Group #15762655 — subgroups

Subgroups — #15762655

Order 3328.

526 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 207 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	#13115798	C₂	1	2	yes	maximal
2	#13115639	C₂	1	2	yes	maximal
3	#13114993	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×#1055	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×#498	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₁₃×#498	—	2	2	no	—
8	C₁₃×#498	C₄	1	4	yes	complemented
9	C₂×C₁₃×#106	—	2	2	no	—
10	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×#88075	—	2	4	no	—
12	C₂×#88057	—	2	2	no	—
13	#11718596	C₂²	1	1	yes	characteristic
14	#11718596	C₂²	1	1	yes	characteristic
15	C₂×C₄×#1055	C₂²	1	2	yes	—
16	#11718121	—	2	4	no	—
17	C₈×#1055	—	2	2	no	—
18	C₂²×Q₈×C₁₃	—	2	2	no	—
19	C₂²×C₈×C₁₃	D₄	1	2	yes	—
20	C₁₃×C₄⋊Q₈	—	2	2	no	—
21	C₁₃×C₄⋊Q₈	—	2	2	no	—
22	C₁₃×C₄⋊Q₈	—	2	2	no	—
23	C₁₃×C₄⋊Q₈	C₂×C₄	1	4	yes	—
24	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
25	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
26	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
27	C₂×C₄⋊Dic₁₃	—	2	1	no	—
28	C₂×C₄⋊Dic₁₃	—	2	1	no	—
29	C₂×C₄⋊Dic₁₃	—	2	2	no	—
30	C₂×C₄⋊Dic₁₃	—	2	2	no	—
31	C₂²×#1055	D₄	1	4	yes	—
32	C₁₃×#106	—	2	8	no	—
33	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₁₃	D₄	1	2	yes	—
35	#88075	—	4	8	no	—
36	#88057	—	4	4	no	—
37	C₄×#1055	—	2	2	no	—
38	C₄×#1055	—	2	2	no	—
39	#9819	—	13	1	no	maximal, Sylow
40	C₂×Q₈×C₁₃	—	2	4	no	—
41	C₂×Q₈×C₁₃	—	4	2	no	—
42	C₂×Q₈×C₁₃	—	4	4	no	—
43	C₂×Q₈×C₁₃	—	4	4	no	—
44	C₂²×C₄×C₁₃	—	4	2	no	—
45	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
46	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×Dic₁₃	—	4	1	no	—
48	C₂²×Dic₁₃	—	4	1	no	—
49	C₂×C₈×C₁₃	C₂²⋊C₄	1	4	yes	—
50	C₂×C₈×C₁₃	SD₁₆	1	4	yes	—
51	C₂×C₈×C₁₃	Q₁₆	1	4	yes	—
52	C₁₃×C₄⋊C₄	—	2	4	no	—
53	C₁₃×C₄⋊C₄	—	2	4	no	—
54	C₁₃×C₄⋊C₄	—	4	2	no	—
55	C₁₃×C₄⋊C₄	—	4	2	no	—
56	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₁₃	#41	1	2	yes	—
59	C₄⋊Dic₁₃	—	4	4	no	—
60	C₄⋊Dic₁₃	—	4	2	no	—
61	C₄⋊Dic₁₃	—	4	2	no	—
62	C₄⋊Dic₁₃	—	4	4	no	—
63	C₂×#1055	—	2	4	no	—
64	C₂×#1055	—	2	4	no	—
65	C₂×#1055	SD₁₆	1	8	yes	—
66	C₂×#498	D₁₃	1	1	yes	characteristic
67	#1990	—	13	2	no	—
68	#1956	—	13	2	no	—
69	#1844	—	13	1	no	—
70	C₂×C₈²	—	13	1	no	—
71	C₂³×C₁₃	#127	1	1	yes	characteristic
72	Q₈×C₁₃	—	4	8	no	—
73	Q₈×C₁₃	—	8	4	no	—
74	C₂×C₄×C₁₃	—	4	4	no	—
75	C₂×C₄×C₁₃	—	8	2	no	—
76	C₂×C₄×C₁₃	—	8	2	no	—
77	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
79	C₂×C₄×C₁₃	#121	1	2	yes	—
80	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
82	C₂×C₄×C₁₃	C₂×SD₁₆	1	4	yes	—
83	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
84	C₂×C₄×C₁₃	#135	1	4	yes	—
85	C₂×Dic₁₃	—	8	1	no	—
86	C₂×Dic₁₃	—	8	2	no	—
87	C₂×Dic₁₃	—	8	1	no	—
88	C₂×Dic₁₃	—	8	2	no	—
89	C₈×C₁₃	#103	1	8	yes	complemented
90	#1055	—	2	8	no	—
91	C₂×C₄⋊Q₈	C₂×D₁₃	1	2	yes	—
92	#498	—	2	2	no	—
93	#498	Dic₁₃	1	4	yes	complemented
94	C₂×#106	—	2	2	no	—
95	C₂×#106	—	26	2	no	—
96	C₂×#103	—	26	4	no	—
97	C₂×C₄×C₈	—	13	2	no	—
98	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
99	#384	—	13	1	no	—
100	#384	—	13	1	no	—
101	#340	—	26	4	no	—
102	C₈²	—	26	2	no	—
103	C₂²×C₁₃	#390	1	1	yes	characteristic
104	C₂²×C₁₃	#492	1	1	yes	characteristic
105	C₂²×C₁₃	#492	1	2	yes	—
106	C₂²×C₁₃	#494	1	1	yes	characteristic
107	C₂²×C₁₃	#495	1	2	yes	—
108	C₄×C₁₃	—	8	4	no	—
109	C₄×C₁₃	C₂×#103	1	2	yes	—
110	C₄×C₁₃	C₄×SD₁₆	1	4	yes	—
111	C₄×C₁₃	#465	1	4	yes	—
112	C₄×C₁₃	#487	1	2	yes	—
113	Dic₁₃	—	16	1	no	—
114	Dic₁₃	—	16	1	no	—
115	C₂²×Q₈	—	2	2	no	—
116	C₂²×C₈	—	13	4	no	—
117	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
118	C₄⋊Q₈	—	2	2	no	—
119	C₄⋊Q₈	—	2	2	no	—
120	C₄⋊Q₈	—	2	2	no	—
121	C₄⋊Q₈	C₂×Dic₁₃	1	4	yes	—
122	C₂×C₄⋊C₄	—	2	2	no	—
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄⋊C₄	—	26	2	no	—
125	C₂×C₄⋊C₄	—	26	2	no	—
126	C₂×C₄⋊C₄	—	26	1	no	—
127	C₂×C₄⋊C₄	—	26	1	no	—
128	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
129	#106	—	2	8	no	—
130	#106	—	52	4	no	—
131	#103	—	52	8	no	—
132	C₄×C₈	—	26	2	no	—
133	C₄×C₈	—	26	2	no	—
134	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
135	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
136	C₂×C₁₃	#1705	1	1	yes	characteristic
137	C₂×C₁₃	#1707	1	2	yes	—
138	C₂×C₁₃	#1708	1	1	yes	characteristic
139	C₂×C₁₃	#1956	1	1	yes	characteristic
140	C₂×C₁₃	#1968	1	2	yes	—
141	C₂×Q₈	—	2	4	no	—
142	C₂×Q₈	—	4	2	no	—
143	C₂×Q₈	—	4	4	no	—
144	C₂×Q₈	—	4	4	no	—
145	C₂²×C₄	—	4	2	no	—
146	C₂²×C₄	—	52	1	no	—
147	C₂²×C₄	—	52	1	no	—
148	C₂²×C₄	D₄×D₁₃	1	2	yes	—
149	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
150	C₂×C₈	—	13	8	no	—
151	C₂×C₈	—	26	4	no	—
152	C₂×C₈	—	26	4	no	—
153	C₂×C₈	#6106	1	4	yes	—
154	C₂×C₈	#6107	1	4	yes	—
155	C₂×C₈	C₂²⋊Dic₁₃	1	4	yes	—
156	C₄⋊C₄	—	2	4	no	—
157	C₄⋊C₄	—	2	4	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	52	4	no	—
161	C₄⋊C₄	—	52	4	no	—
162	C₄⋊C₄	—	52	2	no	—
163	C₄⋊C₄	—	52	2	no	—
164	C₄²	#6129	1	2	yes	—
165	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
166	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
167	C₁₃	#9819	1	1	yes	characteristic, complemented, Sylow
168	C₂³	#88194	1	1	yes	characteristic
169	Q₈	—	4	8	no	—
170	Q₈	—	8	4	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	—	104	1	no	—
175	C₂×C₄	—	104	1	no	—
176	C₂×C₄	—	104	2	no	—
177	C₂×C₄	—	104	2	no	—
178	C₂×C₄	SD₁₆×D₁₃	1	4	yes	—
179	C₂×C₄	#88170	1	4	yes	—
180	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
181	C₂×C₄	#88190	1	1	yes	characteristic
182	C₂×C₄	#88192	1	2	yes	—
183	C₂×C₄	C₂×#6106	1	2	yes	—
184	C₂×C₄	C₂×#6107	1	2	yes	—
185	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
186	C₈	—	26	8	no	—
187	C₈	#88075	1	8	yes	complemented
188	C₂²	#11718746	1	1	yes	characteristic
189	C₂²	#11718753	1	1	yes	characteristic
190	C₂²	#11718829	1	2	yes	—
191	C₂²	#11718836	1	2	yes	—
192	C₂²	#11718882	1	1	yes	characteristic
193	C₄	—	8	4	no	—
194	C₄	—	208	1	no	—
195	C₄	—	208	1	no	—
196	C₄	#11718742	1	2	yes	—
197	C₄	SD₁₆×Dic₁₃	1	4	yes	—
198	C₄	#11718832	1	4	yes	—
199	C₄	C₂×#88075	1	2	yes	—
200	C₂	#13114424	1	1	yes	characteristic
201	C₂	#13114431	1	2	yes	—
202	C₂	#13114438	1	1	yes	characteristic
203	C₂	#13115639	1	1	yes	characteristic
204	C₂	#13115692	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13115798

C₂

yes

maximal

#13115639

C₂

yes

maximal

#13114993

C₂

yes

maximal, characteristic

C₂×C₈×#1055

C₂

yes

maximal, characteristic

C₂×C₁₃×#498

C₂

yes

maximal, characteristic

C₂×C₁₃×C₄⋊Q₈

C₂²

yes

—

C₁₃×#498

—

C₁₃×#498

C₄

yes

complemented

C₂×C₁₃×#106

—

C₂×C₄×C₈×C₁₃

C₂²

yes

—

—

C₂²

yes

characteristic

#11718596

C₂²

yes

characteristic

C₂×C₄×#1055

C₂²

yes

—

#11718121

—

C₈×#1055

—

C₂²×Q₈×C₁₃

—

C₂²×C₈×C₁₃

D₄

yes

—

C₁₃×C₄⋊Q₈

—

C₁₃×C₄⋊Q₈

—

C₁₃×C₄⋊Q₈

—

C₁₃×C₄⋊Q₈

C₂×C₄

yes

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₂²×#1055

D₄

yes

—

C₁₃×#106

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₄×C₈×C₁₃

D₄

yes

—

—

—

—

—

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂×C₈×C₁₃

C₂²⋊C₄

yes

—

C₂×C₈×C₁₃

SD₁₆

yes

—

C₂×C₈×C₁₃

Q₁₆

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂²×C₄

yes

characteristic

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄²×C₁₃

#41

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₂×#1055

—

C₂×#1055

—

C₂×#1055

SD₁₆

yes

—

C₂×#498

D₁₃

yes

—

—

—

C₂×C₈²

—

C₂³×C₁₃

#127

yes

characteristic

Q₈×C₁₃

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₄×D₄

yes

—

C₂×C₄×C₁₃

#121

yes

—

C₂×C₄×C₁₃

#124

yes

characteristic

C₂×C₄×C₁₃

C₂×SD₁₆

yes

—

C₂×C₄×C₁₃

C₂×SD₁₆

yes

—

C₂×C₄×C₁₃

C₂×Q₁₆

yes

—

C₂×C₄×C₁₃

#135

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₈×C₁₃

#103

yes

complemented

#1055

—

C₂×C₄⋊Q₈

C₂×D₁₃

yes

—

#498

—

#498

Dic₁₃

yes

—

—

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

#384

—

100

#384

—

101

#340

—

102

C₈²

—

103

C₂²×C₁₃

#390

yes

characteristic

104

C₂²×C₁₃

#492

yes

characteristic

105

C₂²×C₁₃

#492

yes

—

106

C₂²×C₁₃

#494

yes

characteristic

107

C₂²×C₁₃

#495

yes

—

108

C₄×C₁₃

—

109

C₄×C₁₃

C₂×#103

yes

—

110

C₄×C₁₃

C₄×SD₁₆

yes

—

111

C₄×C₁₃

#465

yes

—

112

C₄×C₁₃

#487

yes

—

113

Dic₁₃

—

114

Dic₁₃

—

115

C₂²×Q₈

—

116

C₂²×C₈

—

117

C₂²×C₈

C₂²⋊D₁₃

yes

—

118

C₄⋊Q₈

—

119

C₄⋊Q₈

—

120

C₄⋊Q₈

—

121

C₄⋊Q₈

C₂×Dic₁₃

yes

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄⋊C₄

—

126

C₂×C₄⋊C₄

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄²

C₂²×D₁₃

yes

characteristic

129

#106

—

130

#106

—

131

#103

—

132

C₄×C₈

—

133

C₄×C₈

—

134

C₄×C₈

C₂×Dic₁₃

yes

—

135

C₄×C₈

C₂²⋊D₁₃

yes

—

136

C₂×C₁₃

#1705

yes

characteristic

137

C₂×C₁₃

#1707

yes

—

138

C₂×C₁₃

#1708

yes

characteristic

139

C₂×C₁₃

#1956

yes

characteristic

140

C₂×C₁₃

#1968

yes

—

141

C₂×Q₈

—

142

C₂×Q₈

—

143

C₂×Q₈

—

144

C₂×Q₈

—

145

C₂²×C₄

—

146

C₂²×C₄

—

147

C₂²×C₄

—

148

C₂²×C₄

D₄×D₁₃

yes

—

149

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

#6106

yes

—

154

C₂×C₈

#6107

yes

—

155

C₂×C₈

C₂²⋊Dic₁₃

yes

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄²

#6129

yes

—

165

C₄²

C₂²×Dic₁₃

yes

characteristic

166

C₄²

C₂×C₂²⋊D₁₃

yes

characteristic

167

C₁₃

#9819

yes

characteristic, complemented, Sylow

168

C₂³

#88194

yes

characteristic

169

Q₈

—

170

Q₈

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

104

—

175

C₂×C₄

—

104

—

176

C₂×C₄

—

104

—

177

C₂×C₄

—

104

—

178

C₂×C₄

SD₁₆×D₁₃

yes

—

179

C₂×C₄

#88170

yes

—

180

C₂×C₄

D₄×Dic₁₃

yes

—

181

C₂×C₄

#88190

yes

characteristic

182

C₂×C₄

#88192

yes

—

183

C₂×C₄

C₂×#6106

yes

—

184

C₂×C₄

C₂×#6107

yes

—

185

C₂×C₄

C₂×C₂²⋊Dic₁₃

yes

characteristic

186

C₈

—

187

C₈

#88075

yes

complemented

188

C₂²

#11718746

yes

characteristic

189

C₂²

#11718753

yes

characteristic

190

C₂²

#11718829

yes

—

191

C₂²

#11718836

yes

—

192

C₂²

#11718882

yes

characteristic

193

C₄

—

194

C₄

—

208

—

195

C₄

—

208

—

196

C₄

#11718742

yes

—

197

C₄

SD₁₆×Dic₁₃

yes

—

198

C₄

#11718832

yes

—

199

C₄

C₂×#88075

yes

—

200

C₂

#13114424

yes

characteristic

201

C₂

#13114431

yes

—

202

C₂

#13114438

yes

characteristic

203

C₂

#13115639

yes

characteristic

204

C₂

#13115692

yes

—

	subgroup	G/H
center	C₂³	#88194
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₁₃×#498	C₂
socle, prime radical	C₂³×C₁₃	#127
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂×#498	D₁₃
hypocenter	C₁₃	#9819
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#498

subgroup

G/H

center

C₂³

#88194

commutator

C₄²×C₁₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₂×C₁₃×#498

C₂

socle, prime radical

C₂³×C₁₃

#127

cosocle

C₂×C₄²×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#498

N	H ≅ G/N	type
8. C₁₃×#498	C₄	semidirect
89. C₈×C₁₃	#103	semidirect
93. #498	Dic₁₃	semidirect
167. C₁₃	#9819	semidirect
187. C₈	#88075	semidirect

H ≅ G/N

type

8. C₁₃×#498

C₄

semidirect

89. C₈×C₁₃

#103

semidirect

93. #498

Dic₁₃

semidirect

167. C₁₃

#9819

semidirect

187. C₈

#88075

semidirect

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

subgroup	index
#13115798	2
#13115639	2
#13114993	2
C₂×C₈×#1055	2
C₂×C₁₃×#498	2
#9819	13

subgroup

index

#13115798

#13115639

#13114993

C₂×C₈×#1055

C₂×C₁₃×#498

#9819

Subgroups — #15762655

Conjugacy classes of subgroups

Complements

Maximal subgroups