Group #16776911 — subgroups

Subgroups — #16776911

Order 3328.

332 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 122 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	2	yes	maximal
2	#13114987	C₂	1	1	yes	maximal, characteristic
3	C₂×#88112	—	2	2	no	—
4	C₂×#88112	C₄	1	4	yes	—
5	C₂×C₄⋊#1057	—	2	2	no	—
6	C₂×C₁₃×C₄⋊C₈	C₄	1	1	yes	characteristic
7	C₂×#88054	—	2	2	no	—
8	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊#1055	C₄	1	1	yes	characteristic
10	C₂²×#1057	—	2	4	no	—
11	C₂²×C₈×C₁₃	—	2	1	no	—
12	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
14	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
15	C₂×C₄×Dic₁₃	D₄	1	2	yes	—
16	C₂×C₄×Dic₁₃	Q₈	1	1	yes	characteristic
17	C₂²×#1055	—	2	1	no	—
18	#88112	—	4	4	no	—
19	#88112	—	4	2	no	—
20	#88112	—	4	2	no	—
21	C₄⋊#1057	—	4	2	no	—
22	C₄⋊#1057	—	4	2	no	—
23	C₁₃×C₄⋊C₈	—	4	1	no	—
24	#88054	—	4	2	no	—
25	#88054	—	4	2	no	—
26	C₄⋊#1055	—	4	1	no	—
27	#7050	—	13	1	no	maximal, Sylow
28	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
29	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×Dic₁₃	C₂²⋊C₄	1	2	yes	—
32	C₂²×Dic₁₃	C₄⋊C₄	1	2	yes	—
33	C₂×#1057	—	4	4	no	—
34	C₂×#1057	—	4	4	no	—
35	C₂×#1057	—	4	4	no	—
36	C₂×C₈×C₁₃	—	4	1	no	—
37	C₂×C₈×C₁₃	—	4	2	no	—
38	C₄²×C₁₃	—	2	1	no	—
39	C₄²×C₁₃	—	2	1	no	—
40	C₄×Dic₁₃	—	2	2	no	—
41	C₄×Dic₁₃	—	2	2	no	—
42	C₄×Dic₁₃	—	4	1	no	—
43	C₄×Dic₁₃	D₈	1	1	yes	characteristic
44	C₄×Dic₁₃	D₈	1	2	yes	—
45	C₄×Dic₁₃	SD₁₆	1	2	yes	—
46	C₄×Dic₁₃	SD₁₆	1	2	yes	—
47	C₄×Dic₁₃	SD₁₆	1	2	yes	—
48	C₄×Dic₁₃	Q₁₆	1	1	yes	characteristic
49	C₄×Dic₁₃	Q₁₆	1	2	yes	—
50	C₂×#1055	—	4	1	no	—
51	C₂×#1055	—	4	2	no	—
52	#1842	—	13	2	no	—
53	#1842	—	13	1	no	—
54	C₂³×C₁₃	#95	1	1	yes	characteristic
55	C₂×C₄×C₁₃	—	2	1	no	—
56	C₂×C₄×C₁₃	—	2	1	no	—
57	C₂×C₄×C₁₃	—	2	2	no	—
58	C₂×C₄×C₁₃	—	2	1	no	—
59	C₂×C₄×C₁₃	#100	1	1	yes	characteristic
60	C₂×C₄×C₁₃	#101	1	1	yes	characteristic
61	C₂×C₄×C₁₃	#102	1	2	yes	—
62	C₂×C₄×C₁₃	#103	1	2	yes	—
63	C₂×C₄×C₁₃	#106	1	1	yes	characteristic
64	C₂×C₄×C₁₃	#107	1	1	yes	characteristic
65	C₂×Dic₁₃	—	2	2	no	—
66	C₂×Dic₁₃	—	2	2	no	—
67	C₂×Dic₁₃	#100	1	2	yes	—
68	C₂×Dic₁₃	#101	1	2	yes	—
69	C₂×Dic₁₃	#102	1	2	yes	—
70	C₂×Dic₁₃	#102	1	2	yes	—
71	C₂×Dic₁₃	#103	1	2	yes	—
72	C₂×Dic₁₃	#103	1	2	yes	—
73	C₂×Dic₁₃	#106	1	2	yes	—
74	C₂×Dic₁₃	#107	1	2	yes	—
75	#1057	—	8	4	no	—
76	C₈×C₁₃	—	8	1	no	—
77	#1055	—	8	1	no	—
78	C₂×C₄⋊C₈	—	13	4	no	—
79	C₂×C₄⋊C₈	—	13	1	no	—
80	C₂×C₄⋊C₈	—	26	2	no	—
81	C₂×C₄⋊C₈	—	26	2	no	—
82	C₂×C₄⋊C₈	—	26	2	no	—
83	C₂×C₄⋊C₈	#261	1	1	yes	characteristic
84	C₄³	—	13	1	no	—
85	C₂²×C₁₃	#338	1	1	yes	characteristic
86	C₂²×C₁₃	#340	1	2	yes	—
87	C₂²×C₁₃	#340	1	1	yes	characteristic
88	C₂²×C₁₃	#343	1	2	yes	—
89	C₂²×C₁₃	#343	1	1	yes	characteristic
90	C₄×C₁₃	—	2	1	no	—
91	C₄×C₁₃	—	2	1	no	—
92	C₄×C₁₃	—	4	1	no	—
93	C₄×C₁₃	#331	1	1	yes	characteristic
94	C₄×C₁₃	#332	1	2	yes	—
95	C₄×C₁₃	#333	1	1	yes	characteristic
96	Dic₁₃	—	2	2	no	—
97	Dic₁₃	—	2	2	no	—
98	Dic₁₃	#331	1	2	yes	—
99	Dic₁₃	#332	1	2	yes	—
100	Dic₁₃	#332	1	2	yes	—
101	Dic₁₃	#333	1	2	yes	—
102	C₂²×C₈	—	2	1	no	—
103	C₂²×C₈	—	26	4	no	—
104	C₂²×C₈	—	26	1	no	—
105	C₂×C₄²	—	13	1	no	—
106	C₂×C₄²	—	13	2	no	—
107	C₂×C₄²	—	13	2	no	—
108	C₂×C₄²	—	13	1	no	—
109	C₂×C₄²	C₂×#261	1	1	yes	characteristic
110	C₄⋊C₈	—	4	1	no	—
111	C₄⋊C₈	—	52	1	no	—
112	C₄⋊C₈	—	52	2	no	—
113	C₄⋊C₈	—	52	2	no	—
114	C₄⋊C₈	—	52	2	no	—
115	C₄⋊C₈	—	52	2	no	—
116	C₄⋊C₈	—	52	2	no	—
117	C₄⋊C₈	—	52	2	no	—
118	C₄⋊C₈	—	52	4	no	—
119	C₂×C₁₃	#1292	1	2	yes	—
120	C₂×C₁₃	#1292	1	1	yes	characteristic
121	C₂×C₁₃	#1296	1	1	yes	characteristic
122	C₂×C₁₃	#1296	1	2	yes	—
123	C₂×C₁₃	#1309	1	1	yes	characteristic
124	C₂²×C₄	—	13	2	no	—
125	C₂²×C₄	—	13	2	no	—
126	C₂²×C₄	C₄×#261	1	1	yes	characteristic
127	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
128	C₂²×C₄	C₂²⋊#261	1	1	yes	characteristic
129	C₂×C₈	—	4	2	no	—
130	C₂×C₈	—	4	1	no	—
131	C₂×C₈	—	52	4	no	—
132	C₂×C₈	—	52	4	no	—
133	C₂×C₈	—	52	4	no	—
134	C₂×C₈	—	52	2	no	—
135	C₂×C₈	—	52	1	no	—
136	C₄²	—	2	1	no	—
137	C₄²	—	2	1	no	—
138	C₄²	—	13	2	no	—
139	C₄²	—	13	1	no	—
140	C₄²	—	13	1	no	—
141	C₄²	—	13	2	no	—
142	C₄²	—	13	2	no	—
143	C₄²	—	13	2	no	—
144	C₄²	—	13	2	no	—
145	C₄²	—	26	2	no	—
146	C₄²	—	26	2	no	—
147	C₄²	—	52	1	no	—
148	C₁₃	#7050	1	1	yes	characteristic, complemented, Sylow
149	C₂³	(C₂×#261)⋊C₄	1	1	yes	characteristic
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	13	2	no	—
155	C₂×C₄	—	13	2	no	—
156	C₂×C₄	—	13	2	no	—
157	C₂×C₄	—	13	2	no	—
158	C₂×C₄	—	13	2	no	—
159	C₂×C₄	—	13	2	no	—
160	C₂×C₄	—	13	2	no	—
161	C₂×C₄	—	13	2	no	—
162	C₂×C₄	—	26	2	no	—
163	C₂×C₄	—	26	2	no	—
164	C₂×C₄	#88101	1	1	yes	characteristic
165	C₂×C₄	#88102	1	1	yes	characteristic
166	C₂×C₄	#88113	1	1	yes	characteristic
167	C₂×C₄	#88115	1	2	yes	—
168	C₂×C₄	#88117	1	2	yes	—
169	C₂×C₄	#88121	1	1	yes	characteristic
170	C₈	—	8	1	no	—
171	C₈	—	104	4	no	—
172	C₈	—	104	1	no	—
173	C₂²	#11718246	1	2	yes	—
174	C₂²	#11718265	1	1	yes	characteristic
175	C₂²	#11718269	1	1	yes	characteristic
176	C₂²	#11718271	1	2	yes	—
177	C₂²	#11718291	1	1	yes	characteristic
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	—	4	1	no	—
181	C₄	—	13	2	no	—
182	C₄	—	13	2	no	—
183	C₄	—	13	2	no	—
184	C₄	—	13	2	no	—
185	C₄	—	26	2	no	—
186	C₄	—	26	2	no	—
187	C₄	#11718245	1	2	yes	—
188	C₄	#11718297	1	1	yes	characteristic
189	C₄	#11718303	1	1	yes	characteristic
190	C₂	#13129705	1	2	yes	—
191	C₂	#13129706	1	1	yes	characteristic
192	C₂	#13129714	1	1	yes	characteristic
193	C₂	#13129715	1	2	yes	—
194	C₂	#13129744	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13130769

C₂

yes

maximal

#13114987

C₂

yes

maximal, characteristic

C₂×#88112

—

C₂×#88112

C₄

yes

—

C₂×C₄⋊#1057

—

C₂×C₁₃×C₄⋊C₈

C₄

yes

characteristic

C₂×#88054

—

C₄²×Dic₁₃

C₂²

yes

characteristic

C₂×C₄⋊#1055

C₄

yes

characteristic

C₂²×#1057

—

C₂²×C₈×C₁₃

—

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

C₂×C₄×Dic₁₃

C₂×C₄

yes

—

C₂×C₄×Dic₁₃

D₄

yes

characteristic

C₂×C₄×Dic₁₃

D₄

yes

—

C₂×C₄×Dic₁₃

Q₈

yes

characteristic

C₂²×#1055

—

—

—

—

—

—

C₁₃×C₄⋊C₈

—

—

—

—

—

C₂²×C₄×C₁₃

C₄²

yes

characteristic

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₃

C₄⋊C₄

yes

characteristic

C₂²×Dic₁₃

C₂²⋊C₄

yes

—

C₂²×Dic₁₃

C₄⋊C₄

yes

—

C₂×#1057

—

C₂×#1057

—

C₂×#1057

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

D₈

yes

characteristic

C₄×Dic₁₃

D₈

yes

—

C₄×Dic₁₃

SD₁₆

yes

—

C₄×Dic₁₃

SD₁₆

yes

—

C₄×Dic₁₃

SD₁₆

yes

—

C₄×Dic₁₃

Q₁₆

yes

characteristic

C₄×Dic₁₃

Q₁₆

yes

—

—

—

—

—

C₂³×C₁₃

#95

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

—

C₂×C₄×C₁₃

#103

yes

—

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

#100

yes

—

C₂×Dic₁₃

#101

yes

—

C₂×Dic₁₃

#102

yes

—

C₂×Dic₁₃

#102

yes

—

C₂×Dic₁₃

#103

yes

—

C₂×Dic₁₃

#103

yes

—

C₂×Dic₁₃

#106

yes

—

C₂×Dic₁₃

#107

yes

—

#1057

—

C₈×C₁₃

—

#1055

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

#261

yes

characteristic

C₄³

—

C₂²×C₁₃

#338

yes

characteristic

C₂²×C₁₃

#340

yes

—

C₂²×C₁₃

#340

yes

characteristic

C₂²×C₁₃

#343

yes

—

C₂²×C₁₃

#343

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#331

yes

characteristic

C₄×C₁₃

#332

yes

—

C₄×C₁₃

#333

yes

characteristic

Dic₁₃

—

Dic₁₃

—

Dic₁₃

#331

yes

—

Dic₁₃

#332

yes

—

100

Dic₁₃

#332

yes

—

101

Dic₁₃

#333

yes

—

102

C₂²×C₈

—

103

C₂²×C₈

—

104

C₂²×C₈

—

105

C₂×C₄²

—

106

C₂×C₄²

—

107

C₂×C₄²

—

108

C₂×C₄²

—

109

C₂×C₄²

C₂×#261

yes

characteristic

110

C₄⋊C₈

—

111

C₄⋊C₈

—

112

C₄⋊C₈

—

113

C₄⋊C₈

—

114

C₄⋊C₈

—

115

C₄⋊C₈

—

116

C₄⋊C₈

—

117

C₄⋊C₈

—

118

C₄⋊C₈

—

119

C₂×C₁₃

#1292

yes

—

120

C₂×C₁₃

#1292

yes

characteristic

121

C₂×C₁₃

#1296

yes

characteristic

122

C₂×C₁₃

#1296

yes

—

123

C₂×C₁₃

#1309

yes

characteristic

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

C₄×#261

yes

characteristic

127

C₂²×C₄

C₄⋊#261

yes

characteristic

128

C₂²×C₄

C₂²⋊#261

yes

characteristic

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₄²

—

137

C₄²

—

138

C₄²

—

139

C₄²

—

140

C₄²

—

141

C₄²

—

142

C₄²

—

143

C₄²

—

144

C₄²

—

145

C₄²

—

146

C₄²

—

147

C₄²

—

148

C₁₃

#7050

yes

characteristic, complemented, Sylow

149

C₂³

(C₂×#261)⋊C₄

yes

characteristic

150

C₂×C₄

—

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

—

163

C₂×C₄

—

164

C₂×C₄

#88101

yes

characteristic

165

C₂×C₄

#88102

yes

characteristic

166

C₂×C₄

#88113

yes

characteristic

167

C₂×C₄

#88115

yes

—

168

C₂×C₄

#88117

yes

—

169

C₂×C₄

#88121

yes

characteristic

170

C₈

—

171

C₈

—

104

—

172

C₈

—

104

—

173

C₂²

#11718246

yes

—

174

C₂²

#11718265

yes

characteristic

175

C₂²

#11718269

yes

characteristic

176

C₂²

#11718271

yes

—

177

C₂²

#11718291

yes

characteristic

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

—

187

C₄

#11718245

yes

—

188

C₄

#11718297

yes

characteristic

189

C₄

#11718303

yes

characteristic

190

C₂

#13129705

yes

—

191

C₂

#13129706

yes

characteristic

192

C₂

#13129714

yes

characteristic

193

C₂

#13129715

yes

—

194

C₂

#13129744

yes

characteristic

	subgroup	G/H
center	C₂³	(C₂×#261)⋊C₄
commutator	C₂²×C₄×C₁₃	C₄²
Frattini	C₂×C₄²	C₂×#261
Fitting	C₂×C₁₃×C₄⋊C₈	C₄
socle, prime radical	C₂³×C₁₃	#95
cosocle	C₄²×Dic₁₃	C₂²
hypercenter	C₂×C₄⋊C₈	#261
hypocenter	C₁₃	#7050
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

(C₂×#261)⋊C₄

commutator

C₂²×C₄×C₁₃

C₄²

Frattini

C₂×C₄²

C₂×#261

Fitting

C₂×C₁₃×C₄⋊C₈

C₄

socle, prime radical

C₂³×C₁₃

#95

cosocle

C₄²×Dic₁₃

C₂²

C₂×C₄⋊C₈

C₁₃

G ▷ C₂²×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
148. C₁₃	#7050	semidirect

H ≅ G/N

type

148. C₁₃

#7050

semidirect

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

subgroup	index
#13130769	2
#13114987	2
#7050	13

subgroup

index

#13130769

#13114987

#7050

Subgroups — #16776911

Conjugacy classes of subgroups

Complements

Maximal subgroups