Group #15779278 — subgroups

Subgroups — #15779278

Order 3328.

342 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 127 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	#13116591	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
3	#13112918	C₂	1	1	yes	maximal, characteristic
4	#13112917	C₂	1	2	yes	maximal
5	C₁₃×#1376	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
12	C₂×#88052	C₂²	1	2	yes	—
13	C₄²×Dic₁₃	C₄	1	1	yes	characteristic
14	C₂×C₄×#1055	C₄	1	1	yes	characteristic
15	#11718093	—	2	4	no	—
16	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
18	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₁₃	—	2	2	no	—
20	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
23	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
24	C₁₃×M₃₂	—	2	2	no	—
25	C₁₃×M₃₂	—	2	2	no	—
26	C₂×C₁₃×C₁₆	—	2	2	no	—
27	C₄×C₈×C₁₃	—	2	2	no	—
28	C₈×Dic₁₃	—	2	2	no	—
29	C₈×Dic₁₃	—	2	2	no	—
30	C₈×Dic₁₃	C₈	1	2	yes	—
31	C₈×Dic₁₃	C₈	1	2	yes	—
32	C₈×Dic₁₃	D₄	1	2	yes	—
33	C₈×Dic₁₃	Q₈	1	2	yes	—
34	#88052	—	2	2	no	—
35	#88052	—	2	2	no	—
36	C₂×#6090	—	2	2	no	—
37	C₄×#1055	—	2	2	no	—
38	#12883	—	13	1	no	maximal, Sylow
39	C₂²×C₄×C₁₃	—	2	1	no	—
40	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×Dic₁₃	—	2	1	no	—
42	C₂²×Dic₁₃	M₁₆	1	2	yes	—
43	C₂×C₈×C₁₃	—	2	1	no	—
44	C₂×C₈×C₁₃	—	2	1	no	—
45	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
46	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈×C₁₃	#41	1	2	yes	—
49	C₂×C₈×C₁₃	#41	1	2	yes	—
50	C₄²×C₁₃	—	2	1	no	—
51	C₄²×C₁₃	M₁₆	1	2	yes	—
52	C₄×Dic₁₃	—	2	8	no	—
53	C₄×Dic₁₃	—	2	1	no	—
54	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
55	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
56	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
57	C₄×Dic₁₃	M₁₆	1	2	yes	—
58	C₂×#1055	—	2	1	no	—
59	C₂×#1055	—	2	1	no	—
60	C₂×#1055	—	2	1	no	—
61	C₂×#1055	—	2	1	no	—
62	C₂×#1055	C₄⋊C₄	1	2	yes	—
63	C₂×#1055	C₂×C₈	1	2	yes	—
64	C₁₃×C₁₆	—	4	2	no	—
65	#6090	—	4	2	no	—
66	#2150	—	13	2	no	—
67	C₄²×C₈	—	13	1	no	—
68	#1376	—	13	2	no	—
69	#1376	—	13	1	no	—
70	#1376	D₁₃	1	1	yes	characteristic
71	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
72	C₂×C₄×C₁₃	—	2	4	no	—
73	C₂×C₄×C₁₃	—	2	2	no	—
74	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
76	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
80	C₂×Dic₁₃	—	2	4	no	—
81	C₂×Dic₁₃	—	2	2	no	—
82	C₂×Dic₁₃	—	2	4	no	—
83	C₂×Dic₁₃	C₄⋊C₈	1	2	yes	—
84	C₂×Dic₁₃	C₄⋊C₈	1	2	yes	—
85	C₈×C₁₃	—	2	2	no	—
86	C₈×C₁₃	#124	1	2	yes	—
87	C₈×C₁₃	#125	1	2	yes	—
88	#1055	—	2	2	no	—
89	#1055	—	2	2	no	—
90	C₂×M₃₂	—	13	2	no	—
91	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
92	C₂×C₄×C₈	—	13	1	no	—
93	C₂×C₄×C₈	—	13	1	no	—
94	C₂×C₄×C₈	—	13	1	no	—
95	C₂×C₄×C₈	—	13	1	no	—
96	C₂×C₄×C₈	—	13	1	no	—
97	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
98	C₄³	—	13	1	no	—
99	C₄⋊C₁₆	—	26	4	no	—
100	C₂²×C₁₃	—	2	2	no	—
101	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
102	C₂²×C₁₃	#432	1	1	yes	characteristic
103	C₂²×C₁₃	#432	1	1	yes	characteristic
104	C₄×C₁₃	—	2	4	no	—
105	C₄×C₁₃	#382	1	1	yes	characteristic
106	C₄×C₁₃	#423	1	1	yes	characteristic
107	C₄×C₁₃	#431	1	1	yes	characteristic
108	C₄×C₁₃	#431	1	1	yes	characteristic
109	Dic₁₃	—	2	4	no	—
110	Dic₁₃	—	2	2	no	—
111	Dic₁₃	#364	1	4	yes	—
112	C₂²×C₈	—	13	1	no	—
113	C₂²×C₈	—	13	1	no	—
114	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
115	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
116	C₂×C₄²	—	13	1	no	—
117	C₂×C₄²	—	13	1	no	—
118	C₂×C₄²	—	26	2	no	—
119	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
120	M₃₂	—	2	2	no	—
121	M₃₂	—	2	2	no	—
122	M₃₂	—	26	2	no	—
123	M₃₂	—	26	2	no	—
124	C₂×C₁₆	—	2	2	no	—
125	C₂×C₁₆	—	26	2	no	—
126	C₄×C₈	—	2	2	no	—
127	C₄×C₈	—	13	2	no	—
128	C₄×C₈	—	13	2	no	—
129	C₄×C₈	—	13	2	no	—
130	C₄×C₈	—	13	2	no	—
131	C₄×C₈	—	26	2	no	—
132	C₄×C₈	—	26	2	no	—
133	C₄×C₈	—	26	2	no	—
134	C₂×C₁₃	—	2	1	no	—
135	C₂×C₁₃	—	2	1	no	—
136	C₂×C₁₃	#1831	1	1	yes	characteristic
137	C₂×C₁₃	C₂×#364	1	2	yes	—
138	C₂²×C₄	—	2	1	no	—
139	C₂²×C₄	—	13	2	no	—
140	C₂²×C₄	—	26	1	no	—
141	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	13	2	no	—
145	C₂×C₈	—	13	2	no	—
146	C₂×C₈	—	26	1	no	—
147	C₂×C₈	—	26	1	no	—
148	C₂×C₈	—	26	1	no	—
149	C₂×C₈	—	26	1	no	—
150	C₂×C₈	C₈×D₁₃	1	2	yes	—
151	C₂×C₈	#6127	1	2	yes	—
152	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
153	C₂×C₈	#6129	1	2	yes	—
154	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
155	C₄²	—	2	1	no	—
156	C₄²	—	13	2	no	—
157	C₄²	—	13	1	no	—
158	C₄²	—	13	1	no	—
159	C₄²	—	13	2	no	—
160	C₄²	—	26	8	no	—
161	C₄²	—	26	1	no	—
162	C₄²	#6094	1	2	yes	—
163	C₁₆	—	4	2	no	—
164	C₁₆	—	52	2	no	—
165	C₁₃	#12883	1	1	yes	characteristic, complemented, Sylow
166	C₂³	M₁₆×D₁₃	1	1	yes	characteristic
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	13	2	no	—
170	C₂×C₄	—	13	2	no	—
171	C₂×C₄	—	26	4	no	—
172	C₂×C₄	—	26	4	no	—
173	C₂×C₄	—	26	2	no	—
174	C₂×C₄	#88126	1	1	yes	characteristic
175	C₂×C₄	#88131	1	1	yes	characteristic
176	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
177	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
178	C₂×C₄	C₂×#6094	1	1	yes	characteristic
179	C₂×C₄	M₁₆×D₁₃	1	1	yes	characteristic
180	C₈	—	2	2	no	—
181	C₈	—	26	2	no	—
182	C₈	—	26	2	no	—
183	C₈	#88139	1	2	yes	—
184	C₈	#88143	1	2	yes	—
185	C₂²	—	2	2	no	—
186	C₂²	#11718366	1	1	yes	characteristic
187	C₂²	#11718380	1	1	yes	characteristic
188	C₂²	D₁₃×C₄⋊C₈	1	1	yes	characteristic
189	C₄	—	2	4	no	—
190	C₄	—	13	4	no	—
191	C₄	—	26	4	no	—
192	C₄	—	26	2	no	—
193	C₄	#11718311	1	1	yes	characteristic
194	C₄	#11718346	1	1	yes	characteristic
195	C₄	#11718379	1	1	yes	characteristic
196	C₄	#11718498	1	1	yes	characteristic
197	C₂	—	2	1	no	—
198	C₂	—	2	1	no	—
199	C₂	#13114951	1	1	yes	characteristic
200	C₂	D₁₃×#364	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13116591

C₂

yes

maximal

C₄×C₈×Dic₁₃

C₂

yes

maximal, characteristic

#13112918

C₂

yes

maximal, characteristic

#13112917

C₂

yes

maximal

C₁₃×#1376

C₂

yes

maximal, characteristic

C₂×C₁₃×M₃₂

C₂²

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₂×C₈×Dic₁₃

C₄

yes

characteristic

C₂×C₈×Dic₁₃

C₄

yes

characteristic

C₂×C₈×Dic₁₃

C₂²

yes

characteristic

C₂×C₈×Dic₁₃

C₂²

yes

characteristic

C₂×#88052

C₂²

yes

—

C₄²×Dic₁₃

C₄

yes

characteristic

C₂×C₄×#1055

C₄

yes

characteristic

#11718093

—

C₂²×C₈×C₁₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₃

C₂³

yes

characteristic

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

C₂×C₄×Dic₁₃

—

C₂×C₄×Dic₁₃

C₂×C₄

yes

characteristic

C₂×C₄×Dic₁₃

C₂×C₄

yes

characteristic

C₂²×#1055

C₂×C₄

yes

characteristic

C₂²×#1055

C₂×C₄

yes

characteristic

C₁₃×M₃₂

—

C₁₃×M₃₂

—

C₂×C₁₃×C₁₆

—

C₄×C₈×C₁₃

—

C₈×Dic₁₃

—

C₈×Dic₁₃

—

C₈×Dic₁₃

C₈

yes

—

C₈×Dic₁₃

C₈

yes

—

C₈×Dic₁₃

D₄

yes

—

C₈×Dic₁₃

Q₈

yes

—

—

—

—

—

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂²×Dic₁₃

—

C₂²×Dic₁₃

M₁₆

yes

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₂×C₈

yes

—

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₃

#41

yes

—

C₂×C₈×C₁₃

#41

yes

—

C₄²×C₁₃

—

C₄²×C₁₃

M₁₆

yes

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

C₄⋊C₄

yes

—

C₄×Dic₁₃

C₂×C₈

yes

characteristic

C₄×Dic₁₃

C₂×C₈

yes

characteristic

C₄×Dic₁₃

M₁₆

yes

—

—

—

—

—

C₄⋊C₄

yes

—

C₂×#1055

C₂×C₈

yes

—

C₁₃×C₁₆

—

#6090

—

#2150

—

C₄²×C₈

—

#1376

—

#1376

—

#1376

D₁₃

yes

characteristic

C₂³×C₁₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₁₃

C₂×M₁₆

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₄⋊C₈

yes

—

C₂×Dic₁₃

C₄⋊C₈

yes

—

C₈×C₁₃

—

C₈×C₁₃

#124

yes

—

C₈×C₁₃

#125

yes

—

#1055

—

#1055

—

C₂×M₃₂

—

C₂×M₃₂

C₂×D₁₃

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

C₄³

—

C₄⋊C₁₆

—

100

C₂²×C₁₃

—

101

C₂²×C₁₃

C₂×C₄⋊C₈

yes

characteristic

102

C₂²×C₁₃

#432

yes

characteristic

103

C₂²×C₁₃

#432

yes

characteristic

104

C₄×C₁₃

—

105

C₄×C₁₃

#382

yes

characteristic

106

C₄×C₁₃

#423

yes

characteristic

107

C₄×C₁₃

#431

yes

characteristic

108

C₄×C₁₃

#431

yes

characteristic

109

Dic₁₃

—

110

Dic₁₃

—

111

Dic₁₃

#364

yes

—

112

C₂²×C₈

—

113

C₂²×C₈

—

114

C₂²×C₈

C₄×D₁₃

yes

characteristic

115

C₂²×C₈

C₂²×D₁₃

yes

characteristic

116

C₂×C₄²

—

117

C₂×C₄²

—

118

C₂×C₄²

—

119

C₂×C₄²

C₄×D₁₃

yes

characteristic

120

M₃₂

—

121

M₃₂

—

122

M₃₂

—

123

M₃₂

—

124

C₂×C₁₆

—

125

C₂×C₁₆

—

126

C₄×C₈

—

127

C₄×C₈

—

128

C₄×C₈

—

129

C₄×C₈

—

130

C₄×C₈

—

131

C₄×C₈

—

132

C₄×C₈

—

133

C₄×C₈

—

134

C₂×C₁₃

—

135

C₂×C₁₃

—

136

C₂×C₁₃

#1831

yes

characteristic

137

C₂×C₁₃

C₂×#364

yes

—

138

C₂²×C₄

—

139

C₂²×C₄

—

140

C₂²×C₄

—

141

C₂²×C₄

C₂×C₄×D₁₃

yes

characteristic

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

C₈×D₁₃

yes

—

151

C₂×C₈

#6127

yes

—

152

C₂×C₈

D₄×D₁₃

yes

characteristic

153

C₂×C₈

#6129

yes

—

154

C₂×C₈

Q₈×D₁₃

yes

characteristic

155

C₄²

—

156

C₄²

—

157

C₄²

—

158

C₄²

—

159

C₄²

—

160

C₄²

—

161

C₄²

—

162

C₄²

#6094

yes

—

163

C₁₆

—

164

C₁₆

—

165

C₁₃

#12883

yes

characteristic, complemented, Sylow

166

C₂³

M₁₆×D₁₃

yes

characteristic

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

#88126

yes

characteristic

175

C₂×C₄

#88131

yes

characteristic

176

C₂×C₄

C₄⋊C₄×D₁₃

yes

characteristic

177

C₂×C₄

C₂×C₈×D₁₃

yes

characteristic

178

C₂×C₄

C₂×#6094

yes

characteristic

179

C₂×C₄

M₁₆×D₁₃

yes

characteristic

180

C₈

—

181

C₈

—

182

C₈

—

183

C₈

#88139

yes

—

184

C₈

#88143

yes

—

185

C₂²

—

186

C₂²

#11718366

yes

characteristic

187

C₂²

#11718380

yes

characteristic

188

C₂²

D₁₃×C₄⋊C₈

yes

characteristic

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

#11718311

yes

characteristic

194

C₄

#11718346

yes

characteristic

195

C₄

#11718379

yes

characteristic

196

C₄

#11718498

yes

characteristic

197

C₂

—

198

C₂

—

199

C₂

#13114951

yes

characteristic

200

C₂

D₁₃×#364

yes

—

	subgroup	G/H
center	C₂×C₈	D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₁₃×#1376	C₂
socle	C₂²×C₁₃	C₂×C₄⋊C₈
prime radical	C₂³×C₁₃	C₂×M₁₆
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#1376	D₁₃
hypocenter	C₁₃	#12883
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1376

subgroup

G/H

center

C₂×C₈

D₄×D₁₃

commutator

C₂×C₄×C₁₃

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₁₃

Fitting

C₁₃×#1376

C₂

socle

C₂²×C₁₃

C₂×C₄⋊C₈

prime radical

C₂³×C₁₃

C₂×M₁₆

cosocle

C₂²×C₈×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1376

N	H ≅ G/N	type
165. C₁₃	#12883	semidirect

H ≅ G/N

type

165. C₁₃

#12883

semidirect

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

subgroup	index
#13116591	2
C₄×C₈×Dic₁₃	2
#13112918	2
#13112917	2
C₁₃×#1376	2
#12883	13

subgroup

index

C₄×C₈×Dic₁₃

C₁₃×#1376

Subgroups — #15779278

Conjugacy classes of subgroups

Complements

Maximal subgroups