Group #14653917 — subgroups

Subgroups — #14653917

Order 2816.

438 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 167 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	#13068768	C₂	1	1	yes	maximal, characteristic
2	#13068469	C₂	1	1	yes	maximal, characteristic
3	#13066499	C₂	1	4	yes	maximal
4	C₁₁×#2859	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₁×#125	—	2	1	no	—
6	C₁₁×#389	—	2	2	no	—
7	C₁₁×#389	C₄	1	4	yes	—
8	C₁₁×#384	C₂²	1	2	yes	—
9	C₁₁×#382	—	2	1	no	—
10	C₄³×C₁₁	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
12	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
13	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
14	C₂×#66867	—	2	1	no	—
15	#10622162	—	2	1	no	—
16	C₁₁×#125	—	4	4	no	—
17	C₂×C₁₁×C₄⋊C₄	—	2	4	no	—
18	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
19	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
20	C₂×C₁₁×C₄⋊C₄	C₂×C₄	1	4	yes	—
21	C₂×C₄²×C₁₁	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₁₁	D₄	1	2	yes	—
23	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
26	C₂²×C₁₁⋊C₈	—	2	2	no	—
27	C₂²×C₁₁⋊C₈	—	2	2	no	—
28	C₁₁×#95	—	2	4	no	—
29	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
30	C₄⋊(C₁₁⋊C₈)	—	2	4	no	—
31	C₄⋊(C₁₁⋊C₈)	—	4	4	no	—
32	C₄⋊(C₁₁⋊C₈)	Q₈	1	4	yes	—
33	#66867	—	4	2	no	—
34	#9902	—	11	1	no	maximal, Sylow
35	C₂²×C₄×C₁₁	—	2	4	no	—
36	C₂²×C₄×C₁₁	C₂²⋊C₄	1	2	yes	—
37	C₂²×C₄×C₁₁	C₂²⋊C₄	1	2	yes	—
38	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
41	C₁₁×C₄⋊C₄	—	4	8	no	—
42	C₁₁×C₄⋊C₄	—	4	4	no	—
43	C₁₁×C₄⋊C₄	—	4	2	no	—
44	C₁₁×C₄⋊C₄	—	4	4	no	—
45	C₁₁×C₄⋊C₄	—	4	2	no	—
46	C₄²×C₁₁	—	2	2	no	—
47	C₄²×C₁₁	—	2	2	no	—
48	C₄²×C₁₁	—	2	2	no	—
49	C₄²×C₁₁	—	4	2	no	—
50	C₄²×C₁₁	SD₁₆	1	2	yes	—
51	C₄²×C₁₁	Q₁₆	1	2	yes	—
52	C₄²×C₁₁	C₂×Q₈	1	2	yes	—
53	C₄²×C₁₁	#41	1	2	yes	—
54	C₂×C₁₁⋊C₈	—	2	4	no	—
55	C₂×C₁₁⋊C₈	—	4	2	no	—
56	C₂×C₁₁⋊C₈	—	4	2	no	—
57	C₂×C₁₁⋊C₈	—	4	2	no	—
58	C₂×C₁₁⋊C₈	—	4	4	no	—
59	#2859	D₁₁	1	1	yes	characteristic
60	#1842	—	11	1	no	—
61	#1767	—	11	1	no	—
62	#1296	—	11	4	no	—
63	C₂³×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₁₁	—	2	2	no	—
65	C₂×C₄×C₁₁	—	2	2	no	—
66	C₂×C₄×C₁₁	—	2	2	no	—
67	C₂×C₄×C₁₁	—	2	2	no	—
68	C₂×C₄×C₁₁	—	2	2	no	—
69	C₂×C₄×C₁₁	—	2	2	no	—
70	C₂×C₄×C₁₁	—	4	4	no	—
71	C₂×C₄×C₁₁	—	4	4	no	—
72	C₂×C₄×C₁₁	—	4	4	no	—
73	C₂×C₄×C₁₁	#103	1	4	yes	—
74	C₂×C₄×C₁₁	C₄×Q₈	1	2	yes	—
75	C₂×C₄×C₁₁	C₂²⋊Q₈	1	2	yes	—
76	C₂×C₄×C₁₁	#124	1	1	yes	characteristic
77	C₂×C₄×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
78	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₁₁	C₂×Q₁₆	1	1	yes	characteristic
80	C₂×C₄×C₁₁	AGL(1,ℤ/8)	1	2	yes	—
81	C₂×C₄×C₁₁	AGL(1,ℤ/8)	1	2	yes	—
82	C₂×C₄×C₁₁	#137	1	2	yes	—
83	C₁₁⋊C₈	—	4	4	no	—
84	C₁₁⋊C₈	—	8	2	no	—
85	C₂×#125	—	2	1	no	—
86	C₂×C₄⋊C₈	—	11	2	no	—
87	C₂×C₄⋊C₈	—	11	2	no	—
88	C₂×C₄⋊C₈	—	22	2	no	—
89	C₂×#97	—	22	1	no	—
90	#389	—	2	2	no	—
91	#389	Dic₁₁	1	4	yes	—
92	#384	C₂×D₁₁	1	2	yes	—
93	#382	—	2	1	no	—
94	C₄³	C₂×D₁₁	1	1	yes	characteristic
95	#336	—	22	1	no	—
96	C₂²×C₁₁	#391	1	1	yes	characteristic
97	C₂²×C₁₁	#413	1	2	yes	—
98	C₂²×C₁₁	C₂×#103	1	1	yes	characteristic
99	C₂²×C₁₁	#417	1	2	yes	—
100	C₂²×C₁₁	#418	1	1	yes	characteristic
101	C₄×C₁₁	—	2	4	no	—
102	C₄×C₁₁	—	2	2	no	—
103	C₄×C₁₁	—	4	2	no	—
104	C₄×C₁₁	—	8	4	no	—
105	C₄×C₁₁	#476	1	2	yes	—
106	C₄×C₁₁	#477	1	2	yes	—
107	C₄×C₁₁	#490	1	1	yes	characteristic
108	C₄×C₁₁	#491	1	1	yes	characteristic
109	C₄×C₁₁	#501	1	2	yes	—
110	C₂²×C₈	—	22	2	no	—
111	C₂²×C₈	—	22	2	no	—
112	#125	—	4	4	no	—
113	C₂×C₄⋊C₄	—	2	4	no	—
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	C₂×Dic₁₁	1	4	yes	—
117	C₂×C₄²	C₂×Dic₁₁	1	2	yes	—
118	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
119	C₂×C₄²	C₂²⋊D₁₁	1	2	yes	—
120	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
121	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
122	C₄⋊C₈	—	11	4	no	—
123	C₄⋊C₈	—	22	2	no	—
124	C₄⋊C₈	—	22	4	no	—
125	C₄⋊C₈	—	44	4	no	—
126	#97	—	44	2	no	—
127	#95	—	2	4	no	—
128	C₂×C₁₁	#1550	1	2	yes	—
129	C₂×C₁₁	#1557	1	2	yes	—
130	C₂×C₁₁	#1987	1	1	yes	characteristic
131	C₂×C₁₁	#1992	1	1	yes	characteristic
132	C₂×C₁₁	#1998	1	1	yes	characteristic
133	C₂²×C₄	—	2	4	no	—
134	C₂²×C₄	C₂²⋊Dic₁₁	1	2	yes	—
135	C₂²×C₄	C₂²⋊Dic₁₁	1	2	yes	—
136	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
137	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
138	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
139	C₂×C₈	—	22	4	no	—
140	C₂×C₈	—	44	2	no	—
141	C₂×C₈	—	44	2	no	—
142	C₂×C₈	—	44	2	no	—
143	C₂×C₈	—	44	4	no	—
144	C₄⋊C₄	—	4	4	no	—
145	C₄⋊C₄	—	4	8	no	—
146	C₄⋊C₄	—	4	4	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄⋊C₄	—	4	2	no	—
149	C₄²	—	2	2	no	—
150	C₄²	—	2	2	no	—
151	C₄²	—	2	2	no	—
152	C₄²	—	4	2	no	—
153	C₄²	#4325	1	2	yes	—
154	C₄²	#4326	1	2	yes	—
155	C₄²	Q₈×D₁₁	1	2	yes	—
156	C₄²	#4343	1	2	yes	—
157	C₁₁	#9902	1	1	yes	characteristic, complemented, Sylow
158	C₂³	C₂×C₂²⋊Dic₁₁	1	1	yes	characteristic
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	4	4	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	—	4	4	no	—
168	C₂×C₄	#66899	1	4	yes	—
169	C₂×C₄	#66985	1	2	yes	—
170	C₂×C₄	C₂×#4325	1	1	yes	characteristic
171	C₂×C₄	C₂×#4326	1	1	yes	characteristic
172	C₂×C₄	#66997	1	1	yes	characteristic
173	C₂×C₄	Q₈×Dic₁₁	1	2	yes	—
174	C₂×C₄	#66999	1	2	yes	—
175	C₂×C₄	#67000	1	1	yes	characteristic
176	C₂×C₄	#67002	1	2	yes	—
177	C₂×C₄	#67004	1	2	yes	—
178	C₈	—	44	4	no	—
179	C₈	—	88	2	no	—
180	C₂²	#10622828	1	1	yes	characteristic
181	C₂²	C₂×#66899	1	1	yes	characteristic
182	C₂²	#10622843	1	1	yes	characteristic
183	C₂²	#10622845	1	2	yes	—
184	C₂²	#10622850	1	2	yes	—
185	C₄	—	2	4	no	—
186	C₄	—	2	2	no	—
187	C₄	—	4	2	no	—
188	C₄	—	8	4	no	—
189	C₄	#10622678	1	2	yes	—
190	C₄	#10622680	1	1	yes	characteristic
191	C₄	#10622682	1	1	yes	characteristic
192	C₄	#10622697	1	2	yes	—
193	C₄	#10622704	1	2	yes	—
194	C₂	#13067306	1	2	yes	—
195	C₂	#13067338	1	2	yes	—
196	C₂	#13069565	1	1	yes	characteristic
197	C₂	#13069586	1	1	yes	characteristic
198	C₂	#13069610	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13068768

C₂

yes

maximal, characteristic

#13068469

C₂

yes

maximal, characteristic

#13066499

C₂

yes

maximal

C₁₁×#2859

C₂

yes

maximal, characteristic

C₂×C₁₁×#125

—

C₁₁×#389

—

C₁₁×#389

C₄

yes

—

C₁₁×#384

C₂²

yes

—

C₁₁×#382

—

C₄³×C₁₁

C₂²

yes

characteristic

C₂×C₄⋊(C₁₁⋊C₈)

—

C₂×C₄⋊(C₁₁⋊C₈)

C₂²

yes

—

C₂×C₄⋊(C₁₁⋊C₈)

C₂²

yes

—

C₂×#66867

—

#10622162

—

C₁₁×#125

—

C₂×C₁₁×C₄⋊C₄

—

C₂×C₁₁×C₄⋊C₄

—

C₂×C₁₁×C₄⋊C₄

—

C₂×C₁₁×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄²×C₁₁

C₂×C₄

yes

—

C₂×C₄²×C₁₁

D₄

yes

—

C₂×C₄²×C₁₁

D₄

yes

characteristic

C₂×C₄²×C₁₁

D₄

yes

characteristic

C₂×C₄²×C₁₁

C₂³

yes

characteristic

C₂²×C₁₁⋊C₈

—

C₂²×C₁₁⋊C₈

—

C₁₁×#95

—

C₄⋊(C₁₁⋊C₈)

—

C₄⋊(C₁₁⋊C₈)

—

C₄⋊(C₁₁⋊C₈)

—

C₄⋊(C₁₁⋊C₈)

Q₈

yes

—

#66867

—

#9902

—

maximal, Sylow

C₂²×C₄×C₁₁

—

C₂²×C₄×C₁₁

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₁

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₁

C₂×D₄

yes

characteristic

C₁₁×C₄⋊C₄

—

C₁₁×C₄⋊C₄

—

C₁₁×C₄⋊C₄

—

C₁₁×C₄⋊C₄

—

C₁₁×C₄⋊C₄

—

C₄²×C₁₁

—

C₄²×C₁₁

—

C₄²×C₁₁

—

C₄²×C₁₁

—

C₄²×C₁₁

SD₁₆

yes

—

C₄²×C₁₁

Q₁₆

yes

—

C₄²×C₁₁

C₂×Q₈

yes

—

C₄²×C₁₁

#41

yes

—

C₂×C₁₁⋊C₈

—

C₂×C₁₁⋊C₈

—

C₂×C₁₁⋊C₈

—

C₂×C₁₁⋊C₈

—

C₂×C₁₁⋊C₈

—

#2859

D₁₁

yes

—

—

—

C₂³×C₁₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

#103

yes

—

C₂×C₄×C₁₁

C₄×Q₈

yes

—

C₂×C₄×C₁₁

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₁

#124

yes

characteristic

C₂×C₄×C₁₁

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₁₁

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₁₁

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₁₁

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₁₁

#137

yes

—

C₁₁⋊C₈

—

C₁₁⋊C₈

—

C₂×#125

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×#97

—

#389

—

#389

Dic₁₁

yes

—

#384

C₂×D₁₁

yes

—

#382

—

C₄³

C₂×D₁₁

yes

characteristic

#336

—

C₂²×C₁₁

#391

yes

characteristic

C₂²×C₁₁

#413

yes

—

C₂²×C₁₁

C₂×#103

yes

characteristic

C₂²×C₁₁

#417

yes

—

100

C₂²×C₁₁

#418

yes

characteristic

101

C₄×C₁₁

—

102

C₄×C₁₁

—

103

C₄×C₁₁

—

104

C₄×C₁₁

—

105

C₄×C₁₁

#476

yes

—

106

C₄×C₁₁

#477

yes

—

107

C₄×C₁₁

#490

yes

characteristic

108

C₄×C₁₁

#491

yes

characteristic

109

C₄×C₁₁

#501

yes

—

110

C₂²×C₈

—

111

C₂²×C₈

—

112

#125

—

113

C₂×C₄⋊C₄

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄⋊C₄

C₂×Dic₁₁

yes

—

117

C₂×C₄²

C₂×Dic₁₁

yes

—

118

C₂×C₄²

C₂²⋊D₁₁

yes

characteristic

119

C₂×C₄²

C₂²⋊D₁₁

yes

—

120

C₂×C₄²

C₂²⋊D₁₁

yes

characteristic

121

C₂×C₄²

C₂²×D₁₁

yes

characteristic

122

C₄⋊C₈

—

123

C₄⋊C₈

—

124

C₄⋊C₈

—

125

C₄⋊C₈

—

126

#97

—

127

#95

—

128

C₂×C₁₁

#1550

yes

—

129

C₂×C₁₁

#1557

yes

—

130

C₂×C₁₁

#1987

yes

characteristic

131

C₂×C₁₁

#1992

yes

characteristic

132

C₂×C₁₁

#1998

yes

characteristic

133

C₂²×C₄

—

134

C₂²×C₄

C₂²⋊Dic₁₁

yes

—

135

C₂²×C₄

C₂²⋊Dic₁₁

yes

—

136

C₂²×C₄

C₂²×Dic₁₁

yes

characteristic

137

C₂²×C₄

C₂×C₂²⋊D₁₁

yes

characteristic

138

C₂²×C₄

C₂×C₂²⋊D₁₁

yes

characteristic

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄²

—

150

C₄²

—

151

C₄²

—

152

C₄²

—

153

C₄²

#4325

yes

—

154

C₄²

#4326

yes

—

155

C₄²

Q₈×D₁₁

yes

—

156

C₄²

#4343

yes

—

157

C₁₁

#9902

yes

characteristic, complemented, Sylow

158

C₂³

C₂×C₂²⋊Dic₁₁

yes

characteristic

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

#66899

yes

—

169

C₂×C₄

#66985

yes

—

170

C₂×C₄

C₂×#4325

yes

characteristic

171

C₂×C₄

C₂×#4326

yes

characteristic

172

C₂×C₄

#66997

yes

characteristic

173

C₂×C₄

Q₈×Dic₁₁

yes

—

174

C₂×C₄

#66999

yes

—

175

C₂×C₄

#67000

yes

characteristic

176

C₂×C₄

#67002

yes

—

177

C₂×C₄

#67004

yes

—

178

C₈

—

179

C₈

—

180

C₂²

#10622828

yes

characteristic

181

C₂²

C₂×#66899

yes

characteristic

182

C₂²

#10622843

yes

characteristic

183

C₂²

#10622845

yes

—

184

C₂²

#10622850

yes

—

185

C₄

—

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

#10622678

yes

—

190

C₄

#10622680

yes

characteristic

191

C₄

#10622682

yes

characteristic

192

C₄

#10622697

yes

—

193

C₄

#10622704

yes

—

194

C₂

#13067306

yes

—

195

C₂

#13067338

yes

—

196

C₂

#13069565

yes

characteristic

197

C₂

#13069586

yes

characteristic

198

C₂

#13069610

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₁
commutator	C₂²×C₄×C₁₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₁
Fitting	C₁₁×#2859	C₂
socle, prime radical	C₂³×C₁₁	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₁	C₂³
hypercenter	#2859	D₁₁
hypocenter	C₁₁	#9902
LCS	G ▷ C₂²×C₄×C₁₁ ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2859

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₁₁

commutator

C₂²×C₄×C₁₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₁

Fitting

C₁₁×#2859

C₂

socle, prime radical

C₂³×C₁₁

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₁₁

C₂³

D₁₁

C₁₁

G ▷ C₂²×C₄×C₁₁ ▷ C₂²×C₁₁ ▷ C₁₁

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #2859

N	H ≅ G/N	type
157. C₁₁	#9902	semidirect

H ≅ G/N

type

157. C₁₁

#9902

semidirect

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

subgroup	index
#13068768	2
#13068469	2
#13066499	2
C₁₁×#2859	2
#9902	11

subgroup

index

C₁₁×#2859

Subgroups — #14653917

Conjugacy classes of subgroups

Complements

Maximal subgroups