Group #15850626 — subgroups

Subgroups — #15850626

Order 3328.

706 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 706 conjugacy classes of subgroups — too many to draw. Only its 187 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	#13125421	C₂	1	4	yes	maximal
2	#13125347	C₂	1	2	yes	maximal
3	#13114453	C₂	1	2	yes	maximal, complemented
4	#13114446	C₂	1	1	yes	maximal, characteristic, complemented
5	#13114432	C₂	1	4	yes	maximal, complemented
6	C₈²×D₁₃	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#1710	C₂	1	1	yes	maximal, characteristic, complemented
8	#11719433	C₂²	1	4	yes	—
9	C₁₃×#500	C₂²	1	2	yes	—
10	C₁₃×#487	C₂²	1	4	yes	—
11	#11718753	C₂²	1	4	yes	—
12	#11718740	C₂²	1	4	yes	—
13	#11718739	C₂²	1	4	yes	—
14	#11718541	—	2	4	no	—
15	#11718539	—	2	4	no	—
16	#11718474	—	2	8	no	—
17	#11718451	—	2	8	no	—
18	C₄×C₈×D₁₃	C₂²	1	2	yes	—
19	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
20	#11718343	C₂²	1	2	yes	—
21	#11718342	C₂²	1	4	yes	—
22	C₈²×C₁₃	C₂²	1	1	yes	characteristic
23	C₈×#1055	C₂²	1	2	yes	—
24	C₈×#1055	C₂²	1	1	yes	characteristic
25	C₁₃×C₄⋊Q₈	C₂³	1	4	yes	—
26	#88200	—	2	4	no	—
27	Q₈×Dic₁₃	—	2	4	no	—
28	C₂×#6107	—	2	8	no	—
29	C₂×C₈×D₁₃	D₄	1	4	yes	—
30	C₂×C₈×D₁₃	D₄	1	2	yes	—
31	#88151	—	2	4	no	—
32	#88146	—	2	4	no	—
33	#88144	—	2	4	no	—
34	#88141	—	2	4	no	—
35	C₄²×D₁₃	C₂³	1	1	yes	characteristic
36	#88123	C₂³	1	4	yes	—
37	C₁₃×#107	—	2	4	no	—
38	C₁₃×#103	—	2	8	no	—
39	C₄×C₈×C₁₃	C₂³	1	2	yes	—
40	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
41	#88058	—	2	4	no	—
42	#88056	—	2	8	no	—
43	C₈×Dic₁₃	D₄	1	4	yes	—
44	C₈×Dic₁₃	D₄	1	2	yes	—
45	#88048	—	2	8	no	—
46	#88047	—	2	8	no	—
47	C₄×#1055	C₂³	1	2	yes	—
48	C₄×#1055	C₂³	1	1	yes	characteristic
49	#19604	—	13	1	no	maximal, Sylow
50	C₂×Q₈×C₁₃	—	2	4	no	—
51	C₂×C₄×D₁₃	C₂×D₄	1	2	yes	—
52	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
53	C₂×Dic₂₆	—	2	4	no	—
54	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
55	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
56	C₁₃×C₄⋊C₄	—	2	4	no	—
57	C₁₃×C₄⋊C₄	—	2	4	no	—
58	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
59	#6107	—	4	16	no	—
60	#6103	—	4	4	no	—
61	C₄⋊Dic₁₃	—	2	4	no	—
62	C₄⋊Dic₁₃	—	2	4	no	—
63	C₄⋊Dic₁₃	—	4	4	no	—
64	#6101	—	4	4	no	—
65	C₄×Dic₁₃	—	4	4	no	—
66	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
67	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
68	C₂×#1055	C₂×D₄	1	2	yes	—
69	C₂×#1055	C₂×D₄	1	4	yes	—
70	C₈×D₁₃	—	2	8	no	—
71	C₈×D₁₃	Q₁₆	1	8	yes	complemented
72	#3170	—	13	4	no	—
73	#3160	—	13	2	no	—
74	#1713	—	13	2	no	—
75	#1710	—	13	1	no	—
76	#1710	D₁₃	1	1	yes	characteristic, complemented
77	#1707	—	13	4	no	—
78	C₂×C₈²	—	13	1	no	—
79	C₂²×D₁₃	#127	1	1	yes	characteristic
80	Q₈×C₁₃	—	4	8	no	—
81	C₂×C₄×C₁₃	—	4	4	no	—
82	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
84	C₂×Dic₁₃	—	4	4	no	—
85	C₂×Dic₁₃	#127	1	2	yes	—
86	C₂×Dic₁₃	#127	1	1	yes	characteristic
87	C₄×D₁₃	—	2	4	no	—
88	C₄×D₁₃	C₂×Q₁₆	1	4	yes	—
89	Dic₂₆	—	4	8	no	—
90	C₈×C₁₃	—	2	4	no	—
91	C₈×C₁₃	C₂×Q₁₆	1	4	yes	—
92	#1055	C₂×Q₁₆	1	4	yes	—
93	#1055	#135	1	8	yes	—
94	#533	—	13	4	no	—
95	#500	—	13	4	no	—
96	#500	—	13	2	no	—
97	#500	C₂×D₁₃	1	2	yes	—
98	#499	—	26	4	no	—
99	#498	—	13	4	no	—
100	#494	—	13	4	no	—
101	#487	—	13	4	no	—
102	#487	C₂×D₁₃	1	4	yes	—
103	#467	—	26	8	no	—
104	C₄×Q₁₆	—	26	8	no	—
105	#427	—	26	4	no	—
106	C₂×C₄×C₈	—	13	1	no	—
107	C₂×C₄×C₈	—	13	2	no	—
108	C₈²	—	13	2	no	—
109	C₈²	—	13	1	no	—
110	C₈²	C₂×D₁₃	1	1	yes	characteristic
111	C₂²×C₁₃	C₂×#127	1	1	yes	characteristic
112	C₂×D₁₃	—	2	2	no	—
113	C₂×D₁₃	#494	1	2	yes	—
114	C₄×C₁₃	—	8	4	no	—
115	C₄×C₁₃	C₂²×Q₁₆	1	2	yes	—
116	C₄×C₁₃	C₂×#135	1	4	yes	—
117	Dic₁₃	—	2	2	no	—
118	Dic₁₃	—	8	4	no	—
119	Dic₁₃	#494	1	2	yes	—
120	C₂×Q₁₆	—	26	8	no	—
121	C₂²×C₈	—	13	2	no	—
122	C₂²×C₈	—	13	4	no	—
123	C₄⋊Q₈	—	13	4	no	—
124	C₄⋊Q₈	C₂²×D₁₃	1	4	yes	—
125	#125	—	26	4	no	—
126	C₂²⋊Q₈	—	26	4	no	—
127	C₂²⋊Q₈	—	26	4	no	—
128	C₄×Q₈	—	26	4	no	—
129	C₄×Q₈	—	26	4	no	—
130	#117	—	26	4	no	—
131	C₂×C₄²	—	13	1	no	—
132	#107	—	2	4	no	—
133	#107	—	26	8	no	—
134	#107	—	26	4	no	—
135	#106	—	26	8	no	—
136	#103	—	2	8	no	—
137	#103	—	26	8	no	—
138	C₄×C₈	—	13	4	no	—
139	C₄×C₈	—	13	2	no	—
140	C₄×C₈	—	13	2	no	—
141	C₄×C₈	—	13	1	no	—
142	C₄×C₈	C₂²×D₁₃	1	2	yes	—
143	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
144	C₂×C₁₃	C₂×#494	1	1	yes	characteristic
145	C₂×C₁₃	#3147	1	2	yes	—
146	D₁₃	—	2	2	no	—
147	C₂×Q₈	—	2	4	no	—
148	C₂×Q₈	—	26	4	no	—
149	C₂²×C₄	—	13	1	no	—
150	C₂²×C₄	—	13	2	no	—
151	Q₁₆	—	52	16	no	—
152	C₂×C₈	—	13	2	no	—
153	C₂×C₈	—	13	4	no	—
154	C₂×C₈	—	13	8	no	—
155	C₂×C₈	—	26	8	no	—
156	C₂×C₈	D₄×D₁₃	1	4	yes	—
157	C₂×C₈	D₄×D₁₃	1	2	yes	—
158	C₄⋊C₄	—	2	4	no	—
159	C₄⋊C₄	—	2	4	no	—
160	C₄⋊C₄	—	26	4	no	—
161	C₄⋊C₄	—	26	4	no	—
162	C₄⋊C₄	—	52	4	no	—
163	C₄⋊C₄	—	52	4	no	—
164	C₂²⋊C₄	—	52	4	no	—
165	C₄²	—	13	2	no	—
166	C₄²	—	13	1	no	—
167	C₄²	—	52	4	no	—
168	C₄²	C₂³×D₁₃	1	1	yes	characteristic
169	C₁₃	#19604	1	1	yes	characteristic, complemented, Sylow
170	C₂³	—	13	1	no	—
171	Q₈	—	4	8	no	—
172	Q₈	—	52	8	no	—
173	C₂×C₄	—	4	4	no	—
174	C₂×C₄	—	13	1	no	—
175	C₂×C₄	—	13	4	no	—
176	C₂×C₄	—	13	2	no	—
177	C₂×C₄	—	26	4	no	—
178	C₂×C₄	—	52	4	no	—
179	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
180	C₂×C₄	C₂×D₄×D₁₃	1	2	yes	—
181	C₈	—	2	4	no	—
182	C₈	—	13	4	no	—
183	C₈	—	13	8	no	—
184	C₈	Q₁₆×D₁₃	1	4	yes	—
185	C₂²	—	13	2	no	—
186	C₂²	—	26	2	no	—
187	C₂²	D₁₃×#127	1	1	yes	characteristic
188	C₄	—	8	4	no	—
189	C₄	—	13	2	no	—
190	C₄	—	26	2	no	—
191	C₄	—	104	4	no	—
192	C₄	C₂×Q₁₆×D₁₃	1	2	yes	—
193	C₄	D₁₃×#135	1	4	yes	—
194	C₂	—	26	2	no	—
195	C₂	D₁₃×#494	1	1	yes	characteristic
196	C₂	#13125261	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13125421

C₂

yes

maximal

#13125347

C₂

yes

maximal

#13114453

C₂

yes

maximal, complemented

#13114446

C₂

yes

maximal, characteristic, complemented

#13114432

C₂

yes

maximal, complemented

C₈²×D₁₃

C₂

yes

maximal, characteristic

C₁₃×#1710

C₂

yes

maximal, characteristic, complemented

#11719433

C₂²

yes

—

C₁₃×#500

C₂²

yes

—

C₁₃×#487

C₂²

yes

—

#11718753

C₂²

yes

—

#11718740

C₂²

yes

—

#11718739

C₂²

yes

—

—

—

—

—

C₄×C₈×D₁₃

C₂²

yes

—

C₄×C₈×D₁₃

C₂²

yes

characteristic

#11718343

C₂²

yes

—

#11718342

C₂²

yes

—

C₈²×C₁₃

C₂²

yes

characteristic

C₈×#1055

C₂²

yes

—

C₈×#1055

C₂²

yes

characteristic

C₁₃×C₄⋊Q₈

C₂³

yes

—

#88200

—

Q₈×Dic₁₃

—

C₂×#6107

—

C₂×C₈×D₁₃

D₄

yes

—

C₂×C₈×D₁₃

D₄

yes

—

—

—

—

—

C₄²×D₁₃

C₂³

yes

characteristic

#88123

C₂³

yes

—

C₁₃×#107

—

C₁₃×#103

—

C₄×C₈×C₁₃

C₂³

yes

—

C₄×C₈×C₁₃

C₂³

yes

characteristic

#88058

—

#88056

—

C₈×Dic₁₃

D₄

yes

—

C₈×Dic₁₃

D₄

yes

—

#88048

—

#88047

—

C₄×#1055

C₂³

yes

—

C₄×#1055

C₂³

yes

characteristic

#19604

—

maximal, Sylow

C₂×Q₈×C₁₃

—

C₂×C₄×D₁₃

C₂×D₄

yes

—

C₂×C₄×D₁₃

C₂×D₄

yes

characteristic

C₂×Dic₂₆

—

C₂×C₈×C₁₃

C₂×D₄

yes

—

C₂×C₈×C₁₃

C₂×D₄

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂⁴

yes

characteristic

#6107

—

#6103

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#6101

—

C₄×Dic₁₃

—

C₄×Dic₁₃

C₂×D₄

yes

—

C₄×Dic₁₃

C₂×D₄

yes

characteristic

C₂×#1055

C₂×D₄

yes

—

C₂×#1055

C₂×D₄

yes

—

C₈×D₁₃

—

C₈×D₁₃

Q₁₆

yes

—

—

—

—

D₁₃

yes

characteristic, complemented

#1707

—

C₂×C₈²

—

C₂²×D₁₃

#127

yes

characteristic

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂²×D₄

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

#127

yes

—

C₂×Dic₁₃

#127

yes

characteristic

C₄×D₁₃

—

C₄×D₁₃

C₂×Q₁₆

yes

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

C₂×Q₁₆

yes

—

#1055

C₂×Q₁₆

yes

—

#1055

#135

yes

—

—

—

—

C₂×D₁₃

yes

—

#499

—

#498

—

100

#494

—

101

#487

—

102

#487

C₂×D₁₃

yes

—

103

#467

—

104

C₄×Q₁₆

—

105

#427

—

106

C₂×C₄×C₈

—

107

C₂×C₄×C₈

—

108

C₈²

—

109

C₈²

—

110

C₈²

C₂×D₁₃

yes

characteristic

111

C₂²×C₁₃

C₂×#127

yes

characteristic

112

C₂×D₁₃

—

113

C₂×D₁₃

#494

yes

—

114

C₄×C₁₃

—

115

C₄×C₁₃

C₂²×Q₁₆

yes

—

116

C₄×C₁₃

C₂×#135

yes

—

117

Dic₁₃

—

118

Dic₁₃

—

119

Dic₁₃

#494

yes

—

120

C₂×Q₁₆

—

121

C₂²×C₈

—

122

C₂²×C₈

—

123

C₄⋊Q₈

—

124

C₄⋊Q₈

C₂²×D₁₃

yes

—

125

#125

—

126

C₂²⋊Q₈

—

127

C₂²⋊Q₈

—

128

C₄×Q₈

—

129

C₄×Q₈

—

130

#117

—

131

C₂×C₄²

—

132

#107

—

133

#107

—

134

#107

—

135

#106

—

136

#103

—

137

#103

—

138

C₄×C₈

—

139

C₄×C₈

—

140

C₄×C₈

—

141

C₄×C₈

—

142

C₄×C₈

C₂²×D₁₃

yes

—

143

C₄×C₈

C₂²×D₁₃

yes

characteristic

144

C₂×C₁₃

C₂×#494

yes

characteristic

145

C₂×C₁₃

#3147

yes

—

146

D₁₃

—

147

C₂×Q₈

—

148

C₂×Q₈

—

149

C₂²×C₄

—

150

C₂²×C₄

—

151

Q₁₆

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

D₄×D₁₃

yes

—

157

C₂×C₈

D₄×D₁₃

yes

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₂²⋊C₄

—

165

C₄²

—

166

C₄²

—

167

C₄²

—

168

C₄²

C₂³×D₁₃

yes

characteristic

169

C₁₃

#19604

yes

characteristic, complemented, Sylow

170

C₂³

—

171

Q₈

—

172

Q₈

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

C₂×D₄×D₁₃

yes

characteristic

180

C₂×C₄

C₂×D₄×D₁₃

yes

—

181

C₈

—

182

C₈

—

183

C₈

—

184

C₈

Q₁₆×D₁₃

yes

—

185

C₂²

—

186

C₂²

—

187

C₂²

D₁₃×#127

yes

characteristic

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

—

104

—

192

C₄

C₂×Q₁₆×D₁₃

yes

—

193

C₄

D₁₃×#135

yes

—

194

C₂

—

195

C₂

D₁₃×#494

yes

characteristic

196

C₂

#13125261

yes

—

	subgroup	G/H
center	C₂²	D₁₃×#127
commutator, cosocle	C₄²×C₁₃	C₂⁴
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#1710	C₂
socle	C₂²×C₁₃	C₂×#127
prime radical	C₂²×D₁₃	#127
hypercenter	#1710	D₁₃
hypocenter	C₁₃	#19604
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1710

subgroup

G/H

center

C₂²

D₁₃×#127

commutator, cosocle

C₄²×C₁₃

C₂⁴

Frattini

C₄²

C₂³×D₁₃

Fitting

C₁₃×#1710

C₂

C₂²×C₁₃

C₂²×D₁₃

D₁₃

C₁₃

G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #1710

N	H ≅ G/N	type
3. #13114453	C₂	semidirect
4. #13114446	C₂	semidirect
5. #13114432	C₂	semidirect
7. C₁₃×#1710	C₂	semidirect
71. C₈×D₁₃	Q₁₆	semidirect
76. #1710	D₁₃	semidirect
169. C₁₃	#19604	semidirect

H ≅ G/N

type

3. #13114453

C₂

semidirect

4. #13114446

C₂

semidirect

5. #13114432

C₂

semidirect

7. C₁₃×#1710

C₂

semidirect

71. C₈×D₁₃

Q₁₆

semidirect

76. #1710

D₁₃

semidirect

169. C₁₃

#19604

semidirect

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

subgroup	index
#13125421	2
#13125347	2
#13114453	2
#13114446	2
#13114432	2
C₈²×D₁₃	2
C₁₃×#1710	2
#19604	13

subgroup

index

C₈²×D₁₃

C₁₃×#1710

#19604

Subgroups — #15850626

Conjugacy classes of subgroups

Complements

Maximal subgroups