Group #13119872 — subgroups

Subgroups — #13119872

Order 1664.

452 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 181 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×#88144	C₂	1	2	yes	maximal
2	C₁₃×#384	C₂	1	1	yes	maximal, characteristic
3	#11718646	C₂	1	2	yes	maximal
4	#11718645	C₂	1	4	yes	maximal
5	#11718637	C₂	1	2	yes	maximal
6	#11718634	C₂	1	2	yes	maximal
7	C₄×C₄⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic
8	C₄×C₄⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic
9	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₁₃	C₂²	1	4	yes	—
14	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
18	C₂×#6101	C₂²	1	4	yes	—
19	C₂×#6101	C₂²	1	4	yes	—
20	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
21	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
22	#88144	—	2	8	no	—
23	#88071	C₂²	1	2	yes	—
24	#88071	C₂²	1	2	yes	—
25	#88071	C₂²	1	4	yes	—
26	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
27	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
30	C₂²×Dic₁₃	C₂³	1	2	yes	—
31	C₂²×Dic₁₃	C₂³	1	2	yes	—
32	C₂²×Dic₁₃	C₂³	1	2	yes	—
33	C₂²×Dic₁₃	C₂³	1	2	yes	—
34	C₁₃×C₄⋊C₄	—	2	4	no	—
35	C₁₃×C₄⋊C₄	—	2	4	no	—
36	C₁₃×C₄⋊C₄	—	4	2	no	—
37	C₄²×C₁₃	Q₈	1	4	yes	—
38	C₄⋊Dic₁₃	—	2	2	no	—
39	C₄⋊Dic₁₃	—	2	2	no	—
40	C₄⋊Dic₁₃	—	2	8	no	—
41	C₄⋊Dic₁₃	D₄	1	4	yes	—
42	C₄⋊Dic₁₃	Q₈	1	4	yes	—
43	#6101	—	2	8	no	—
44	#6101	—	4	4	no	—
45	C₄×Dic₁₃	—	2	4	no	—
46	C₄×Dic₁₃	—	2	4	no	—
47	#2552	—	13	1	no	maximal, Sylow
48	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
49	C₂×C₄×C₁₃	—	2	1	no	—
50	C₂×C₄×C₁₃	—	2	1	no	—
51	C₂×C₄×C₁₃	—	2	4	no	—
52	C₂×C₄×C₁₃	—	2	4	no	—
53	C₂×C₄×C₁₃	—	2	4	no	—
54	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
56	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
57	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
58	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
59	C₂×C₄×C₁₃	#41	1	2	yes	—
60	C₂×C₄×C₁₃	#41	1	2	yes	—
61	C₂×Dic₁₃	—	2	2	no	—
62	C₂×Dic₁₃	—	2	2	no	—
63	C₂×Dic₁₃	—	2	2	no	—
64	C₂×Dic₁₃	—	2	2	no	—
65	C₂×Dic₁₃	—	2	2	no	—
66	C₂×Dic₁₃	—	2	2	no	—
67	C₂×Dic₁₃	—	2	2	no	—
68	C₂×Dic₁₃	—	2	2	no	—
69	C₂×Dic₁₃	—	2	2	no	—
70	C₂×Dic₁₃	—	2	2	no	—
71	C₂×Dic₁₃	C₂×D₄	1	4	yes	—
72	C₂×Dic₁₃	C₂×Q₈	1	4	yes	—
73	C₂×#125	—	13	2	no	—
74	#398	—	13	4	no	—
75	#389	—	13	2	no	—
76	#389	—	13	2	no	—
77	#384	—	13	2	no	—
78	#384	D₁₃	1	1	yes	characteristic
79	C₄×C₄⋊C₄	—	13	1	no	—
80	C₄×C₄⋊C₄	—	13	1	no	—
81	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
82	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
83	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
84	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
85	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
86	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
87	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
88	C₄×C₁₃	—	2	2	no	—
89	C₄×C₁₃	—	4	4	no	—
90	C₄×C₁₃	#125	1	4	yes	complemented
91	C₄×C₁₃	C₄⋊Q₈	1	4	yes	—
92	Dic₁₃	—	2	4	no	—
93	Dic₁₃	—	2	4	no	—
94	Dic₁₃	—	4	2	no	—
95	Dic₁₃	—	4	2	no	—
96	#125	—	26	8	no	—
97	C₂×C₄⋊C₄	—	13	1	no	—
98	C₂×C₄⋊C₄	—	13	1	no	—
99	C₂×C₄⋊C₄	—	13	4	no	—
100	C₂×C₄⋊C₄	—	13	4	no	—
101	C₂×C₄⋊C₄	—	13	1	no	—
102	C₂×C₄⋊C₄	—	13	1	no	—
103	C₂×C₄⋊C₄	—	13	4	no	—
104	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
105	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
106	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
107	C₂×C₄²	—	13	2	no	—
108	C₂×C₄²	—	13	2	no	—
109	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
110	#95	—	13	2	no	—
111	#95	—	13	2	no	—
112	#95	—	13	4	no	—
113	C₂×C₁₃	C₂×#125	1	1	yes	characteristic
114	C₂×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
115	C₂×C₁₃	#544	1	1	yes	characteristic
116	C₂×C₁₃	#547	1	1	yes	characteristic
117	C₂×C₁₃	#550	1	1	yes	characteristic
118	C₂×C₁₃	#554	1	1	yes	characteristic
119	C₂×C₁₃	#557	1	1	yes	characteristic
120	C₂²×C₄	—	13	2	no	—
121	C₂²×C₄	—	13	2	no	—
122	C₂²×C₄	—	13	2	no	—
123	C₂²×C₄	—	13	2	no	—
124	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
125	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
126	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
127	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
128	C₄⋊C₄	—	2	4	no	—
129	C₄⋊C₄	—	2	4	no	—
130	C₄⋊C₄	—	4	2	no	—
131	C₄⋊C₄	—	13	4	no	—
132	C₄⋊C₄	—	13	4	no	—
133	C₄⋊C₄	—	26	8	no	—
134	C₄⋊C₄	—	26	8	no	—
135	C₄⋊C₄	—	26	2	no	—
136	C₄⋊C₄	—	26	2	no	—
137	C₄⋊C₄	—	52	4	no	—
138	C₄²	—	26	4	no	—
139	C₄²	—	26	4	no	—
140	C₄²	Dic₂₆	1	4	yes	—
141	C₁₃	#2552	1	1	yes	characteristic, complemented, Sylow
142	C₂³	C₂³×D₁₃	1	1	yes	characteristic
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	13	4	no	—
149	C₂×C₄	—	13	4	no	—
150	C₂×C₄	—	26	2	no	—
151	C₂×C₄	—	26	2	no	—
152	C₂×C₄	—	26	2	no	—
153	C₂×C₄	—	26	2	no	—
154	C₂×C₄	—	26	2	no	—
155	C₂×C₄	—	26	2	no	—
156	C₂×C₄	—	26	2	no	—
157	C₂×C₄	—	26	2	no	—
158	C₂×C₄	—	26	2	no	—
159	C₂×C₄	—	26	2	no	—
160	C₂×C₄	C₂×Dic₂₆	1	2	yes	—
161	C₂×C₄	C₂×Dic₂₆	1	2	yes	—
162	C₂×C₄	C₂×Dic₂₆	1	2	yes	—
163	C₂×C₄	D₄×D₁₃	1	2	yes	—
164	C₂×C₄	#6129	1	2	yes	—
165	C₂×C₄	Q₈×D₁₃	1	2	yes	—
166	C₂×C₄	#6131	1	2	yes	—
167	C₂²	C₂²×Dic₂₆	1	1	yes	characteristic
168	C₂²	C₂×D₄×D₁₃	1	1	yes	characteristic
169	C₂²	C₂×#6129	1	1	yes	characteristic
170	C₂²	#88251	1	1	yes	characteristic
171	C₂²	C₂×Q₈×D₁₃	1	1	yes	characteristic
172	C₂²	C₂×#6131	1	1	yes	characteristic
173	C₂²	#88254	1	1	yes	characteristic
174	C₄	—	2	2	no	—
175	C₄	—	4	4	no	—
176	C₄	—	26	4	no	—
177	C₄	—	26	4	no	—
178	C₄	—	52	2	no	—
179	C₄	—	52	2	no	—
180	C₄	#88142	1	4	yes	—
181	C₄	#88144	1	4	yes	complemented
182	C₂	C₂×#88142	1	1	yes	characteristic
183	C₂	C₂×#88144	1	1	yes	characteristic
184	C₂	#11719205	1	1	yes	characteristic
185	C₂	#11719423	1	1	yes	characteristic
186	C₂	#11719426	1	1	yes	characteristic
187	C₂	#11719431	1	1	yes	characteristic
188	C₂	#11719435	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88144

C₂

yes

maximal

C₁₃×#384

C₂

yes

maximal, characteristic

#11718646

C₂

yes

maximal

#11718645

C₂

yes

maximal

#11718637

C₂

yes

maximal

#11718634

C₂

yes

maximal

C₄×C₄⋊Dic₁₃

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₁₃

C₂

yes

maximal, characteristic

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₂²

yes

characteristic

C₂×#6101

C₂²

yes

—

C₂×#6101

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

#88144

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

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

—

C₂²×Dic₁₃

C₂³

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

Q₈

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

D₄

yes

—

C₄⋊Dic₁₃

Q₈

yes

—

#6101

—

#6101

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

#2552

—

maximal, Sylow

C₂³×C₁₃

C₂⁴

yes

characteristic

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×Q₈

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₃

#41

yes

—

C₂×C₄×C₁₃

#41

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂×D₄

yes

—

C₂×Dic₁₃

C₂×Q₈

yes

—

—

—

—

—

—

D₁₃

yes

characteristic

C₄×C₄⋊C₄

—

C₄×C₄⋊C₄

—

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₂²×C₁₃

C₂²×Q₈

yes

characteristic

C₂²×C₁₃

C₂²×Q₈

yes

characteristic

C₂²×C₁₃

C₂×#41

yes

characteristic

C₂²×C₁₃

C₂×#41

yes

characteristic

C₂²×C₁₃

H(5,𝔽₂)

yes

characteristic

C₂²×C₁₃

2₋¹⁺⁴

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#125

yes

complemented

C₄×C₁₃

C₄⋊Q₈

yes

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

100

C₂×C₄⋊C₄

—

101

C₂×C₄⋊C₄

—

102

C₂×C₄⋊C₄

—

103

C₂×C₄⋊C₄

—

104

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

105

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

106

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

107

C₂×C₄²

—

108

C₂×C₄²

—

109

C₂×C₄²

C₂×D₁₃

yes

characteristic

110

#95

—

111

#95

—

112

#95

—

113

C₂×C₁₃

C₂×#125

yes

characteristic

114

C₂×C₁₃

C₂×C₄⋊Q₈

yes

characteristic

115

C₂×C₁₃

#544

yes

characteristic

116

C₂×C₁₃

#547

yes

characteristic

117

C₂×C₁₃

#550

yes

characteristic

118

C₂×C₁₃

#554

yes

characteristic

119

C₂×C₁₃

#557

yes

characteristic

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

C₂²×D₁₃

yes

—

125

C₂²×C₄

C₂²×D₁₃

yes

characteristic

126

C₂²×C₄

C₂²×D₁₃

yes

characteristic

127

C₂²×C₄

C₂²×D₁₃

yes

characteristic

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄²

—

139

C₄²

—

140

C₄²

Dic₂₆

yes

—

141

C₁₃

#2552

yes

characteristic, complemented, Sylow

142

C₂³

C₂³×D₁₃

yes

characteristic

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

C₂×Dic₂₆

yes

—

161

C₂×C₄

C₂×Dic₂₆

yes

—

162

C₂×C₄

C₂×Dic₂₆

yes

—

163

C₂×C₄

D₄×D₁₃

yes

—

164

C₂×C₄

#6129

yes

—

165

C₂×C₄

Q₈×D₁₃

yes

—

166

C₂×C₄

#6131

yes

—

167

C₂²

C₂²×Dic₂₆

yes

characteristic

168

C₂²

C₂×D₄×D₁₃

yes

characteristic

169

C₂²

C₂×#6129

yes

characteristic

170

C₂²

#88251

yes

characteristic

171

C₂²

C₂×Q₈×D₁₃

yes

characteristic

172

C₂²

C₂×#6131

yes

characteristic

173

C₂²

#88254

yes

characteristic

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

#88142

yes

—

181

C₄

#88144

yes

complemented

182

C₂

C₂×#88142

yes

characteristic

183

C₂

C₂×#88144

yes

characteristic

184

C₂

#11719205

yes

characteristic

185

C₂

#11719423

yes

characteristic

186

C₂

#11719426

yes

characteristic

187

C₂

#11719431

yes

characteristic

188

C₂

#11719435

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, prime radical, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₁₃×#384	C₂
hypercenter	#384	D₁₃
hypocenter	C₁₃	#2552
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, prime radical, cosocle

C₂³×C₁₃

C₂⁴

Fitting

C₁₃×#384

C₂

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
90. C₄×C₁₃	#125	semidirect
141. C₁₃	#2552	semidirect
181. C₄	#88144	semidirect

H ≅ G/N

type

90. C₄×C₁₃

#125

semidirect

141. C₁₃

#2552

semidirect

181. C₄

#88144

semidirect

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

subgroup	index
C₂×#88144	2
C₁₃×#384	2
#11718646	2
#11718645	2
#11718637	2
#11718634	2
C₄×C₄⋊Dic₁₃	2
#2552	13

subgroup

index

C₁₃×#384

C₄×C₄⋊Dic₁₃

Subgroups — #13119872

Conjugacy classes of subgroups

Complements

Maximal subgroups