Group #15767694 — subgroups

Subgroups — #15767694

Order 3328.

292 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 93 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	#13115817	C₂	1	1	yes	maximal, characteristic
2	#13114990	C₂	1	1	yes	maximal, characteristic
3	#13113491	C₂	1	2	yes	maximal
4	#13112732	C₂	1	2	yes	maximal
5	C₁₃×#1274	C₂	1	1	yes	maximal, characteristic
6	#11719175	C₂²	1	1	yes	characteristic
7	C₁₃×#431	C₂²	1	2	yes	—
8	C₁₃×#376	C₂²	1	1	yes	characteristic
9	#11718585	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊#1055	—	2	1	no	—
11	C₂×C₄⋊#1055	—	2	1	no	—
12	C₂×C₄⋊#1055	C₂²	1	2	yes	—
13	#11718115	—	2	1	no	—
14	#11718115	—	2	1	no	—
15	#11718074	—	2	2	no	—
16	#11718073	—	2	2	no	—
17	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
18	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
19	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
20	#88180	—	2	1	no	—
21	#88178	—	2	2	no	—
22	C₂×#6099	—	2	1	no	—
23	C₂²×#1055	—	2	2	no	—
24	#88124	—	2	1	no	—
25	#88123	D₄	1	2	yes	—
26	C₄×Dic₂₆	—	2	2	no	—
27	C₁₃×C₄⋊C₈	—	2	2	no	—
28	C₁₃×C₂²⋊C₈	—	2	2	no	—
29	C₄×C₈×C₁₃	—	2	2	no	—
30	C₁₃×#95	—	2	2	no	—
31	C₄⋊#1055	—	2	2	no	—
32	C₄⋊#1055	—	2	2	no	—
33	C₄⋊#1055	—	2	2	no	—
34	C₄⋊#1055	—	4	2	no	—
35	C₄⋊#1055	—	4	2	no	—
36	#10628	—	13	1	no	maximal, Sylow
37	C₂²×C₄×C₁₃	—	2	1	no	—
38	C₂²×C₄×C₁₃	—	2	1	no	—
39	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	#41	1	2	yes	—
41	C₂×Dic₂₆	—	2	2	no	—
42	C₂×C₈×C₁₃	—	4	2	no	—
43	C₄²×C₁₃	—	2	1	no	—
44	C₄²×C₁₃	—	2	1	no	—
45	C₄²×C₁₃	SD₁₆	1	2	yes	—
46	C₄²×C₁₃	Q₁₆	1	2	yes	—
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	C₂²⋊Dic₁₃	—	4	1	no	—
51	C₄⋊Dic₁₃	—	2	2	no	—
52	C₄⋊Dic₁₃	—	2	2	no	—
53	#6101	—	4	2	no	—
54	C₄×Dic₁₃	—	4	1	no	—
55	#6099	—	4	1	no	—
56	#6099	—	4	1	no	—
57	C₂×#1055	—	4	2	no	—
58	C₂×#1055	—	4	1	no	—
59	C₂×#1055	—	4	2	no	—
60	C₂×#1055	—	4	2	no	—
61	#1995	—	13	1	no	—
62	#1843	—	13	1	no	—
63	#1542	—	13	2	no	—
64	#1301	—	13	2	no	—
65	#1274	D₁₃	1	1	yes	characteristic
66	C₂³×C₁₃	#123	1	1	yes	characteristic
67	C₂×C₄×C₁₃	—	2	2	no	—
68	C₂×C₄×C₁₃	—	2	2	no	—
69	C₂×C₄×C₁₃	—	4	2	no	—
70	C₂×C₄×C₁₃	—	4	2	no	—
71	C₂×C₄×C₁₃	—	4	2	no	—
72	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
73	C₂×C₄×C₁₃	#123	1	2	yes	—
74	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
75	C₂×C₄×C₁₃	#126	1	2	yes	—
76	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
78	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
80	C₂×Dic₁₃	—	4	2	no	—
81	Dic₂₆	—	8	2	no	—
82	C₈×C₁₃	—	8	2	no	—
83	#1055	—	8	2	no	—
84	#1055	—	8	1	no	—
85	#533	—	13	1	no	—
86	#431	C₂×D₁₃	1	2	yes	—
87	#423	—	13	1	no	—
88	C₂×C₄⋊C₈	—	13	2	no	—
89	C₂×C₄⋊C₈	—	26	1	no	—
90	C₂×C₄⋊C₈	—	26	1	no	—
91	#376	C₂×D₁₃	1	1	yes	characteristic
92	#339	—	26	1	no	—
93	#339	—	26	1	no	—
94	#333	—	26	2	no	—
95	#332	—	26	2	no	—
96	C₂²×C₁₃	—	2	2	no	—
97	C₂²×C₁₃	#397	1	1	yes	characteristic
98	C₂²×C₁₃	#483	1	1	yes	characteristic
99	C₂²×C₁₃	#484	1	1	yes	characteristic
100	C₄×C₁₃	—	2	2	no	—
101	C₄×C₁₃	—	2	2	no	—
102	C₄×C₁₃	—	4	2	no	—
103	C₄×C₁₃	—	4	2	no	—
104	C₄×C₁₃	#460	1	1	yes	characteristic
105	C₄×C₁₃	#462	1	1	yes	characteristic
106	C₄×C₁₃	#487	1	1	yes	characteristic
107	C₄×C₁₃	#489	1	1	yes	characteristic
108	Dic₁₃	—	8	2	no	—
109	C₂×M₁₆	—	26	1	no	—
110	C₂²×C₈	—	26	2	no	—
111	C₄⋊Q₈	—	13	2	no	—
112	#125	—	26	1	no	—
113	C₂²⋊Q₈	—	26	2	no	—
114	C₄×Q₈	—	26	2	no	—
115	#117	—	26	1	no	—
116	C₂×C₄²	D₅₂	1	1	yes	characteristic
117	C₂×C₄²	D₅₂	1	1	yes	characteristic
118	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	26	2	no	—
121	C₄⋊C₈	—	26	2	no	—
122	C₄⋊C₈	—	26	2	no	—
123	C₄⋊C₈	—	52	2	no	—
124	C₄⋊C₈	—	52	2	no	—
125	C₂²⋊C₈	—	2	2	no	—
126	C₄×C₈	—	2	2	no	—
127	#95	—	2	2	no	—
128	C₂×C₁₃	—	2	1	no	—
129	C₂×C₁₃	—	2	1	no	—
130	C₂×C₁₃	#2072	1	1	yes	characteristic
131	C₂×C₁₃	#2080	1	2	yes	—
132	C₂×Q₈	—	26	2	no	—
133	C₂²×C₄	—	2	1	no	—
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
136	C₂²×C₄	#6129	1	2	yes	—
137	M₁₆	—	52	1	no	—
138	M₁₆	—	52	1	no	—
139	C₂×C₈	—	4	2	no	—
140	C₂×C₈	—	52	1	no	—
141	C₂×C₈	—	52	2	no	—
142	C₂×C₈	—	52	2	no	—
143	C₂×C₈	—	52	2	no	—
144	C₄⋊C₄	—	26	2	no	—
145	C₄⋊C₄	—	26	2	no	—
146	C₄⋊C₄	—	52	2	no	—
147	C₂²⋊C₄	—	52	1	no	—
148	C₄²	—	2	1	no	—
149	C₄²	—	2	1	no	—
150	C₄²	—	52	1	no	—
151	C₄²	#6095	1	2	yes	—
152	C₄²	Dic₅₂	1	2	yes	—
153	C₄²	#6127	1	2	yes	—
154	C₄²	D₄×D₁₃	1	1	yes	characteristic
155	C₄²	#6131	1	1	yes	characteristic
156	C₁₃	#10628	1	1	yes	characteristic, complemented, Sylow
157	C₂³	#88140	1	1	yes	characteristic
158	Q₈	—	104	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	52	2	no	—
165	C₂×C₄	#88129	1	1	yes	characteristic
166	C₂×C₄	#88137	1	2	yes	—
167	C₂×C₄	#88149	1	1	yes	characteristic
168	C₂×C₄	#88152	1	2	yes	—
169	C₂×C₄	C₂×#6095	1	1	yes	characteristic
170	C₂×C₄	C₂×Dic₅₂	1	1	yes	characteristic
171	C₂×C₄	#88162	1	1	yes	characteristic
172	C₂×C₄	#88163	1	1	yes	characteristic
173	C₈	—	8	2	no	—
174	C₈	—	104	2	no	—
175	C₈	—	104	1	no	—
176	C₂²	—	2	2	no	—
177	C₂²	#11718335	1	1	yes	characteristic
178	C₂²	#11718382	1	1	yes	characteristic
179	C₂²	#11718383	1	1	yes	characteristic
180	C₄	—	2	2	no	—
181	C₄	—	2	2	no	—
182	C₄	—	4	2	no	—
183	C₄	—	4	2	no	—
184	C₄	—	104	2	no	—
185	C₄	#11718342	1	1	yes	characteristic
186	C₄	#11718375	1	1	yes	characteristic
187	C₄	#11718509	1	1	yes	characteristic
188	C₄	#11718510	1	1	yes	characteristic
189	C₂	—	2	1	no	—
190	C₂	—	2	1	no	—
191	C₂	#13116246	1	1	yes	characteristic
192	C₂	#13116288	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13115817

C₂

yes

maximal, characteristic

#13114990

C₂

yes

maximal, characteristic

#13113491

C₂

yes

maximal

#13112732

C₂

yes

maximal

C₁₃×#1274

C₂

yes

maximal, characteristic

#11719175

C₂²

yes

characteristic

C₁₃×#431

C₂²

yes

—

C₁₃×#376

C₂²

yes

characteristic

#11718585

C₂²

yes

characteristic

C₂×C₄⋊#1055

—

C₂×C₄⋊#1055

—

C₂×C₄⋊#1055

C₂²

yes

—

—

—

—

—

C₂×C₄²×C₁₃

D₄

yes

characteristic

C₂×C₄²×C₁₃

D₄

yes

characteristic

C₂×C₄²×C₁₃

C₂³

yes

—

—

—

C₂²×#1055

—

#88124

—

#88123

D₄

yes

—

C₄×Dic₂₆

—

C₁₃×C₄⋊C₈

—

C₁₃×C₂²⋊C₈

—

C₄×C₈×C₁₃

—

C₁₃×#95

—

—

—

—

—

—

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

#41

yes

—

C₂×Dic₂₆

—

C₂×C₈×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

SD₁₆

yes

—

C₄²×C₁₃

Q₁₆

yes

—

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄²×C₁₃

#41

yes

—

C₄²×C₁₃

#41

yes

characteristic

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#6101

—

C₄×Dic₁₃

—

—

—

—

—

—

—

—

—

—

—

D₁₃

yes

characteristic

C₂³×C₁₃

#123

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#121

yes

characteristic

C₂×C₄×C₁₃

#123

yes

—

C₂×C₄×C₁₃

#124

yes

characteristic

C₂×C₄×C₁₃

#126

yes

—

C₂×C₄×C₁₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₁₃

C₂×Q₁₆

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×Dic₁₃

—

Dic₂₆

—

C₈×C₁₃

—

—

—

—

C₂×D₁₃

yes

—

#423

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

#376

C₂×D₁₃

yes

—

—

—

—

C₂²×C₁₃

—

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

100

C₄×C₁₃

—

101

C₄×C₁₃

—

102

C₄×C₁₃

—

103

C₄×C₁₃

—

104

C₄×C₁₃

#460

yes

characteristic

105

C₄×C₁₃

#462

yes

characteristic

106

C₄×C₁₃

#487

yes

characteristic

107

C₄×C₁₃

#489

yes

characteristic

108

Dic₁₃

—

109

C₂×M₁₆

—

110

C₂²×C₈

—

111

C₄⋊Q₈

—

112

#125

—

113

C₂²⋊Q₈

—

114

C₄×Q₈

—

115

#117

—

116

C₂×C₄²

D₅₂

yes

characteristic

117

C₂×C₄²

D₅₂

yes

characteristic

118

C₂×C₄²

C₂²×D₁₃

yes

characteristic

119

C₄⋊C₈

—

120

C₄⋊C₈

—

121

C₄⋊C₈

—

122

C₄⋊C₈

—

123

C₄⋊C₈

—

124

C₄⋊C₈

—

125

C₂²⋊C₈

—

126

C₄×C₈

—

127

#95

—

128

C₂×C₁₃

—

129

C₂×C₁₃

—

130

C₂×C₁₃

#2072

yes

characteristic

131

C₂×C₁₃

#2080

yes

—

132

C₂×Q₈

—

133

C₂²×C₄

—

134

C₂²×C₄

—

135

C₂²×C₄

C₂×D₅₂

yes

characteristic

136

C₂²×C₄

#6129

yes

—

137

M₁₆

—

138

M₁₆

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₂²⋊C₄

—

148

C₄²

—

149

C₄²

—

150

C₄²

—

151

C₄²

#6095

yes

—

152

C₄²

Dic₅₂

yes

—

153

C₄²

#6127

yes

—

154

C₄²

D₄×D₁₃

yes

characteristic

155

C₄²

#6131

yes

characteristic

156

C₁₃

#10628

yes

characteristic, complemented, Sylow

157

C₂³

#88140

yes

characteristic

158

Q₈

—

104

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

#88129

yes

characteristic

166

C₂×C₄

#88137

yes

—

167

C₂×C₄

#88149

yes

characteristic

168

C₂×C₄

#88152

yes

—

169

C₂×C₄

C₂×#6095

yes

characteristic

170

C₂×C₄

C₂×Dic₅₂

yes

characteristic

171

C₂×C₄

#88162

yes

characteristic

172

C₂×C₄

#88163

yes

characteristic

173

C₈

—

174

C₈

—

104

—

175

C₈

—

104

—

176

C₂²

—

177

C₂²

#11718335

yes

characteristic

178

C₂²

#11718382

yes

characteristic

179

C₂²

#11718383

yes

characteristic

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

104

—

185

C₄

#11718342

yes

characteristic

186

C₄

#11718375

yes

characteristic

187

C₄

#11718509

yes

characteristic

188

C₄

#11718510

yes

characteristic

189

C₂

—

190

C₂

—

191

C₂

#13116246

yes

characteristic

192

C₂

#13116288

yes

—

	subgroup	G/H
center	C₂×C₄	#88129
commutator, cosocle	C₂×C₄²×C₁₃	C₂³
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1274	C₂
socle	C₂²×C₁₃	#397
prime radical	C₂³×C₁₃	#123
hypercenter	#1274	D₁₃
hypocenter	C₁₃	#10628
LCS	G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1274

subgroup

G/H

center

C₂×C₄

#88129

commutator, cosocle

C₂×C₄²×C₁₃

C₂³

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1274

C₂

C₂²×C₁₃

C₂³×C₁₃

D₁₃

C₁₃

G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1274

N	H ≅ G/N	type
156. C₁₃	#10628	semidirect

H ≅ G/N

type

156. C₁₃

#10628

semidirect

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

subgroup	index
#13115817	2
#13114990	2
#13113491	2
#13112732	2
C₁₃×#1274	2
#10628	13

subgroup

index

C₁₃×#1274

Subgroups — #15767694

Conjugacy classes of subgroups

Complements

Maximal subgroups