Group #15696638 — subgroups

Subgroups — C₁₃×#10299 #15696638

Order 3328.

576 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 130 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×#3229	C₂	1	2	yes	maximal
2	C₁₃×#2868	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1994	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1546	C₂	1	2	yes	maximal
5	C₁₃×#1276	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#533	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×C₄⋊Q₈	—	2	2	no	—
8	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₁₃×#498	—	2	2	no	—
10	C₁₃×#487	—	2	2	no	—
11	C₁₃×#483	—	2	4	no	—
12	C₁₃×#475	—	2	4	no	—
13	C₁₃×#431	C₂²	1	2	yes	—
14	C₁₃×#423	C₂²	1	1	yes	characteristic
15	C₁₃×#384	—	2	4	no	—
16	C₄³×C₁₃	C₂²	1	1	yes	characteristic
17	C₁₃×#339	—	2	2	no	—
18	C₁₃×#332	—	2	4	no	—
19	C₂²×Q₈×C₁₃	—	2	2	no	—
20	C₂×C₁₃×M₁₆	—	2	1	no	—
21	C₁₃×C₄⋊Q₈	—	2	2	no	—
22	C₁₃×C₄⋊Q₈	—	2	2	no	—
23	C₁₃×C₄⋊Q₈	—	2	2	no	—
24	C₁₃×C₄⋊Q₈	—	2	2	no	—
25	C₁₃×C₄⋊Q₈	—	4	4	no	—
26	C₁₃×C₄⋊Q₈	—	4	4	no	—
27	C₁₃×C₄⋊Q₈	D₄	1	2	yes	—
28	C₁₃×#125	—	2	1	no	—
29	C₁₃×C₂²⋊Q₈	—	2	2	no	—
30	C₄×Q₈×C₁₃	—	2	2	no	—
31	C₁₃×#117	—	2	1	no	—
32	C₂×C₁₃×C₄⋊C₄	—	2	8	no	—
33	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
34	C₂×C₁₃×C₄⋊C₄	D₄	1	4	yes	—
35	C₂×C₄²×C₁₃	—	2	2	no	—
36	C₂×C₄²×C₁₃	D₄	1	2	yes	—
37	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
38	C₁₃×#106	—	2	8	no	—
39	C₁₃×C₄⋊C₈	—	2	2	no	—
40	C₁₃×C₄⋊C₈	D₄	1	4	yes	—
41	C₁₃×#103	—	2	8	no	—
42	C₁₃×C₂²⋊C₈	—	2	2	no	—
43	C₄×C₈×C₁₃	—	2	2	no	—
44	#10299	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	C₂×Q₈×C₁₃	—	2	2	no	—
46	C₂×Q₈×C₁₃	—	4	2	no	—
47	C₂×Q₈×C₁₃	—	4	2	no	—
48	C₂×Q₈×C₁₃	—	4	4	no	—
49	C₂×Q₈×C₁₃	—	4	4	no	—
50	C₂²×C₄×C₁₃	—	2	2	no	—
51	C₂²×C₄×C₁₃	—	2	4	no	—
52	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
53	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
54	C₁₃×M₁₆	—	4	2	no	—
55	C₂×C₈×C₁₃	—	2	4	no	—
56	C₂×C₈×C₁₃	—	4	1	no	—
57	C₁₃×C₄⋊C₄	—	2	4	no	—
58	C₁₃×C₄⋊C₄	—	2	2	no	—
59	C₁₃×C₄⋊C₄	—	2	4	no	—
60	C₁₃×C₄⋊C₄	—	2	2	no	—
61	C₁₃×C₄⋊C₄	—	4	16	no	—
62	C₁₃×C₄⋊C₄	—	4	2	no	—
63	C₁₃×C₄⋊C₄	—	4	2	no	—
64	C₁₃×C₄⋊C₄	—	4	2	no	—
65	C₁₃×C₂²⋊C₄	—	4	1	no	—
66	C₄²×C₁₃	—	2	8	no	—
67	C₄²×C₁₃	—	2	2	no	—
68	C₄²×C₁₃	—	4	1	no	—
69	C₄²×C₁₃	SD₁₆	1	4	yes	—
70	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₁₃	C₂×D₄	1	2	yes	—
72	C₄²×C₁₃	#41	1	1	yes	characteristic
73	#3229	C₂×C₁₃	1	2	yes	—
74	#2868	C₂×C₁₃	1	1	yes	characteristic
75	#1994	C₂×C₁₃	1	1	yes	characteristic
76	#1546	C₂×C₁₃	1	2	yes	—
77	#1276	C₂×C₁₃	1	1	yes	characteristic
78	C₂³×C₁₃	#120	1	1	yes	characteristic
79	Q₈×C₁₃	—	4	4	no	—
80	Q₈×C₁₃	—	8	4	no	—
81	Q₈×C₁₃	—	8	2	no	—
82	Q₈×C₁₃	—	8	2	no	—
83	C₂×C₄×C₁₃	—	2	8	no	—
84	C₂×C₄×C₁₃	—	2	4	no	—
85	C₂×C₄×C₁₃	—	2	4	no	—
86	C₂×C₄×C₁₃	—	4	2	no	—
87	C₂×C₄×C₁₃	—	4	4	no	—
88	C₂×C₄×C₁₃	—	4	8	no	—
89	C₂×C₄×C₁₃	#120	1	2	yes	—
90	C₂×C₄×C₁₃	#121	1	2	yes	—
91	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
92	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
94	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
95	C₈×C₁₃	—	4	4	no	—
96	C₈×C₁₃	—	8	1	no	—
97	#533	C₂²×C₁₃	1	1	yes	characteristic
98	C₂×C₄⋊Q₈	—	2	2	no	—
99	C₂×C₄⋊Q₈	C₂²×C₁₃	1	2	yes	—
100	#498	—	2	2	no	—
101	#487	—	2	2	no	—
102	#483	—	2	4	no	—
103	#475	—	2	4	no	—
104	#431	C₂²×C₁₃	1	2	yes	—
105	#423	C₂²×C₁₃	1	1	yes	characteristic
106	#384	—	2	4	no	—
107	C₄³	C₂²×C₁₃	1	1	yes	characteristic
108	#339	—	2	2	no	—
109	#332	—	2	4	no	—
110	C₂²×C₁₃	—	2	2	no	—
111	C₂²×C₁₃	#392	1	1	yes	characteristic
112	C₂²×C₁₃	#450	1	2	yes	—
113	C₄×C₁₃	—	2	8	no	—
114	C₄×C₁₃	—	2	2	no	—
115	C₄×C₁₃	—	8	2	no	—
116	C₄×C₁₃	—	8	4	no	—
117	C₄×C₁₃	#456	1	4	yes	—
118	C₄×C₁₃	#460	1	2	yes	—
119	C₄×C₁₃	#492	1	1	yes	characteristic
120	C₄×C₁₃	#496	1	1	yes	characteristic
121	C₂²×Q₈	—	2	2	no	—
122	C₂×M₁₆	—	2	1	no	—
123	C₄⋊Q₈	—	2	2	no	—
124	C₄⋊Q₈	—	2	2	no	—
125	C₄⋊Q₈	—	2	2	no	—
126	C₄⋊Q₈	—	2	2	no	—
127	C₄⋊Q₈	—	4	4	no	—
128	C₄⋊Q₈	—	4	4	no	—
129	C₄⋊Q₈	D₄×C₁₃	1	2	yes	—
130	#125	—	2	1	no	—
131	C₂²⋊Q₈	—	2	2	no	—
132	C₄×Q₈	—	2	2	no	—
133	#117	—	2	1	no	—
134	C₂×C₄⋊C₄	—	2	8	no	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	D₄×C₁₃	1	4	yes	—
137	C₂×C₄²	—	2	2	no	—
138	C₂×C₄²	D₄×C₁₃	1	2	yes	—
139	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
140	#106	—	2	8	no	—
141	C₄⋊C₈	—	2	2	no	—
142	C₄⋊C₈	D₄×C₁₃	1	4	yes	—
143	#103	—	2	8	no	—
144	C₂²⋊C₈	—	2	2	no	—
145	C₄×C₈	—	2	2	no	—
146	C₂×C₁₃	—	2	2	no	—
147	C₂×C₁₃	#2005	1	2	yes	—
148	C₂×C₁₃	#2014	1	1	yes	characteristic
149	C₂×Q₈	—	2	2	no	—
150	C₂×Q₈	—	4	2	no	—
151	C₂×Q₈	—	4	2	no	—
152	C₂×Q₈	—	4	4	no	—
153	C₂×Q₈	—	4	4	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	2	4	no	—
156	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
157	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
158	M₁₆	—	4	2	no	—
159	C₂×C₈	—	2	4	no	—
160	C₂×C₈	—	4	1	no	—
161	C₄⋊C₄	—	2	4	no	—
162	C₄⋊C₄	—	2	2	no	—
163	C₄⋊C₄	—	2	4	no	—
164	C₄⋊C₄	—	2	2	no	—
165	C₄⋊C₄	—	4	16	no	—
166	C₄⋊C₄	—	4	2	no	—
167	C₄⋊C₄	—	4	2	no	—
168	C₄⋊C₄	—	4	2	no	—
169	C₂²⋊C₄	—	4	1	no	—
170	C₄²	—	2	8	no	—
171	C₄²	—	2	2	no	—
172	C₄²	—	4	1	no	—
173	C₄²	C₁₃×SD₁₆	1	4	yes	—
174	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
175	C₄²	C₂×D₄×C₁₃	1	2	yes	—
176	C₄²	C₁₃×#41	1	1	yes	characteristic
177	C₁₃	#10299	1	1	yes	characteristic, direct product factor, Sylow
178	C₂³	C₁₃×#120	1	1	yes	characteristic
179	Q₈	—	4	4	no	—
180	Q₈	—	8	4	no	—
181	Q₈	—	8	2	no	—
182	Q₈	—	8	2	no	—
183	C₂×C₄	—	2	8	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	4	no	—
188	C₂×C₄	—	4	8	no	—
189	C₂×C₄	C₁₃×#120	1	2	yes	—
190	C₂×C₄	C₁₃×#121	1	2	yes	—
191	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
192	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
193	C₂×C₄	C₂×C₁₃×SD₁₆	1	2	yes	—
194	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
195	C₈	—	4	4	no	—
196	C₈	—	8	1	no	—
197	C₂²	—	2	2	no	—
198	C₂²	C₁₃×#392	1	1	yes	characteristic
199	C₂²	C₁₃×#450	1	2	yes	—
200	C₄	—	2	8	no	—
201	C₄	—	2	2	no	—
202	C₄	—	8	2	no	—
203	C₄	—	8	4	no	—
204	C₄	C₁₃×#456	1	4	yes	—
205	C₄	C₁₃×#460	1	2	yes	—
206	C₄	C₁₃×#492	1	1	yes	characteristic
207	C₄	C₁₃×#496	1	1	yes	characteristic
208	C₂	—	2	2	no	—
209	C₂	C₁₃×#2005	1	2	yes	—
210	C₂	C₁₃×#2014	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#3229

C₂

yes

maximal

C₁₃×#2868

C₂

yes

maximal, characteristic

C₁₃×#1994

C₂

yes

maximal, characteristic

C₁₃×#1546

C₂

yes

maximal

C₁₃×#1276

C₂

yes

maximal, characteristic

C₁₃×#533

C₂²

yes

characteristic

C₂×C₁₃×C₄⋊Q₈

—

C₂×C₁₃×C₄⋊Q₈

C₂²

yes

—

C₁₃×#498

—

C₁₃×#487

—

C₁₃×#483

—

C₁₃×#475

—

C₁₃×#431

C₂²

yes

—

C₁₃×#423

C₂²

yes

characteristic

C₁₃×#384

—

C₄³×C₁₃

C₂²

yes

characteristic

C₁₃×#339

—

C₁₃×#332

—

C₂²×Q₈×C₁₃

—

C₂×C₁₃×M₁₆

—

C₁₃×C₄⋊Q₈

—

C₁₃×C₄⋊Q₈

—

C₁₃×C₄⋊Q₈

—

C₁₃×C₄⋊Q₈

—

C₁₃×C₄⋊Q₈

—

C₁₃×C₄⋊Q₈

—

C₁₃×C₄⋊Q₈

D₄

yes

—

C₁₃×#125

—

C₁₃×C₂²⋊Q₈

—

C₄×Q₈×C₁₃

—

C₁₃×#117

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

D₄

yes

—

C₂×C₄²×C₁₃

—

C₂×C₄²×C₁₃

D₄

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₁₃×#106

—

C₁₃×C₄⋊C₈

—

C₁₃×C₄⋊C₈

D₄

yes

—

C₁₃×#103

—

C₁₃×C₂²⋊C₈

—

C₄×C₈×C₁₃

—

#10299

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₁₃×M₁₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

SD₁₆

yes

—

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄²×C₁₃

C₂×D₄

yes

—

C₄²×C₁₃

#41

yes

characteristic

#3229

C₂×C₁₃

yes

—

#2868

C₂×C₁₃

yes

characteristic

#1994

C₂×C₁₃

yes

characteristic

#1546

C₂×C₁₃

yes

—

#1276

C₂×C₁₃

yes

characteristic

C₂³×C₁₃

#120

yes

characteristic

Q₈×C₁₃

—

Q₈×C₁₃

—

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

#120

yes

—

C₂×C₄×C₁₃

#121

yes

—

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

C₂×SD₁₆

yes

—

C₂×C₄×C₁₃

AGL(1,ℤ/8)

yes

—

C₈×C₁₃

—

C₈×C₁₃

—

#533

C₂²×C₁₃

yes

characteristic

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

C₂²×C₁₃

yes

—

100

#498

—

101

#487

—

102

#483

—

103

#475

—

104

#431

C₂²×C₁₃

yes

—

105

#423

C₂²×C₁₃

yes

characteristic

106

#384

—

107

C₄³

C₂²×C₁₃

yes

characteristic

108

#339

—

109

#332

—

110

C₂²×C₁₃

—

111

C₂²×C₁₃

#392

yes

characteristic

112

C₂²×C₁₃

#450

yes

—

113

C₄×C₁₃

—

114

C₄×C₁₃

—

115

C₄×C₁₃

—

116

C₄×C₁₃

—

117

C₄×C₁₃

#456

yes

—

118

C₄×C₁₃

#460

yes

—

119

C₄×C₁₃

#492

yes

characteristic

120

C₄×C₁₃

#496

yes

characteristic

121

C₂²×Q₈

—

122

C₂×M₁₆

—

123

C₄⋊Q₈

—

124

C₄⋊Q₈

—

125

C₄⋊Q₈

—

126

C₄⋊Q₈

—

127

C₄⋊Q₈

—

128

C₄⋊Q₈

—

129

C₄⋊Q₈

D₄×C₁₃

yes

—

130

#125

—

131

C₂²⋊Q₈

—

132

C₄×Q₈

—

133

#117

—

134

C₂×C₄⋊C₄

—

135

C₂×C₄⋊C₄

—

136

C₂×C₄⋊C₄

D₄×C₁₃

yes

—

137

C₂×C₄²

—

138

C₂×C₄²

D₄×C₁₃

yes

—

139

C₂×C₄²

C₂³×C₁₃

yes

characteristic

140

#106

—

141

C₄⋊C₈

—

142

C₄⋊C₈

D₄×C₁₃

yes

—

143

#103

—

144

C₂²⋊C₈

—

145

C₄×C₈

—

146

C₂×C₁₃

—

147

C₂×C₁₃

#2005

yes

—

148

C₂×C₁₃

#2014

yes

characteristic

149

C₂×Q₈

—

150

C₂×Q₈

—

151

C₂×Q₈

—

152

C₂×Q₈

—

153

C₂×Q₈

—

154

C₂²×C₄

—

155

C₂²×C₄

—

156

C₂²×C₄

C₂×D₄×C₁₃

yes

—

157

C₂²×C₄

C₂×D₄×C₁₃

yes

characteristic

158

M₁₆

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

168

C₄⋊C₄

—

169

C₂²⋊C₄

—

170

C₄²

—

171

C₄²

—

172

C₄²

—

173

C₄²

C₁₃×SD₁₆

yes

—

174

C₄²

C₂×D₄×C₁₃

yes

characteristic

175

C₄²

C₂×D₄×C₁₃

yes

—

176

C₄²

C₁₃×#41

yes

characteristic

177

C₁₃

#10299

yes

characteristic, direct product factor, Sylow

178

C₂³

C₁₃×#120

yes

characteristic

179

Q₈

—

180

Q₈

—

181

Q₈

—

182

Q₈

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

C₁₃×#120

yes

—

190

C₂×C₄

C₁₃×#121

yes

—

191

C₂×C₄

C₁₃×#121

yes

characteristic

192

C₂×C₄

C₁₃×#127

yes

characteristic

193

C₂×C₄

C₂×C₁₃×SD₁₆

yes

—

194

C₂×C₄

C₁₃×AGL(1,ℤ/8)

yes

—

195

C₈

—

196

C₈

—

197

C₂²

—

198

C₂²

C₁₃×#392

yes

characteristic

199

C₂²

C₁₃×#450

yes

—

200

C₄

—

201

C₄

—

202

C₄

—

203

C₄

—

204

C₄

C₁₃×#456

yes

—

205

C₄

C₁₃×#460

yes

—

206

C₄

C₁₃×#492

yes

characteristic

207

C₄

C₁₃×#496

yes

characteristic

208

C₂

—

209

C₂

C₁₃×#2005

yes

—

210

C₂

C₁₃×#2014

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
prime radical	C₂³×C₁₃	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₃

#392

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₁₃

prime radical

C₂³×C₁₃

#120

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

N	H ≅ G/N	type
44. #10299	C₁₃	direct (N × H)
177. C₁₃	#10299	direct (N × H)

H ≅ G/N

type

44. #10299

C₁₃

direct (N × H)

177. C₁₃

#10299

direct (N × H)

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

subgroup	index
C₁₃×#3229	2
C₁₃×#2868	2
C₁₃×#1994	2
C₁₃×#1546	2
C₁₃×#1276	2
#10299	13

subgroup

index

C₁₃×#3229

C₁₃×#2868

C₁₃×#1994

C₁₃×#1546

C₁₃×#1276

#10299

Subgroups — C₁₃×#10299 #15696638

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C13×#10299 #15696638

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#10299 #15696638