Group #15697025 — subgroups

Subgroups — C₁₃×#10686 #15697025

Order 3328.

410 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 162 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	C₁₃×#1990	C₂	1	2	yes	maximal
2	C₁₃×#1956	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1844	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1834	C₂	1	2	yes	maximal
5	C₁₃×#1273	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×#106	—	2	2	no	—
8	C₂×C₁₃×#103	—	2	2	no	—
9	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₁₃×#384	C₂²	1	2	yes	—
12	C₁₃×#384	C₂²	1	1	yes	characteristic
13	C₁₃×#340	—	2	4	no	—
14	C₁₃×#340	—	2	4	no	—
15	C₂²×Q₈×C₁₃	—	2	1	no	—
16	C₂²×C₈×C₁₃	D₄	1	4	yes	—
17	C₂²×C₈×C₁₃	D₄	1	2	yes	—
18	C₁₃×C₄⋊Q₈	—	2	2	no	—
19	C₁₃×C₄⋊Q₈	—	2	1	no	—
20	C₁₃×C₄⋊Q₈	Q₈	1	2	yes	—
21	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
22	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
23	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
24	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
25	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
26	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
27	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
28	C₁₃×#106	—	4	4	no	—
29	C₁₃×#103	—	4	4	no	—
30	C₄×C₈×C₁₃	—	2	2	no	—
31	C₄×C₈×C₁₃	—	2	2	no	—
32	C₄×C₈×C₁₃	—	2	2	no	—
33	#10686	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂×Q₈×C₁₃	—	2	2	no	—
35	C₂×Q₈×C₁₃	—	4	1	no	—
36	C₂×Q₈×C₁₃	—	4	2	no	—
37	C₂×Q₈×C₁₃	—	4	2	no	—
38	C₂²×C₄×C₁₃	—	4	1	no	—
39	C₂²×C₄×C₁₃	—	4	1	no	—
40	C₂²×C₄×C₁₃	—	4	2	no	—
41	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
43	C₂×C₈×C₁₃	—	2	2	no	—
44	C₂×C₈×C₁₃	—	2	2	no	—
45	C₂×C₈×C₁₃	—	2	4	no	—
46	C₂×C₈×C₁₃	—	2	4	no	—
47	C₂×C₈×C₁₃	—	2	4	no	—
48	C₂×C₈×C₁₃	SD₁₆	1	4	yes	—
49	C₁₃×C₄⋊C₄	—	2	4	no	—
50	C₁₃×C₄⋊C₄	—	4	2	no	—
51	C₁₃×C₄⋊C₄	—	4	2	no	—
52	C₁₃×C₄⋊C₄	—	4	4	no	—
53	C₁₃×C₄⋊C₄	—	4	4	no	—
54	C₁₃×C₄⋊C₄	—	4	4	no	—
55	C₁₃×C₄⋊C₄	—	4	4	no	—
56	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
57	C₄²×C₁₃	#41	1	1	yes	characteristic
58	C₄²×C₁₃	#41	1	2	yes	—
59	#1990	C₂×C₁₃	1	2	yes	—
60	#1956	C₂×C₁₃	1	1	yes	characteristic
61	#1844	C₂×C₁₃	1	1	yes	characteristic
62	#1834	C₂×C₁₃	1	2	yes	—
63	#1273	C₂×C₁₃	1	1	yes	characteristic
64	C₂³×C₁₃	#127	1	1	yes	characteristic
65	Q₈×C₁₃	—	4	4	no	—
66	Q₈×C₁₃	—	8	2	no	—
67	C₂×C₄×C₁₃	—	4	2	no	—
68	C₂×C₄×C₁₃	—	8	1	no	—
69	C₂×C₄×C₁₃	—	8	1	no	—
70	C₂×C₄×C₁₃	—	8	1	no	—
71	C₂×C₄×C₁₃	—	8	2	no	—
72	C₂×C₄×C₁₃	—	8	2	no	—
73	C₂×C₄×C₁₃	—	8	2	no	—
74	C₂×C₄×C₁₃	—	8	2	no	—
75	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
77	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
78	C₂×C₄×C₁₃	#123	1	2	yes	—
79	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
80	C₂×C₄×C₁₃	#135	1	4	yes	—
81	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
82	C₂×C₄×C₁₃	#137	1	4	yes	—
83	C₈×C₁₃	—	2	4	no	—
84	C₈×C₁₃	—	4	4	no	—
85	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
86	C₂×#106	—	2	2	no	—
87	C₂×#103	—	2	2	no	—
88	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
89	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
90	#384	C₂²×C₁₃	1	1	yes	characteristic
91	#384	C₂²×C₁₃	1	2	yes	—
92	#340	—	2	4	no	—
93	#340	—	2	4	no	—
94	C₂²×C₁₃	#399	1	1	yes	characteristic
95	C₂²×C₁₃	#492	1	1	yes	characteristic
96	C₂²×C₁₃	#495	1	2	yes	—
97	C₂²×C₁₃	#496	1	1	yes	characteristic
98	C₂²×C₁₃	#497	1	2	yes	—
99	C₄×C₁₃	—	8	2	no	—
100	C₄×C₁₃	—	16	1	no	—
101	C₄×C₁₃	—	16	2	no	—
102	C₄×C₁₃	#476	1	2	yes	—
103	C₄×C₁₃	#479	1	4	yes	—
104	C₄×C₁₃	#481	1	2	yes	—
105	C₄×C₁₃	#485	1	4	yes	—
106	C₂²×Q₈	—	2	1	no	—
107	C₂²×C₈	D₄×C₁₃	1	4	yes	—
108	C₂²×C₈	D₄×C₁₃	1	2	yes	—
109	C₄⋊Q₈	—	2	2	no	—
110	C₄⋊Q₈	—	2	1	no	—
111	C₄⋊Q₈	Q₈×C₁₃	1	2	yes	—
112	C₂×C₄⋊C₄	—	2	2	no	—
113	C₂×C₄⋊C₄	—	2	2	no	—
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	—	2	1	no	—
117	C₂×C₄⋊C₄	—	2	2	no	—
118	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
119	#106	—	4	4	no	—
120	#103	—	4	4	no	—
121	C₄×C₈	—	2	2	no	—
122	C₄×C₈	—	2	2	no	—
123	C₄×C₈	—	2	2	no	—
124	C₂×C₁₃	#1708	1	1	yes	characteristic
125	C₂×C₁₃	#1716	1	1	yes	characteristic
126	C₂×C₁₃	#1719	1	2	yes	—
127	C₂×C₁₃	#2094	1	1	yes	characteristic
128	C₂×C₁₃	#2097	1	2	yes	—
129	C₂×Q₈	—	2	2	no	—
130	C₂×Q₈	—	4	1	no	—
131	C₂×Q₈	—	4	2	no	—
132	C₂×Q₈	—	4	2	no	—
133	C₂²×C₄	—	4	1	no	—
134	C₂²×C₄	—	4	1	no	—
135	C₂²×C₄	—	4	2	no	—
136	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
137	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	4	no	—
141	C₂×C₈	—	2	4	no	—
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	C₁₃×SD₁₆	1	4	yes	—
144	C₄⋊C₄	—	2	4	no	—
145	C₄⋊C₄	—	4	2	no	—
146	C₄⋊C₄	—	4	2	no	—
147	C₄⋊C₄	—	4	4	no	—
148	C₄⋊C₄	—	4	4	no	—
149	C₄⋊C₄	—	4	4	no	—
150	C₄⋊C₄	—	4	4	no	—
151	C₄²	C₂×Q₈×C₁₃	1	1	yes	characteristic
152	C₄²	C₁₃×#41	1	1	yes	characteristic
153	C₄²	C₁₃×#41	1	2	yes	—
154	C₁₃	#10686	1	1	yes	characteristic, direct product factor, Sylow
155	C₂³	C₁₃×#127	1	1	yes	characteristic
156	Q₈	—	4	4	no	—
157	Q₈	—	8	2	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	8	1	no	—
160	C₂×C₄	—	8	1	no	—
161	C₂×C₄	—	8	1	no	—
162	C₂×C₄	—	8	2	no	—
163	C₂×C₄	—	8	2	no	—
164	C₂×C₄	—	8	2	no	—
165	C₂×C₄	—	8	2	no	—
166	C₂×C₄	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
167	C₂×C₄	C₁₃×C₂²⋊Q₈	1	2	yes	—
168	C₂×C₄	C₁₃×#123	1	1	yes	characteristic
169	C₂×C₄	C₁₃×#123	1	2	yes	—
170	C₂×C₄	C₂×C₁₃×SD₁₆	1	2	yes	—
171	C₂×C₄	C₁₃×#135	1	4	yes	—
172	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
173	C₂×C₄	C₁₃×#137	1	4	yes	—
174	C₈	—	2	4	no	—
175	C₈	—	4	4	no	—
176	C₂²	C₁₃×#399	1	1	yes	characteristic
177	C₂²	C₁₃×#492	1	1	yes	characteristic
178	C₂²	C₁₃×#495	1	2	yes	—
179	C₂²	C₁₃×#496	1	1	yes	characteristic
180	C₂²	C₁₃×#497	1	2	yes	—
181	C₄	—	8	2	no	—
182	C₄	—	16	1	no	—
183	C₄	—	16	2	no	—
184	C₄	C₁₃×#476	1	2	yes	—
185	C₄	C₁₃×#479	1	4	yes	—
186	C₄	C₁₃×#481	1	2	yes	—
187	C₄	C₁₃×#485	1	4	yes	—
188	C₂	C₁₃×#1708	1	1	yes	characteristic
189	C₂	C₁₃×#1716	1	1	yes	characteristic
190	C₂	C₁₃×#1719	1	2	yes	—
191	C₂	C₁₃×#2094	1	1	yes	characteristic
192	C₂	C₁₃×#2097	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#1990

C₂

yes

maximal

C₁₃×#1956

C₂

yes

maximal, characteristic

C₁₃×#1844

C₂

yes

maximal, characteristic

C₁₃×#1834

C₂

yes

maximal

C₁₃×#1273

C₂

yes

maximal, characteristic

C₂×C₁₃×C₄⋊Q₈

C₂²

yes

characteristic

C₂×C₁₃×#106

—

C₂×C₁₃×#103

—

C₂×C₄×C₈×C₁₃

C₂²

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₁₃×#384

C₂²

yes

—

C₁₃×#384

C₂²

yes

characteristic

C₁₃×#340

—

C₁₃×#340

—

C₂²×Q₈×C₁₃

—

C₂²×C₈×C₁₃

D₄

yes

—

C₂²×C₈×C₁₃

D₄

yes

—

C₁₃×C₄⋊Q₈

—

C₁₃×C₄⋊Q₈

—

C₁₃×C₄⋊Q₈

Q₈

yes

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₁₃×#106

—

C₁₃×#103

—

C₄×C₈×C₁₃

—

C₄×C₈×C₁₃

—

C₄×C₈×C₁₃

—

#10686

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

SD₁₆

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂×Q₈

yes

C₄²×C₁₃

yes

C₄²×C₁₃

yes

—

#1990

C₂×C₁₃

yes

—

#1956

C₂×C₁₃

yes

characteristic

#1844

C₂×C₁₃

yes

characteristic

#1834

C₂×C₁₃

yes

—

#1273

C₂×C₁₃

yes

characteristic

C₂³×C₁₃

#127

yes

characteristic

Q₈×C₁₃

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₃

#123

yes

characteristic

C₂×C₄×C₁₃

#123

yes

—

C₂×C₄×C₁₃

C₂×SD₁₆

yes

—

C₂×C₄×C₁₃

#135

yes

—

C₂×C₄×C₁₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₁₃

#137

yes

—

C₈×C₁₃

—

C₈×C₁₃

—

C₂×C₄⋊Q₈

C₂²×C₁₃

yes

characteristic

C₂×#106

—

C₂×#103

—

C₂×C₄×C₈

C₂²×C₁₃

yes

—

C₂×C₄×C₈

C₂²×C₁₃

yes

characteristic

#384

C₂²×C₁₃

yes

characteristic

#384

C₂²×C₁₃

yes

—

#340

—

#340

—

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

—

C₂²×C₁₃

#496

yes

characteristic

C₂²×C₁₃

#497

yes

—

C₄×C₁₃

—

100

C₄×C₁₃

—

101

C₄×C₁₃

—

102

C₄×C₁₃

#476

yes

—

103

C₄×C₁₃

#479

yes

—

104

C₄×C₁₃

#481

yes

—

105

C₄×C₁₃

#485

yes

—

106

C₂²×Q₈

—

107

C₂²×C₈

D₄×C₁₃

yes

—

108

C₂²×C₈

D₄×C₁₃

yes

—

109

C₄⋊Q₈

—

110

C₄⋊Q₈

—

111

C₄⋊Q₈

Q₈×C₁₃

yes

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄⋊C₄

—

117

C₂×C₄⋊C₄

—

118

C₂×C₄²

C₂³×C₁₃

yes

characteristic

119

#106

—

120

#103

—

121

C₄×C₈

—

122

C₄×C₈

—

123

C₄×C₈

—

124

C₂×C₁₃

#1708

yes

characteristic

125

C₂×C₁₃

#1716

yes

characteristic

126

C₂×C₁₃

#1719

yes

—

127

C₂×C₁₃

#2094

yes

characteristic

128

C₂×C₁₃

#2097

yes

—

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂²×C₄

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

C₂×D₄×C₁₃

yes

characteristic

137

C₂²×C₄

C₂×D₄×C₁₃

yes

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

C₁₃×SD₁₆

yes

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄²

C₂×Q₈×C₁₃

yes

characteristic

152

C₄²

C₁₃×#41

yes

characteristic

153

C₄²

C₁₃×#41

yes

—

154

C₁₃

#10686

yes

characteristic, direct product factor, Sylow

155

C₂³

C₁₃×#127

yes

characteristic

156

Q₈

—

157

Q₈

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

C₁₃×C₂²⋊Q₈

yes

characteristic

167

C₂×C₄

C₁₃×C₂²⋊Q₈

yes

—

168

C₂×C₄

C₁₃×#123

yes

characteristic

169

C₂×C₄

C₁₃×#123

yes

—

170

C₂×C₄

C₂×C₁₃×SD₁₆

yes

—

171

C₂×C₄

C₁₃×#135

yes

—

172

C₂×C₄

C₁₃×AGL(1,ℤ/8)

yes

—

173

C₂×C₄

C₁₃×#137

yes

—

174

C₈

—

175

C₈

—

176

C₂²

C₁₃×#399

yes

characteristic

177

C₂²

C₁₃×#492

yes

characteristic

178

C₂²

C₁₃×#495

yes

—

179

C₂²

C₁₃×#496

yes

characteristic

180

C₂²

C₁₃×#497

yes

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

C₁₃×#476

yes

—

185

C₄

C₁₃×#479

yes

—

186

C₄

C₁₃×#481

yes

—

187

C₄

C₁₃×#485

yes

—

188

C₂

C₁₃×#1708

yes

characteristic

189

C₂

C₁₃×#1716

yes

characteristic

190

C₂

C₁₃×#1719

yes

—

191

C₂

C₁₃×#2094

yes

characteristic

192

C₂

C₁₃×#2097

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₁₃

#127

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₁₃

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

N	H ≅ G/N	type
33. #10686	C₁₃	direct (N × H)
154. C₁₃	#10686	direct (N × H)

H ≅ G/N

type

33. #10686

C₁₃

direct (N × H)

154. C₁₃

#10686

direct (N × H)

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

subgroup	index
C₁₃×#1990	2
C₁₃×#1956	2
C₁₃×#1844	2
C₁₃×#1834	2
C₁₃×#1273	2
#10686	13

subgroup

index

C₁₃×#1990

C₁₃×#1956

C₁₃×#1844

C₁₃×#1834

C₁₃×#1273

#10686

Subgroups — C₁₃×#10686 #15697025

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C13×#10686 #15697025

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#10686 #15697025