Group #15767930 — subgroups

Subgroups — #15767930

Order 3328.

354 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 101 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	#13115668	C₂	1	1	yes	maximal, characteristic
2	#13114990	C₂	1	1	yes	maximal, characteristic, complemented
3	#13113607	C₂	1	2	yes	maximal, complemented
4	#13113236	C₂	1	2	yes	maximal
5	C₁₃×#1274	C₂	1	1	yes	maximal, characteristic, complemented
6	#11719182	C₂²	1	1	yes	characteristic
7	C₁₃×#431	C₂²	1	2	yes	—
8	C₁₃×#376	C₂²	1	1	yes	characteristic
9	C₂×#88045	—	2	2	no	—
10	#11718585	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊#1055	—	2	2	no	—
12	C₂×C₄⋊#1055	C₂²	1	2	yes	—
13	#11718051	—	2	4	no	—
14	#11718049	—	2	4	no	—
15	C₂×#6127	—	2	1	no	—
16	C₂×C₄²×C₁₃	D₄	1	2	yes	—
17	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
18	#88184	—	2	2	no	—
19	C₂×#6099	—	2	1	no	—
20	C₂²×#1055	—	2	2	no	—
21	#88129	—	2	1	no	—
22	#88128	Q₈	1	1	yes	characteristic
23	C₄×D₅₂	—	2	2	no	—
24	#88123	Q₈	1	1	yes	characteristic
25	C₁₃×C₄⋊C₈	—	2	2	no	—
26	C₁₃×C₂²⋊C₈	—	2	2	no	—
27	C₄×C₈×C₁₃	D₄	1	4	yes	—
28	C₁₃×#95	—	2	2	no	—
29	#88045	—	4	4	no	—
30	C₄⋊#1055	—	2	2	no	—
31	C₄⋊#1055	—	2	2	no	—
32	C₄⋊#1055	—	2	2	no	—
33	C₄⋊#1055	—	4	4	no	—
34	#10666	—	13	1	no	maximal, Sylow
35	C₂²×C₄×C₁₃	—	2	2	no	—
36	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₃	#41	1	2	yes	—
38	C₂×C₂²⋊D₁₃	—	4	1	no	—
39	#6127	—	4	2	no	—
40	#6127	—	4	2	no	—
41	C₂×D₅₂	—	2	2	no	—
42	C₂×D₅₂	—	2	1	no	—
43	C₂×C₄×D₁₃	—	4	1	no	—
44	C₂×Dic₂₆	—	2	1	no	—
45	C₂×C₈×C₁₃	—	2	4	no	—
46	C₄²×C₁₃	—	2	2	no	—
47	C₄²×C₁₃	SD₁₆	1	4	yes	—
48	C₄²×C₁₃	C₂×D₄	1	2	yes	—
49	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
50	C₄²×C₁₃	#41	1	1	yes	characteristic
51	#6103	—	4	2	no	—
52	C₄⋊Dic₁₃	—	2	2	no	—
53	#6099	—	4	2	no	—
54	C₂×#1055	—	4	2	no	—
55	C₂×#1055	—	4	1	no	—
56	C₂×#1055	—	4	4	no	—
57	#1963	—	13	1	no	—
58	#1843	—	13	1	no	—
59	#1570	—	13	2	no	—
60	#1480	—	13	2	no	—
61	#1274	D₁₃	1	1	yes	characteristic, complemented
62	C₂³×C₁₃	#123	1	1	yes	characteristic
63	C₂²×D₁₃	—	4	1	no	—
64	C₂×C₄×C₁₃	—	2	4	no	—
65	C₂×C₄×C₁₃	—	4	2	no	—
66	C₂×C₄×C₁₃	—	4	4	no	—
67	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
68	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
69	C₂×C₄×C₁₃	#123	1	2	yes	—
70	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
71	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
72	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
73	C₂²⋊D₁₃	—	8	2	no	—
74	C₂×Dic₁₃	—	4	1	no	—
75	D₅₂	—	4	4	no	—
76	D₅₂	—	4	2	no	—
77	C₄×D₁₃	—	8	1	no	—
78	C₄×D₁₃	—	8	1	no	—
79	Dic₂₆	—	4	2	no	—
80	C₈×C₁₃	—	4	4	no	—
81	#1055	—	8	1	no	—
82	#1055	—	8	2	no	—
83	#532	—	13	1	no	—
84	#431	C₂×D₁₃	1	2	yes	—
85	#423	—	13	1	no	—
86	C₂×C₄⋊C₈	—	13	2	no	—
87	C₂×C₄⋊C₈	—	26	2	no	—
88	C₂×#104	—	26	2	no	—
89	#376	C₂×D₁₃	1	1	yes	characteristic
90	#334	—	26	4	no	—
91	D₄⋊C₈	—	26	4	no	—
92	C₂²×C₁₃	—	2	2	no	—
93	C₂²×C₁₃	#399	1	1	yes	characteristic
94	C₂²×C₁₃	#481	1	2	yes	—
95	C₂×D₁₃	—	8	1	no	—
96	C₂×D₁₃	—	8	2	no	—
97	C₄×C₁₃	—	2	2	no	—
98	C₄×C₁₃	—	2	2	no	—
99	C₄×C₁₃	—	4	4	no	—
100	C₄×C₁₃	#476	1	2	yes	—
101	C₄×C₁₃	#492	1	1	yes	characteristic
102	C₄×C₁₃	#496	1	1	yes	characteristic
103	Dic₁₃	—	8	1	no	—
104	C₂×#41	—	26	1	no	—
105	C₂×M₁₆	—	26	1	no	—
106	C₂²×C₈	—	26	2	no	—
107	C₄⋊Q₈	—	13	1	no	—
108	#127	—	13	1	no	—
109	#124	—	26	1	no	—
110	#121	—	26	2	no	—
111	C₄×D₄	—	26	2	no	—
112	C₂×C₄²	D₅₂	1	2	yes	—
113	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
114	C₄⋊C₈	—	2	2	no	—
115	C₄⋊C₈	—	26	2	no	—
116	C₄⋊C₈	—	26	2	no	—
117	C₄⋊C₈	—	26	2	no	—
118	C₄⋊C₈	—	52	4	no	—
119	#104	—	52	4	no	—
120	C₂²⋊C₈	—	2	2	no	—
121	C₄×C₈	D₅₂	1	4	yes	—
122	#95	—	2	2	no	—
123	C₂×C₁₃	—	2	2	no	—
124	C₂×C₁₃	#2094	1	1	yes	characteristic
125	C₂×C₁₃	#2098	1	2	yes	—
126	D₁₃	—	16	1	no	—
127	#41	—	52	2	no	—
128	#41	—	52	2	no	—
129	C₂×Q₈	—	26	1	no	—
130	C₂×D₄	—	26	2	no	—
131	C₂×D₄	—	26	1	no	—
132	C₂×D₄	—	52	1	no	—
133	C₂²×C₄	—	2	2	no	—
134	C₂²×C₄	—	52	1	no	—
135	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
136	C₂²×C₄	#6129	1	2	yes	—
137	M₁₆	—	52	2	no	—
138	C₂×C₈	—	2	4	no	—
139	C₂×C₈	—	52	2	no	—
140	C₂×C₈	—	52	1	no	—
141	C₂×C₈	—	52	4	no	—
142	C₄⋊C₄	—	26	2	no	—
143	C₂²⋊C₄	—	52	2	no	—
144	C₄²	—	2	2	no	—
145	C₄²	#6095	1	4	yes	—
146	C₄²	C₂×D₅₂	1	2	yes	—
147	C₄²	#6129	1	1	yes	characteristic
148	C₄²	Q₈×D₁₃	1	1	yes	characteristic
149	C₁₃	#10666	1	1	yes	characteristic, complemented, Sylow
150	C₂³	—	52	1	no	—
151	C₂³	#88140	1	1	yes	characteristic
152	Q₈	—	52	2	no	—
153	D₄	—	52	4	no	—
154	D₄	—	52	2	no	—
155	D₄	—	104	2	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	—	4	4	no	—
159	C₂×C₄	—	52	1	no	—
160	C₂×C₄	—	104	1	no	—
161	C₂×C₄	—	104	1	no	—
162	C₂×C₄	#88128	1	1	yes	characteristic
163	C₂×C₄	#88140	1	2	yes	—
164	C₂×C₄	#88151	1	2	yes	—
165	C₂×C₄	#88151	1	1	yes	characteristic
166	C₂×C₄	C₂×#6095	1	2	yes	—
167	C₂×C₄	#88162	1	2	yes	—
168	C₈	—	4	4	no	—
169	C₈	—	104	1	no	—
170	C₈	—	104	2	no	—
171	C₂²	—	2	2	no	—
172	C₂²	—	104	1	no	—
173	C₂²	—	104	2	no	—
174	C₂²	#11718338	1	1	yes	characteristic
175	C₂²	#11718396	1	2	yes	—
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	4	4	no	—
179	C₄	—	104	1	no	—
180	C₄	#11718354	1	1	yes	characteristic
181	C₄	#11718373	1	1	yes	characteristic
182	C₄	#11718503	1	2	yes	—
183	C₂	—	2	2	no	—
184	C₂	—	208	1	no	—
185	C₂	#13116368	1	1	yes	characteristic
186	C₂	#13116390	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13115668

C₂

yes

maximal, characteristic

#13114990

C₂

yes

maximal, characteristic, complemented

#13113607

C₂

yes

maximal, complemented

#13113236

C₂

yes

maximal

C₁₃×#1274

C₂

yes

maximal, characteristic, complemented

#11719182

C₂²

yes

characteristic

C₁₃×#431

C₂²

yes

—

C₁₃×#376

C₂²

yes

characteristic

C₂×#88045

—

#11718585

C₂²

yes

characteristic

C₂×C₄⋊#1055

—

C₂×C₄⋊#1055

C₂²

yes

—

#11718051

—

#11718049

—

C₂×#6127

—

C₂×C₄²×C₁₃

D₄

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

#88184

—

C₂×#6099

—

C₂²×#1055

—

#88129

—

#88128

Q₈

yes

characteristic

C₄×D₅₂

—

#88123

Q₈

yes

characteristic

C₁₃×C₄⋊C₈

—

C₁₃×C₂²⋊C₈

—

C₄×C₈×C₁₃

D₄

yes

—

C₁₃×#95

—

—

—

—

—

—

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

#41

yes

—

C₂×C₂²⋊D₁₃

—

#6127

—

#6127

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×C₄×D₁₃

—

C₂×Dic₂₆

—

C₂×C₈×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

SD₁₆

yes

—

C₄²×C₁₃

C₂×D₄

yes

—

C₄²×C₁₃

C₂×Q₈

yes

C₄²×C₁₃

yes

—

C₄⋊Dic₁₃

—

—

—

—

—

—

—

—

—

D₁₃

yes

characteristic, complemented

C₂³×C₁₃

#123

yes

characteristic

C₂²×D₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

#123

yes

—

C₂×C₄×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

C₂×SD₁₆

yes

—

C₂×C₄×C₁₃

AGL(1,ℤ/8)

yes

—

C₂²⋊D₁₃

—

C₂×Dic₁₃

—

D₅₂

—

D₅₂

—

C₄×D₁₃

—

C₄×D₁₃

—

Dic₂₆

—

C₈×C₁₃

—

—

—

—

C₂×D₁₃

yes

—

#423

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×#104

—

#376

C₂×D₁₃

yes

characteristic

#334

—

D₄⋊C₈

—

C₂²×C₁₃

—

C₂²×C₁₃

#399

yes

characteristic

C₂²×C₁₃

#481

yes

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

100

C₄×C₁₃

#476

yes

—

101

C₄×C₁₃

#492

yes

characteristic

102

C₄×C₁₃

#496

yes

characteristic

103

Dic₁₃

—

104

C₂×#41

—

105

C₂×M₁₆

—

106

C₂²×C₈

—

107

C₄⋊Q₈

—

108

#127

—

109

#124

—

110

#121

—

111

C₄×D₄

—

112

C₂×C₄²

D₅₂

yes

—

113

C₂×C₄²

C₂²×D₁₃

yes

characteristic

114

C₄⋊C₈

—

115

C₄⋊C₈

—

116

C₄⋊C₈

—

117

C₄⋊C₈

—

118

C₄⋊C₈

—

119

#104

—

120

C₂²⋊C₈

—

121

C₄×C₈

D₅₂

yes

—

122

#95

—

123

C₂×C₁₃

—

124

C₂×C₁₃

#2094

yes

characteristic

125

C₂×C₁₃

#2098

yes

—

126

D₁₃

—

127

#41

—

128

#41

—

129

C₂×Q₈

—

130

C₂×D₄

—

131

C₂×D₄

—

132

C₂×D₄

—

133

C₂²×C₄

—

134

C₂²×C₄

—

135

C₂²×C₄

C₂×D₅₂

yes

characteristic

136

C₂²×C₄

#6129

yes

—

137

M₁₆

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₄⋊C₄

—

143

C₂²⋊C₄

—

144

C₄²

—

145

C₄²

#6095

yes

—

146

C₄²

C₂×D₅₂

yes

—

147

C₄²

#6129

yes

characteristic

148

C₄²

Q₈×D₁₃

yes

characteristic

149

C₁₃

#10666

yes

characteristic, complemented, Sylow

150

C₂³

—

151

C₂³

#88140

yes

characteristic

152

Q₈

—

153

D₄

—

154

D₄

—

155

D₄

—

104

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

104

—

161

C₂×C₄

—

104

—

162

C₂×C₄

#88128

yes

characteristic

163

C₂×C₄

#88140

yes

—

164

C₂×C₄

#88151

yes

—

165

C₂×C₄

#88151

yes

characteristic

166

C₂×C₄

C₂×#6095

yes

—

167

C₂×C₄

#88162

yes

—

168

C₈

—

169

C₈

—

104

—

170

C₈

—

104

—

171

C₂²

—

172

C₂²

—

104

—

173

C₂²

—

104

—

174

C₂²

#11718338

yes

characteristic

175

C₂²

#11718396

yes

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

104

—

180

C₄

#11718354

yes

characteristic

181

C₄

#11718373

yes

characteristic

182

C₄

#11718503

yes

—

183

C₂

—

184

C₂

—

208

—

185

C₂

#13116368

yes

characteristic

186

C₂

#13116390

yes

—

	subgroup	G/H
center	C₂×C₄	#88128
commutator, cosocle	C₂×C₄²×C₁₃	C₂³
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1274	C₂
socle	C₂²×C₁₃	#399
prime radical	#11719182	C₂²
hypercenter	#1274	D₁₃
hypocenter	C₁₃	#10666
LCS	G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1274

subgroup

G/H

center

C₂×C₄

#88128

commutator, cosocle

C₂×C₄²×C₁₃

C₂³

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1274

C₂

C₂²×C₁₃

C₂²

D₁₃

C₁₃

G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1274

N	H ≅ G/N	type
2. #13114990	C₂	semidirect
3. #13113607	C₂	semidirect
5. C₁₃×#1274	C₂	semidirect
61. #1274	D₁₃	semidirect
149. C₁₃	#10666	semidirect

H ≅ G/N

type

2. #13114990

C₂

semidirect

3. #13113607

C₂

semidirect

5. C₁₃×#1274

C₂

semidirect

61. #1274

D₁₃

semidirect

149. C₁₃

#10666

semidirect

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

subgroup	index
#13115668	2
#13114990	2
#13113607	2
#13113236	2
C₁₃×#1274	2
#10666	13

subgroup

index

C₁₃×#1274

Subgroups — #15767930

Conjugacy classes of subgroups

Complements

Maximal subgroups