Group #16782363 — subgroups

Subgroups — #16782363

Order 3328.

378 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 144 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	#13130768	C₂	1	1	yes	maximal, characteristic
2	#13130751	C₂	1	1	yes	maximal, characteristic
3	#13129666	C₂	1	4	yes	maximal
4	#97×Dic₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#1057	—	2	2	no	—
6	C₂×C₄⋊#1057	C₂²	1	2	yes	—
7	C₂×C₄⋊#1057	C₂²	1	2	yes	—
8	C₂×C₁₃×#97	C₄	1	1	yes	characteristic
9	C₂×#88055	C₄	1	2	yes	—
10	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
11	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
12	C₂×#88043	C₄	1	1	yes	characteristic
13	#11718290	—	2	2	no	—
14	C₂²×#1057	—	2	2	no	—
15	C₂²×#1057	—	2	2	no	—
16	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
17	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₁₃	D₄	1	2	yes	—
20	C₂×C₄×Dic₁₃	Q₈	1	2	yes	—
21	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
22	C₂²×#1055	C₂×C₄	1	2	yes	—
23	C₄⋊#1057	—	2	4	no	—
24	C₄⋊#1057	—	2	4	no	—
25	C₄⋊#1057	—	4	4	no	—
26	C₁₃×#97	—	2	2	no	—
27	#88055	—	2	4	no	—
28	C₈×Dic₁₃	—	2	2	no	—
29	C₈×Dic₁₃	—	2	2	no	—
30	#88043	C₈	1	4	yes	complemented
31	#8786	—	13	1	no	maximal, Sylow
32	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
33	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
34	C₂²×Dic₁₃	C₄⋊C₄	1	2	yes	—
35	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
36	C₂²×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
37	C₂×#1057	—	4	2	no	—
38	C₂×#1057	—	4	2	no	—
39	C₂×#1057	—	4	4	no	—
40	C₂×#1057	—	4	2	no	—
41	C₂×#1057	—	4	2	no	—
42	C₂×C₈×C₁₃	—	2	2	no	—
43	C₂×C₈×C₁₃	—	2	2	no	—
44	C₂×C₈×C₁₃	M₁₆	1	4	yes	—
45	C₄²×C₁₃	C₂×C₈	1	2	yes	—
46	C₄²×C₁₃	M₁₆	1	2	yes	—
47	C₄×Dic₁₃	—	2	2	no	—
48	C₄×Dic₁₃	—	2	2	no	—
49	C₄×Dic₁₃	—	2	2	no	—
50	C₄×Dic₁₃	—	4	2	no	—
51	C₄×Dic₁₃	#41	1	2	yes	—
52	C₄×Dic₁₃	#41	1	2	yes	—
53	C₂×#1055	—	2	2	no	—
54	C₂×#1055	—	2	2	no	—
55	C₂×#1055	C₂×C₈	1	4	yes	—
56	#1842	—	13	1	no	—
57	#1831	—	13	1	no	—
58	C₄×#97	—	13	1	no	—
59	#1277	—	13	4	no	—
60	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₃	—	2	2	no	—
62	C₂×C₄×C₁₃	—	2	2	no	—
63	C₂×C₄×C₁₃	C₄⋊C₈	1	4	yes	—
64	C₂×C₄×C₁₃	#117	1	2	yes	—
65	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
66	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
67	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
68	C₂×Dic₁₃	—	2	2	no	—
69	C₂×Dic₁₃	—	2	2	no	—
70	C₂×Dic₁₃	—	2	2	no	—
71	C₂×Dic₁₃	—	2	2	no	—
72	C₂×Dic₁₃	#124	1	1	yes	characteristic
73	C₂×Dic₁₃	#124	1	1	yes	characteristic
74	C₂×Dic₁₃	#125	1	2	yes	—
75	C₂×Dic₁₃	AGL(1,ℤ/8)	1	2	yes	—
76	C₂×Dic₁₃	#137	1	2	yes	—
77	#1057	—	8	2	no	—
78	#1057	—	8	2	no	—
79	C₈×C₁₃	—	2	4	no	—
80	#1055	—	2	4	no	—
81	C₂×C₄⋊C₈	—	13	2	no	—
82	C₂×C₄⋊C₈	—	13	2	no	—
83	C₂×C₄⋊C₈	—	26	2	no	—
84	C₂×#97	—	13	1	no	—
85	C₂×#97	—	13	2	no	—
86	C₂×#97	#261	1	1	yes	characteristic
87	C₂×C₄×C₈	—	13	2	no	—
88	C₄³	—	13	1	no	—
89	#336	—	26	2	no	—
90	C₂²×C₁₃	#382	1	1	yes	characteristic
91	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
92	C₂²×C₁₃	#423	1	1	yes	characteristic
93	C₂²×C₁₃	#428	1	2	yes	—
94	C₂²×C₁₃	#430	1	2	yes	—
95	C₄×C₁₃	—	2	4	no	—
96	C₄×C₁₃	#431	1	2	yes	—
97	C₄×C₁₃	#432	1	2	yes	—
98	Dic₁₃	—	2	2	no	—
99	Dic₁₃	—	4	2	no	—
100	Dic₁₃	#489	1	2	yes	—
101	Dic₁₃	#490	1	1	yes	characteristic
102	Dic₁₃	#491	1	1	yes	characteristic
103	C₂²×C₈	—	13	2	no	—
104	C₂²×C₈	—	26	2	no	—
105	C₂²×C₈	—	26	2	no	—
106	C₂²×C₈	C₂×#261	1	2	yes	—
107	C₂×C₄²	—	13	1	no	—
108	C₂×C₄²	—	13	2	no	—
109	C₂×C₄²	—	13	1	no	—
110	C₂×C₄²	—	13	2	no	—
111	C₂×C₄²	C₂×#261	1	1	yes	characteristic
112	C₄⋊C₈	—	26	4	no	—
113	C₄⋊C₈	—	26	4	no	—
114	C₄⋊C₈	—	52	4	no	—
115	#97	—	2	2	no	—
116	#97	—	13	4	no	—
117	#97	—	26	4	no	—
118	C₄×C₈	—	26	2	no	—
119	C₄×C₈	—	26	2	no	—
120	C₂×C₁₃	#1565	1	2	yes	—
121	C₂×C₁₃	#1569	1	2	yes	—
122	C₂×C₁₃	#1831	1	1	yes	characteristic
123	C₂×C₁₃	#1836	1	1	yes	characteristic
124	C₂×C₁₃	#1838	1	1	yes	characteristic
125	C₂²×C₄	—	13	1	no	—
126	C₂²×C₄	—	13	1	no	—
127	C₂²×C₄	—	13	2	no	—
128	C₂²×C₄	C₄⋊#261	1	2	yes	—
129	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	13	4	no	—
133	C₂×C₈	—	26	2	no	—
134	C₂×C₈	—	26	2	no	—
135	C₂×C₈	—	52	2	no	—
136	C₂×C₈	—	52	2	no	—
137	C₂×C₈	—	52	4	no	—
138	C₂×C₈	—	52	2	no	—
139	C₂×C₈	—	52	2	no	—
140	C₂×C₈	#6118	1	4	yes	—
141	C₄²	—	13	2	no	—
142	C₄²	—	13	2	no	—
143	C₄²	—	26	2	no	—
144	C₄²	—	26	2	no	—
145	C₄²	—	26	2	no	—
146	C₄²	—	52	2	no	—
147	C₄²	D₁₃⋊C₈	1	2	yes	—
148	C₄²	#6122	1	2	yes	—
149	C₁₃	#8786	1	1	yes	characteristic, complemented, Sylow
150	C₂³	C₂×C₄⋊#261	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	13	1	no	—
154	C₂×C₄	—	13	1	no	—
155	C₂×C₄	—	13	2	no	—
156	C₂×C₄	—	13	2	no	—
157	C₂×C₄	—	13	2	no	—
158	C₂×C₄	—	26	2	no	—
159	C₂×C₄	—	26	2	no	—
160	C₂×C₄	—	26	2	no	—
161	C₂×C₄	—	26	2	no	—
162	C₂×C₄	#88112	1	4	yes	—
163	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
164	C₂×C₄	C₂×#6118	1	2	yes	—
165	C₂×C₄	#88237	1	2	yes	—
166	C₂×C₄	C₂×#6122	1	1	yes	characteristic
167	C₈	—	2	4	no	—
168	C₈	—	26	4	no	—
169	C₈	—	104	2	no	—
170	C₈	—	104	2	no	—
171	C₂²	#11719052	1	1	yes	characteristic
172	C₂²	#11719060	1	1	yes	characteristic
173	C₂²	#11719098	1	2	yes	—
174	C₂²	#11719100	1	2	yes	—
175	C₂²	C₂×#88112	1	1	yes	characteristic
176	C₄	—	2	4	no	—
177	C₄	—	13	2	no	—
178	C₄	—	13	1	no	—
179	C₄	—	13	1	no	—
180	C₄	—	26	2	no	—
181	C₄	—	52	2	no	—
182	C₄	#11719051	1	2	yes	—
183	C₄	#11719130	1	2	yes	—
184	C₂	#13130389	1	2	yes	—
185	C₂	#13130401	1	2	yes	—
186	C₂	#13130753	1	1	yes	characteristic
187	C₂	#13130759	1	1	yes	characteristic
188	C₂	#13130762	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13130768

C₂

yes

maximal, characteristic

#13130751

C₂

yes

maximal, characteristic

#13129666

C₂

yes

maximal

#97×Dic₁₃

C₂

yes

maximal, characteristic

C₂×C₄⋊#1057

—

C₂×C₄⋊#1057

C₂²

yes

—

C₂×C₄⋊#1057

C₂²

yes

—

C₂×C₁₃×#97

C₄

yes

characteristic

C₂×#88055

C₄

yes

—

C₂×C₈×Dic₁₃

C₂²

yes

—

C₄²×Dic₁₃

C₂²

yes

characteristic

C₂×#88043

C₄

yes

characteristic

#11718290

—

C₂²×#1057

—

C₂²×#1057

—

C₂²×C₈×C₁₃

C₂×C₄

yes

—

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

C₂×C₄×Dic₁₃

C₂×C₄

yes

characteristic

C₂×C₄×Dic₁₃

D₄

yes

—

C₂×C₄×Dic₁₃

Q₈

yes

—

C₂×C₄×Dic₁₃

C₂³

yes

characteristic

C₂²×#1055

C₂×C₄

yes

—

C₄⋊#1057

—

C₄⋊#1057

—

C₄⋊#1057

—

C₁₃×#97

—

#88055

—

C₈×Dic₁₃

—

C₈×Dic₁₃

—

#88043

C₈

yes

complemented

#8786

—

maximal, Sylow

C₂²×C₄×C₁₃

C₄⋊C₄

yes

—

C₂²×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂²×Dic₁₃

C₄⋊C₄

yes

—

C₂²×Dic₁₃

C₂×D₄

yes

characteristic

C₂²×Dic₁₃

C₂×Q₈

yes

—

—

—

—

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

M₁₆

yes

—

C₄²×C₁₃

C₂×C₈

yes

—

C₄²×C₁₃

M₁₆

yes

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

#41

yes

—

C₄×Dic₁₃

#41

yes

—

C₂×#1055

—

C₂×#1055

—

C₂×#1055

C₂×C₈

yes

—

#1842

—

#1831

—

C₄×#97

—

#1277

—

C₂³×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₄⋊C₈

yes

—

C₂×C₄×C₁₃

#117

yes

—

C₂×C₄×C₁₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₃

C₂×M₁₆

yes

—

C₂×C₄×C₁₃

C₂×M₁₆

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

yes

C₂×Dic₁₃

yes

C₂×Dic₁₃

yes

—

C₂×Dic₁₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₁₃

#137

yes

—

#1057

—

#1057

—

C₈×C₁₃

—

#1055

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×#97

—

C₂×#97

—

C₂×#97

#261

yes

characteristic

C₂×C₄×C₈

—

C₄³

—

#336

—

C₂²×C₁₃

#382

yes

characteristic

C₂²×C₁₃

C₂×C₄⋊C₈

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

—

C₂²×C₁₃

#430

yes

—

C₄×C₁₃

—

C₄×C₁₃

#431

yes

—

C₄×C₁₃

#432

yes

—

Dic₁₃

—

Dic₁₃

—

100

Dic₁₃

#489

yes

—

101

Dic₁₃

#490

yes

characteristic

102

Dic₁₃

#491

yes

characteristic

103

C₂²×C₈

—

104

C₂²×C₈

—

105

C₂²×C₈

—

106

C₂²×C₈

C₂×#261

yes

—

107

C₂×C₄²

—

108

C₂×C₄²

—

109

C₂×C₄²

—

110

C₂×C₄²

—

111

C₂×C₄²

C₂×#261

yes

characteristic

112

C₄⋊C₈

—

113

C₄⋊C₈

—

114

C₄⋊C₈

—

115

#97

—

116

#97

—

117

#97

—

118

C₄×C₈

—

119

C₄×C₈

—

120

C₂×C₁₃

#1565

yes

—

121

C₂×C₁₃

#1569

yes

—

122

C₂×C₁₃

#1831

yes

characteristic

123

C₂×C₁₃

#1836

yes

characteristic

124

C₂×C₁₃

#1838

yes

characteristic

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

—

128

C₂²×C₄

C₄⋊#261

yes

—

129

C₂²×C₄

C₂²×#261

yes

characteristic

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

#6118

yes

—

141

C₄²

—

142

C₄²

—

143

C₄²

—

144

C₄²

—

145

C₄²

—

146

C₄²

—

147

C₄²

D₁₃⋊C₈

yes

—

148

C₄²

#6122

yes

—

149

C₁₃

#8786

yes

characteristic, complemented, Sylow

150

C₂³

C₂×C₄⋊#261

yes

characteristic

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

#88112

yes

—

163

C₂×C₄

C₂×D₁₃⋊C₈

yes

characteristic

164

C₂×C₄

C₂×#6118

yes

—

165

C₂×C₄

#88237

yes

—

166

C₂×C₄

C₂×#6122

yes

characteristic

167

C₈

—

168

C₈

—

169

C₈

—

104

—

170

C₈

—

104

—

171

C₂²

#11719052

yes

characteristic

172

C₂²

#11719060

yes

characteristic

173

C₂²

#11719098

yes

—

174

C₂²

#11719100

yes

—

175

C₂²

C₂×#88112

yes

characteristic

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

—

182

C₄

#11719051

yes

—

183

C₄

#11719130

yes

—

184

C₂

#13130389

yes

—

185

C₂

#13130401

yes

—

186

C₂

#13130753

yes

characteristic

187

C₂

#13130759

yes

characteristic

188

C₂

#13130762

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#261
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₂×C₁₃×#97	C₄
socle, prime radical	C₂³×C₁₃	C₂×C₄⋊C₄
cosocle	C₂×C₄×Dic₁₃	C₂³
hypercenter	C₂×#97	#261
hypocenter	C₁₃	#8786
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×C₄⋊#261

commutator

C₂×C₄×C₁₃

C₂²×C₈

Frattini

C₂²×C₄

C₂²×#261

Fitting

C₂×C₁₃×#97

C₄

socle, prime radical

C₂³×C₁₃

C₂×C₄⋊C₄

cosocle

C₂×C₄×Dic₁₃

C₂³

C₂×#97

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
30. #88043	C₈	semidirect
149. C₁₃	#8786	semidirect

H ≅ G/N

type

30. #88043

C₈

semidirect

149. C₁₃

#8786

semidirect

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

subgroup	index
#13130768	2
#13130751	2
#13129666	2
#97×Dic₁₃	2
#8786	13

subgroup

index

#97×Dic₁₃

Subgroups — #16782363

Conjugacy classes of subgroups

Complements

Maximal subgroups