Group #15767726 — subgroups

Subgroups — #15767726

Order 3328.

346 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 85 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	#13115660	C₂	1	1	yes	maximal, characteristic
2	#13114977	C₂	1	1	yes	maximal, characteristic, complemented
3	#13113503	C₂	1	2	yes	maximal
4	#13112695	C₂	1	2	yes	maximal, complemented
5	C₁₃×#1281	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#88128	C₂²	1	1	yes	characteristic
7	C₁₃×#431	C₂²	1	2	yes	—
8	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
9	C₂×#88061	—	2	2	no	—
10	#11718597	C₂²	1	1	yes	characteristic
11	#11718585	C₂²	1	2	yes	—
12	#11718144	—	2	2	no	—
13	#11718077	—	2	4	no	—
14	C₂²×D₅₂	—	2	1	no	—
15	C₂²×D₅₂	—	2	2	no	—
16	C₂²×C₈×C₁₃	D₄	1	2	yes	—
17	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
18	C₂×C₄⋊Dic₁₃	—	2	1	no	—
19	C₂×#6099	—	2	2	no	—
20	#88128	—	2	2	no	—
21	#88128	—	2	1	no	—
22	#88128	D₄	1	2	yes	—
23	C₁₃×C₄⋊C₈	—	2	2	no	—
24	C₁₃×C₂²⋊C₈	—	2	2	no	—
25	C₁₃×#97	—	2	2	no	—
26	C₄×C₈×C₁₃	—	2	2	no	—
27	#88071	—	2	2	no	—
28	#88061	—	4	4	no	—
29	C₄⋊#1055	—	2	2	no	—
30	C₄⋊#1055	—	2	2	no	—
31	#10634	—	13	1	no	maximal, Sylow
32	C₂³×D₁₃	—	4	1	no	—
33	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₃	#41	1	2	yes	—
35	C₂²×Dic₁₃	—	4	1	no	—
36	C₂×D₅₂	—	2	2	no	—
37	C₂×D₅₂	—	2	4	no	—
38	C₂×D₅₂	—	4	2	no	—
39	C₂×D₅₂	—	4	2	no	—
40	C₂×D₅₂	—	4	1	no	—
41	C₂×D₅₂	—	4	2	no	—
42	C₂×D₅₂	—	8	4	no	—
43	C₂×C₈×C₁₃	—	2	2	no	—
44	C₂×C₈×C₁₃	—	2	2	no	—
45	C₂×C₈×C₁₃	—	2	2	no	—
46	C₂×C₈×C₁₃	—	4	2	no	—
47	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
48	C₄²×C₁₃	#41	1	2	yes	—
49	C₄²×C₁₃	#41	1	1	yes	characteristic
50	C₄⋊Dic₁₃	—	4	2	no	—
51	#6099	—	4	2	no	—
52	#6099	—	4	2	no	—
53	C₂×#1055	—	4	2	no	—
54	#1961	—	13	1	no	—
55	#1838	—	13	1	no	—
56	#1545	—	13	2	no	—
57	#1286	—	13	2	no	—
58	#1281	D₁₃	1	1	yes	characteristic, complemented
59	C₂³×C₁₃	#124	1	1	yes	characteristic
60	C₂²×D₁₃	—	4	2	no	—
61	C₂²×D₁₃	—	8	4	no	—
62	C₂²×D₁₃	—	8	1	no	—
63	C₂²×D₁₃	—	8	1	no	—
64	C₂×C₄×C₁₃	—	4	2	no	—
65	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
66	C₂×C₄×C₁₃	#123	1	2	yes	—
67	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
68	C₂×C₄×C₁₃	#126	1	2	yes	—
69	C₂×C₄×C₁₃	#135	1	2	yes	—
70	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
71	C₂×Dic₁₃	—	8	2	no	—
72	C₂×Dic₁₃	—	8	1	no	—
73	D₅₂	—	4	4	no	—
74	D₅₂	—	8	4	no	—
75	D₅₂	—	8	4	no	—
76	D₅₂	—	8	2	no	—
77	C₈×C₁₃	—	4	2	no	—
78	C₈×C₁₃	—	8	2	no	—
79	#1055	—	8	2	no	—
80	C₂×#127	—	13	1	no	—
81	#431	C₂×D₁₃	1	2	yes	—
82	#423	—	13	2	no	—
83	C₂×#102	—	26	2	no	—
84	C₂×#97	C₂×D₁₃	1	1	yes	characteristic
85	#382	—	13	1	no	—
86	#341	—	26	2	no	—
87	#331	—	26	4	no	—
88	C₂²×C₁₃	—	2	2	no	—
89	C₂²×C₁₃	#397	1	1	yes	characteristic
90	C₂²×C₁₃	#488	1	1	yes	characteristic
91	C₂²×C₁₃	#490	1	1	yes	characteristic
92	C₂×D₁₃	—	8	2	no	—
93	C₂×D₁₃	—	8	4	no	—
94	C₂×D₁₃	—	16	2	no	—
95	C₂×D₁₃	—	16	2	no	—
96	C₄×C₁₃	—	2	2	no	—
97	C₄×C₁₃	—	2	2	no	—
98	C₄×C₁₃	#463	1	2	yes	—
99	C₄×C₁₃	#482	1	2	yes	—
100	Dic₁₃	—	16	1	no	—
101	C₂²×D₄	—	26	1	no	—
102	C₂²×D₄	—	26	2	no	—
103	C₂×M₁₆	—	26	2	no	—
104	C₂²×C₈	D₅₂	1	2	yes	—
105	#127	—	13	2	no	—
106	#127	—	26	2	no	—
107	#127	—	26	1	no	—
108	C₂×C₄⋊C₄	—	26	1	no	—
109	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
110	C₄⋊C₈	—	2	2	no	—
111	C₄⋊C₈	—	26	2	no	—
112	C₄⋊C₈	—	26	2	no	—
113	#102	—	52	4	no	—
114	C₂²⋊C₈	—	2	2	no	—
115	#97	—	2	2	no	—
116	C₄×C₈	—	2	2	no	—
117	#95	—	26	2	no	—
118	C₂×C₁₃	—	2	1	no	—
119	C₂×C₁₃	—	2	1	no	—
120	C₂×C₁₃	#2073	1	1	yes	characteristic
121	C₂×C₁₃	#2081	1	2	yes	—
122	D₁₃	—	16	2	no	—
123	C₂⁴	—	52	1	no	—
124	C₂×D₄	—	26	4	no	—
125	C₂×D₄	—	26	2	no	—
126	C₂×D₄	—	52	1	no	—
127	C₂×D₄	—	52	2	no	—
128	C₂×D₄	—	52	2	no	—
129	C₂×D₄	—	52	2	no	—
130	C₂×D₄	—	104	4	no	—
131	C₂²×C₄	—	52	1	no	—
132	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
133	C₂²×C₄	#6127	1	2	yes	—
134	M₁₆	—	52	2	no	—
135	M₁₆	—	52	2	no	—
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	4	2	no	—
140	C₂×C₈	—	52	2	no	—
141	C₄⋊C₄	—	52	2	no	—
142	C₄²	D₄×D₁₃	1	1	yes	characteristic
143	C₄²	#6129	1	2	yes	—
144	C₄²	#6131	1	1	yes	characteristic
145	C₁₃	#10634	1	1	yes	characteristic, complemented, Sylow
146	C₂³	—	52	2	no	—
147	C₂³	—	104	4	no	—
148	C₂³	—	104	1	no	—
149	C₂³	—	104	1	no	—
150	C₂³	#88129	1	1	yes	characteristic
151	D₄	—	52	4	no	—
152	D₄	—	104	4	no	—
153	D₄	—	104	4	no	—
154	D₄	—	104	2	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	—	104	2	no	—
157	C₂×C₄	—	104	1	no	—
158	C₂×C₄	#88137	1	2	yes	—
159	C₂×C₄	#88140	1	1	yes	characteristic
160	C₂×C₄	#88149	1	1	yes	characteristic
161	C₂×C₄	#88152	1	2	yes	—
162	C₂×C₄	#88158	1	2	yes	—
163	C₂×C₄	#88162	1	2	yes	—
164	C₈	—	4	2	no	—
165	C₈	—	8	2	no	—
166	C₈	—	104	2	no	—
167	C₂²	—	2	2	no	—
168	C₂²	—	104	2	no	—
169	C₂²	—	104	4	no	—
170	C₂²	—	208	2	no	—
171	C₂²	—	208	2	no	—
172	C₂²	#11718335	1	1	yes	characteristic
173	C₂²	#11718358	1	1	yes	characteristic
174	C₂²	#11718374	1	1	yes	characteristic
175	C₄	—	2	2	no	—
176	C₄	—	2	2	no	—
177	C₄	—	208	1	no	—
178	C₄	#11718398	1	2	yes	—
179	C₄	#11718505	1	2	yes	—
180	C₂	—	2	1	no	—
181	C₂	—	2	1	no	—
182	C₂	—	208	2	no	—
183	C₂	#13116251	1	1	yes	characteristic
184	C₂	#13116295	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13115660

C₂

yes

maximal, characteristic

#13114977

C₂

yes

maximal, characteristic, complemented

#13113503

C₂

yes

maximal

#13112695

C₂

yes

maximal, complemented

C₁₃×#1281

C₂

yes

maximal, characteristic, complemented

C₂×#88128

C₂²

yes

characteristic

C₁₃×#431

C₂²

yes

—

C₂×C₁₃×#97

C₂²

yes

characteristic

C₂×#88061

—

#11718597

C₂²

yes

characteristic

#11718585

C₂²

yes

—

#11718144

—

#11718077

—

C₂²×D₅₂

—

C₂²×D₅₂

—

C₂²×C₈×C₁₃

D₄

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₃

—

—

—

—

D₄

yes

—

C₁₃×C₄⋊C₈

—

C₁₃×C₂²⋊C₈

—

C₁₃×#97

—

C₄×C₈×C₁₃

—

—

—

—

—

—

C₂³×D₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

#41

yes

—

C₂²×Dic₁₃

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄²×C₁₃

#41

yes

—

C₄²×C₁₃

#41

yes

characteristic

C₄⋊Dic₁₃

—

—

—

—

—

—

—

—

D₁₃

yes

characteristic, complemented

C₂³×C₁₃

#124

yes

characteristic

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#121

yes

characteristic

C₂×C₄×C₁₃

#123

yes

—

C₂×C₄×C₁₃

#123

yes

characteristic

C₂×C₄×C₁₃

#126

yes

—

C₂×C₄×C₁₃

#135

yes

—

C₂×C₄×C₁₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

D₅₂

—

D₅₂

—

D₅₂

—

D₅₂

—

C₈×C₁₃

—

C₈×C₁₃

—

#1055

—

C₂×#127

—

#431

C₂×D₁₃

yes

—

#423

—

C₂×#102

—

C₂×#97

C₂×D₁₃

yes

—

—

—

C₂²×C₁₃

—

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#463

yes

—

C₄×C₁₃

#482

yes

—

100

Dic₁₃

—

101

C₂²×D₄

—

102

C₂²×D₄

—

103

C₂×M₁₆

—

104

C₂²×C₈

D₅₂

yes

—

105

#127

—

106

#127

—

107

#127

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄²

C₂²×D₁₃

yes

characteristic

110

C₄⋊C₈

—

111

C₄⋊C₈

—

112

C₄⋊C₈

—

113

#102

—

114

C₂²⋊C₈

—

115

#97

—

116

C₄×C₈

—

117

#95

—

118

C₂×C₁₃

—

119

C₂×C₁₃

—

120

C₂×C₁₃

#2073

yes

characteristic

121

C₂×C₁₃

#2081

yes

—

122

D₁₃

—

123

C₂⁴

—

124

C₂×D₄

—

125

C₂×D₄

—

126

C₂×D₄

—

127

C₂×D₄

—

128

C₂×D₄

—

129

C₂×D₄

—

130

C₂×D₄

—

104

—

131

C₂²×C₄

—

132

C₂²×C₄

C₂×D₅₂

yes

characteristic

133

C₂²×C₄

#6127

yes

—

134

M₁₆

—

135

M₁₆

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₄⋊C₄

—

142

C₄²

D₄×D₁₃

yes

characteristic

143

C₄²

#6129

yes

—

144

C₄²

#6131

yes

characteristic

145

C₁₃

#10634

yes

characteristic, complemented, Sylow

146

C₂³

—

147

C₂³

—

104

—

148

C₂³

—

104

—

149

C₂³

—

104

—

150

C₂³

#88129

yes

characteristic

151

D₄

—

152

D₄

—

104

—

153

D₄

—

104

—

154

D₄

—

104

—

155

C₂×C₄

—

156

C₂×C₄

—

104

—

157

C₂×C₄

—

104

—

158

C₂×C₄

#88137

yes

—

159

C₂×C₄

#88140

yes

characteristic

160

C₂×C₄

#88149

yes

characteristic

161

C₂×C₄

#88152

yes

—

162

C₂×C₄

#88158

yes

—

163

C₂×C₄

#88162

yes

—

164

C₈

—

165

C₈

—

166

C₈

—

104

—

167

C₂²

—

168

C₂²

—

104

—

169

C₂²

—

104

—

170

C₂²

—

208

—

171

C₂²

—

208

—

172

C₂²

#11718335

yes

characteristic

173

C₂²

#11718358

yes

characteristic

174

C₂²

#11718374

yes

characteristic

175

C₄

—

176

C₄

—

177

C₄

—

208

—

178

C₄

#11718398

yes

—

179

C₄

#11718505

yes

—

180

C₂

—

181

C₂

—

182

C₂

—

208

—

183

C₂

#13116251

yes

characteristic

184

C₂

#13116295

yes

—

	subgroup	G/H
center	C₂²	#11718335
commutator, cosocle	C₂×C₄²×C₁₃	C₂³
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1281	C₂
socle	C₂²×C₁₃	#397
prime radical	C₂×#88128	C₂²
hypercenter	#1281	D₁₃
hypocenter	C₁₃	#10634
LCS	G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1281

subgroup

G/H

center

C₂²

#11718335

commutator, cosocle

C₂×C₄²×C₁₃

C₂³

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1281

C₂

C₂²×C₁₃

C₂²

D₁₃

C₁₃

G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1281

N	H ≅ G/N	type
2. #13114977	C₂	semidirect
4. #13112695	C₂	semidirect
5. C₁₃×#1281	C₂	semidirect
58. #1281	D₁₃	semidirect
145. C₁₃	#10634	semidirect

H ≅ G/N

type

2. #13114977

C₂

semidirect

4. #13112695

C₂

semidirect

5. C₁₃×#1281

C₂

semidirect

58. #1281

D₁₃

semidirect

145. C₁₃

#10634

semidirect

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

subgroup	index
#13115660	2
#13114977	2
#13113503	2
#13112695	2
C₁₃×#1281	2
#10634	13

subgroup

index

C₁₃×#1281

Subgroups — #15767726

Conjugacy classes of subgroups

Complements

Maximal subgroups