	no	—
146	C₂×C₄⋊C₄	—	26	1	no	—
147	C₂×C₄²	—	26	1	no	—
148	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
149	#107	—	26	8	no	—
150	#107	—	26	8	no	—
151	C₄⋊C₈	—	2	2	no	—
152	C₄⋊C₈	—	26	2	no	—
153	C₄⋊C₈	D₅₂	1	4	yes	—
154	#103	—	26	8	no	—
155	C₄×C₈	—	2	2	no	—
156	C₂×C₁₃	#1676	1	2	yes	—
157	C₂×C₁₃	#1696	1	2	yes	—
158	C₂×C₁₃	#2001	1	1	yes	characteristic
159	C₂×C₁₃	#2016	1	1	yes	characteristic
160	C₂×C₁₃	#2018	1	1	yes	characteristic
161	C₂×Q₈	—	13	2	no	—
162	C₂×Q₈	—	26	1	no	—
163	C₂×Q₈	—	26	2	no	—
164	C₂×Q₈	—	26	4	no	—
165	C₂×Q₈	—	26	2	no	—
166	C₂×Q₈	—	52	2	no	—
167	C₂×Q₈	—	104	4	no	—
168	C₂²×C₄	—	26	1	no	—
169	C₂²×C₄	—	26	1	no	—
170	C₂²×C₄	—	52	2	no	—
171	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
172	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
173	C₂²×C₄	#6131	1	1	yes	characteristic
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	—	4	2	no	—
178	C₂×C₈	—	4	2	no	—
179	C₂×C₈	—	52	1	no	—
180	C₂×C₈	—	52	2	no	—
181	C₂×C₈	Dic₅₂	1	4	yes	—
182	C₄⋊C₄	—	26	4	no	—
183	C₄⋊C₄	—	26	2	no	—
184	C₄⋊C₄	—	26	4	no	—
185	C₄⋊C₄	—	52	2	no	—
186	C₄⋊C₄	—	52	2	no	—
187	C₄⋊C₄	—	52	2	no	—
188	C₄²	—	52	1	no	—
189	C₄²	—	52	1	no	—
190	C₄²	C₂×D₅₂	1	2	yes	—
191	C₄²	D₄×D₁₃	1	2	yes	—
192	C₁₃	#10279	1	1	yes	characteristic, complemented, Sylow
193	C₂³	#88149	1	1	yes	characteristic
194	Q₈	—	26	4	no	—
195	Q₈	—	52	2	no	—
196	Q₈	—	104	4	no	—
197	Q₈	—	104	4	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	26	2	no	—
200	C₂×C₄	—	52	1	no	—
201	C₂×C₄	—	52	1	no	—
202	C₂×C₄	—	52	2	no	—
203	C₂×C₄	—	52	1	no	—
204	C₂×C₄	—	52	4	no	—
205	C₂×C₄	—	104	4	no	—
206	C₂×C₄	#88128	1	1	yes	characteristic
207	C₂×C₄	#88136	1	1	yes	characteristic
208	C₂×C₄	#88136	1	2	yes	—
209	C₂×C₄	#88149	1	2	yes	—
210	C₂×C₄	C₂×Dic₅₂	1	2	yes	—
211	C₂×C₄	#88163	1	2	yes	—
212	C₂×C₄	#88166	1	1	yes	characteristic
213	C₂×C₄	#88169	1	2	yes	—
214	C₂×C₄	Q₁₆×D₁₃	1	1	yes	characteristic
215	C₈	—	2	4	no	—
216	C₈	—	4	4	no	—
217	C₈	—	104	1	no	—
218	C₂²	#11718332	1	1	yes	characteristic
219	C₂²	#11718510	1	2	yes	—
220	C₂²	#11718529	1	1	yes	characteristic
221	C₂²	#11718548	1	1	yes	characteristic
222	C₂²	#11718558	1	2	yes	—
223	C₄	—	2	2	no	—
224	C₄	—	52	2	no	—
225	C₄	—	104	1	no	—
226	C₄	—	104	4	no	—
227	C₄	#11718360	1	1	yes	characteristic
228	C₄	#11718376	1	1	yes	characteristic
229	C₄	#11718400	1	2	yes	—
230	C₄	#11718444	1	2	yes	—
231	C₄	#11718469	1	2	yes	—
232	C₂	#13114241	1	2	yes	—
233	C₂	#13114365	1	2	yes	—
234	C₂	#13115847	1	1	yes	characteristic
235	C₂	#13115938	1	1	yes	characteristic
236	C₂	#13115946	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11718343

C₂

yes

maximal, characteristic

C₂×#11718511

C₂

yes

maximal

#13115772

C₂

yes

maximal, characteristic

C₂×#11718073

C₂

yes

maximal

C₁₃×#1277

C₂

yes

maximal, characteristic

C₂×#88123

C₂²

yes

—

C₂×C₄×Dic₂₆

C₂²

yes

characteristic

C₂×C₁₃×C₄⋊C₈

C₂²

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

—

—

—

C₂×C₄⋊#1055

C₂²

yes

—

—

—

—

—

C₂²×Dic₂₆

—

C₂²×Dic₂₆

D₄

yes

characteristic

C₂²×C₈×C₁₃

—

C₂²×C₈×C₁₃

D₄

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

D₄

yes

characteristic

C₂×#6101

—

C₂×C₄×Dic₁₃

—

C₂²×#1055

—

—

—

—

D₄

yes

—

C₄×Dic₂₆

—

C₄×Dic₂₆

—

C₁₃×C₄⋊C₈

—

C₁₃×C₄⋊C₈

D₄

yes

—

C₄×C₈×C₁₃

—

—

—

—

—

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

#41

yes

characteristic

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

Q₁₆

yes

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

Q₁₆

yes

—

C₄²×C₁₃

C₂×D₄

yes

—

C₄²×C₁₃

C₂×D₄

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#6101

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

—

—

—

—

—

—

D₁₃

yes

characteristic

C₂³×C₁₃

#121

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#120

yes

characteristic

C₂×C₄×C₁₃

#120

yes

—

C₂×C₄×C₁₃

#121

yes

—

C₂×C₄×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₁₃

C₂×Q₁₆

yes

—

C₂×C₄×C₁₃

#135

yes

characteristic

C₂×C₄×C₁₃

#137

yes

—

C₂×C₄×C₁₃

#137

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

100

Dic₂₆

—

101

Dic₂₆

—

102

C₈×C₁₃

—

103

C₈×C₁₃

—

104

#1055

—

105

C₂×C₄⋊Q₈

—

106

C₂×C₄×Q₈

—

107

#500

—

108

#477

—

109

#477

—

110

C₂×#107

—

111

C₂×#107

—

112

C₂×C₄⋊C₈

—

113

C₂×C₄⋊C₈

C₂×D₁₃

yes

—

114

C₂×#103

—

115

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

116

#336

—

117

#333

—

118

C₂²×C₁₃

#392

yes

characteristic

119

C₂²×C₁₃

#462

yes

—

120

C₂²×C₁₃

#467

yes

characteristic

121

C₂²×C₁₃

#470

yes

characteristic

122

C₂²×C₁₃

#472

yes

—

123

C₄×C₁₃

—

124

C₄×C₁₃

#451

yes

—

125

C₄×C₁₃

#451

yes

—

126

C₄×C₁₃

#452

yes

—

127

C₄×C₁₃

#494

yes

characteristic

128

C₄×C₁₃

#497

yes

characteristic

129

Dic₁₃

—

130

Dic₁₃

—

131

Dic₁₃

—

132

C₂²×Q₈

—

133

C₂²×Q₈

—

134

C₂²×C₈

—

135

C₂²×C₈

—

136

C₂²×C₈

D₅₂

yes

—

137

C₄⋊Q₈

—

138

C₄⋊Q₈

—

139

C₄⋊Q₈

—

140

C₄⋊Q₈

—

141

C₄×Q₈

—

142

C₄×Q₈

—

143

C₂×C₄⋊C₄

—

144

C₂×C₄⋊C₄

—

145

C₂×C₄⋊C₄

—

146

C₂×C₄⋊C₄

—

147

C₂×C₄²

—

148

C₂×C₄²

C₂²×D₁₃

yes

characteristic

149

#107

—

150

#107

—

151

C₄⋊C₈

—

152

C₄⋊C₈

—

153

C₄⋊C₈

D₅₂

yes

—

154

#103

—

155

C₄×C₈

—

156

C₂×C₁₃

#1676

yes

—

157

C₂×C₁₃

#1696

yes

—

158

C₂×C₁₃

#2001

yes

characteristic

159

C₂×C₁₃

#2016

yes

characteristic

160

C₂×C₁₃

#2018

yes

characteristic

161

C₂×Q₈

—

162

C₂×Q₈

—

163

C₂×Q₈

—

164

C₂×Q₈

—

165

C₂×Q₈

—

166

C₂×Q₈

—

167

C₂×Q₈

—

104

—

168

C₂²×C₄

—

169

C₂²×C₄

—

170

C₂²×C₄

—

171

C₂²×C₄

C₂×D₅₂

yes

characteristic

172

C₂²×C₄

D₄×D₁₃

yes

characteristic

173

C₂²×C₄

#6131

yes

characteristic

174

C₂×C₈

—

175

C₂×C₈

—

176

C₂×C₈

—

177

C₂×C₈

—

178

C₂×C₈

—

179

C₂×C₈

—

180

C₂×C₈

—

181

C₂×C₈

Dic₅₂

yes

—

182

C₄⋊C₄

—

183

C₄⋊C₄

—

184

C₄⋊C₄

—

185

C₄⋊C₄

—

186

C₄⋊C₄

—

187

C₄⋊C₄

—

188

C₄²

—

189

C₄²

—

190

C₄²

C₂×D₅₂

yes

—

191

C₄²

D₄×D₁₃

yes

—

192

C₁₃

#10279

yes

characteristic, complemented, Sylow

193

C₂³

#88149

yes

characteristic

194

Q₈

—

195

Q₈

—

196

Q₈

—

104

—

197

Q₈

—

104

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

202

C₂×C₄

—

203

C₂×C₄

—

204

C₂×C₄

—

205

C₂×C₄

—

104

—

206

C₂×C₄

#88128

yes

characteristic

207

C₂×C₄

#88136

yes

characteristic

208

C₂×C₄

#88136

yes

—

209

C₂×C₄

#88149

yes

—

210

C₂×C₄

C₂×Dic₅₂

yes

—

211

C₂×C₄

#88163

yes

—

212

C₂×C₄

#88166

yes

characteristic

213

C₂×C₄

#88169

yes

—

214

C₂×C₄

Q₁₆×D₁₃

yes

characteristic

215

C₈

—

216

C₈

—

217

C₈

—

104

—

218

C₂²

#11718332

yes

characteristic

219

C₂²

#11718510

yes

—

220

C₂²

#11718529

yes

characteristic

221

C₂²

#11718548

yes

characteristic

222

C₂²

#11718558

yes

—

223

C₄

—

224

C₄

—

225

C₄

—

104

—

226

C₄

—

104

—

227

C₄

#11718360

yes

characteristic

228

C₄

#11718376

yes

characteristic

229

C₄

#11718400

yes

—

230

C₄

#11718444

yes

—

231

C₄

#11718469

yes

—

232

C₂

#13114241

yes

—

233

C₂

#13114365

yes

—

234

C₂

#13115847

yes

characteristic

235

C₂

#13115938

yes

characteristic

236

C₂

#13115946

yes

characteristic

	subgroup	G/H
center	C₂³	#88149
commutator, cosocle	C₂×C₄²×C₁₃	C₂³
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1277	C₂
socle, prime radical	C₂³×C₁₃	#121
hypercenter	#1277	D₁₃
hypocenter	C₁₃	#10279
LCS	G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1277

subgroup

G/H

center

C₂³

#88149

commutator, cosocle

C₂×C₄²×C₁₃

C₂³

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1277

C₂

C₂³×C₁₃

D₁₃

C₁₃

G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1277

N	H ≅ G/N	type
192. C₁₃	#10279	semidirect

H ≅ G/N

type

192. C₁₃

#10279

semidirect

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

subgroup	index
C₂×#11718343	2
C₂×#11718511	2
#13115772	2
C₂×#11718073	2
C₁₃×#1277	2
#10279	13

subgroup

index

C₁₃×#1277

Subgroups — #15765545

Conjugacy classes of subgroups

Complements

Maximal subgroups