Group #15696195 — subgroups

Subgroups — C₁₃×#9856 #15696195

Order 3328.

530 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 174 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×#3147	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1848	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#1492	C₂	1	4	yes	maximal, complemented
4	C₂×C₈²×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₃×#135	—	2	2	no	—
6	C₁₃×#532	C₂²	1	2	yes	—
7	C₁₃×#495	—	2	2	no	—
8	C₁₃×#494	C₄	1	1	yes	characteristic
9	C₁₃×#493	C₄	1	1	yes	characteristic
10	C₁₃×#492	C₄	1	2	yes	—
11	C₂×C₁₃×#108	—	2	2	no	—
12	C₁₃×#423	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₁₃×D₄⋊C₈	—	2	8	no	—
16	C₈²×C₁₃	—	2	2	no	—
17	C₂×C₁₃×#41	—	2	2	no	—
18	C₁₃×#135	—	4	4	no	—
19	C₁₃×#135	—	4	4	no	—
20	C₂×C₁₃×Q₁₆	—	2	2	no	—
21	C₂×C₁₃×SD₁₆	—	2	4	no	—
22	C₂×C₁₃×D₈	—	2	2	no	—
23	C₂×C₁₃×M₁₆	—	2	2	no	—
24	C₂²×C₈×C₁₃	—	2	2	no	—
25	C₂²×C₈×C₁₃	D₄	1	2	yes	—
26	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
27	C₁₃×#127	C₂×C₄	1	2	yes	—
28	C₁₃×#124	—	2	2	no	—
29	C₁₃×#121	—	2	4	no	—
30	C₄×D₄×C₁₃	—	2	4	no	—
31	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
32	C₁₃×#108	—	4	4	no	—
33	C₁₃×C₄⋊C₈	—	2	4	no	—
34	C₄×C₈×C₁₃	—	2	2	no	—
35	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₁₃	D₄	1	4	yes	—
38	C₄×C₈×C₁₃	D₄	1	2	yes	—
39	#9856	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₁₃×#41	—	4	4	no	—
41	C₁₃×#41	—	4	4	no	—
42	C₂×Q₈×C₁₃	—	2	2	no	—
43	C₂×D₄×C₁₃	—	2	2	no	—
44	C₂×D₄×C₁₃	—	2	4	no	—
45	C₂×D₄×C₁₃	—	4	2	no	—
46	C₂²×C₄×C₁₃	—	4	2	no	—
47	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	#41	1	2	yes	—
49	C₁₃×Q₁₆	—	4	4	no	—
50	C₁₃×SD₁₆	—	4	8	no	—
51	C₁₃×D₈	—	4	4	no	—
52	C₁₃×M₁₆	—	4	4	no	—
53	C₂×C₈×C₁₃	—	2	8	no	—
54	C₂×C₈×C₁₃	—	2	4	no	—
55	C₂×C₈×C₁₃	—	4	2	no	—
56	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
57	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
58	C₂×C₈×C₁₃	D₈	1	4	yes	—
59	C₂×C₈×C₁₃	SD₁₆	1	4	yes	—
60	C₁₃×C₄⋊C₄	—	2	4	no	—
61	C₁₃×C₂²⋊C₄	—	4	4	no	—
62	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
63	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₁₃	C₂×D₄	1	2	yes	—
65	#3147	C₂×C₁₃	1	1	yes	characteristic
66	#1848	C₂×C₁₃	1	1	yes	characteristic, complemented
67	#1492	C₂×C₁₃	1	4	yes	complemented
68	C₂×C₈²	C₂×C₁₃	1	1	yes	characteristic, complemented
69	C₂³×C₁₃	—	4	2	no	—
70	C₂³×C₁₃	#124	1	1	yes	characteristic
71	Q₈×C₁₃	—	4	4	no	—
72	D₄×C₁₃	—	4	4	no	—
73	D₄×C₁₃	—	4	8	no	—
74	D₄×C₁₃	—	8	4	no	—
75	C₂×C₄×C₁₃	—	2	4	no	—
76	C₂×C₄×C₁₃	—	4	2	no	—
77	C₂×C₄×C₁₃	—	8	2	no	—
78	C₂×C₄×C₁₃	—	8	2	no	—
79	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
81	C₂×C₄×C₁₃	#121	1	2	yes	—
82	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
84	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
85	C₈×C₁₃	—	2	8	no	—
86	C₈×C₁₃	—	8	2	no	—
87	C₈×C₁₃	#102	1	4	yes	—
88	C₈×C₁₃	#102	1	4	yes	—
89	C₂×#135	—	2	2	no	—
90	#532	C₂²×C₁₃	1	2	yes	—
91	#495	—	2	2	no	—
92	#494	C₄×C₁₃	1	1	yes	characteristic
93	#493	C₄×C₁₃	1	1	yes	characteristic
94	#492	C₄×C₁₃	1	2	yes	—
95	C₂×#108	—	2	2	no	—
96	#423	C₂²×C₁₃	1	2	yes	—
97	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
98	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
99	D₄⋊C₈	—	2	8	no	—
100	C₈²	—	2	2	no	—
101	C₂²×C₁₃	—	2	2	no	—
102	C₂²×C₁₃	—	8	2	no	—
103	C₂²×C₁₃	—	8	4	no	—
104	C₂²×C₁₃	#390	1	1	yes	characteristic
105	C₂²×C₁₃	#486	1	2	yes	—
106	C₄×C₁₃	—	2	2	no	—
107	C₄×C₁₃	—	8	2	no	—
108	C₄×C₁₃	C₂×#102	1	2	yes	—
109	C₄×C₁₃	#443	1	4	yes	—
110	C₄×C₁₃	#492	1	1	yes	characteristic
111	C₄×C₁₃	#493	1	1	yes	characteristic
112	C₂×#41	—	2	2	no	—
113	#135	—	4	4	no	—
114	#135	—	4	4	no	—
115	C₂×Q₁₆	—	2	2	no	—
116	C₂×SD₁₆	—	2	4	no	—
117	C₂×D₈	—	2	2	no	—
118	C₂×M₁₆	—	2	2	no	—
119	C₂²×C₈	—	2	2	no	—
120	C₂²×C₈	D₄×C₁₃	1	2	yes	—
121	C₄⋊Q₈	C₂×C₄×C₁₃	1	2	yes	—
122	#127	C₂×C₄×C₁₃	1	2	yes	—
123	#124	—	2	2	no	—
124	#121	—	2	4	no	—
125	C₄×D₄	—	2	4	no	—
126	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
127	#108	—	4	4	no	—
128	C₄⋊C₈	—	2	4	no	—
129	C₄×C₈	—	2	2	no	—
130	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
131	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
132	C₄×C₈	D₄×C₁₃	1	4	yes	—
133	C₄×C₈	D₄×C₁₃	1	2	yes	—
134	C₂×C₁₃	—	2	2	no	—
135	C₂×C₁₃	—	16	2	no	—
136	C₂×C₁₃	#1957	1	1	yes	characteristic
137	C₂×C₁₃	#1974	1	2	yes	—
138	#41	—	4	4	no	—
139	#41	—	4	4	no	—
140	C₂×Q₈	—	2	2	no	—
141	C₂×D₄	—	2	2	no	—
142	C₂×D₄	—	2	4	no	—
143	C₂×D₄	—	4	2	no	—
144	C₂²×C₄	—	4	2	no	—
145	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
146	C₂²×C₄	C₁₃×#41	1	2	yes	—
147	Q₁₆	—	4	4	no	—
148	SD₁₆	—	4	8	no	—
149	D₈	—	4	4	no	—
150	M₁₆	—	4	4	no	—
151	C₂×C₈	—	2	8	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	—	4	2	no	—
154	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
155	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
156	C₂×C₈	C₁₃×D₈	1	4	yes	—
157	C₂×C₈	C₁₃×SD₁₆	1	4	yes	—
158	C₄⋊C₄	—	2	4	no	—
159	C₂²⋊C₄	—	4	4	no	—
160	C₄²	C₂²×C₄×C₁₃	1	1	yes	characteristic
161	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
162	C₄²	C₂×D₄×C₁₃	1	2	yes	—
163	C₁₃	#9856	1	1	yes	characteristic, direct product factor, Sylow
164	C₂³	—	4	2	no	—
165	C₂³	C₁₃×#124	1	1	yes	characteristic
166	Q₈	—	4	4	no	—
167	D₄	—	4	4	no	—
168	D₄	—	4	8	no	—
169	D₄	—	8	4	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
175	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
176	C₂×C₄	C₁₃×#121	1	2	yes	—
177	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
178	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
179	C₂×C₄	C₂×C₁₃×SD₁₆	1	2	yes	—
180	C₈	—	2	8	no	—
181	C₈	—	8	2	no	—
182	C₈	C₁₃×#102	1	4	yes	—
183	C₈	C₁₃×#102	1	4	yes	—
184	C₂²	—	2	2	no	—
185	C₂²	—	8	2	no	—
186	C₂²	—	8	4	no	—
187	C₂²	C₁₃×#390	1	1	yes	characteristic
188	C₂²	C₁₃×#486	1	2	yes	—
189	C₄	—	2	2	no	—
190	C₄	—	8	2	no	—
191	C₄	C₂×C₁₃×#102	1	2	yes	—
192	C₄	C₁₃×#443	1	4	yes	—
193	C₄	C₁₃×#492	1	1	yes	characteristic
194	C₄	C₁₃×#493	1	1	yes	characteristic
195	C₂	—	2	2	no	—
196	C₂	—	16	2	no	—
197	C₂	C₁₃×#1957	1	1	yes	characteristic
198	C₂	C₁₃×#1974	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#3147

C₂

yes

maximal, characteristic

C₁₃×#1848

C₂

yes

maximal, characteristic, complemented

C₁₃×#1492

C₂

yes

maximal, complemented

C₂×C₈²×C₁₃

C₂

yes

maximal, characteristic, complemented

C₂×C₁₃×#135

—

C₁₃×#532

C₂²

yes

—

C₁₃×#495

—

C₁₃×#494

C₄

yes

characteristic

C₁₃×#493

C₄

yes

characteristic

C₁₃×#492

C₄

yes

—

C₂×C₁₃×#108

—

C₁₃×#423

C₂²

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₁₃×D₄⋊C₈

—

C₈²×C₁₃

—

C₂×C₁₃×#41

—

C₁₃×#135

—

C₁₃×#135

—

C₂×C₁₃×Q₁₆

—

C₂×C₁₃×SD₁₆

—

C₂×C₁₃×D₈

—

C₂×C₁₃×M₁₆

—

C₂²×C₈×C₁₃

—

C₂²×C₈×C₁₃

D₄

yes

—

C₁₃×C₄⋊Q₈

C₂×C₄

yes

—

C₁₃×#127

C₂×C₄

yes

—

C₁₃×#124

—

C₁₃×#121

—

C₄×D₄×C₁₃

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₁₃×#108

—

C₁₃×C₄⋊C₈

—

C₄×C₈×C₁₃

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

D₄

yes

—

C₄×C₈×C₁₃

D₄

yes

—

#9856

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₁₃×#41

—

C₁₃×#41

—

C₂×Q₈×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

#41

yes

—

C₁₃×Q₁₆

—

C₁₃×SD₁₆

—

C₁₃×D₈

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₂²⋊C₄

yes

—

C₂×C₈×C₁₃

C₂²⋊C₄

yes

—

C₂×C₈×C₁₃

D₈

yes

—

C₂×C₈×C₁₃

SD₁₆

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

C₂²×C₄

yes

characteristic

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄²×C₁₃

C₂×D₄

yes

—

#3147

C₂×C₁₃

yes

characteristic

#1848

C₂×C₁₃

yes

characteristic, complemented

#1492

C₂×C₁₃

yes

complemented

C₂×C₈²

C₂×C₁₃

yes

characteristic, complemented

C₂³×C₁₃

—

C₂³×C₁₃

#124

yes

characteristic

Q₈×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×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

#127

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

—

C₂×C₄×C₁₃

C₂×SD₁₆

yes

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

#102

yes

—

C₈×C₁₃

#102

yes

—

C₂×#135

—

#532

C₂²×C₁₃

yes

—

#495

—

#494

C₄×C₁₃

yes

characteristic

#493

C₄×C₁₃

yes

characteristic

#492

C₄×C₁₃

yes

—

C₂×#108

—

#423

C₂²×C₁₃

yes

—

C₂×C₄×C₈

C₂²×C₁₃

yes

—

C₂×C₄×C₈

C₂²×C₁₃

yes

characteristic

D₄⋊C₈

—

100

C₈²

—

101

C₂²×C₁₃

—

102

C₂²×C₁₃

—

103

C₂²×C₁₃

—

104

C₂²×C₁₃

#390

yes

characteristic

105

C₂²×C₁₃

#486

yes

—

106

C₄×C₁₃

—

107

C₄×C₁₃

—

108

C₄×C₁₃

C₂×#102

yes

—

109

C₄×C₁₃

#443

yes

—

110

C₄×C₁₃

#492

yes

characteristic

111

C₄×C₁₃

#493

yes

characteristic

112

C₂×#41

—

113

#135

—

114

#135

—

115

C₂×Q₁₆

—

116

C₂×SD₁₆

—

117

C₂×D₈

—

118

C₂×M₁₆

—

119

C₂²×C₈

—

120

C₂²×C₈

D₄×C₁₃

yes

—

121

C₄⋊Q₈

C₂×C₄×C₁₃

yes

—

122

#127

C₂×C₄×C₁₃

yes

—

123

#124

—

124

#121

—

125

C₄×D₄

—

126

C₂×C₄²

C₂³×C₁₃

yes

characteristic

127

#108

—

128

C₄⋊C₈

—

129

C₄×C₈

—

130

C₄×C₈

C₂×C₄×C₁₃

yes

characteristic

131

C₄×C₈

C₂×C₄×C₁₃

yes

characteristic

132

C₄×C₈

D₄×C₁₃

yes

—

133

C₄×C₈

D₄×C₁₃

yes

—

134

C₂×C₁₃

—

135

C₂×C₁₃

—

136

C₂×C₁₃

#1957

yes

characteristic

137

C₂×C₁₃

#1974

yes

—

138

#41

—

139

#41

—

140

C₂×Q₈

—

141

C₂×D₄

—

142

C₂×D₄

—

143

C₂×D₄

—

144

C₂²×C₄

—

145

C₂²×C₄

C₂×D₄×C₁₃

yes

characteristic

146

C₂²×C₄

C₁₃×#41

yes

—

147

Q₁₆

—

148

SD₁₆

—

149

D₈

—

150

M₁₆

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

C₁₃×C₂²⋊C₄

yes

—

155

C₂×C₈

C₁₃×C₂²⋊C₄

yes

—

156

C₂×C₈

C₁₃×D₈

yes

—

157

C₂×C₈

C₁₃×SD₁₆

yes

—

158

C₄⋊C₄

—

159

C₂²⋊C₄

—

160

C₄²

C₂²×C₄×C₁₃

yes

characteristic

161

C₄²

C₂×D₄×C₁₃

yes

characteristic

162

C₄²

C₂×D₄×C₁₃

yes

—

163

C₁₃

#9856

yes

characteristic, direct product factor, Sylow

164

C₂³

—

165

C₂³

C₁₃×#124

yes

characteristic

166

Q₈

—

167

D₄

—

168

D₄

—

169

D₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

C₂×C₁₃×C₂²⋊C₄

yes

characteristic

175

C₂×C₄

C₄×D₄×C₁₃

yes

—

176

C₂×C₄

C₁₃×#121

yes

—

177

C₂×C₄

C₁₃×#127

yes

characteristic

178

C₂×C₄

C₂×C₁₃×D₈

yes

—

179

C₂×C₄

C₂×C₁₃×SD₁₆

yes

—

180

C₈

—

181

C₈

—

182

C₈

C₁₃×#102

yes

—

183

C₈

C₁₃×#102

yes

—

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

C₁₃×#390

yes

characteristic

188

C₂²

C₁₃×#486

yes

—

189

C₄

—

190

C₄

—

191

C₄

C₂×C₁₃×#102

yes

—

192

C₄

C₁₃×#443

yes

—

193

C₄

C₁₃×#492

yes

characteristic

194

C₄

C₁₃×#493

yes

characteristic

195

C₂

—

196

C₂

—

197

C₂

C₁₃×#1957

yes

characteristic

198

C₂

C₁₃×#1974

yes

—

	subgroup	G/H
center	C₂×C₄×C₁₃	#127
commutator	C₄²	C₂²×C₄×C₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
socle	C₂²×C₁₃	#390
prime radical	C₁₃×#3147	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

subgroup

G/H

center

C₂×C₄×C₁₃

#127

commutator

C₄²

C₂²×C₄×C₁₃

Frattini, cosocle

C₂×C₄²

C₂³×C₁₃

socle

C₂²×C₁₃

#390

prime radical

C₁₃×#3147

C₂

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

N	H ≅ G/N	type
2. C₁₃×#1848	C₂	semidirect
3. C₁₃×#1492	C₂	semidirect
4. C₂×C₈²×C₁₃	C₂	semidirect
39. #9856	C₁₃	direct (N × H)
66. #1848	C₂×C₁₃	semidirect
67. #1492	C₂×C₁₃	semidirect
68. C₂×C₈²	C₂×C₁₃	semidirect
163. C₁₃	#9856	direct (N × H)

H ≅ G/N

type

2. C₁₃×#1848

C₂

semidirect

3. C₁₃×#1492

C₂

semidirect

4. C₂×C₈²×C₁₃

C₂

semidirect

39. #9856

C₁₃

direct (N × H)

66. #1848

C₂×C₁₃

semidirect

67. #1492

C₂×C₁₃

semidirect

68. C₂×C₈²

C₂×C₁₃

semidirect

163. C₁₃

#9856

direct (N × H)

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

subgroup	index
C₁₃×#3147	2
C₁₃×#1848	2
C₁₃×#1492	2
C₂×C₈²×C₁₃	2
#9856	13

subgroup

index

C₁₃×#3147

C₁₃×#1848

C₁₃×#1492

C₂×C₈²×C₁₃

#9856

Subgroups — C₁₃×#9856 #15696195

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C13×#9856 #15696195

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#9856 #15696195