Group #16018420 — subgroups

Subgroups — #16018420

Order 3328.

482 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 187 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	#13125348	C₂	1	2	yes	maximal
2	#13117121	C₂	1	4	yes	maximal, complemented
3	#13117120	C₂	1	4	yes	maximal, complemented
4	#13116620	C₂	1	1	yes	maximal, characteristic
5	#13116618	C₂	1	2	yes	maximal
6	C₂×#11718094	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#2158	C₂	1	1	yes	maximal, characteristic
8	#11719222	—	2	2	no	—
9	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
10	C₁₃×#427	C₂²	1	2	yes	—
11	#11718767	C₂²	1	2	yes	—
12	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
13	C₂²×#6090	C₂²	1	1	yes	characteristic
14	#11718659	—	2	2	no	—
15	#11718539	C₂²	1	4	yes	—
16	#11718535	C₂²	1	4	yes	—
17	#11718519	—	2	2	no	—
18	#11718516	—	2	2	no	—
19	C₁₃×#367	C₂²	1	2	yes	—
20	C₁₃×#366	C₂²	1	2	yes	—
21	#11718097	C₂²	1	2	yes	—
22	#11718096	C₂²	1	2	yes	—
23	#11718094	C₂²	1	2	yes	—
24	#11718094	C₂²	1	2	yes	—
25	#11718082	C₂²	1	4	yes	—
26	#11718081	C₂²	1	4	yes	—
27	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
28	C₁₃×#117	—	2	2	no	—
29	#88180	—	2	2	no	—
30	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
31	C₂²×#1055	D₄	1	1	yes	characteristic
32	#88144	—	2	4	no	—
33	#88142	—	2	4	no	—
34	#88132	—	4	4	no	—
35	C₄×Dic₂₆	—	4	4	no	—
36	C₂×C₁₃×C₁₆	Q₈	1	4	yes	complemented
37	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
38	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
39	C₁₃×#107	C₂³	1	4	yes	—
40	C₁₃×#106	—	2	2	no	—
41	#88058	C₂³	1	4	yes	—
42	#88057	—	2	2	no	—
43	C₈×Dic₁₃	D₄	1	2	yes	—
44	C₈×Dic₁₃	C₂³	1	2	yes	—
45	C₂×#6090	Q₈	1	4	yes	complemented
46	C₂×#6090	C₂³	1	2	yes	—
47	#88048	—	2	4	no	—
48	#88047	—	2	4	no	—
49	#33728	—	13	1	no	maximal, Sylow
50	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×Dic₁₃	D₈	1	2	yes	—
52	C₂×Dic₂₆	—	4	4	no	—
53	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
55	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
56	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
57	C₁₃×C₄⋊C₄	—	2	4	no	—
58	C₁₃×C₂²⋊C₄	—	4	2	no	—
59	C₄²×C₁₃	—	4	2	no	—
60	C₂²⋊Dic₁₃	—	4	2	no	—
61	C₄⋊Dic₁₃	—	2	4	no	—
62	C₄⋊Dic₁₃	—	4	4	no	—
63	#6101	—	4	4	no	—
64	#6101	—	4	4	no	—
65	C₄×Dic₁₃	—	4	2	no	—
66	C₄×Dic₁₃	D₈	1	2	yes	—
67	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
68	C₂×#1055	Q₁₆	1	4	yes	—
69	C₂×#1055	C₂×D₄	1	2	yes	—
70	C₁₃×C₁₆	C₂×Q₈	1	2	yes	—
71	C₁₃×C₁₆	C₂×Q₈	1	2	yes	—
72	#6090	C₂×Q₈	1	4	yes	—
73	#3160	—	13	2	no	—
74	#2255	—	13	4	no	—
75	#2255	—	13	4	no	—
76	#2158	—	13	1	no	—
77	#2158	—	13	2	no	—
78	#2158	D₁₃	1	1	yes	characteristic
79	C₂×#346	—	13	1	no	—
80	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₁₃	—	4	4	no	—
82	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
83	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
85	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
86	C₂×Dic₁₃	—	4	2	no	—
87	C₂×Dic₁₃	—	4	4	no	—
88	C₂×Dic₁₃	C₂×D₈	1	4	yes	—
89	Dic₂₆	—	8	8	no	—
90	C₈×C₁₃	C₄⋊Q₈	1	2	yes	—
91	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
92	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
93	#1055	C₂×Q₁₆	1	4	yes	—
94	#533	—	26	2	no	—
95	C₂²×C₁₆	—	13	1	no	—
96	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
97	#500	—	13	4	no	—
98	#499	—	13	4	no	—
99	#499	—	26	2	no	—
100	#498	—	26	2	no	—
101	#427	—	13	2	no	—
102	#427	—	26	2	no	—
103	#427	C₂×D₁₃	1	2	yes	—
104	C₂×C₄×C₈	—	13	1	no	—
105	#367	—	13	2	no	—
106	#367	—	13	4	no	—
107	#367	C₂×D₁₃	1	2	yes	—
108	#366	—	13	2	no	—
109	#366	—	13	4	no	—
110	#366	C₂×D₁₃	1	2	yes	—
111	#346	—	13	2	no	—
112	#346	—	13	2	no	—
113	C₂²×C₁₃	—	2	2	no	—
114	C₂²×C₁₃	#500	1	2	yes	—
115	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
116	C₄×C₁₃	—	8	4	no	—
117	C₄×C₁₃	#500	1	2	yes	—
118	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
119	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
120	Dic₁₃	—	2	4	no	—
121	Dic₁₃	—	8	4	no	—
122	C₂²×C₈	—	13	1	no	—
123	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
124	C₄⋊Q₈	—	26	4	no	—
125	#125	—	26	4	no	—
126	C₂²⋊Q₈	—	52	4	no	—
127	C₄×Q₈	—	52	4	no	—
128	#117	—	2	2	no	—
129	#117	—	26	2	no	—
130	C₂×C₄²	—	13	1	no	—
131	C₂×C₁₆	—	13	4	no	—
132	C₂×C₁₆	—	13	2	no	—
133	C₂×C₁₆	Dic₂₆	1	4	yes	complemented
134	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
135	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
136	#107	—	13	4	no	—
137	#107	—	26	4	no	—
138	#107	C₂²×D₁₃	1	4	yes	—
139	#106	—	2	2	no	—
140	#106	—	26	4	no	—
141	#106	—	26	2	no	—
142	C₄×C₈	—	13	2	no	—
143	C₄×C₈	—	13	2	no	—
144	C₂×C₁₃	—	2	2	no	—
145	C₂×C₁₃	C₂×#500	1	1	yes	characteristic
146	C₂×C₁₃	#3414	1	2	yes	—
147	C₂×Q₈	—	52	4	no	—
148	C₂²×C₄	—	13	2	no	—
149	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
150	C₂×C₈	—	13	4	no	—
151	C₂×C₈	—	13	2	no	—
152	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
153	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
154	C₂×C₈	Q₈×D₁₃	1	2	yes	—
155	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
156	C₄⋊C₄	—	2	4	no	—
157	C₄⋊C₄	—	26	4	no	—
158	C₄⋊C₄	—	52	4	no	—
159	C₄⋊C₄	—	52	4	no	—
160	C₄⋊C₄	—	52	4	no	—
161	C₂²⋊C₄	—	4	2	no	—
162	C₂²⋊C₄	—	52	2	no	—
163	C₄²	—	4	2	no	—
164	C₄²	—	13	2	no	—
165	C₄²	—	13	2	no	—
166	C₄²	—	52	2	no	—
167	C₁₆	—	13	4	no	—
168	C₁₆	C₂×Dic₂₆	1	2	yes	—
169	C₁₆	C₂×Dic₂₆	1	2	yes	—
170	C₁₃	#33728	1	1	yes	characteristic, complemented, Sylow
171	C₂³	D₈×D₁₃	1	1	yes	characteristic
172	Q₈	—	104	8	no	—
173	C₂×C₄	—	4	4	no	—
174	C₂×C₄	—	13	4	no	—
175	C₂×C₄	—	52	2	no	—
176	C₂×C₄	—	52	4	no	—
177	C₂×C₄	#88142	1	2	yes	—
178	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
179	C₂×C₄	Q₁₆×D₁₃	1	2	yes	—
180	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
181	C₈	—	13	4	no	—
182	C₈	#88142	1	2	yes	—
183	C₈	C₂²×Dic₂₆	1	1	yes	characteristic
184	C₈	C₂×Q₈×D₁₃	1	1	yes	characteristic
185	C₂²	—	2	2	no	—
186	C₂²	#11718535	1	2	yes	—
187	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
188	C₄	—	8	4	no	—
189	C₄	—	26	4	no	—
190	C₄	—	104	4	no	—
191	C₄	#11718535	1	2	yes	—
192	C₄	C₂×#88142	1	1	yes	characteristic
193	C₄	C₂×Q₁₆×D₁₃	1	1	yes	characteristic
194	C₂	—	2	2	no	—
195	C₂	C₂×#11718535	1	1	yes	characteristic
196	C₂	#13127969	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13125348

C₂

yes

maximal

#13117121

C₂

yes

maximal, complemented

#13117120

C₂

yes

maximal, complemented

#13116620

C₂

yes

maximal, characteristic

#13116618

C₂

yes

maximal

C₂×#11718094

C₂

yes

maximal, characteristic

C₁₃×#2158

C₂

yes

maximal, characteristic

#11719222

—

C₂²×C₁₃×C₁₆

C₂²

yes

characteristic

C₁₃×#427

C₂²

yes

—

#11718767

C₂²

yes

—

C₂×C₈×Dic₁₃

C₂²

yes

characteristic

C₂²×#6090

C₂²

yes

characteristic

#11718659

—

#11718539

C₂²

yes

—

#11718535

C₂²

yes

—

#11718519

—

#11718516

—

C₁₃×#367

C₂²

yes

—

C₁₃×#366

C₂²

yes

—

#11718097

C₂²

yes

—

#11718096

C₂²

yes

—

#11718094

C₂²

yes

—

#11718094

C₂²

yes

—

#11718082

C₂²

yes

—

#11718081

C₂²

yes

—

C₂²×C₈×C₁₃

C₂³

yes

characteristic

C₁₃×#117

—

#88180

—

C₂×C₄×Dic₁₃

D₄

yes

characteristic

C₂²×#1055

D₄

yes

—

—

—

C₄×Dic₂₆

—

C₂×C₁₃×C₁₆

Q₈

yes

complemented

C₂×C₁₃×C₁₆

C₂³

yes

characteristic

C₂×C₁₃×C₁₆

C₂³

yes

characteristic

C₁₃×#107

C₂³

yes

—

C₁₃×#106

—

#88058

C₂³

yes

—

#88057

—

C₈×Dic₁₃

D₄

yes

—

C₈×Dic₁₃

C₂³

yes

—

C₂×#6090

Q₈

yes

complemented

C₂×#6090

C₂³

yes

—

—

—

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×Dic₁₃

D₈

yes

—

C₂×Dic₂₆

—

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃

C₂×Q₈

yes

—

C₂×C₈×C₁₃

C₂×Q₈

yes

—

C₂×C₈×C₁₃

C₂⁴

yes

characteristic

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

—

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#6101

—

#6101

—

C₄×Dic₁₃

—

C₄×Dic₁₃

D₈

yes

—

C₄×Dic₁₃

C₂×D₄

yes

—

C₂×#1055

Q₁₆

yes

—

C₂×#1055

C₂×D₄

yes

—

C₁₃×C₁₆

C₂×Q₈

yes

—

C₁₃×C₁₆

C₂×Q₈

yes

—

#6090

C₂×Q₈

yes

—

—

—

—

—

—

D₁₃

yes

characteristic

C₂×#346

—

C₂³×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₄⋊Q₈

yes

—

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

C₂×Q₁₆

yes

—

C₂×C₄×C₁₃

C₂²×D₄

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂×D₈

yes

—

Dic₂₆

—

C₈×C₁₃

C₄⋊Q₈

yes

—

C₈×C₁₃

C₂²×Q₈

yes

characteristic

C₈×C₁₃

C₂²×Q₈

yes

characteristic

#1055

C₂×Q₁₆

yes

—

#533

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₁₃

yes

—

—

—

100

#498

—

101

#427

—

102

#427

—

103

#427

C₂×D₁₃

yes

—

104

C₂×C₄×C₈

—

105

#367

—

106

#367

—

107

#367

C₂×D₁₃

yes

—

108

#366

—

109

#366

—

110

#366

C₂×D₁₃

yes

—

111

#346

—

112

#346

—

113

C₂²×C₁₃

—

114

C₂²×C₁₃

#500

yes

—

115

C₂²×C₁₃

C₂²×D₈

yes

characteristic

116

C₄×C₁₃

—

117

C₄×C₁₃

#500

yes

—

118

C₄×C₁₃

C₂×C₄⋊Q₈

yes

characteristic

119

C₄×C₁₃

C₂²×Q₁₆

yes

characteristic

120

Dic₁₃

—

121

Dic₁₃

—

122

C₂²×C₈

—

123

C₂²×C₈

C₂²×D₁₃

yes

characteristic

124

C₄⋊Q₈

—

125

#125

—

126

C₂²⋊Q₈

—

127

C₄×Q₈

—

128

#117

—

129

#117

—

130

C₂×C₄²

—

131

C₂×C₁₆

—

132

C₂×C₁₆

—

133

C₂×C₁₆

Dic₂₆

yes

complemented

134

C₂×C₁₆

C₂²×D₁₃

yes

characteristic

135

C₂×C₁₆

C₂²×D₁₃

yes

characteristic

136

#107

—

137

#107

—

138

#107

C₂²×D₁₃

yes

—

139

#106

—

140

#106

—

141

#106

—

142

C₄×C₈

—

143

C₄×C₈

—

144

C₂×C₁₃

—

145

C₂×C₁₃

C₂×#500

yes

characteristic

146

C₂×C₁₃

#3414

yes

—

147

C₂×Q₈

—

148

C₂²×C₄

—

149

C₂²×C₄

D₄×D₁₃

yes

characteristic

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

C₂×Dic₂₆

yes

—

153

C₂×C₈

D₄×D₁₃

yes

characteristic

154

C₂×C₈

Q₈×D₁₃

yes

—

155

C₂×C₈

C₂³×D₁₃

yes

characteristic

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₂²⋊C₄

—

162

C₂²⋊C₄

—

163

C₄²

—

164

C₄²

—

165

C₄²

—

166

C₄²

—

167

C₁₆

—

168

C₁₆

C₂×Dic₂₆

yes

—

169

C₁₆

C₂×Dic₂₆

yes

—

170

C₁₃

#33728

yes

characteristic, complemented, Sylow

171

C₂³

D₈×D₁₃

yes

characteristic

172

Q₈

—

104

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

#88142

yes

—

178

C₂×C₄

D₈×D₁₃

yes

characteristic

179

C₂×C₄

Q₁₆×D₁₃

yes

—

180

C₂×C₄

C₂×D₄×D₁₃

yes

characteristic

181

C₈

—

182

C₈

#88142

yes

—

183

C₈

C₂²×Dic₂₆

yes

characteristic

184

C₈

C₂×Q₈×D₁₃

yes

characteristic

185

C₂²

—

186

C₂²

#11718535

yes

—

187

C₂²

C₂×D₈×D₁₃

yes

characteristic

188

C₄

—

189

C₄

—

190

C₄

—

104

—

191

C₄

#11718535

yes

—

192

C₄

C₂×#88142

yes

characteristic

193

C₄

C₂×Q₁₆×D₁₃

yes

characteristic

194

C₂

—

195

C₂

C₂×#11718535

yes

characteristic

196

C₂

#13127969

yes

—

	subgroup	G/H
center	C₂×C₄	D₈×D₁₃
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2158	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂³×C₁₃	C₂×D₈
hypercenter	#2158	D₁₃
hypocenter	C₁₃	#33728
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

subgroup

G/H

center

C₂×C₄

D₈×D₁₃

commutator, cosocle

C₂×C₈×C₁₃

C₂⁴

Frattini

C₂×C₈

C₂³×D₁₃

Fitting

C₁₃×#2158

C₂

socle

C₂²×C₁₃

C₂²×D₈

prime radical

C₂³×C₁₃

C₂×D₈

D₁₃

C₁₃

G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

N	H ≅ G/N	type
2. #13117121	C₂	semidirect
3. #13117120	C₂	semidirect
36. C₂×C₁₃×C₁₆	Q₈	semidirect
45. C₂×#6090	Q₈	semidirect
133. C₂×C₁₆	Dic₂₆	semidirect
170. C₁₃	#33728	semidirect

H ≅ G/N

type

2. #13117121

C₂

semidirect

3. #13117120

C₂

semidirect

36. C₂×C₁₃×C₁₆

Q₈

semidirect

45. C₂×#6090

Q₈

semidirect

133. C₂×C₁₆

Dic₂₆

semidirect

170. C₁₃

#33728

semidirect

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

subgroup	index
#13125348	2
#13117121	2
#13117120	2
#13116620	2
#13116618	2
C₂×#11718094	2
C₁₃×#2158	2
#33728	13

subgroup

index

C₁₃×#2158

Subgroups — #16018420

Conjugacy classes of subgroups

Complements

Maximal subgroups