Group #16782367 — subgroups

Subgroups — #16782367

Order 3328.

342 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 116 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	#13130769	C₂	1	1	yes	maximal, characteristic
2	#13130751	C₂	1	1	yes	maximal, characteristic
3	#13129666	C₂	1	4	yes	maximal
4	#13114580	C₂	1	1	yes	maximal, characteristic
5	C₂×#88112	—	2	2	no	—
6	C₂×C₄⋊#1057	C₂²	1	2	yes	—
7	C₂×C₄⋊#1057	C₂²	1	2	yes	—
8	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
9	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
10	#11718290	—	2	2	no	—
11	C₁₃×#336	C₄	1	1	yes	characteristic
12	#11718142	C₄	1	2	yes	—
13	#11718116	C₄	1	1	yes	characteristic
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	#88112	—	4	4	no	—
24	C₄⋊#1057	—	2	4	no	—
25	C₄⋊#1057	—	2	4	no	—
26	C₈×Dic₁₃	—	2	2	no	—
27	C₈×Dic₁₃	—	2	2	no	—
28	#8789	—	13	1	no	maximal, Sylow
29	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
30	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
31	C₂²×Dic₁₃	C₄⋊C₄	1	2	yes	—
32	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
33	C₂²×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
34	C₂×#1057	—	4	2	no	—
35	C₂×#1057	—	4	2	no	—
36	C₂×#1057	—	4	4	no	—
37	C₂×#1057	—	4	2	no	—
38	C₂×#1057	—	4	2	no	—
39	C₂×C₈×C₁₃	—	2	2	no	—
40	C₂×C₈×C₁₃	—	2	2	no	—
41	C₂×C₈×C₁₃	—	2	2	no	—
42	C₄²×C₁₃	—	2	2	no	—
43	C₄×Dic₁₃	—	2	1	no	—
44	C₄×Dic₁₃	—	2	1	no	—
45	C₄×Dic₁₃	—	2	4	no	—
46	C₄×Dic₁₃	—	2	2	no	—
47	C₄×Dic₁₃	—	2	2	no	—
48	C₄×Dic₁₃	#41	1	2	yes	—
49	C₄×Dic₁₃	#41	1	2	yes	—
50	C₂×#1055	—	2	2	no	—
51	C₂×#1055	—	2	2	no	—
52	C₂×#1055	—	2	2	no	—
53	#1842	—	13	1	no	—
54	#1831	—	13	1	no	—
55	#1744	—	13	1	no	—
56	#1277	—	13	4	no	—
57	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₃	—	2	2	no	—
59	C₂×C₄×C₁₃	—	2	2	no	—
60	C₂×C₄×C₁₃	—	2	2	no	—
61	C₂×C₄×C₁₃	#117	1	2	yes	—
62	C₂×C₄×C₁₃	#131	1	2	yes	—
63	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
64	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
65	C₂×Dic₁₃	—	2	2	no	—
66	C₂×Dic₁₃	—	2	2	no	—
67	C₂×Dic₁₃	—	2	2	no	—
68	C₂×Dic₁₃	#108	1	4	yes	—
69	C₂×Dic₁₃	#124	1	1	yes	characteristic
70	C₂×Dic₁₃	#124	1	1	yes	characteristic
71	C₂×Dic₁₃	#125	1	2	yes	—
72	C₂×Dic₁₃	AGL(1,ℤ/8)	1	2	yes	—
73	C₂×Dic₁₃	#137	1	2	yes	—
74	#1057	—	8	2	no	—
75	#1057	—	8	2	no	—
76	C₈×C₁₃	—	4	2	no	—
77	#1055	—	4	2	no	—
78	C₂×C₄⋊C₈	—	13	2	no	—
79	C₂×C₄⋊C₈	—	13	2	no	—
80	C₂×C₄⋊C₈	—	26	2	no	—
81	C₂×C₄×C₈	—	13	2	no	—
82	C₄³	—	13	1	no	—
83	#336	—	13	2	no	—
84	#336	—	13	1	no	—
85	#336	—	26	2	no	—
86	#336	#261	1	1	yes	characteristic
87	C₂²×C₁₃	#382	1	1	yes	characteristic
88	C₂²×C₁₃	#424	1	1	yes	characteristic
89	C₂²×C₁₃	#424	1	1	yes	characteristic
90	C₂²×C₁₃	#428	1	2	yes	—
91	C₂²×C₁₃	C₂×#108	1	2	yes	—
92	C₄×C₁₃	—	4	2	no	—
93	C₄×C₁₃	#433	1	2	yes	—
94	C₄×C₁₃	#433	1	2	yes	—
95	Dic₁₃	—	2	4	no	—
96	Dic₁₃	—	2	2	no	—
97	Dic₁₃	#489	1	2	yes	—
98	Dic₁₃	#490	1	1	yes	characteristic
99	Dic₁₃	#491	1	1	yes	characteristic
100	C₂²×C₈	—	13	2	no	—
101	C₂²×C₈	—	26	2	no	—
102	C₂²×C₈	—	26	2	no	—
103	C₂²×C₈	C₂×#261	1	2	yes	—
104	C₂×C₄²	—	13	1	no	—
105	C₂×C₄²	—	13	2	no	—
106	C₂×C₄²	—	13	2	no	—
107	C₂×C₄²	—	13	1	no	—
108	C₂×C₄²	C₂×#261	1	1	yes	characteristic
109	C₄⋊C₈	—	26	4	no	—
110	C₄⋊C₈	—	26	4	no	—
111	C₄⋊C₈	—	52	4	no	—
112	C₄×C₈	—	26	2	no	—
113	C₄×C₈	—	26	2	no	—
114	C₂×C₁₃	#1573	1	2	yes	—
115	C₂×C₁₃	#1574	1	2	yes	—
116	C₂×C₁₃	#1832	1	1	yes	characteristic
117	C₂×C₁₃	#1836	1	1	yes	characteristic
118	C₂×C₁₃	#1837	1	1	yes	characteristic
119	C₂²×C₄	—	13	1	no	—
120	C₂²×C₄	—	13	2	no	—
121	C₂²×C₄	—	13	1	no	—
122	C₂²×C₄	C₄⋊#261	1	2	yes	—
123	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	26	2	no	—
128	C₂×C₈	—	26	2	no	—
129	C₂×C₈	—	26	2	no	—
130	C₂×C₈	—	52	2	no	—
131	C₂×C₈	—	52	2	no	—
132	C₂×C₈	—	52	4	no	—
133	C₂×C₈	—	52	2	no	—
134	C₂×C₈	—	52	2	no	—
135	C₄²	—	2	2	no	—
136	C₄²	—	13	2	no	—
137	C₄²	—	13	2	no	—
138	C₄²	—	26	2	no	—
139	C₄²	—	26	4	no	—
140	C₄²	—	26	2	no	—
141	C₄²	—	26	1	no	—
142	C₄²	—	26	1	no	—
143	C₁₃	#8789	1	1	yes	characteristic, complemented, Sylow
144	C₂³	C₂×C₄⋊#261	1	1	yes	characteristic
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	13	1	no	—
149	C₂×C₄	—	13	1	no	—
150	C₂×C₄	—	13	2	no	—
151	C₂×C₄	—	13	2	no	—
152	C₂×C₄	—	13	2	no	—
153	C₂×C₄	—	13	4	no	—
154	C₂×C₄	—	26	2	no	—
155	C₂×C₄	—	26	2	no	—
156	C₂×C₄	—	26	2	no	—
157	C₂×C₄	#88237	1	2	yes	—
158	C₂×C₄	#88238	1	1	yes	characteristic
159	C₂×C₄	#88238	1	2	yes	—
160	C₂×C₄	#88240	1	1	yes	characteristic
161	C₈	—	4	2	no	—
162	C₈	—	52	2	no	—
163	C₈	—	104	2	no	—
164	C₈	—	104	2	no	—
165	C₂²	#11719060	1	1	yes	characteristic
166	C₂²	#11719067	1	1	yes	characteristic
167	C₂²	#11719079	1	1	yes	characteristic
168	C₂²	D₁₃⋊#108	1	2	yes	—
169	C₂²	#11719098	1	2	yes	—
170	C₄	—	4	2	no	—
171	C₄	—	13	2	no	—
172	C₄	—	13	1	no	—
173	C₄	—	13	1	no	—
174	C₄	—	26	2	no	—
175	C₄	—	26	4	no	—
176	C₄	#11719068	1	2	yes	—
177	C₄	#11719077	1	2	yes	—
178	C₂	#13130410	1	2	yes	—
179	C₂	#13130412	1	2	yes	—
180	C₂	#13130754	1	1	yes	characteristic
181	C₂	#13130759	1	1	yes	characteristic
182	C₂	#13130761	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13130769

C₂

yes

maximal, characteristic

#13130751

C₂

yes

maximal, characteristic

#13129666

C₂

yes

maximal

#13114580

C₂

yes

maximal, characteristic

C₂×#88112

—

C₂×C₄⋊#1057

C₂²

yes

—

C₂×C₄⋊#1057

C₂²

yes

—

C₂×C₈×Dic₁₃

C₂²

yes

—

C₄²×Dic₁₃

C₂²

yes

characteristic

#11718290

—

C₁₃×#336

C₄

yes

characteristic

#11718142

C₄

yes

—

#11718116

C₄

yes

characteristic

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

—

#88112

—

C₄⋊#1057

—

C₄⋊#1057

—

C₈×Dic₁₃

—

C₈×Dic₁₃

—

#8789

—

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

—

C₄²×C₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

#41

yes

—

C₄×Dic₁₃

#41

yes

—

—

—

—

—

—

—

—

C₂³×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#117

yes

—

C₂×C₄×C₁₃

#131

yes

—

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

#108

yes

—

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×C₄×C₈

—

C₄³

—

—

—

—

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

—

C₂²×C₁₃

C₂×#108

yes

—

C₄×C₁₃

—

C₄×C₁₃

#433

yes

—

C₄×C₁₃

#433

yes

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

#489

yes

—

Dic₁₃

yes

Dic₁₃

yes

100

C₂²×C₈

—

101

C₂²×C₈

—

102

C₂²×C₈

—

103

C₂²×C₈

C₂×#261

yes

—

104

C₂×C₄²

—

105

C₂×C₄²

—

106

C₂×C₄²

—

107

C₂×C₄²

—

108

C₂×C₄²

C₂×#261

yes

characteristic

109

C₄⋊C₈

—

110

C₄⋊C₈

—

111

C₄⋊C₈

—

112

C₄×C₈

—

113

C₄×C₈

—

114

C₂×C₁₃

#1573

yes

—

115

C₂×C₁₃

#1574

yes

—

116

C₂×C₁₃

#1832

yes

characteristic

117

C₂×C₁₃

#1836

yes

characteristic

118

C₂×C₁₃

#1837

yes

characteristic

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

C₄⋊#261

yes

—

123

C₂²×C₄

C₂²×#261

yes

characteristic

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₄²

—

136

C₄²

—

137

C₄²

—

138

C₄²

—

139

C₄²

—

140

C₄²

—

141

C₄²

—

142

C₄²

—

143

C₁₃

#8789

yes

characteristic, complemented, Sylow

144

C₂³

C₂×C₄⋊#261

yes

characteristic

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

#88237

yes

—

158

C₂×C₄

#88238

yes

characteristic

159

C₂×C₄

#88238

yes

—

160

C₂×C₄

#88240

yes

characteristic

161

C₈

—

162

C₈

—

163

C₈

—

104

—

164

C₈

—

104

—

165

C₂²

#11719060

yes

characteristic

166

C₂²

#11719067

yes

characteristic

167

C₂²

#11719079

yes

characteristic

168

C₂²

D₁₃⋊#108

yes

—

169

C₂²

#11719098

yes

—

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

#11719068

yes

—

177

C₄

#11719077

yes

—

178

C₂

#13130410

yes

—

179

C₂

#13130412

yes

—

180

C₂

#13130754

yes

characteristic

181

C₂

#13130759

yes

characteristic

182

C₂

#13130761

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#261
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₁₃×#336	C₄
socle, prime radical	C₂³×C₁₃	C₂×C₄⋊C₄
cosocle	C₂×C₄×Dic₁₃	C₂³
hypercenter	#336	#261
hypocenter	C₁₃	#8789
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×C₄⋊#261

commutator

C₂²×C₄×C₁₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×#261

Fitting

C₁₃×#336

C₄

socle, prime radical

C₂³×C₁₃

C₂×C₄⋊C₄

cosocle

C₂×C₄×Dic₁₃

C₂³

C₁₃

G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
143. C₁₃	#8789	semidirect

H ≅ G/N

type

143. C₁₃

#8789

semidirect

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

subgroup	index
#13130769	2
#13130751	2
#13129666	2
#13114580	2
#8789	13

subgroup

index

Subgroups — #16782367

Conjugacy classes of subgroups

Complements

Maximal subgroups