Group #15765713 — subgroups

Subgroups — #15765713

Order 3328.

618 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 161 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×#11718739	C₂	1	2	yes	maximal
2	#13115798	C₂	1	2	yes	maximal
3	#13115638	C₂	1	1	yes	maximal, characteristic
4	#13112663	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×#494	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×Q₁₆	—	2	2	no	—
7	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×#88123	C₂²	1	1	yes	characteristic
9	C₁₃×#494	—	2	4	no	—
10	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×#88056	—	2	2	no	—
12	#11718739	—	2	4	no	—
13	#11718739	—	2	4	no	—
14	C₂×#88047	—	2	4	no	—
15	#11718596	C₂²	1	1	yes	characteristic
16	C₂×C₄×#1055	C₂²	1	2	yes	—
17	#11718121	—	2	4	no	—
18	C₂²×Q₈×C₁₃	—	2	2	no	—
19	C₂²×Dic₂₆	—	2	1	no	—
20	C₂×C₁₃×Q₁₆	—	2	8	no	—
21	C₂×C₁₃×Q₁₆	—	4	4	no	—
22	C₂×C₁₃×Q₁₆	—	4	4	no	—
23	C₂²×C₈×C₁₃	D₄	1	2	yes	—
24	C₁₃×C₄⋊Q₈	—	2	2	no	—
25	C₁₃×C₄⋊Q₈	—	2	2	no	—
26	C₁₃×C₄⋊Q₈	—	2	2	no	—
27	C₁₃×C₄⋊Q₈	D₄	1	4	yes	—
28	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
29	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
30	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
31	C₂×C₄⋊Dic₁₃	—	2	2	no	—
32	C₂×C₄⋊Dic₁₃	—	2	1	no	—
33	C₂×C₄⋊Dic₁₃	—	2	2	no	—
34	C₂²×#1055	D₄	1	4	yes	—
35	#88123	—	2	2	no	—
36	#88123	—	2	1	no	—
37	#88123	D₄	1	2	yes	—
38	C₄×C₈×C₁₃	—	2	2	no	—
39	#88056	—	4	4	no	—
40	#88047	—	2	16	no	—
41	C₄×#1055	—	2	2	no	—
42	C₄×#1055	—	2	2	no	—
43	#10309	—	13	1	no	maximal, Sylow
44	C₂×Q₈×C₁₃	—	2	4	no	—
45	C₂×Q₈×C₁₃	—	4	2	no	—
46	C₂×Q₈×C₁₃	—	4	4	no	—
47	C₂×Q₈×C₁₃	—	4	4	no	—
48	C₂²×C₄×C₁₃	—	4	2	no	—
49	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×Dic₁₃	—	4	1	no	—
52	C₂²×Dic₁₃	—	4	1	no	—
53	C₂×Dic₂₆	—	2	2	no	—
54	C₂×Dic₂₆	—	4	2	no	—
55	C₂×Dic₂₆	—	4	2	no	—
56	C₂×Dic₂₆	—	4	1	no	—
57	C₁₃×Q₁₆	—	4	16	no	—
58	C₂×C₈×C₁₃	—	2	2	no	—
59	C₂×C₈×C₁₃	—	2	2	no	—
60	C₂×C₈×C₁₃	D₈	1	4	yes	—
61	C₁₃×C₄⋊C₄	—	2	4	no	—
62	C₁₃×C₄⋊C₄	—	2	4	no	—
63	C₁₃×C₄⋊C₄	—	4	2	no	—
64	C₁₃×C₄⋊C₄	—	4	2	no	—
65	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₁₃	C₂×D₄	1	2	yes	—
67	C₄²×C₁₃	#41	1	1	yes	characteristic
68	C₄⋊Dic₁₃	—	2	4	no	—
69	C₄⋊Dic₁₃	—	4	2	no	—
70	C₄⋊Dic₁₃	—	4	2	no	—
71	C₄⋊Dic₁₃	—	4	4	no	—
72	C₂×#1055	—	2	4	no	—
73	C₂×#1055	—	2	4	no	—
74	C₂×#1055	Q₁₆	1	8	yes	complemented
75	C₂×#500	—	13	2	no	—
76	C₂×#494	D₁₃	1	1	yes	characteristic
77	#1990	—	13	2	no	—
78	#1956	—	13	1	no	—
79	#1273	—	13	1	no	—
80	C₂³×C₁₃	#127	1	1	yes	characteristic
81	Q₈×C₁₃	—	4	8	no	—
82	Q₈×C₁₃	—	8	4	no	—
83	C₂×C₄×C₁₃	—	4	4	no	—
84	C₂×C₄×C₁₃	—	8	2	no	—
85	C₂×C₄×C₁₃	—	8	2	no	—
86	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
87	C₂×C₄×C₁₃	#120	1	2	yes	—
88	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
89	C₂×C₄×C₁₃	#121	1	2	yes	—
90	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
91	C₂×C₄×C₁₃	C₂×Q₁₆	1	4	yes	—
92	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
93	C₂×C₄×C₁₃	#137	1	4	yes	—
94	C₂×Dic₁₃	—	4	2	no	—
95	C₂×Dic₁₃	—	8	1	no	—
96	C₂×Dic₁₃	—	8	1	no	—
97	C₂×Dic₁₃	—	8	1	no	—
98	C₂×Dic₁₃	—	8	2	no	—
99	Dic₂₆	—	4	4	no	—
100	Dic₂₆	—	8	2	no	—
101	C₈×C₁₃	—	2	4	no	—
102	#1055	—	2	8	no	—
103	C₂²×Q₁₆	—	2	2	no	—
104	C₂×C₄⋊Q₈	—	13	1	no	—
105	C₂×C₄⋊Q₈	C₂×D₁₃	1	2	yes	—
106	#500	—	26	4	no	—
107	#500	—	26	4	no	—
108	#494	—	2	4	no	—
109	C₂×#107	—	26	4	no	—
110	C₂×#103	—	26	2	no	—
111	C₂×C₄×C₈	—	13	2	no	—
112	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
113	#384	—	13	1	no	—
114	#340	—	26	4	no	—
115	C₂²×C₁₃	#392	1	1	yes	characteristic
116	C₂²×C₁₃	#493	1	1	yes	characteristic
117	C₂²×C₁₃	#494	1	2	yes	—
118	C₂²×C₁₃	#496	1	1	yes	characteristic
119	C₂²×C₁₃	#497	1	2	yes	—
120	C₄×C₁₃	—	8	4	no	—
121	C₄×C₁₃	#447	1	2	yes	—
122	C₄×C₁₃	#451	1	4	yes	—
123	C₄×C₁₃	#459	1	2	yes	—
124	C₄×C₁₃	#462	1	4	yes	—
125	Dic₁₃	—	8	2	no	—
126	Dic₁₃	—	16	1	no	—
127	C₂²×Q₈	—	2	2	no	—
128	C₂²×Q₈	—	26	1	no	—
129	C₂×Q₁₆	—	2	8	no	—
130	C₂×Q₁₆	—	4	4	no	—
131	C₂×Q₁₆	—	4	4	no	—
132	C₂²×C₈	—	13	4	no	—
133	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
134	C₄⋊Q₈	—	2	2	no	—
135	C₄⋊Q₈	—	2	2	no	—
136	C₄⋊Q₈	—	2	2	no	—
137	C₄⋊Q₈	—	13	2	no	—
138	C₄⋊Q₈	—	26	1	no	—
139	C₄⋊Q₈	—	26	2	no	—
140	C₄⋊Q₈	C₂²⋊D₁₃	1	4	yes	—
141	C₂×C₄⋊C₄	—	2	2	no	—
142	C₂×C₄⋊C₄	—	2	2	no	—
143	C₂×C₄⋊C₄	—	26	2	no	—
144	C₂×C₄⋊C₄	—	26	2	no	—
145	C₂×C₄⋊C₄	—	26	1	no	—
146	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
147	#107	—	26	16	no	—
148	#103	—	52	4	no	—
149	C₄×C₈	—	2	2	no	—
150	C₄×C₈	—	26	2	no	—
151	C₄×C₈	—	26	2	no	—
152	C₂×C₁₃	#1713	1	1	yes	characteristic
153	C₂×C₁₃	#1722	1	1	yes	characteristic
154	C₂×C₁₃	#1723	1	2	yes	—
155	C₂×C₁₃	#2011	1	1	yes	characteristic
156	C₂×C₁₃	#2016	1	2	yes	—
157	C₂×Q₈	—	2	4	no	—
158	C₂×Q₈	—	4	2	no	—
159	C₂×Q₈	—	4	4	no	—
160	C₂×Q₈	—	4	4	no	—
161	C₂×Q₈	—	26	2	no	—
162	C₂×Q₈	—	52	1	no	—
163	C₂×Q₈	—	52	2	no	—
164	C₂×Q₈	—	52	2	no	—
165	C₂²×C₄	—	4	2	no	—
166	C₂²×C₄	—	52	1	no	—
167	C₂²×C₄	—	52	1	no	—
168	C₂²×C₄	D₄×D₁₃	1	2	yes	—
169	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
170	Q₁₆	—	4	16	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	13	8	no	—
174	C₂×C₈	—	26	4	no	—
175	C₂×C₈	—	26	4	no	—
176	C₂×C₈	#6104	1	4	yes	—
177	C₄⋊C₄	—	2	4	no	—
178	C₄⋊C₄	—	2	4	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	—	26	4	no	—
182	C₄⋊C₄	—	52	2	no	—
183	C₄⋊C₄	—	52	4	no	—
184	C₄⋊C₄	—	52	2	no	—
185	C₄²	D₄×D₁₃	1	1	yes	characteristic
186	C₄²	#6129	1	1	yes	characteristic
187	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
188	C₁₃	#10309	1	1	yes	characteristic, complemented, Sylow
189	C₂³	#88194	1	1	yes	characteristic
190	Q₈	—	4	8	no	—
191	Q₈	—	8	4	no	—
192	Q₈	—	52	4	no	—
193	Q₈	—	104	2	no	—
194	C₂×C₄	—	4	4	no	—
195	C₂×C₄	—	8	2	no	—
196	C₂×C₄	—	8	2	no	—
197	C₂×C₄	—	52	2	no	—
198	C₂×C₄	—	104	1	no	—
199	C₂×C₄	—	104	1	no	—
200	C₂×C₄	—	104	1	no	—
201	C₂×C₄	—	104	2	no	—
202	C₂×C₄	Q₁₆×D₁₃	1	4	yes	—
203	C₂×C₄	#88172	1	4	yes	—
204	C₂×C₄	C₂×#6104	1	2	yes	—
205	C₂×C₄	#88186	1	2	yes	—
206	C₂×C₄	#88191	1	2	yes	—
207	C₂×C₄	#88192	1	1	yes	characteristic
208	C₂×C₄	#88193	1	2	yes	—
209	C₂×C₄	#88207	1	1	yes	characteristic
210	C₈	—	2	4	no	—
211	C₈	—	26	8	no	—
212	C₂²	#11718734	1	1	yes	characteristic
213	C₂²	#11718735	1	1	yes	characteristic
214	C₂²	#11718842	1	2	yes	—
215	C₂²	#11718851	1	2	yes	—
216	C₂²	#11718884	1	1	yes	characteristic
217	C₄	—	8	4	no	—
218	C₄	—	104	2	no	—
219	C₄	—	208	1	no	—
220	C₄	#11718733	1	2	yes	—
221	C₄	#11718843	1	4	yes	—
222	C₄	#11718847	1	4	yes	—
223	C₄	#11718878	1	2	yes	—
224	C₂	#13114452	1	1	yes	characteristic
225	C₂	#13114508	1	1	yes	characteristic
226	C₂	#13114511	1	2	yes	—
227	C₂	#13115907	1	1	yes	characteristic
228	C₂	#13115934	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11718739

C₂

yes

maximal

#13115798

C₂

yes

maximal

#13115638

C₂

yes

maximal, characteristic

#13112663

C₂

yes

maximal, characteristic

C₂×C₁₃×#494

C₂

yes

maximal, characteristic

C₂²×C₁₃×Q₁₆

—

C₂×C₁₃×C₄⋊Q₈

C₂²

yes

—

C₂×#88123

C₂²

yes

characteristic

C₁₃×#494

—

C₂×C₄×C₈×C₁₃

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₂×C₄×#1055

C₂²

yes

—

#11718121

—

C₂²×Q₈×C₁₃

—

C₂²×Dic₂₆

—

C₂×C₁₃×Q₁₆

—

C₂×C₁₃×Q₁₆

—

C₂×C₁₃×Q₁₆

—

C₂²×C₈×C₁₃

D₄

yes

—

C₁₃×C₄⋊Q₈

—

C₁₃×C₄⋊Q₈

—

C₁₃×C₄⋊Q₈

—

C₁₃×C₄⋊Q₈

D₄

yes

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₂²×#1055

D₄

yes

—

#88123

—

#88123

—

#88123

D₄

yes

—

C₄×C₈×C₁₃

—

—

—

—

—

—

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×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₁₃×Q₁₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

D₈

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄²×C₁₃

C₂×D₄

yes

—

C₄²×C₁₃

#41

yes

characteristic

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₂×#1055

—

C₂×#1055

—

C₂×#1055

Q₁₆

yes

complemented

C₂×#500

—

C₂×#494

D₁₃

yes

—

—

—

C₂³×C₁₃

#127

yes

characteristic

Q₈×C₁₃

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#120

yes

characteristic

C₂×C₄×C₁₃

#120

yes

—

C₂×C₄×C₁₃

#121

yes

characteristic

C₂×C₄×C₁₃

#121

yes

—

C₂×C₄×C₁₃

C₂×D₈

yes

—

C₂×C₄×C₁₃

C₂×Q₁₆

yes

—

C₂×C₄×C₁₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₁₃

#137

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

100

Dic₂₆

—

101

C₈×C₁₃

—

102

#1055

—

103

C₂²×Q₁₆

—

104

C₂×C₄⋊Q₈

—

105

C₂×C₄⋊Q₈

C₂×D₁₃

yes

—

106

#500

—

107

#500

—

108

#494

—

109

C₂×#107

—

110

C₂×#103

—

111

C₂×C₄×C₈

—

112

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

113

#384

—

114

#340

—

115

C₂²×C₁₃

#392

yes

characteristic

116

C₂²×C₁₃

#493

yes

characteristic

117

C₂²×C₁₃

#494

yes

—

118

C₂²×C₁₃

#496

yes

characteristic

119

C₂²×C₁₃

#497

yes

—

120

C₄×C₁₃

—

121

C₄×C₁₃

#447

yes

—

122

C₄×C₁₃

#451

yes

—

123

C₄×C₁₃

#459

yes

—

124

C₄×C₁₃

#462

yes

—

125

Dic₁₃

—

126

Dic₁₃

—

127

C₂²×Q₈

—

128

C₂²×Q₈

—

129

C₂×Q₁₆

—

130

C₂×Q₁₆

—

131

C₂×Q₁₆

—

132

C₂²×C₈

—

133

C₂²×C₈

C₂²⋊D₁₃

yes

—

134

C₄⋊Q₈

—

135

C₄⋊Q₈

—

136

C₄⋊Q₈

—

137

C₄⋊Q₈

—

138

C₄⋊Q₈

—

139

C₄⋊Q₈

—

140

C₄⋊Q₈

C₂²⋊D₁₃

yes

—

141

C₂×C₄⋊C₄

—

142

C₂×C₄⋊C₄

—

143

C₂×C₄⋊C₄

—

144

C₂×C₄⋊C₄

—

145

C₂×C₄⋊C₄

—

146

C₂×C₄²

C₂²×D₁₃

yes

characteristic

147

#107

—

148

#103

—

149

C₄×C₈

—

150

C₄×C₈

—

151

C₄×C₈

—

152

C₂×C₁₃

#1713

yes

characteristic

153

C₂×C₁₃

#1722

yes

characteristic

154

C₂×C₁₃

#1723

yes

—

155

C₂×C₁₃

#2011

yes

characteristic

156

C₂×C₁₃

#2016

yes

—

157

C₂×Q₈

—

158

C₂×Q₈

—

159

C₂×Q₈

—

160

C₂×Q₈

—

161

C₂×Q₈

—

162

C₂×Q₈

—

163

C₂×Q₈

—

164

C₂×Q₈

—

165

C₂²×C₄

—

166

C₂²×C₄

—

167

C₂²×C₄

—

168

C₂²×C₄

D₄×D₁₃

yes

—

169

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

170

Q₁₆

—

171

C₂×C₈

—

172

C₂×C₈

—

173

C₂×C₈

—

174

C₂×C₈

—

175

C₂×C₈

—

176

C₂×C₈

#6104

yes

—

177

C₄⋊C₄

—

178

C₄⋊C₄

—

179

C₄⋊C₄

—

180

C₄⋊C₄

—

181

C₄⋊C₄

—

182

C₄⋊C₄

—

183

C₄⋊C₄

—

184

C₄⋊C₄

—

185

C₄²

D₄×D₁₃

yes

characteristic

186

C₄²

#6129

yes

characteristic

187

C₄²

C₂×C₂²⋊D₁₃

yes

—

188

C₁₃

#10309

yes

characteristic, complemented, Sylow

189

C₂³

#88194

yes

characteristic

190

Q₈

—

191

Q₈

—

192

Q₈

—

193

Q₈

—

104

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

104

—

199

C₂×C₄

—

104

—

200

C₂×C₄

—

104

—

201

C₂×C₄

—

104

—

202

C₂×C₄

Q₁₆×D₁₃

yes

—

203

C₂×C₄

#88172

yes

—

204

C₂×C₄

C₂×#6104

yes

—

205

C₂×C₄

#88186

yes

—

206

C₂×C₄

#88191

yes

—

207

C₂×C₄

#88192

yes

characteristic

208

C₂×C₄

#88193

yes

—

209

C₂×C₄

#88207

yes

characteristic

210

C₈

—

211

C₈

—

212

C₂²

#11718734

yes

characteristic

213

C₂²

#11718735

yes

characteristic

214

C₂²

#11718842

yes

—

215

C₂²

#11718851

yes

—

216

C₂²

#11718884

yes

characteristic

217

C₄

—

218

C₄

—

104

—

219

C₄

—

208

—

220

C₄

#11718733

yes

—

221

C₄

#11718843

yes

—

222

C₄

#11718847

yes

—

223

C₄

#11718878

yes

—

224

C₂

#13114452

yes

characteristic

225

C₂

#13114508

yes

characteristic

226

C₂

#13114511

yes

—

227

C₂

#13115907

yes

characteristic

228

C₂

#13115934

yes

—

	subgroup	G/H
center	C₂³	#88194
commutator, cosocle	C₂×C₄²×C₁₃	C₂³
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₁₃×#494	C₂
socle, prime radical	C₂³×C₁₃	#127
hypercenter	C₂×#494	D₁₃
hypocenter	C₁₃	#10309
LCS	G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#494

subgroup

G/H

center

C₂³

#88194

commutator, cosocle

C₂×C₄²×C₁₃

C₂³

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₂×C₁₃×#494

C₂

C₂³×C₁₃

D₁₃

C₁₃

G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#494

N	H ≅ G/N	type
74. C₂×#1055	Q₁₆	semidirect
188. C₁₃	#10309	semidirect

H ≅ G/N

type

74. C₂×#1055

Q₁₆

semidirect

188. C₁₃

#10309

semidirect

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

subgroup	index
C₂×#11718739	2
#13115798	2
#13115638	2
#13112663	2
C₂×C₁₃×#494	2
#10309	13

subgroup

index

C₂×C₁₃×#494

Subgroups — #15765713

Conjugacy classes of subgroups

Complements

Maximal subgroups