Group #14653233 — subgroups

Subgroups — #14653233

Order 2816.

414 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 159 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	#13068788	C₂	1	1	yes	maximal, characteristic
2	#13067301	C₂	1	2	yes	maximal
3	#13067297	C₂	1	2	yes	maximal
4	#13066899	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₁×#500	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₁×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₁₁×#500	—	2	1	no	—
8	C₁₁×#500	—	2	1	no	—
9	C₁₁×#500	C₄	1	4	yes	—
10	C₂×C₁₁×#107	—	2	2	no	—
11	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
12	C₂×#66883	—	2	2	no	—
13	#10622538	C₂²	1	1	yes	characteristic
14	#10622538	C₂²	1	1	yes	characteristic
15	#10622537	C₂²	1	2	yes	—
16	#10622185	—	2	4	no	—
17	#10622184	—	2	4	no	—
18	#10622093	—	2	1	no	—
19	C₈×C₁₁⋊C₈	—	2	1	no	—
20	C₂²×Q₈×C₁₁	—	2	2	no	—
21	C₂²×C₈×C₁₁	D₄	1	2	yes	—
22	C₁₁×C₄⋊Q₈	—	2	2	no	—
23	C₁₁×C₄⋊Q₈	—	2	2	no	—
24	C₁₁×C₄⋊Q₈	—	2	2	no	—
25	C₁₁×C₄⋊Q₈	C₂×C₄	1	4	yes	—
26	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
27	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
28	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
29	C₂×#4318	—	2	2	no	—
30	C₁₁×#107	—	2	4	no	—
31	C₁₁×#107	—	2	4	no	—
32	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
34	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
35	C₂²⋊(C₁₁⋊C₈)	—	2	1	no	—
36	C₂²⋊(C₁₁⋊C₈)	—	2	1	no	—
37	#66883	—	4	4	no	—
38	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
39	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
40	C₄⋊(C₁₁⋊C₈)	D₄	1	2	yes	—
41	C₄⋊(C₁₁⋊C₈)	D₄	1	2	yes	—
42	C₄×C₁₁⋊C₈	—	2	1	no	—
43	C₄×C₁₁⋊C₈	—	2	1	no	—
44	#9771	—	11	1	no	maximal, Sylow
45	C₂×Q₈×C₁₁	—	2	4	no	—
46	C₂×Q₈×C₁₁	—	4	2	no	—
47	C₂×Q₈×C₁₁	—	8	4	no	—
48	C₂²×C₄×C₁₁	—	4	2	no	—
49	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
51	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
52	C₂×C₈×C₁₁	C₂²⋊C₄	1	2	yes	—
53	C₂×C₈×C₁₁	C₂²⋊C₄	1	2	yes	—
54	C₂×C₈×C₁₁	SD₁₆	1	4	yes	—
55	C₂×C₈×C₁₁	Q₁₆	1	4	yes	—
56	C₁₁×C₄⋊C₄	—	2	4	no	—
57	C₁₁×C₄⋊C₄	—	2	4	no	—
58	C₁₁×C₄⋊C₄	—	4	2	no	—
59	C₁₁×C₄⋊C₄	—	4	2	no	—
60	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
61	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
63	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
64	#4318	—	4	4	no	—
65	C₂×C₁₁⋊C₈	—	2	2	no	—
66	C₂×C₁₁⋊C₈	—	2	2	no	—
67	C₂×C₁₁⋊C₈	—	4	2	no	—
68	C₂×#500	D₁₁	1	1	yes	characteristic
69	#1848	—	11	1	no	—
70	#1549	—	11	2	no	—
71	#1548	—	11	2	no	—
72	#1450	—	11	1	no	—
73	C₂³×C₁₁	#124	1	1	yes	characteristic
74	Q₈×C₁₁	—	8	4	no	—
75	Q₈×C₁₁	—	8	4	no	—
76	C₂×C₄×C₁₁	—	2	2	no	—
77	C₂×C₄×C₁₁	—	2	2	no	—
78	C₂×C₄×C₁₁	—	4	4	no	—
79	C₂×C₄×C₁₁	—	8	4	no	—
80	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
82	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
83	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
84	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
85	C₂×C₄×C₁₁	#127	1	1	yes	characteristic
86	C₂×C₄×C₁₁	C₂×SD₁₆	1	2	yes	—
87	C₂×C₄×C₁₁	C₂×Q₁₆	1	2	yes	—
88	C₈×C₁₁	#103	1	4	yes	—
89	C₈×C₁₁	#103	1	4	yes	—
90	C₁₁⋊C₈	—	4	2	no	—
91	C₁₁⋊C₈	—	4	2	no	—
92	C₁₁⋊C₈	—	8	2	no	—
93	C₂×C₄⋊Q₈	C₂×D₁₁	1	2	yes	—
94	#500	—	2	1	no	—
95	#500	—	2	1	no	—
96	#500	Dic₁₁	1	4	yes	—
97	#431	—	11	1	no	—
98	#431	—	11	1	no	—
99	C₂×#108	—	22	2	no	—
100	C₂×#107	—	2	2	no	—
101	#423	—	11	2	no	—
102	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
103	#333	—	22	4	no	—
104	#332	—	22	4	no	—
105	#322	—	22	1	no	—
106	C₈²	—	22	1	no	—
107	C₂²×C₁₁	—	2	1	no	—
108	C₂²×C₁₁	—	2	1	no	—
109	C₂²×C₁₁	#390	1	1	yes	characteristic
110	C₂²×C₁₁	#487	1	2	yes	—
111	C₄×C₁₁	—	2	1	no	—
112	C₄×C₁₁	—	2	1	no	—
113	C₄×C₁₁	—	8	4	no	—
114	C₄×C₁₁	C₂×#103	1	2	yes	—
115	C₄×C₁₁	#472	1	2	yes	—
116	C₄×C₁₁	#473	1	2	yes	—
117	C₄×C₁₁	#492	1	1	yes	characteristic
118	C₄×C₁₁	#494	1	1	yes	characteristic
119	C₂²×Q₈	—	2	2	no	—
120	C₂×M₁₆	—	22	2	no	—
121	C₂²×C₈	C₂²⋊D₁₁	1	2	yes	—
122	C₄⋊Q₈	—	2	2	no	—
123	C₄⋊Q₈	—	2	2	no	—
124	C₄⋊Q₈	—	2	2	no	—
125	C₄⋊Q₈	C₂×Dic₁₁	1	4	yes	—
126	C₂×C₄⋊C₄	—	2	2	no	—
127	C₂×C₄⋊C₄	—	2	2	no	—
128	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
129	#108	—	44	4	no	—
130	#107	—	2	4	no	—
131	#107	—	2	4	no	—
132	C₄⋊C₈	—	11	2	no	—
133	C₄⋊C₈	—	11	2	no	—
134	C₄⋊C₈	—	22	2	no	—
135	C₄⋊C₈	—	22	2	no	—
136	C₂²⋊C₈	—	22	1	no	—
137	C₂²⋊C₈	—	22	1	no	—
138	C₄×C₈	—	22	1	no	—
139	C₄×C₈	—	22	1	no	—
140	C₄×C₈	C₂×Dic₁₁	1	2	yes	—
141	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
142	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
143	C₂×C₁₁	—	2	2	no	—
144	C₂×C₁₁	#1956	1	1	yes	characteristic
145	C₂×C₁₁	#1978	1	1	yes	characteristic
146	C₂×C₁₁	#1979	1	1	yes	characteristic
147	C₂×Q₈	—	2	4	no	—
148	C₂×Q₈	—	4	2	no	—
149	C₂×Q₈	—	8	4	no	—
150	C₂²×C₄	—	4	2	no	—
151	C₂²×C₄	#4341	1	1	yes	characteristic
152	C₂²×C₄	#4341	1	1	yes	characteristic
153	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
154	M₁₆	—	44	4	no	—
155	C₂×C₈	—	22	2	no	—
156	C₂×C₈	—	22	2	no	—
157	C₂×C₈	—	44	2	no	—
158	C₂×C₈	#4325	1	4	yes	—
159	C₂×C₈	#4326	1	4	yes	—
160	C₂×C₈	C₂²⋊Dic₁₁	1	2	yes	—
161	C₂×C₈	C₂²⋊Dic₁₁	1	2	yes	—
162	C₄⋊C₄	—	2	4	no	—
163	C₄⋊C₄	—	2	4	no	—
164	C₄⋊C₄	—	4	2	no	—
165	C₄⋊C₄	—	4	2	no	—
166	C₄²	D₄×D₁₁	1	1	yes	characteristic
167	C₄²	D₄×D₁₁	1	1	yes	characteristic
168	C₄²	C₂²×Dic₁₁	1	1	yes	characteristic
169	C₄²	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
170	C₁₁	#9771	1	1	yes	characteristic, complemented, Sylow
171	C₂³	#66989	1	1	yes	characteristic
172	Q₈	—	8	4	no	—
173	Q₈	—	8	4	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	4	4	no	—
177	C₂×C₄	—	8	4	no	—
178	C₂×C₄	D₄×Dic₁₁	1	1	yes	characteristic
179	C₂×C₄	D₄×Dic₁₁	1	1	yes	characteristic
180	C₂×C₄	#66991	1	1	yes	characteristic
181	C₂×C₄	#66991	1	1	yes	characteristic
182	C₂×C₄	#66993	1	1	yes	characteristic
183	C₂×C₄	C₂×#4325	1	2	yes	—
184	C₂×C₄	C₂×#4326	1	2	yes	—
185	C₂×C₄	C₂×C₂²⋊Dic₁₁	1	1	yes	characteristic
186	C₈	—	44	2	no	—
187	C₈	—	44	2	no	—
188	C₈	—	88	2	no	—
189	C₈	#66899	1	4	yes	—
190	C₈	#66899	1	4	yes	—
191	C₂²	—	2	1	no	—
192	C₂²	—	2	1	no	—
193	C₂²	#10622694	1	2	yes	—
194	C₂²	#10622834	1	1	yes	characteristic
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	8	4	no	—
198	C₄	#10622698	1	1	yes	characteristic
199	C₄	#10622705	1	1	yes	characteristic
200	C₄	#10622773	1	2	yes	—
201	C₄	#10622805	1	2	yes	—
202	C₄	C₂×#66899	1	2	yes	—
203	C₂	—	2	2	no	—
204	C₂	#13069419	1	1	yes	characteristic
205	C₂	#13069527	1	1	yes	characteristic
206	C₂	#13069532	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13068788

C₂

yes

maximal, characteristic

#13067301

C₂

yes

maximal

#13067297

C₂

yes

maximal

#13066899

C₂

yes

maximal, characteristic

C₂×C₁₁×#500

C₂

yes

maximal, characteristic

C₂×C₁₁×C₄⋊Q₈

C₂²

yes

—

C₁₁×#500

—

C₁₁×#500

—

C₁₁×#500

C₄

yes

—

C₂×C₁₁×#107

—

C₂×C₄×C₈×C₁₁

C₂²

yes

characteristic

C₂×#66883

—

#10622538

C₂²

yes

characteristic

#10622538

C₂²

yes

characteristic

#10622537

C₂²

yes

—

#10622185

—

#10622184

—

#10622093

—

C₈×C₁₁⋊C₈

—

C₂²×Q₈×C₁₁

—

C₂²×C₈×C₁₁

D₄

yes

—

C₁₁×C₄⋊Q₈

—

C₁₁×C₄⋊Q₈

—

C₁₁×C₄⋊Q₈

—

C₁₁×C₄⋊Q₈

C₂×C₄

yes

—

C₂×C₁₁×C₄⋊C₄

—

C₂×C₁₁×C₄⋊C₄

—

C₂×C₄²×C₁₁

C₂³

yes

characteristic

C₂×#4318

—

C₁₁×#107

—

C₁₁×#107

—

C₄×C₈×C₁₁

C₂×C₄

yes

—

C₄×C₈×C₁₁

D₄

yes

characteristic

C₄×C₈×C₁₁

D₄

yes

characteristic

C₂²⋊(C₁₁⋊C₈)

—

C₂²⋊(C₁₁⋊C₈)

—

#66883

—

C₄⋊(C₁₁⋊C₈)

—

C₄⋊(C₁₁⋊C₈)

—

C₄⋊(C₁₁⋊C₈)

D₄

yes

—

C₄⋊(C₁₁⋊C₈)

D₄

yes

—

C₄×C₁₁⋊C₈

—

C₄×C₁₁⋊C₈

—

#9771

—

maximal, Sylow

C₂×Q₈×C₁₁

—

C₂×Q₈×C₁₁

—

C₂×Q₈×C₁₁

—

C₂²×C₄×C₁₁

—

C₂²×C₄×C₁₁

C₂×D₄

yes

C₂²×C₄×C₁₁

yes

C₂²×C₄×C₁₁

yes

C₂×C₈×C₁₁

C₂²⋊C₄

yes

—

C₂×C₈×C₁₁

C₂²⋊C₄

yes

—

C₂×C₈×C₁₁

SD₁₆

yes

—

C₂×C₈×C₁₁

Q₁₆

yes

—

C₁₁×C₄⋊C₄

—

C₁₁×C₄⋊C₄

—

C₁₁×C₄⋊C₄

—

C₁₁×C₄⋊C₄

—

C₄²×C₁₁

C₂²×C₄

yes

characteristic

C₄²×C₁₁

C₂×D₄

yes

characteristic

C₄²×C₁₁

C₂×D₄

yes

characteristic

C₄²×C₁₁

C₂×D₄

yes

characteristic

#4318

—

C₂×C₁₁⋊C₈

—

C₂×C₁₁⋊C₈

—

C₂×C₁₁⋊C₈

—

C₂×#500

D₁₁

yes

—

—

—

—

C₂³×C₁₁

#124

yes

characteristic

Q₈×C₁₁

—

Q₈×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₁

C₄×D₄

yes

characteristic

C₂×C₄×C₁₁

C₄×D₄

yes

C₂×C₄×C₁₁

yes

C₂×C₄×C₁₁

yes

C₂×C₄×C₁₁

yes

C₂×C₄×C₁₁

C₂×SD₁₆

yes

—

C₂×C₄×C₁₁

C₂×Q₁₆

yes

—

C₈×C₁₁

#103

yes

—

C₈×C₁₁

#103

yes

—

C₁₁⋊C₈

—

C₁₁⋊C₈

—

C₁₁⋊C₈

—

C₂×C₄⋊Q₈

C₂×D₁₁

yes

—

#500

—

#500

—

#500

Dic₁₁

yes

—

#431

—

#431

—

C₂×#108

—

100

C₂×#107

—

101

#423

—

102

C₂×C₄×C₈

C₂×D₁₁

yes

characteristic

103

#333

—

104

#332

—

105

#322

—

106

C₈²

—

107

C₂²×C₁₁

—

108

C₂²×C₁₁

—

109

C₂²×C₁₁

#390

yes

characteristic

110

C₂²×C₁₁

#487

yes

—

111

C₄×C₁₁

—

112

C₄×C₁₁

—

113

C₄×C₁₁

—

114

C₄×C₁₁

C₂×#103

yes

—

115

C₄×C₁₁

#472

yes

—

116

C₄×C₁₁

#473

yes

—

117

C₄×C₁₁

#492

yes

characteristic

118

C₄×C₁₁

#494

yes

characteristic

119

C₂²×Q₈

—

120

C₂×M₁₆

—

121

C₂²×C₈

C₂²⋊D₁₁

yes

—

122

C₄⋊Q₈

—

123

C₄⋊Q₈

—

124

C₄⋊Q₈

—

125

C₄⋊Q₈

C₂×Dic₁₁

yes

—

126

C₂×C₄⋊C₄

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄²

C₂²×D₁₁

yes

characteristic

129

#108

—

130

#107

—

131

#107

—

132

C₄⋊C₈

—

133

C₄⋊C₈

—

134

C₄⋊C₈

—

135

C₄⋊C₈

—

136

C₂²⋊C₈

—

137

C₂²⋊C₈

—

138

C₄×C₈

—

139

C₄×C₈

—

140

C₄×C₈

C₂×Dic₁₁

yes

—

141

C₄×C₈

C₂²⋊D₁₁

yes

characteristic

142

C₄×C₈

C₂²⋊D₁₁

yes

characteristic

143

C₂×C₁₁

—

144

C₂×C₁₁

#1956

yes

characteristic

145

C₂×C₁₁

#1978

yes

characteristic

146

C₂×C₁₁

#1979

yes

characteristic

147

C₂×Q₈

—

148

C₂×Q₈

—

149

C₂×Q₈

—

150

C₂²×C₄

—

151

C₂²×C₄

#4341

yes

characteristic

152

C₂²×C₄

#4341

yes

characteristic

153

C₂²×C₄

C₂×C₂²⋊D₁₁

yes

characteristic

154

M₁₆

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

#4325

yes

—

159

C₂×C₈

#4326

yes

—

160

C₂×C₈

C₂²⋊Dic₁₁

yes

—

161

C₂×C₈

C₂²⋊Dic₁₁

yes

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄²

D₄×D₁₁

yes

characteristic

167

C₄²

D₄×D₁₁

yes

characteristic

168

C₄²

C₂²×Dic₁₁

yes

characteristic

169

C₄²

C₂×C₂²⋊D₁₁

yes

characteristic

170

C₁₁

#9771

yes

characteristic, complemented, Sylow

171

C₂³

#66989

yes

characteristic

172

Q₈

—

173

Q₈

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

D₄×Dic₁₁

yes

characteristic

179

C₂×C₄

D₄×Dic₁₁

yes

characteristic

180

C₂×C₄

#66991

yes

characteristic

181

C₂×C₄

#66991

yes

characteristic

182

C₂×C₄

#66993

yes

characteristic

183

C₂×C₄

C₂×#4325

yes

—

184

C₂×C₄

C₂×#4326

yes

—

185

C₂×C₄

C₂×C₂²⋊Dic₁₁

yes

characteristic

186

C₈

—

187

C₈

—

188

C₈

—

189

C₈

#66899

yes

—

190

C₈

#66899

yes

—

191

C₂²

—

192

C₂²

—

193

C₂²

#10622694

yes

—

194

C₂²

#10622834

yes

characteristic

195

C₄

—

196

C₄

—

197

C₄

—

198

C₄

#10622698

yes

characteristic

199

C₄

#10622705

yes

characteristic

200

C₄

#10622773

yes

—

201

C₄

#10622805

yes

—

202

C₄

C₂×#66899

yes

—

203

C₂

—

204

C₂

#13069419

yes

characteristic

205

C₂

#13069527

yes

characteristic

206

C₂

#13069532

yes

characteristic

	subgroup	G/H
center	C₂²	#10622834
commutator	C₄²×C₁₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₁
Fitting	C₂×C₁₁×#500	C₂
socle	C₂²×C₁₁	#390
prime radical	C₂³×C₁₁	#124
cosocle	C₂×C₄²×C₁₁	C₂³
hypercenter	C₂×#500	D₁₁
hypocenter	C₁₁	#9771
LCS	G ▷ C₄²×C₁₁ ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#500

subgroup

G/H

center

C₂²

#10622834

commutator

C₄²×C₁₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₁

Fitting

C₂×C₁₁×#500

C₂

C₂²×C₁₁

C₂³×C₁₁

C₂×C₄²×C₁₁

C₂³

D₁₁

C₁₁

G ▷ C₄²×C₁₁ ▷ C₂²×C₁₁ ▷ C₁₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#500

N	H ≅ G/N	type
170. C₁₁	#9771	semidirect

H ≅ G/N

type

170. C₁₁

#9771

semidirect

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

subgroup	index
#13068788	2
#13067301	2
#13067297	2
#13066899	2
C₂×C₁₁×#500	2
#9771	11

subgroup

index

C₂×C₁₁×#500

Subgroups — #14653233

Conjugacy classes of subgroups

Complements

Maximal subgroups