Group #13119839 — subgroups

Subgroups — #13119839

Order 1664.

676 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 173 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×#127	C₂	1	1	yes	maximal, characteristic
2	C₂×#88190	C₂	1	2	yes	maximal, complemented
3	C₂×C₄×Dic₂₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718883	C₂	1	4	yes	maximal, complemented
5	#11718882	C₂	1	4	yes	maximal, complemented
6	#11718882	C₂	1	2	yes	maximal, complemented
7	C₄×C₄⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₄×C₁₃	C₂²	1	2	yes	—
9	C₂²×D₄×C₁₃	C₂²	1	2	yes	—
10	C₂²×D₄×C₁₃	C₂²	1	2	yes	—
11	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic
12	C₁₃×#127	—	2	4	no	—
13	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊Dic₁₃	C₂²	1	8	yes	—
15	C₂×C₂²⋊Dic₁₃	C₂²	1	8	yes	—
16	#88190	—	2	8	no	—
17	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
19	C₂×#6101	C₂²	1	2	yes	—
20	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
21	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
22	C₄×Dic₂₆	—	2	4	no	—
23	#88071	C₂²	1	2	yes	—
24	C₂⁴×C₁₃	C₂³	1	4	yes	—
25	C₂×D₄×C₁₃	—	2	4	no	—
26	C₂×D₄×C₁₃	—	2	4	no	—
27	C₂×D₄×C₁₃	—	2	4	no	—
28	C₂×D₄×C₁₃	—	2	4	no	—
29	C₂×D₄×C₁₃	—	2	4	no	—
30	C₂×D₄×C₁₃	—	2	8	no	—
31	C₂×D₄×C₁₃	—	4	4	no	—
32	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
35	C₂²×Dic₁₃	C₂³	1	4	yes	—
36	C₂²×Dic₁₃	C₂³	1	2	yes	—
37	C₂²×Dic₁₃	C₂³	1	2	yes	—
38	C₂×Dic₂₆	—	2	2	no	—
39	C₂×Dic₂₆	—	2	2	no	—
40	C₂×Dic₂₆	D₄	1	4	yes	complemented
41	C₄²×C₁₃	D₄	1	4	yes	—
42	C₂²⋊Dic₁₃	—	2	16	no	—
43	C₂²⋊Dic₁₃	—	4	8	no	—
44	C₄⋊Dic₁₃	—	2	2	no	—
45	C₄⋊Dic₁₃	—	2	4	no	—
46	#6101	—	2	4	no	—
47	C₄×Dic₁₃	—	2	4	no	—
48	C₄×Dic₁₃	—	2	4	no	—
49	#2548	—	13	1	no	maximal, Sylow
50	C₂³×C₁₃	—	2	4	no	—
51	C₂³×C₁₃	—	2	4	no	—
52	C₂³×C₁₃	—	2	4	no	—
53	C₂³×C₁₃	—	2	8	no	—
54	C₂³×C₁₃	—	2	8	no	—
55	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
56	D₄×C₁₃	—	4	8	no	—
57	D₄×C₁₃	—	4	8	no	—
58	D₄×C₁₃	—	4	8	no	—
59	C₂×C₄×C₁₃	—	2	2	no	—
60	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
63	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
64	C₂×C₄×C₁₃	#41	1	2	yes	—
65	C₂×C₄×C₁₃	#41	1	4	yes	—
66	C₂×Dic₁₃	—	2	2	no	—
67	C₂×Dic₁₃	—	2	2	no	—
68	C₂×Dic₁₃	—	2	2	no	—
69	C₂×Dic₁₃	—	2	4	no	—
70	C₂×Dic₁₃	—	2	4	no	—
71	C₂×Dic₁₃	—	2	4	no	—
72	C₂×Dic₁₃	—	2	4	no	—
73	C₂×Dic₁₃	C₂×D₄	1	4	yes	—
74	Dic₂₆	—	2	8	no	—
75	C₂×#127	D₁₃	1	1	yes	characteristic
76	C₂×#124	—	13	2	no	—
77	C₂×C₄×Q₈	—	13	1	no	—
78	#394	—	13	4	no	—
79	#390	—	13	4	no	—
80	#390	—	13	2	no	—
81	C₄×C₄⋊C₄	—	13	1	no	—
82	C₂²×C₁₃	—	4	4	no	—
83	C₂²×C₁₃	—	4	4	no	—
84	C₂²×C₁₃	—	4	4	no	—
85	C₂²×C₁₃	—	4	8	no	—
86	C₂²×C₁₃	—	4	8	no	—
87	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
88	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
89	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
90	C₂²×C₁₃	C₂×#41	1	2	yes	—
91	C₂²×C₁₃	H(5,𝔽₂)	1	2	yes	—
92	C₄×C₁₃	—	2	2	no	—
93	C₄×C₁₃	#121	1	4	yes	—
94	C₄×C₁₃	#124	1	4	yes	complemented
95	Dic₁₃	—	2	4	no	—
96	Dic₁₃	—	4	2	no	—
97	Dic₁₃	—	4	4	no	—
98	C₂²×Q₈	—	13	1	no	—
99	C₂²×D₄	C₂×D₁₃	1	2	yes	—
100	C₂²×D₄	C₂×D₁₃	1	2	yes	—
101	C₂²×D₄	C₂×D₁₃	1	2	yes	—
102	#127	—	2	4	no	—
103	#124	—	26	8	no	—
104	C₄×Q₈	—	26	4	no	—
105	C₂×C₄⋊C₄	—	13	2	no	—
106	C₂×C₄⋊C₄	—	13	1	no	—
107	C₂×C₄⋊C₄	—	13	2	no	—
108	C₂×C₂²⋊C₄	—	13	8	no	—
109	C₂×C₂²⋊C₄	—	13	8	no	—
110	C₂×C₄²	—	13	2	no	—
111	C₂×C₄²	—	13	2	no	—
112	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
113	#95	—	13	2	no	—
114	C₂×C₁₃	—	8	4	no	—
115	C₂×C₁₃	C₂×#121	1	1	yes	characteristic
116	C₂×C₁₃	C₂×#124	1	1	yes	characteristic
117	C₂×C₁₃	#535	1	1	yes	characteristic
118	C₂×C₁₃	#550	1	2	yes	—
119	C₂×C₁₃	#559	1	2	yes	—
120	C₂⁴	C₂²×D₁₃	1	4	yes	—
121	C₂×Q₈	—	13	4	no	—
122	C₂×Q₈	—	26	2	no	—
123	C₂×Q₈	—	26	2	no	—
124	C₂×D₄	—	2	4	no	—
125	C₂×D₄	—	2	8	no	—
126	C₂×D₄	—	2	4	no	—
127	C₂×D₄	—	2	4	no	—
128	C₂×D₄	—	2	4	no	—
129	C₂×D₄	—	2	4	no	—
130	C₂×D₄	—	4	4	no	—
131	C₂²×C₄	—	13	2	no	—
132	C₂²×C₄	—	13	4	no	—
133	C₂²×C₄	—	13	2	no	—
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	C₂²⋊C₄	—	26	16	no	—
141	C₂²⋊C₄	—	52	8	no	—
142	C₄²	—	26	4	no	—
143	C₄²	—	26	4	no	—
144	C₄²	C₂²⋊D₁₃	1	4	yes	—
145	C₁₃	#2548	1	1	yes	characteristic, complemented, Sylow
146	C₂³	—	2	4	no	—
147	C₂³	—	2	8	no	—
148	C₂³	—	2	4	no	—
149	C₂³	—	2	8	no	—
150	C₂³	—	2	4	no	—
151	C₂³	C₂³×D₁₃	1	1	yes	characteristic
152	Q₈	—	26	8	no	—
153	D₄	—	4	8	no	—
154	D₄	—	4	8	no	—
155	D₄	—	4	8	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	13	4	no	—
158	C₂×C₄	—	26	2	no	—
159	C₂×C₄	—	26	2	no	—
160	C₂×C₄	—	26	4	no	—
161	C₂×C₄	—	26	4	no	—
162	C₂×C₄	—	26	4	no	—
163	C₂×C₄	—	26	4	no	—
164	C₂×C₄	—	26	2	no	—
165	C₂×C₄	D₄×D₁₃	1	2	yes	—
166	C₂×C₄	#6129	1	4	yes	—
167	C₂×C₄	#6129	1	2	yes	—
168	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
169	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
170	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
171	C₂²	—	4	4	no	—
172	C₂²	—	4	4	no	—
173	C₂²	—	4	4	no	—
174	C₂²	—	4	8	no	—
175	C₂²	—	4	8	no	—
176	C₂²	C₂×D₄×D₁₃	1	1	yes	characteristic
177	C₂²	C₂×#6129	1	2	yes	—
178	C₂²	C₂×#6129	1	1	yes	characteristic
179	C₂²	#88251	1	2	yes	—
180	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
181	C₄	—	2	2	no	—
182	C₄	—	26	4	no	—
183	C₄	—	52	2	no	—
184	C₄	—	52	4	no	—
185	C₄	#88190	1	4	yes	complemented
186	C₄	#88192	1	4	yes	—
187	C₂	—	8	4	no	—
188	C₂	#11719418	1	2	yes	—
189	C₂	#11719423	1	2	yes	—
190	C₂	C₂×#88190	1	1	yes	characteristic
191	C₂	C₂×#88192	1	1	yes	characteristic
192	C₂	#11719510	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₃×#127

C₂

yes

maximal, characteristic

C₂×#88190

C₂

yes

maximal, complemented

C₂×C₄×Dic₂₆

C₂

yes

maximal, characteristic, complemented

#11718883

C₂

yes

maximal, complemented

#11718882

C₂

yes

maximal, complemented

#11718882

C₂

yes

maximal, complemented

C₄×C₄⋊Dic₁₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₄×C₁₃

C₂²

yes

—

C₂²×D₄×C₁₃

C₂²

yes

—

C₂²×D₄×C₁₃

C₂²

yes

—

C₂²×Dic₂₆

C₂²

yes

characteristic

C₁₃×#127

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₂²⋊Dic₁₃

C₂²

yes

—

C₂×C₂²⋊Dic₁₃

C₂²

yes

—

#88190

—

C₂×C₄⋊Dic₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×#6101

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

C₄×Dic₂₆

—

#88071

C₂²

yes

—

C₂⁴×C₁₃

C₂³

yes

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

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

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

D₄

yes

complemented

C₄²×C₁₃

D₄

yes

—

C₂²⋊Dic₁₃

—

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#6101

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

#2548

—

maximal, Sylow

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

C₂⁴

yes

characteristic

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

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×D₄

yes

—

Dic₂₆

—

C₂×#127

D₁₃

yes

characteristic

C₂×#124

—

C₂×C₄×Q₈

—

#394

—

#390

—

#390

—

C₄×C₄⋊C₄

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₂²×C₁₃

C₂×#41

yes

characteristic

C₂²×C₁₃

C₂×#41

yes

—

C₂²×C₁₃

H(5,𝔽₂)

yes

—

C₄×C₁₃

—

C₄×C₁₃

#121

yes

—

C₄×C₁₃

#124

yes

complemented

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂²×Q₈

—

C₂²×D₄

C₂×D₁₃

yes

—

100

C₂²×D₄

C₂×D₁₃

yes

—

101

C₂²×D₄

C₂×D₁₃

yes

—

102

#127

—

103

#124

—

104

C₄×Q₈

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₂²⋊C₄

—

109

C₂×C₂²⋊C₄

—

110

C₂×C₄²

—

111

C₂×C₄²

—

112

C₂×C₄²

C₂×D₁₃

yes

characteristic

113

#95

—

114

C₂×C₁₃

—

115

C₂×C₁₃

C₂×#121

yes

characteristic

116

C₂×C₁₃

C₂×#124

yes

characteristic

117

C₂×C₁₃

#535

yes

characteristic

118

C₂×C₁₃

#550

yes

—

119

C₂×C₁₃

#559

yes

—

120

C₂⁴

C₂²×D₁₃

yes

—

121

C₂×Q₈

—

122

C₂×Q₈

—

123

C₂×Q₈

—

124

C₂×D₄

—

125

C₂×D₄

—

126

C₂×D₄

—

127

C₂×D₄

—

128

C₂×D₄

—

129

C₂×D₄

—

130

C₂×D₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

—

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

C₂²⋊C₄

—

141

C₂²⋊C₄

—

142

C₄²

—

143

C₄²

—

144

C₄²

C₂²⋊D₁₃

yes

—

145

C₁₃

#2548

yes

characteristic, complemented, Sylow

146

C₂³

—

147

C₂³

—

148

C₂³

—

149

C₂³

—

150

C₂³

—

151

C₂³

C₂³×D₁₃

yes

characteristic

152

Q₈

—

153

D₄

—

154

D₄

—

155

D₄

—

156

C₂×C₄

—

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

D₄×D₁₃

yes

—

166

C₂×C₄

#6129

yes

—

167

C₂×C₄

#6129

yes

—

168

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

169

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

170

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

171

C₂²

—

172

C₂²

—

173

C₂²

—

174

C₂²

—

175

C₂²

—

176

C₂²

C₂×D₄×D₁₃

yes

characteristic

177

C₂²

C₂×#6129

yes

—

178

C₂²

C₂×#6129

yes

characteristic

179

C₂²

#88251

yes

—

180

C₂²

C₂²×C₂²⋊D₁₃

yes

characteristic

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

#88190

yes

complemented

186

C₄

#88192

yes

—

187

C₂

—

188

C₂

#11719418

yes

—

189

C₂

#11719423

yes

—

190

C₂

C₂×#88190

yes

characteristic

191

C₂

C₂×#88192

yes

characteristic

192

C₂

#11719510

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, cosocle	C₂³×C₁₃	C₂⁴
Fitting, prime radical	C₂×C₁₃×#127	C₂
hypercenter	C₂×#127	D₁₃
hypocenter	C₁₃	#2548
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×#127

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, cosocle

C₂³×C₁₃

C₂⁴

Fitting, prime radical

C₂×C₁₃×#127

C₂

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ C₂×#127

N	H ≅ G/N	type
2. C₂×#88190	C₂	semidirect
3. C₂×C₄×Dic₂₆	C₂	semidirect
4. #11718883	C₂	semidirect
5. #11718882	C₂	semidirect
6. #11718882	C₂	semidirect
7. C₄×C₄⋊Dic₁₃	C₂	semidirect
40. C₂×Dic₂₆	D₄	semidirect
94. C₄×C₁₃	#124	semidirect
145. C₁₃	#2548	semidirect
185. C₄	#88190	semidirect

H ≅ G/N

type

2. C₂×#88190

C₂

semidirect

3. C₂×C₄×Dic₂₆

C₂

semidirect

4. #11718883

C₂

semidirect

5. #11718882

C₂

semidirect

6. #11718882

C₂

semidirect

7. C₄×C₄⋊Dic₁₃

C₂

semidirect

40. C₂×Dic₂₆

D₄

semidirect

94. C₄×C₁₃

#124

semidirect

145. C₁₃

#2548

semidirect

185. C₄

#88190

semidirect

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

subgroup	index
C₂×C₁₃×#127	2
C₂×#88190	2
C₂×C₄×Dic₂₆	2
#11718883	2
#11718882	2
C₄×C₄⋊Dic₁₃	2
#2548	13

subgroup

index

C₂×C₁₃×#127

C₂×C₄×Dic₂₆

C₄×C₄⋊Dic₁₃

Subgroups — #13119839

Conjugacy classes of subgroups

Complements

Maximal subgroups