Group #13122887 — subgroups

Subgroups — #13122887

Order 1664.

464 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 133 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×C₁₃×#124	C₂	1	1	yes	maximal, characteristic
2	C₂×#88124	C₂	1	1	yes	maximal, characteristic, complemented
3	#11718890	C₂	1	2	yes	maximal, complemented
4	#11718883	C₂	1	2	yes	maximal
5	#11718614	C₂	1	2	yes	maximal, complemented
6	#11718610	C₂	1	2	yes	maximal, complemented
7	#11718609	C₂	1	4	yes	maximal, complemented
8	#11718597	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
11	C₁₃×#124	—	2	4	no	—
12	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
13	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊Dic₁₃	C₂²	1	4	yes	—
16	C₂×C₂²⋊Dic₁₃	C₂²	1	4	yes	—
17	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
20	C₂×#6101	C₂²	1	2	yes	—
21	C₂×#6101	C₂²	1	2	yes	—
22	C₂×#6101	C₂²	1	4	yes	—
23	#88124	—	2	4	no	—
24	#88071	C₂²	1	2	yes	—
25	#88071	C₂²	1	2	yes	—
26	#88071	C₂²	1	4	yes	—
27	C₂⁴×C₁₃	C₂³	1	2	yes	—
28	C₂×Q₈×C₁₃	—	2	2	no	—
29	C₂×Q₈×C₁₃	—	4	2	no	—
30	C₂×D₄×C₁₃	—	2	2	no	—
31	C₂×D₄×C₁₃	—	4	2	no	—
32	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
33	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
36	C₂²×Dic₁₃	C₂³	1	4	yes	—
37	C₂²×Dic₁₃	C₂³	1	2	yes	—
38	C₂²×Dic₁₃	C₂³	1	2	yes	—
39	C₁₃×C₂²⋊C₄	—	2	4	no	—
40	C₁₃×C₂²⋊C₄	—	2	4	no	—
41	C₄²×C₁₃	D₄	1	4	yes	—
42	C₂²⋊Dic₁₃	—	4	4	no	—
43	C₂²⋊Dic₁₃	—	4	4	no	—
44	C₄⋊Dic₁₃	—	2	2	no	—
45	C₄⋊Dic₁₃	—	2	2	no	—
46	C₄⋊Dic₁₃	—	4	2	no	—
47	#6101	—	2	4	no	—
48	#6101	—	2	4	no	—
49	#6101	—	4	4	no	—
50	#2820	—	13	1	no	maximal, Sylow
51	C₂³×C₁₃	—	2	2	no	—
52	C₂³×C₁₃	—	2	2	no	—
53	C₂³×C₁₃	—	2	4	no	—
54	C₂³×C₁₃	—	2	4	no	—
55	C₂³×C₁₃	—	2	2	no	—
56	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
57	Q₈×C₁₃	—	4	4	no	—
58	D₄×C₁₃	—	4	4	no	—
59	C₂×C₄×C₁₃	—	2	1	no	—
60	C₂×C₄×C₁₃	—	2	2	no	—
61	C₂×C₄×C₁₃	—	2	2	no	—
62	C₂×C₄×C₁₃	—	2	1	no	—
63	C₂×C₄×C₁₃	—	2	2	no	—
64	C₂×C₄×C₁₃	—	2	4	no	—
65	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
66	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
67	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
68	C₂×Dic₁₃	—	2	2	no	—
69	C₂×Dic₁₃	—	2	2	no	—
70	C₂×Dic₁₃	—	2	4	no	—
71	C₂×Dic₁₃	—	2	4	no	—
72	C₂×Dic₁₃	—	2	4	no	—
73	C₂×Dic₁₃	—	2	2	no	—
74	C₂×Dic₁₃	—	2	2	no	—
75	C₂×Dic₁₃	—	2	4	no	—
76	C₂×#125	—	13	1	no	—
77	C₂×#124	D₁₃	1	1	yes	characteristic
78	#399	—	13	2	no	—
79	#397	—	13	2	no	—
80	#396	—	13	4	no	—
81	#395	—	13	2	no	—
82	#394	—	13	2	no	—
83	#382	—	13	1	no	—
84	C₂²×C₁₃	—	4	2	no	—
85	C₂²×C₁₃	—	4	2	no	—
86	C₂²×C₁₃	—	4	2	no	—
87	C₂²×C₁₃	—	4	4	no	—
88	C₂²×C₁₃	—	4	4	no	—
89	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
90	C₂²×C₁₃	H(5,𝔽₂)	1	2	yes	—
91	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
92	C₂²×C₁₃	2₋¹⁺⁴	1	2	yes	—
93	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
94	C₄×C₁₃	—	2	2	no	—
95	C₄×C₁₃	—	2	2	no	—
96	C₄×C₁₃	—	2	2	no	—
97	C₄×C₁₃	—	4	2	no	—
98	Dic₁₃	—	4	2	no	—
99	Dic₁₃	—	4	2	no	—
100	Dic₁₃	—	4	4	no	—
101	C₂²×Q₈	C₂×D₁₃	1	1	yes	characteristic
102	C₂²×D₄	C₂×D₁₃	1	1	yes	characteristic
103	#125	—	26	4	no	—
104	#124	—	2	4	no	—
105	C₂×C₄⋊C₄	—	13	2	no	—
106	C₂×C₄⋊C₄	—	13	2	no	—
107	C₂×C₄⋊C₄	—	13	1	no	—
108	C₂×C₄⋊C₄	—	13	1	no	—
109	C₂×C₄⋊C₄	—	13	4	no	—
110	C₂×C₄⋊C₄	—	13	2	no	—
111	C₂×C₂²⋊C₄	—	13	4	no	—
112	C₂×C₂²⋊C₄	—	13	4	no	—
113	C₂×C₂²⋊C₄	C₂×D₁₃	1	2	yes	—
114	C₂×C₂²⋊C₄	C₂×D₁₃	1	2	yes	—
115	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
116	#95	—	13	2	no	—
117	#95	—	13	2	no	—
118	#95	—	13	4	no	—
119	C₂×C₁₃	—	8	2	no	—
120	C₂×C₁₃	#535	1	1	yes	characteristic
121	C₂×C₁₃	#536	1	1	yes	characteristic
122	C₂×C₁₃	#537	1	1	yes	characteristic
123	C₂×C₁₃	#562	1	2	yes	—
124	C₂×C₁₃	#563	1	2	yes	—
125	C₂⁴	C₂²×D₁₃	1	2	yes	—
126	C₂×Q₈	—	2	2	no	—
127	C₂×Q₈	—	4	2	no	—
128	C₂×D₄	—	2	2	no	—
129	C₂×D₄	—	4	2	no	—
130	C₂²×C₄	—	13	4	no	—
131	C₂²×C₄	—	13	2	no	—
132	C₂²×C₄	—	13	2	no	—
133	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
134	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
135	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
136	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
137	C₄⋊C₄	—	26	4	no	—
138	C₄⋊C₄	—	26	4	no	—
139	C₄⋊C₄	—	26	2	no	—
140	C₄⋊C₄	—	26	2	no	—
141	C₄⋊C₄	—	52	4	no	—
142	C₄⋊C₄	—	52	2	no	—
143	C₂²⋊C₄	—	2	4	no	—
144	C₂²⋊C₄	—	2	4	no	—
145	C₂²⋊C₄	—	52	4	no	—
146	C₂²⋊C₄	—	52	4	no	—
147	C₄²	C₂²⋊D₁₃	1	4	yes	—
148	C₁₃	#2820	1	1	yes	characteristic, complemented, Sylow
149	C₂³	—	2	4	no	—
150	C₂³	—	2	2	no	—
151	C₂³	—	2	4	no	—
152	C₂³	—	2	2	no	—
153	C₂³	—	2	2	no	—
154	C₂³	C₂³×D₁₃	1	1	yes	characteristic
155	Q₈	—	4	4	no	—
156	D₄	—	4	4	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	26	4	no	—
164	C₂×C₄	—	26	4	no	—
165	C₂×C₄	—	26	2	no	—
166	C₂×C₄	—	26	4	no	—
167	C₂×C₄	—	26	2	no	—
168	C₂×C₄	—	26	4	no	—
169	C₂×C₄	—	26	2	no	—
170	C₂×C₄	—	26	2	no	—
171	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
172	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
173	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
174	C₂²	—	4	2	no	—
175	C₂²	—	4	2	no	—
176	C₂²	—	4	2	no	—
177	C₂²	—	4	4	no	—
178	C₂²	—	4	4	no	—
179	C₂²	#88251	1	2	yes	—
180	C₂²	#88256	1	1	yes	characteristic
181	C₂²	#88257	1	2	yes	—
182	C₂²	#88257	1	1	yes	characteristic
183	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	2	2	no	—
186	C₄	—	2	2	no	—
187	C₄	—	4	2	no	—
188	C₄	—	52	2	no	—
189	C₄	—	52	2	no	—
190	C₄	—	52	4	no	—
191	C₂	—	8	2	no	—
192	C₂	#11719377	1	2	yes	—
193	C₂	#11719389	1	2	yes	—
194	C₂	#11719510	1	1	yes	characteristic
195	C₂	#11719530	1	1	yes	characteristic
196	C₂	#11719538	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₃×#124

C₂

yes

maximal, characteristic

C₂×#88124

C₂

yes

maximal, characteristic, complemented

#11718890

C₂

yes

maximal, complemented

#11718883

C₂

yes

maximal

#11718614

C₂

yes

maximal, complemented

#11718610

C₂

yes

maximal, complemented

#11718609

C₂

yes

maximal, complemented

#11718597

C₂

yes

maximal, characteristic, complemented

C₂²×Q₈×C₁₃

C₂²

yes

characteristic

C₂²×D₄×C₁₃

C₂²

yes

characteristic

C₁₃×#124

—

C₂×C₁₃×C₂²⋊C₄

C₂²

yes

—

C₂×C₁₃×C₂²⋊C₄

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₂²⋊Dic₁₃

C₂²

yes

—

C₂×C₂²⋊Dic₁₃

C₂²

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×#6101

C₂²

yes

—

C₂×#6101

C₂²

yes

—

C₂×#6101

C₂²

yes

—

#88124

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

C₂⁴×C₁₃

C₂³

yes

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₁₃×C₂²⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

D₄

yes

—

C₂²⋊Dic₁₃

—

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

—

—

—

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

C₂⁴

yes

characteristic

Q₈×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×#125

—

C₂×#124

D₁₃

yes

—

—

—

—

—

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₂²×C₁₃

H(5,𝔽₂)

yes

—

C₂²×C₁₃

H(5,𝔽₂)

yes

characteristic

C₂²×C₁₃

2₋¹⁺⁴

yes

—

C₂²×C₁₃

2₋¹⁺⁴

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

Dic₁₃

—

Dic₁₃

—

100

Dic₁₃

—

101

C₂²×Q₈

C₂×D₁₃

yes

characteristic

102

C₂²×D₄

C₂×D₁₃

yes

characteristic

103

#125

—

104

#124

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₂²⋊C₄

—

112

C₂×C₂²⋊C₄

—

113

C₂×C₂²⋊C₄

C₂×D₁₃

yes

—

114

C₂×C₂²⋊C₄

C₂×D₁₃

yes

—

115

C₂×C₄²

C₂×D₁₃

yes

characteristic

116

#95

—

117

#95

—

118

#95

—

119

C₂×C₁₃

—

120

C₂×C₁₃

#535

yes

characteristic

121

C₂×C₁₃

#536

yes

characteristic

122

C₂×C₁₃

#537

yes

characteristic

123

C₂×C₁₃

#562

yes

—

124

C₂×C₁₃

#563

yes

—

125

C₂⁴

C₂²×D₁₃

yes

—

126

C₂×Q₈

—

127

C₂×Q₈

—

128

C₂×D₄

—

129

C₂×D₄

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

C₂²×D₁₃

yes

—

134

C₂²×C₄

C₂²×D₁₃

yes

characteristic

135

C₂²×C₄

C₂²×D₁₃

yes

characteristic

136

C₂²×C₄

C₂²×D₁₃

yes

characteristic

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₂²⋊C₄

—

144

C₂²⋊C₄

—

145

C₂²⋊C₄

—

146

C₂²⋊C₄

—

147

C₄²

C₂²⋊D₁₃

yes

—

148

C₁₃

#2820

yes

characteristic, complemented, Sylow

149

C₂³

—

150

C₂³

—

151

C₂³

—

152

C₂³

—

153

C₂³

—

154

C₂³

C₂³×D₁₃

yes

characteristic

155

Q₈

—

156

D₄

—

157

C₂×C₄

—

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

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

172

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

173

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

174

C₂²

—

175

C₂²

—

176

C₂²

—

177

C₂²

—

178

C₂²

—

179

C₂²

#88251

yes

—

180

C₂²

#88256

yes

characteristic

181

C₂²

#88257

yes

—

182

C₂²

#88257

yes

characteristic

183

C₂²

C₂²×C₂²⋊D₁₃

yes

characteristic

184

C₄

—

185

C₄

—

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

—

190

C₄

—

191

C₂

—

192

C₂

#11719377

yes

—

193

C₂

#11719389

yes

—

194

C₂

#11719510

yes

characteristic

195

C₂

#11719530

yes

characteristic

196

C₂

#11719538

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₂×C₁₃×#124	C₂
prime radical	C₂²×D₄×C₁₃	C₂²
hypercenter	C₂×#124	D₁₃
hypocenter	C₁₃	#2820
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×#124

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, cosocle

C₂³×C₁₃

C₂⁴

Fitting

C₂×C₁₃×#124

C₂

prime radical

C₂²×D₄×C₁₃

C₂²

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ C₂×#124

N	H ≅ G/N	type
2. C₂×#88124	C₂	semidirect
3. #11718890	C₂	semidirect
5. #11718614	C₂	semidirect
6. #11718610	C₂	semidirect
7. #11718609	C₂	semidirect
8. #11718597	C₂	semidirect
148. C₁₃	#2820	semidirect

H ≅ G/N

type

2. C₂×#88124

C₂

semidirect

3. #11718890

C₂

semidirect

5. #11718614

C₂

semidirect

6. #11718610

C₂

semidirect

7. #11718609

C₂

semidirect

8. #11718597

C₂

semidirect

148. C₁₃

#2820

semidirect

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

subgroup	index
C₂×C₁₃×#124	2
C₂×#88124	2
#11718890	2
#11718883	2
#11718614	2
#11718610	2
#11718609	2
#11718597	2
#2820	13

subgroup

index

C₂×C₁₃×#124

Subgroups — #13122887

Conjugacy classes of subgroups

Complements

Maximal subgroups