Group #16785021 — subgroups

Subgroups — #16785021

Order 3328.

406 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 112 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	#13130777	C₂	1	1	yes	maximal, characteristic
2	#13130186	C₂	1	2	yes	maximal
3	#13130185	C₂	1	2	yes	maximal
4	#13130030	C₂	1	1	yes	maximal, characteristic
5	#13125345	C₂	1	1	yes	maximal, characteristic
6	#11719433	C₂²	1	2	yes	—
7	C₂×#88104	—	2	2	no	—
8	#11719052	C₂²	1	1	yes	characteristic
9	#11719052	C₂²	1	1	yes	characteristic
10	C₄×D₁₃⋊C₈	C₂²	1	2	yes	—
11	C₁₃×#498	C₄	1	1	yes	characteristic
12	#11718739	C₄	1	2	yes	—
13	#11718521	—	2	2	no	—
14	#11718519	—	2	2	no	—
15	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
16	#11718341	C₄	1	1	yes	characteristic
17	Q₈⋊#1057	—	2	4	no	—
18	#11718251	—	2	2	no	—
19	#11718244	—	2	4	no	—
20	C₂×#6118	—	2	1	no	—
21	C₂×#6118	—	2	1	no	—
22	C₂×D₁₃⋊C₈	—	2	2	no	—
23	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
24	#88200	—	2	2	no	—
25	Q₈×Dic₁₃	—	2	2	no	—
26	C₂×C₈×D₁₃	D₄	1	2	yes	—
27	#88151	—	2	2	no	—
28	#88146	—	2	2	no	—
29	#88144	—	2	2	no	—
30	#88141	—	2	2	no	—
31	C₄²×D₁₃	C₂³	1	1	yes	characteristic
32	#88123	C₂×C₄	1	2	yes	—
33	C₄⋊#1057	—	2	2	no	—
34	C₄⋊#1057	—	2	2	no	—
35	C₄×#1057	—	2	2	no	—
36	C₄×#1057	D₄	1	4	yes	—
37	#88104	—	4	4	no	—
38	C₁₃×#106	—	2	2	no	—
39	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
40	#88057	—	2	2	no	—
41	C₈×Dic₁₃	D₄	1	2	yes	—
42	#88047	—	2	4	no	—
43	C₄×#1055	C₂×C₄	1	1	yes	characteristic
44	#9861	—	13	1	no	maximal, Sylow
45	C₂×Q₈×C₁₃	—	2	2	no	—
46	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×D₁₃	#41	1	2	yes	—
48	C₂×Dic₂₆	—	2	2	no	—
49	C₂×#1057	—	2	4	no	—
50	C₂×#1057	—	4	1	no	—
51	C₂×#1057	—	4	1	no	—
52	#6118	—	4	2	no	—
53	#6118	—	4	2	no	—
54	D₁₃⋊C₈	—	4	4	no	—
55	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
56	C₁₃×C₄⋊C₄	—	2	2	no	—
57	C₁₃×C₄⋊C₄	—	2	2	no	—
58	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
59	#6103	—	4	2	no	—
60	C₄⋊Dic₁₃	—	2	2	no	—
61	C₄⋊Dic₁₃	—	2	2	no	—
62	C₄⋊Dic₁₃	—	4	2	no	—
63	#6101	—	4	2	no	—
64	C₄×Dic₁₃	—	4	2	no	—
65	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
66	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
67	C₂×#1055	C₂²⋊C₄	1	2	yes	—
68	C₈×D₁₃	SD₁₆	1	4	yes	—
69	C₈×D₁₃	Q₁₆	1	4	yes	—
70	#3160	—	13	1	no	—
71	#1848	—	13	1	no	—
72	#1493	—	13	2	no	—
73	#1493	—	13	2	no	—
74	C₂×#322	—	13	1	no	—
75	C₂²×D₁₃	#124	1	1	yes	characteristic
76	Q₈×C₁₃	—	4	4	no	—
77	C₂×C₄×C₁₃	—	4	2	no	—
78	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
80	C₂×Dic₁₃	—	4	2	no	—
81	C₂×Dic₁₃	#121	1	2	yes	—
82	C₂×Dic₁₃	#127	1	1	yes	characteristic
83	C₄×D₁₃	—	2	4	no	—
84	C₄×D₁₃	C₂×SD₁₆	1	2	yes	—
85	C₄×D₁₃	C₂×Q₁₆	1	2	yes	—
86	Dic₂₆	—	4	4	no	—
87	#1057	—	4	4	no	—
88	#1057	—	8	1	no	—
89	#1057	—	8	1	no	—
90	C₈×C₁₃	#103	1	4	yes	—
91	#1055	#103	1	4	yes	—
92	#533	—	13	2	no	—
93	#500	—	13	2	no	—
94	#499	—	26	2	no	—
95	#498	—	13	1	no	—
96	#498	#261	1	1	yes	characteristic
97	C₂×#108	—	26	2	no	—
98	#427	—	26	2	no	—
99	#423	—	13	1	no	—
100	#423	—	13	1	no	—
101	C₂×C₄×C₈	—	13	1	no	—
102	C₂×C₄×C₈	—	13	2	no	—
103	Q₈⋊C₈	—	26	4	no	—
104	Q₈⋊C₈	—	26	4	no	—
105	#322	—	26	2	no	—
106	C₂²×C₁₃	#390	1	1	yes	characteristic
107	C₂×D₁₃	—	2	2	no	—
108	C₂×D₁₃	#487	1	2	yes	—
109	C₄×C₁₃	—	8	2	no	—
110	C₄×C₁₃	C₂×#103	1	2	yes	—
111	C₄×C₁₃	#444	1	4	yes	—
112	Dic₁₃	—	2	2	no	—
113	Dic₁₃	—	8	2	no	—
114	Dic₁₃	#492	1	1	yes	characteristic
115	Dic₁₃	#494	1	1	yes	characteristic
116	C₂×M₁₆	—	26	1	no	—
117	C₂×M₁₆	—	26	1	no	—
118	C₂²×C₈	—	13	2	no	—
119	C₂²×C₈	—	26	2	no	—
120	C₄⋊Q₈	—	13	2	no	—
121	C₄⋊Q₈	C₂×#261	1	2	yes	—
122	#125	—	26	2	no	—
123	C₂²⋊Q₈	—	26	2	no	—
124	C₂²⋊Q₈	—	26	2	no	—
125	C₄×Q₈	—	26	2	no	—
126	C₄×Q₈	—	26	2	no	—
127	#117	—	26	2	no	—
128	C₂×C₄²	—	13	1	no	—
129	#108	—	52	4	no	—
130	#107	—	26	4	no	—
131	#106	—	2	2	no	—
132	#106	—	26	2	no	—
133	C₄⋊C₈	—	26	2	no	—
134	C₄⋊C₈	—	26	2	no	—
135	C₄×C₈	—	13	2	no	—
136	C₄×C₈	—	13	4	no	—
137	C₄×C₈	—	13	1	no	—
138	C₄×C₈	—	26	2	no	—
139	C₄×C₈	C₂×#261	1	1	yes	characteristic
140	C₂×C₁₃	#1956	1	1	yes	characteristic
141	C₂×C₁₃	#1975	1	2	yes	—
142	D₁₃	—	2	2	no	—
143	C₂×Q₈	—	2	2	no	—
144	C₂×Q₈	—	26	2	no	—
145	C₂²×C₄	—	13	1	no	—
146	C₂²×C₄	—	13	2	no	—
147	M₁₆	—	52	2	no	—
148	M₁₆	—	52	2	no	—
149	C₂×C₈	—	13	2	no	—
150	C₂×C₈	—	13	4	no	—
151	C₂×C₈	—	13	4	no	—
152	C₂×C₈	—	26	4	no	—
153	C₂×C₈	—	52	4	no	—
154	C₂×C₈	—	52	1	no	—
155	C₂×C₈	—	52	1	no	—
156	C₂×C₈	C₂²⋊#261	1	2	yes	—
157	C₄⋊C₄	—	2	2	no	—
158	C₄⋊C₄	—	2	2	no	—
159	C₄⋊C₄	—	26	2	no	—
160	C₄⋊C₄	—	26	2	no	—
161	C₄⋊C₄	—	52	2	no	—
162	C₄⋊C₄	—	52	2	no	—
163	C₂²⋊C₄	—	52	2	no	—
164	C₄²	—	13	2	no	—
165	C₄²	—	13	1	no	—
166	C₄²	—	52	2	no	—
167	C₄²	C₂²×#261	1	1	yes	characteristic
168	C₁₃	#9861	1	1	yes	characteristic, complemented, Sylow
169	C₂³	—	13	1	no	—
170	Q₈	—	4	4	no	—
171	Q₈	—	52	4	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	—	13	1	no	—
174	C₂×C₄	—	13	2	no	—
175	C₂×C₄	—	13	2	no	—
176	C₂×C₄	—	13	2	no	—
177	C₂×C₄	—	26	4	no	—
178	C₂×C₄	—	52	2	no	—
179	C₂×C₄	D₄×#261	1	2	yes	—
180	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
181	C₈	—	13	4	no	—
182	C₈	—	52	4	no	—
183	C₈	—	104	1	no	—
184	C₈	—	104	1	no	—
185	C₈	#88117	1	4	yes	—
186	C₂²	—	13	2	no	—
187	C₂²	—	26	2	no	—
188	C₂²	#11719151	1	1	yes	characteristic
189	C₄	—	8	2	no	—
190	C₄	—	13	1	no	—
191	C₄	—	13	1	no	—
192	C₄	—	26	2	no	—
193	C₄	—	104	2	no	—
194	C₄	#11719108	1	4	yes	—
195	C₄	C₂×#88117	1	2	yes	—
196	C₂	—	26	2	no	—
197	C₂	#13130963	1	1	yes	characteristic
198	C₂	#13130992	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13130777

C₂

yes

maximal, characteristic

#13130186

C₂

yes

maximal

#13130185

C₂

yes

maximal

#13130030

C₂

yes

maximal, characteristic

#13125345

C₂

yes

maximal, characteristic

#11719433

C₂²

yes

—

C₂×#88104

—

#11719052

C₂²

yes

characteristic

#11719052

C₂²

yes

characteristic

C₄×D₁₃⋊C₈

C₂²

yes

—

C₁₃×#498

C₄

yes

characteristic

#11718739

C₄

yes

—

#11718521

—

#11718519

—

C₄×C₈×D₁₃

C₂²

yes

characteristic

#11718341

C₄

yes

—

—

—

—

—

C₂×D₁₃⋊C₈

—

C₁₃×C₄⋊Q₈

C₂×C₄

yes

—

#88200

—

Q₈×Dic₁₃

—

C₂×C₈×D₁₃

D₄

yes

—

—

—

—

—

C₄²×D₁₃

C₂³

yes

characteristic

#88123

C₂×C₄

yes

—

—

—

—

D₄

yes

—

#88104

—

C₁₃×#106

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

#88057

—

C₈×Dic₁₃

D₄

yes

—

#88047

—

C₄×#1055

C₂×C₄

yes

characteristic

#9861

—

maximal, Sylow

C₂×Q₈×C₁₃

—

C₂×C₄×D₁₃

C₂×D₄

yes

characteristic

C₂×C₄×D₁₃

#41

yes

—

C₂×Dic₂₆

—

—

—

—

—

—

D₁₃⋊C₈

—

C₂×C₈×C₁₃

C₂²⋊C₄

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂²×C₄

yes

characteristic

#6103

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#6101

—

C₄×Dic₁₃

—

C₄×Dic₁₃

C₂×D₄

yes

characteristic

C₄×Dic₁₃

C₂×D₄

yes

—

C₂×#1055

C₂²⋊C₄

yes

—

C₈×D₁₃

SD₁₆

yes

—

C₈×D₁₃

Q₁₆

yes

—

—

—

—

—

—

C₂²×D₁₃

#124

yes

characteristic

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₄×D₄

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

#121

yes

—

C₂×Dic₁₃

#127

yes

characteristic

C₄×D₁₃

—

C₄×D₁₃

C₂×SD₁₆

yes

—

C₄×D₁₃

C₂×Q₁₆

yes

—

Dic₂₆

—

#1057

—

#1057

—

#1057

—

C₈×C₁₃

#103

yes

—

#1055

#103

yes

—

—

—

—

—

yes

—

—

—

100

#423

—

101

C₂×C₄×C₈

—

102

C₂×C₄×C₈

—

103

Q₈⋊C₈

—

104

Q₈⋊C₈

—

105

#322

—

106

C₂²×C₁₃

#390

yes

characteristic

107

C₂×D₁₃

—

108

C₂×D₁₃

#487

yes

—

109

C₄×C₁₃

—

110

C₄×C₁₃

C₂×#103

yes

—

111

C₄×C₁₃

#444

yes

—

112

Dic₁₃

—

113

Dic₁₃

—

114

Dic₁₃

#492

yes

characteristic

115

Dic₁₃

#494

yes

characteristic

116

C₂×M₁₆

—

117

C₂×M₁₆

—

118

C₂²×C₈

—

119

C₂²×C₈

—

120

C₄⋊Q₈

—

121

C₄⋊Q₈

C₂×#261

yes

—

122

#125

—

123

C₂²⋊Q₈

—

124

C₂²⋊Q₈

—

125

C₄×Q₈

—

126

C₄×Q₈

—

127

#117

—

128

C₂×C₄²

—

129

#108

—

130

#107

—

131

#106

—

132

#106

—

133

C₄⋊C₈

—

134

C₄⋊C₈

—

135

C₄×C₈

—

136

C₄×C₈

—

137

C₄×C₈

—

138

C₄×C₈

—

139

C₄×C₈

C₂×#261

yes

characteristic

140

C₂×C₁₃

#1956

yes

characteristic

141

C₂×C₁₃

#1975

yes

—

142

D₁₃

—

143

C₂×Q₈

—

144

C₂×Q₈

—

145

C₂²×C₄

—

146

C₂²×C₄

—

147

M₁₆

—

148

M₁₆

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

C₂²⋊#261

yes

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₂²⋊C₄

—

164

C₄²

—

165

C₄²

—

166

C₄²

—

167

C₄²

C₂²×#261

yes

characteristic

168

C₁₃

#9861

yes

characteristic, complemented, Sylow

169

C₂³

—

170

Q₈

—

171

Q₈

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

D₄×#261

yes

—

180

C₂×C₄

C₂×C₂²⋊#261

yes

characteristic

181

C₈

—

182

C₈

—

183

C₈

—

104

—

184

C₈

—

104

—

185

C₈

#88117

yes

—

186

C₂²

—

187

C₂²

—

188

C₂²

#11719151

yes

characteristic

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

—

104

—

194

C₄

#11719108

yes

—

195

C₄

C₂×#88117

yes

—

196

C₂

—

197

C₂

#13130963

yes

characteristic

198

C₂

#13130992

yes

—

	subgroup	G/H
center	C₂²	#11719151
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₄²	C₂²×#261
Fitting	C₁₃×#498	C₄
socle	C₂²×C₁₃	#390
prime radical	C₂²×D₁₃	#124
cosocle	C₄²×D₁₃	C₂³
hypercenter	#498	#261
hypocenter	C₁₃	#9861
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

#11719151

commutator

C₄²×C₁₃

C₂²×C₄

Frattini

C₄²

C₂²×#261

Fitting

C₁₃×#498

C₄

C₂²×C₁₃

C₂²×D₁₃

C₄²×D₁₃

C₂³

C₁₃

G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
168. C₁₃	#9861	semidirect

H ≅ G/N

type

168. C₁₃

#9861

semidirect

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

subgroup	index
#13130777	2
#13130186	2
#13130185	2
#13130030	2
#13125345	2
#9861	13

subgroup

index

Subgroups — #16785021

Conjugacy classes of subgroups

Complements

Maximal subgroups