Group #15767906 — subgroups

Subgroups — #15767906

Order 3328.

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

normal subgroups only

This group has 320 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	#13115776	C₂	1	1	yes	maximal, characteristic
2	#13115011	C₂	1	1	yes	maximal, characteristic
3	#13113611	C₂	1	2	yes	maximal
4	#13113266	C₂	1	2	yes	maximal
5	C₁₃×#1275	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₂₆	C₂²	1	1	yes	characteristic
7	C₁₃×#431	C₂²	1	2	yes	—
8	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
9	C₂×#88059	—	2	2	no	—
10	#11718585	C₂²	1	2	yes	—
11	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
12	#11718142	—	2	2	no	—
13	#11718050	—	2	4	no	—
14	#11718048	—	2	4	no	—
15	C₂²×Dic₂₆	Q₈	1	1	yes	characteristic
16	C₂²×C₈×C₁₃	D₄	1	2	yes	—
17	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
18	C₂×C₄⋊Dic₁₃	Q₈	1	1	yes	characteristic
19	C₂×#6101	—	2	1	no	—
20	C₂×C₄×Dic₁₃	—	2	1	no	—
21	C₂×#6099	—	2	2	no	—
22	C₂²×#1055	—	2	1	no	—
23	C₄×Dic₂₆	—	2	2	no	—
24	C₄×Dic₂₆	—	2	2	no	—
25	C₁₃×C₄⋊C₈	—	2	2	no	—
26	C₁₃×C₂²⋊C₈	—	2	2	no	—
27	C₁₃×#97	—	2	2	no	—
28	C₄×C₈×C₁₃	D₄	1	4	yes	—
29	#88059	—	4	4	no	—
30	C₄⋊#1055	—	2	2	no	—
31	C₄⋊#1055	—	2	2	no	—
32	C₄⋊#1055	—	2	2	no	—
33	#10662	—	13	1	no	maximal, Sylow
34	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
36	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
37	C₂²×Dic₁₃	—	2	1	no	—
38	C₂²×Dic₁₃	—	2	1	no	—
39	C₂×Dic₂₆	—	2	2	no	—
40	C₂×Dic₂₆	—	4	2	no	—
41	C₂×C₈×C₁₃	—	2	2	no	—
42	C₂×C₈×C₁₃	—	2	2	no	—
43	C₂×C₈×C₁₃	—	2	4	no	—
44	C₂×C₈×C₁₃	Q₁₆	1	4	yes	—
45	C₄²×C₁₃	C₂×D₄	1	2	yes	—
46	C₄²×C₁₃	#41	1	2	yes	—
47	C₄⋊Dic₁₃	—	2	2	no	—
48	#6101	—	4	2	no	—
49	C₄×Dic₁₃	—	4	1	no	—
50	C₄×Dic₁₃	—	4	1	no	—
51	#6099	—	4	4	no	—
52	C₂×#1055	—	4	1	no	—
53	C₂×#1055	—	4	2	no	—
54	C₂×#1055	—	4	2	no	—
55	#1985	—	13	1	no	—
56	#1849	—	13	1	no	—
57	#1571	—	13	2	no	—
58	#1487	—	13	2	no	—
59	#1275	D₁₃	1	1	yes	characteristic
60	C₂³×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
61	C₂×C₄×C₁₃	—	4	1	no	—
62	C₂×C₄×C₁₃	—	4	1	no	—
63	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
64	C₂×C₄×C₁₃	#123	1	2	yes	—
65	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
66	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
67	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
68	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
69	C₂×Dic₁₃	—	2	2	no	—
70	C₂×Dic₁₃	—	4	1	no	—
71	C₂×Dic₁₃	—	4	1	no	—
72	C₂×Dic₁₃	—	4	2	no	—
73	C₂×Dic₁₃	—	4	1	no	—
74	Dic₂₆	—	2	4	no	—
75	Dic₂₆	—	4	2	no	—
76	C₈×C₁₃	—	2	4	no	—
77	C₈×C₁₃	—	4	4	no	—
78	#1055	—	8	1	no	—
79	#1055	—	8	2	no	—
80	C₂×C₄×Q₈	—	13	1	no	—
81	#431	C₂×D₁₃	1	2	yes	—
82	C₂×#108	—	26	2	no	—
83	#423	—	13	2	no	—
84	C₂×C₄⋊C₈	—	13	1	no	—
85	C₂×#97	C₂×D₁₃	1	1	yes	characteristic
86	#336	—	26	2	no	—
87	#335	—	26	4	no	—
88	Q₈⋊C₈	—	26	4	no	—
89	C₂²×C₁₃	—	2	1	no	—
90	C₂²×C₁₃	—	2	1	no	—
91	C₂²×C₁₃	#399	1	1	yes	characteristic
92	C₂²×C₁₃	#477	1	2	yes	—
93	C₄×C₁₃	—	2	2	no	—
94	C₄×C₁₃	—	2	2	no	—
95	C₄×C₁₃	#484	1	2	yes	—
96	C₄×C₁₃	#494	1	1	yes	characteristic
97	C₄×C₁₃	#496	1	1	yes	characteristic
98	Dic₁₃	—	4	2	no	—
99	Dic₁₃	—	8	1	no	—
100	C₂²×Q₈	—	13	1	no	—
101	C₂×M₁₆	—	26	2	no	—
102	C₂²×C₈	—	26	1	no	—
103	C₂²×C₈	D₅₂	1	2	yes	—
104	C₄×Q₈	—	26	2	no	—
105	C₄×Q₈	—	26	2	no	—
106	C₂×C₄⋊C₄	—	13	1	no	—
107	C₂×C₄⋊C₄	—	26	1	no	—
108	C₂×C₄²	—	26	1	no	—
109	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
110	#108	—	52	4	no	—
111	C₄⋊C₈	—	2	2	no	—
112	C₄⋊C₈	—	26	2	no	—
113	C₄⋊C₈	—	26	2	no	—
114	C₄⋊C₈	—	26	2	no	—
115	C₂²⋊C₈	—	2	2	no	—
116	#97	—	2	2	no	—
117	C₄×C₈	D₅₂	1	4	yes	—
118	C₂×C₁₃	—	2	2	no	—
119	C₂×C₁₃	#2091	1	1	yes	characteristic
120	C₂×C₁₃	#2095	1	1	yes	characteristic
121	C₂×C₁₃	#2098	1	1	yes	characteristic
122	C₂×Q₈	—	26	2	no	—
123	C₂×Q₈	—	52	2	no	—
124	C₂²×C₄	—	26	1	no	—
125	C₂²×C₄	—	26	1	no	—
126	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
127	C₂²×C₄	#6129	1	1	yes	characteristic
128	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
129	M₁₆	—	52	4	no	—
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	4	no	—
133	C₂×C₈	—	52	1	no	—
134	C₂×C₈	—	52	2	no	—
135	C₂×C₈	—	52	2	no	—
136	C₂×C₈	Dic₅₂	1	4	yes	—
137	C₄⋊C₄	—	26	2	no	—
138	C₄⋊C₄	—	52	2	no	—
139	C₄²	—	52	1	no	—
140	C₄²	—	52	1	no	—
141	C₄²	C₂×D₅₂	1	2	yes	—
142	C₄²	#6129	1	2	yes	—
143	C₁₃	#10662	1	1	yes	characteristic, complemented, Sylow
144	C₂³	#88151	1	1	yes	characteristic
145	Q₈	—	26	4	no	—
146	Q₈	—	52	2	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	26	2	no	—
150	C₂×C₄	—	52	1	no	—
151	C₂×C₄	—	52	1	no	—
152	C₂×C₄	—	52	2	no	—
153	C₂×C₄	—	52	1	no	—
154	C₂×C₄	#88128	1	1	yes	characteristic
155	C₂×C₄	#88140	1	2	yes	—
156	C₂×C₄	#88140	1	1	yes	characteristic
157	C₂×C₄	#88151	1	2	yes	—
158	C₂×C₄	C₂×Dic₅₂	1	2	yes	—
159	C₂×C₄	#88162	1	2	yes	—
160	C₈	—	2	4	no	—
161	C₈	—	4	4	no	—
162	C₈	—	104	1	no	—
163	C₈	—	104	2	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	#11718338	1	1	yes	characteristic
167	C₂²	#11718511	1	2	yes	—
168	C₄	—	2	2	no	—
169	C₄	—	2	2	no	—
170	C₄	—	52	2	no	—
171	C₄	—	104	1	no	—
172	C₄	#11718360	1	1	yes	characteristic
173	C₄	#11718373	1	1	yes	characteristic
174	C₄	#11718383	1	2	yes	—
175	C₂	—	2	2	no	—
176	C₂	#13116353	1	1	yes	characteristic
177	C₂	#13116373	1	1	yes	characteristic
178	C₂	#13116390	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13115776

C₂

yes

maximal, characteristic

#13115011

C₂

yes

maximal, characteristic

#13113611

C₂

yes

maximal

#13113266

C₂

yes

maximal

C₁₃×#1275

C₂

yes

maximal, characteristic

C₂×C₄×Dic₂₆

C₂²

yes

characteristic

C₁₃×#431

C₂²

yes

—

C₂×C₁₃×#97

C₂²

yes

characteristic

C₂×#88059

—

#11718585

C₂²

yes

—

C₂×C₄⋊#1055

C₂²

yes

—

—

—

C₂²×Dic₂₆

Q₈

yes

characteristic

C₂²×C₈×C₁₃

D₄

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₃

Q₈

yes

characteristic

C₂×#6101

—

C₂×C₄×Dic₁₃

—

C₂×#6099

—

C₂²×#1055

—

C₄×Dic₂₆

—

C₄×Dic₂₆

—

C₁₃×C₄⋊C₈

—

C₁₃×C₂²⋊C₈

—

C₁₃×#97

—

C₄×C₈×C₁₃

D₄

yes

—

—

—

—

—

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×Q₈

yes

characteristic

C₂²×C₄×C₁₃

#41

yes

characteristic

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

Q₁₆

yes

—

C₄²×C₁₃

C₂×D₄

yes

—

C₄²×C₁₃

#41

yes

—

C₄⋊Dic₁₃

—

#6101

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

—

—

—

—

—

—

—

—

D₁₃

yes

characteristic

C₂³×C₁₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₃

#123

yes

—

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

C₂×Q₁₆

yes

—

C₂×C₄×C₁₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

—

#1055

—

#1055

—

C₂×C₄×Q₈

—

#431

C₂×D₁₃

yes

—

C₂×#108

—

#423

—

C₂×C₄⋊C₈

—

C₂×#97

C₂×D₁₃

yes

characteristic

#336

—

#335

—

Q₈⋊C₈

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#399

yes

characteristic

C₂²×C₁₃

#477

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#484

yes

—

C₄×C₁₃

yes

C₄×C₁₃

yes

Dic₁₃

—

Dic₁₃

—

100

C₂²×Q₈

—

101

C₂×M₁₆

—

102

C₂²×C₈

—

103

C₂²×C₈

D₅₂

yes

—

104

C₄×Q₈

—

105

C₄×Q₈

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄²

—

109

C₂×C₄²

C₂²×D₁₃

yes

characteristic

110

#108

—

111

C₄⋊C₈

—

112

C₄⋊C₈

—

113

C₄⋊C₈

—

114

C₄⋊C₈

—

115

C₂²⋊C₈

—

116

#97

—

117

C₄×C₈

D₅₂

yes

—

118

C₂×C₁₃

—

119

C₂×C₁₃

#2091

yes

characteristic

120

C₂×C₁₃

#2095

yes

characteristic

121

C₂×C₁₃

#2098

yes

characteristic

122

C₂×Q₈

—

123

C₂×Q₈

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

C₂×D₅₂

yes

characteristic

127

C₂²×C₄

#6129

yes

characteristic

128

C₂²×C₄

Q₈×D₁₃

yes

characteristic

129

M₁₆

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

Dic₅₂

yes

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄²

—

140

C₄²

—

141

C₄²

C₂×D₅₂

yes

—

142

C₄²

#6129

yes

—

143

C₁₃

#10662

yes

characteristic, complemented, Sylow

144

C₂³

#88151

yes

characteristic

145

Q₈

—

146

Q₈

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

#88128

yes

characteristic

155

C₂×C₄

#88140

yes

—

156

C₂×C₄

#88140

yes

characteristic

157

C₂×C₄

#88151

yes

—

158

C₂×C₄

C₂×Dic₅₂

yes

—

159

C₂×C₄

#88162

yes

—

160

C₈

—

161

C₈

—

162

C₈

—

104

—

163

C₈

—

104

—

164

C₂²

—

165

C₂²

—

166

C₂²

#11718338

yes

characteristic

167

C₂²

#11718511

yes

—

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

—

104

—

172

C₄

#11718360

yes

characteristic

173

C₄

#11718373

yes

characteristic

174

C₄

#11718383

yes

—

175

C₂

—

176

C₂

#13116353

yes

characteristic

177

C₂

#13116373

yes

characteristic

178

C₂

#13116390

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#88128
commutator, cosocle	C₂×C₄²×C₁₃	C₂³
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1275	C₂
socle	C₂²×C₁₃	#399
prime radical	C₂³×C₁₃	C₂²⋊Q₈
hypercenter	#1275	D₁₃
hypocenter	C₁₃	#10662
LCS	G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1275

subgroup

G/H

center

C₂×C₄

#88128

commutator, cosocle

C₂×C₄²×C₁₃

C₂³

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1275

C₂

socle

C₂²×C₁₃

#399

prime radical

C₂³×C₁₃

C₂²⋊Q₈

D₁₃

C₁₃

G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1275

N	H ≅ G/N	type
143. C₁₃	#10662	semidirect

H ≅ G/N

type

143. C₁₃

#10662

semidirect

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

subgroup	index
#13115776	2
#13115011	2
#13113611	2
#13113266	2
C₁₃×#1275	2
#10662	13

subgroup

index

C₁₃×#1275

Subgroups — #15767906

Conjugacy classes of subgroups

Complements

Maximal subgroups