Group #15765642 — subgroups

Subgroups — #15765642

Order 3328.

504 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 111 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	#13125984	C₂	1	2	yes	maximal
2	#13123214	C₂	1	1	yes	maximal, characteristic
3	#13113522	C₂	1	2	yes	maximal
4	#13112669	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1994	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#533	C₂²	1	1	yes	characteristic
7	C₂×#88123	C₂²	1	2	yes	—
8	C₁₃×#423	C₂²	1	1	yes	characteristic
9	C₄³×C₁₃	C₂²	1	1	yes	characteristic
10	#11718742	—	2	2	no	—
11	#11718739	—	2	2	no	—
12	#11718705	—	2	4	no	—
13	#11718682	—	2	4	no	—
14	#11718597	—	2	2	no	—
15	#11718590	—	2	4	no	—
16	#11718586	C₂²	1	2	yes	—
17	C₁₃×#339	—	2	2	no	—
18	#11718078	—	2	4	no	—
19	C₂²×Dic₂₆	—	2	2	no	—
20	C₂×C₁₃×M₁₆	—	2	1	no	—
21	C₁₃×C₄⋊Q₈	D₄	1	2	yes	—
22	C₁₃×#125	—	2	1	no	—
23	C₁₃×C₂²⋊Q₈	—	2	2	no	—
24	C₄×Q₈×C₁₃	—	2	2	no	—
25	C₁₃×#117	—	2	1	no	—
26	C₂×C₄²×C₁₃	—	2	2	no	—
27	C₂×C₄²×C₁₃	D₄	1	2	yes	—
28	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊Dic₁₃	—	2	2	no	—
30	C₂×C₄⋊Dic₁₃	D₄	1	4	yes	—
31	#88123	—	2	2	no	—
32	#88123	—	2	2	no	—
33	#88123	—	2	2	no	—
34	#88123	—	2	2	no	—
35	C₁₃×C₄⋊C₈	—	2	2	no	—
36	#88075	—	2	8	no	—
37	#88071	—	2	8	no	—
38	C₂²⋊#1055	—	2	2	no	—
39	#88047	—	2	8	no	—
40	C₄⋊#1055	D₄	1	4	yes	—
41	C₄×#1055	—	2	2	no	—
42	#10295	—	13	1	no	maximal, Sylow
43	C₂×Q₈×C₁₃	—	2	2	no	—
44	C₂²×C₄×C₁₃	—	2	2	no	—
45	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
46	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×Dic₁₃	—	2	4	no	—
48	C₂×Dic₂₆	—	4	8	no	—
49	C₂×Dic₂₆	—	4	2	no	—
50	C₂×Dic₂₆	—	4	2	no	—
51	C₁₃×M₁₆	—	4	2	no	—
52	C₂×C₈×C₁₃	—	4	1	no	—
53	C₁₃×C₄⋊C₄	—	2	2	no	—
54	C₁₃×C₄⋊C₄	—	2	2	no	—
55	C₁₃×C₄⋊C₄	—	4	2	no	—
56	C₁₃×C₂²⋊C₄	—	4	1	no	—
57	C₄²×C₁₃	—	2	2	no	—
58	C₄²×C₁₃	—	4	4	no	—
59	C₄²×C₁₃	—	4	1	no	—
60	C₄²×C₁₃	Q₁₆	1	4	yes	—
61	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₁₃	C₂×D₄	1	2	yes	—
63	C₄²×C₁₃	#41	1	1	yes	characteristic
64	C₄⋊Dic₁₃	—	2	4	no	—
65	C₄⋊Dic₁₃	—	2	4	no	—
66	C₄⋊Dic₁₃	—	4	2	no	—
67	C₄⋊Dic₁₃	—	4	2	no	—
68	C₂×#1055	—	2	4	no	—
69	#3230	—	13	2	no	—
70	#2862	—	13	1	no	—
71	#1994	D₁₃	1	1	yes	characteristic
72	#1549	—	13	2	no	—
73	#1276	—	13	1	no	—
74	C₂³×C₁₃	#120	1	1	yes	characteristic
75	Q₈×C₁₃	—	4	4	no	—
76	C₂×C₄×C₁₃	—	2	4	no	—
77	C₂×C₄×C₁₃	—	2	4	no	—
78	C₂×C₄×C₁₃	—	4	4	no	—
79	C₂×C₄×C₁₃	—	4	2	no	—
80	C₂×C₄×C₁₃	#120	1	2	yes	—
81	C₂×C₄×C₁₃	#121	1	2	yes	—
82	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
83	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
85	C₂×C₄×C₁₃	#137	1	2	yes	—
86	C₂×Dic₁₃	—	4	8	no	—
87	C₂×Dic₁₃	—	4	4	no	—
88	Dic₂₆	—	8	4	no	—
89	Dic₂₆	—	8	4	no	—
90	C₈×C₁₃	—	8	1	no	—
91	#1055	—	4	4	no	—
92	#533	C₂×D₁₃	1	1	yes	characteristic
93	C₂×C₄⋊Q₈	—	13	2	no	—
94	#500	—	26	2	no	—
95	#487	—	26	2	no	—
96	#484	—	26	4	no	—
97	#477	—	26	4	no	—
98	#431	—	13	2	no	—
99	#423	C₂×D₁₃	1	1	yes	characteristic
100	#391	—	26	4	no	—
101	#382	—	26	2	no	—
102	C₄³	C₂×D₁₃	1	1	yes	characteristic
103	#339	—	2	2	no	—
104	#332	—	26	4	no	—
105	C₂²×C₁₃	—	2	2	no	—
106	C₂²×C₁₃	#392	1	1	yes	characteristic
107	C₂²×C₁₃	#451	1	2	yes	—
108	C₄×C₁₃	—	2	2	no	—
109	C₄×C₁₃	—	4	4	no	—
110	C₄×C₁₃	—	8	2	no	—
111	C₄×C₁₃	#454	1	4	yes	—
112	C₄×C₁₃	#462	1	2	yes	—
113	C₄×C₁₃	#494	1	1	yes	characteristic
114	C₄×C₁₃	#497	1	1	yes	characteristic
115	Dic₁₃	—	8	4	no	—
116	C₂²×Q₈	—	26	2	no	—
117	C₂×M₁₆	—	2	1	no	—
118	C₄⋊Q₈	—	26	2	no	—
119	C₄⋊Q₈	—	26	2	no	—
120	C₄⋊Q₈	—	26	2	no	—
121	C₄⋊Q₈	—	26	2	no	—
122	C₄⋊Q₈	D₅₂	1	2	yes	—
123	#125	—	2	1	no	—
124	C₂²⋊Q₈	—	2	2	no	—
125	C₄×Q₈	—	2	2	no	—
126	#117	—	2	1	no	—
127	C₂×C₄⋊C₄	—	13	4	no	—
128	C₂×C₄⋊C₄	—	26	2	no	—
129	C₂×C₄²	—	2	2	no	—
130	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
131	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
132	#107	—	26	8	no	—
133	C₄⋊C₈	—	2	2	no	—
134	C₄⋊C₈	—	13	4	no	—
135	#103	—	26	8	no	—
136	C₂²⋊C₈	—	26	2	no	—
137	C₄×C₈	—	26	2	no	—
138	#95	—	26	8	no	—
139	C₂×C₁₃	—	2	2	no	—
140	C₂×C₁₃	#2003	1	2	yes	—
141	C₂×C₁₃	#2016	1	1	yes	characteristic
142	C₂×Q₈	—	2	2	no	—
143	C₂×Q₈	—	52	8	no	—
144	C₂×Q₈	—	52	2	no	—
145	C₂×Q₈	—	52	2	no	—
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	—	26	4	no	—
148	C₂²×C₄	D₄×D₁₃	1	2	yes	—
149	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
150	M₁₆	—	4	2	no	—
151	C₂×C₈	—	4	1	no	—
152	C₂×C₈	—	26	4	no	—
153	C₄⋊C₄	—	2	2	no	—
154	C₄⋊C₄	—	2	2	no	—
155	C₄⋊C₄	—	4	2	no	—
156	C₄⋊C₄	—	26	4	no	—
157	C₄⋊C₄	—	26	4	no	—
158	C₄⋊C₄	—	52	2	no	—
159	C₄⋊C₄	—	52	2	no	—
160	C₂²⋊C₄	—	4	1	no	—
161	C₄²	—	2	2	no	—
162	C₄²	—	4	4	no	—
163	C₄²	—	4	1	no	—
164	C₄²	#6107	1	4	yes	—
165	C₄²	C₂×D₅₂	1	1	yes	characteristic
166	C₄²	#6127	1	1	yes	characteristic
167	C₄²	D₄×D₁₃	1	2	yes	—
168	C₁₃	#10295	1	1	yes	characteristic, complemented, Sylow
169	C₂³	#88191	1	1	yes	characteristic
170	Q₈	—	4	4	no	—
171	Q₈	—	104	4	no	—
172	Q₈	—	104	4	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	52	8	no	—
178	C₂×C₄	—	52	4	no	—
179	C₂×C₄	#88136	1	2	yes	—
180	C₂×C₄	#88138	1	2	yes	—
181	C₂×C₄	#88184	1	1	yes	characteristic
182	C₂×C₄	#88194	1	1	yes	characteristic
183	C₂×C₄	C₂×#6107	1	2	yes	—
184	C₂×C₄	#88205	1	2	yes	—
185	C₈	—	8	1	no	—
186	C₈	—	52	4	no	—
187	C₂²	—	2	2	no	—
188	C₂²	#11718621	1	1	yes	characteristic
189	C₂²	#11718711	1	2	yes	—
190	C₄	—	2	2	no	—
191	C₄	—	4	4	no	—
192	C₄	—	8	2	no	—
193	C₄	—	104	4	no	—
194	C₄	#11718488	1	4	yes	—
195	C₄	#11718692	1	2	yes	—
196	C₄	#11718721	1	1	yes	characteristic
197	C₄	#11718753	1	1	yes	characteristic
198	C₂	—	2	2	no	—
199	C₂	#13115857	1	2	yes	—
200	C₂	#13115935	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13125984

C₂

yes

maximal

#13123214

C₂

yes

maximal, characteristic

#13113522

C₂

yes

maximal

#13112669

C₂

yes

maximal, characteristic

C₁₃×#1994

C₂

yes

maximal, characteristic

C₁₃×#533

C₂²

yes

characteristic

C₂×#88123

C₂²

yes

—

C₁₃×#423

C₂²

yes

characteristic

C₄³×C₁₃

C₂²

yes

—

—

—

—

—

—

C₂²

yes

—

C₁₃×#339

—

#11718078

—

C₂²×Dic₂₆

—

C₂×C₁₃×M₁₆

—

C₁₃×C₄⋊Q₈

D₄

yes

—

C₁₃×#125

—

C₁₃×C₂²⋊Q₈

—

C₄×Q₈×C₁₃

—

C₁₃×#117

—

C₂×C₄²×C₁₃

—

C₂×C₄²×C₁₃

D₄

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

D₄

yes

—

—

—

—

—

C₁₃×C₄⋊C₈

—

#88075

—

#88071

—

C₂²⋊#1055

—

#88047

—

C₄⋊#1055

D₄

yes

—

C₄×#1055

—

#10295

—

maximal, Sylow

C₂×Q₈×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×Dic₁₃

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

Q₁₆

yes

—

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄²×C₁₃

C₂×D₄

yes

—

C₄²×C₁₃

#41

yes

characteristic

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

—

—

—

D₁₃

yes

characteristic

#1549

—

#1276

—

C₂³×C₁₃

#120

yes

characteristic

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#120

yes

—

C₂×C₄×C₁₃

#121

yes

—

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

C₂×Q₁₆

yes

—

C₂×C₄×C₁₃

#137

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

#1055

—

#533

C₂×D₁₃

yes

characteristic

C₂×C₄⋊Q₈

—

—

—

—

—

—

C₂×D₁₃

yes

characteristic

100

#391

—

101

#382

—

102

C₄³

C₂×D₁₃

yes

characteristic

103

#339

—

104

#332

—

105

C₂²×C₁₃

—

106

C₂²×C₁₃

#392

yes

characteristic

107

C₂²×C₁₃

#451

yes

—

108

C₄×C₁₃

—

109

C₄×C₁₃

—

110

C₄×C₁₃

—

111

C₄×C₁₃

#454

yes

—

112

C₄×C₁₃

#462

yes

—

113

C₄×C₁₃

#494

yes

characteristic

114

C₄×C₁₃

#497

yes

characteristic

115

Dic₁₃

—

116

C₂²×Q₈

—

117

C₂×M₁₆

—

118

C₄⋊Q₈

—

119

C₄⋊Q₈

—

120

C₄⋊Q₈

—

121

C₄⋊Q₈

—

122

C₄⋊Q₈

D₅₂

yes

—

123

#125

—

124

C₂²⋊Q₈

—

125

C₄×Q₈

—

126

#117

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄²

—

130

C₂×C₄²

C₂²⋊D₁₃

yes

—

131

C₂×C₄²

C₂²×D₁₃

yes

characteristic

132

#107

—

133

C₄⋊C₈

—

134

C₄⋊C₈

—

135

#103

—

136

C₂²⋊C₈

—

137

C₄×C₈

—

138

#95

—

139

C₂×C₁₃

—

140

C₂×C₁₃

#2003

yes

—

141

C₂×C₁₃

#2016

yes

characteristic

142

C₂×Q₈

—

143

C₂×Q₈

—

144

C₂×Q₈

—

145

C₂×Q₈

—

146

C₂²×C₄

—

147

C₂²×C₄

—

148

C₂²×C₄

D₄×D₁₃

yes

—

149

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

150

M₁₆

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₂²⋊C₄

—

161

C₄²

—

162

C₄²

—

163

C₄²

—

164

C₄²

#6107

yes

—

165

C₄²

C₂×D₅₂

yes

characteristic

166

C₄²

#6127

yes

characteristic

167

C₄²

D₄×D₁₃

yes

—

168

C₁₃

#10295

yes

characteristic, complemented, Sylow

169

C₂³

#88191

yes

characteristic

170

Q₈

—

171

Q₈

—

104

—

172

Q₈

—

104

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

#88136

yes

—

180

C₂×C₄

#88138

yes

—

181

C₂×C₄

#88184

yes

characteristic

182

C₂×C₄

#88194

yes

characteristic

183

C₂×C₄

C₂×#6107

yes

—

184

C₂×C₄

#88205

yes

—

185

C₈

—

186

C₈

—

187

C₂²

—

188

C₂²

#11718621

yes

characteristic

189

C₂²

#11718711

yes

—

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

—

104

—

194

C₄

#11718488

yes

—

195

C₄

#11718692

yes

—

196

C₄

#11718721

yes

characteristic

197

C₄

#11718753

yes

characteristic

198

C₂

—

199

C₂

#13115857

yes

—

200

C₂

#13115935

yes

characteristic

	subgroup	G/H
center	C₂²	#11718621
commutator, cosocle	C₂×C₄²×C₁₃	C₂³
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1994	C₂
socle	C₂²×C₁₃	#392
prime radical	C₂³×C₁₃	#120
hypercenter	#1994	D₁₃
hypocenter	C₁₃	#10295
LCS	G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1994

subgroup

G/H

center

C₂²

#11718621

commutator, cosocle

C₂×C₄²×C₁₃

C₂³

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1994

C₂

C₂²×C₁₃

C₂³×C₁₃

D₁₃

C₁₃

G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1994

N	H ≅ G/N	type
168. C₁₃	#10295	semidirect

H ≅ G/N

type

168. C₁₃

#10295

semidirect

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

subgroup	index
#13125984	2
#13123214	2
#13113522	2
#13112669	2
C₁₃×#1994	2
#10295	13

subgroup

index

C₁₃×#1994

Subgroups — #15765642

Conjugacy classes of subgroups

Complements

Maximal subgroups