Group #14652855 — subgroups

Subgroups — #14652855

Order 2816.

538 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 175 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	#13068733	C₂	1	1	yes	maximal, characteristic
2	#13068469	C₂	1	1	yes	maximal, characteristic
3	C₂×#10622184	C₂	1	4	yes	maximal
4	C₁₁×#2868	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₁×C₄⋊Q₈	C₄	1	4	yes	—
6	C₂×C₁₁×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₁₁×#384	—	2	1	no	—
8	C₁₁×#384	—	2	2	no	—
9	C₁₁×#384	—	2	1	no	—
10	C₄³×C₁₁	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
13	C₂×#66867	—	2	1	no	—
14	#10622184	—	2	8	no	—
15	#10622162	—	2	1	no	—
16	#10622162	—	2	2	no	—
17	C₂²×Q₈×C₁₁	C₂×C₄	1	4	yes	—
18	C₁₁×C₄⋊Q₈	—	2	4	no	—
19	C₁₁×C₄⋊Q₈	—	2	4	no	—
20	C₁₁×C₄⋊Q₈	—	4	8	no	—
21	C₂×C₁₁×C₄⋊C₄	—	2	4	no	—
22	C₂×C₁₁×C₄⋊C₄	—	2	4	no	—
23	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
24	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
25	C₂×C₄²×C₁₁	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₁	D₄	1	2	yes	—
28	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
30	C₂²×C₁₁⋊C₈	—	2	2	no	—
31	C₂²×C₁₁⋊C₈	—	2	2	no	—
32	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
33	C₄⋊(C₁₁⋊C₈)	—	2	4	no	—
34	C₄⋊(C₁₁⋊C₈)	D₄	1	4	yes	—
35	#66867	—	4	2	no	—
36	#9702	—	11	1	no	maximal, Sylow
37	C₂×Q₈×C₁₁	—	2	8	no	—
38	C₂×Q₈×C₁₁	—	4	8	no	—
39	C₂²×C₄×C₁₁	—	2	4	no	—
40	C₂²×C₄×C₁₁	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₁₁	C₂²⋊C₄	1	2	yes	—
42	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
45	C₁₁×C₄⋊C₄	—	4	8	no	—
46	C₁₁×C₄⋊C₄	—	4	8	no	—
47	C₁₁×C₄⋊C₄	—	4	4	no	—
48	C₁₁×C₄⋊C₄	—	4	2	no	—
49	C₁₁×C₄⋊C₄	—	4	2	no	—
50	C₄²×C₁₁	—	2	4	no	—
51	C₄²×C₁₁	—	2	2	no	—
52	C₄²×C₁₁	—	2	4	no	—
53	C₄²×C₁₁	SD₁₆	1	2	yes	—
54	C₄²×C₁₁	Q₁₆	1	2	yes	—
55	C₄²×C₁₁	C₂×D₄	1	2	yes	—
56	C₄²×C₁₁	#41	1	2	yes	—
57	C₂×C₁₁⋊C₈	—	2	4	no	—
58	C₂×C₁₁⋊C₈	—	4	2	no	—
59	C₂×C₁₁⋊C₈	—	4	2	no	—
60	C₂×C₁₁⋊C₈	—	4	2	no	—
61	C₂×C₁₁⋊C₈	—	4	4	no	—
62	#2868	D₁₁	1	1	yes	characteristic
63	#1831	—	11	1	no	—
64	#1767	—	11	1	no	—
65	C₂×#333	—	11	4	no	—
66	C₂³×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	Q₈×C₁₁	—	8	8	no	—
68	C₂×C₄×C₁₁	—	2	2	no	—
69	C₂×C₄×C₁₁	—	2	2	no	—
70	C₂×C₄×C₁₁	—	2	2	no	—
71	C₂×C₄×C₁₁	—	2	2	no	—
72	C₂×C₄×C₁₁	—	2	2	no	—
73	C₂×C₄×C₁₁	—	4	4	no	—
74	C₂×C₄×C₁₁	—	4	8	no	—
75	C₂×C₄×C₁₁	#100	1	4	yes	—
76	C₂×C₄×C₁₁	#103	1	4	yes	—
77	C₂×C₄×C₁₁	C₄×D₄	1	2	yes	—
78	C₂×C₄×C₁₁	#121	1	2	yes	—
79	C₂×C₄×C₁₁	#124	1	1	yes	characteristic
80	C₂×C₄×C₁₁	#127	1	1	yes	characteristic
81	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
82	C₂×C₄×C₁₁	C₂×Q₁₆	1	1	yes	characteristic
83	C₂×C₄×C₁₁	#137	1	2	yes	—
84	C₂×C₄×C₁₁	#137	1	4	yes	—
85	C₁₁⋊C₈	—	4	4	no	—
86	C₁₁⋊C₈	—	8	2	no	—
87	C₂×C₄⋊Q₈	Dic₁₁	1	4	yes	—
88	C₂×C₄⋊Q₈	C₂×D₁₁	1	2	yes	—
89	C₂×C₄⋊C₈	—	11	2	no	—
90	C₂×C₄⋊C₈	—	11	2	no	—
91	C₂×#97	—	22	1	no	—
92	#384	—	2	1	no	—
93	#384	—	2	2	no	—
94	#384	—	2	1	no	—
95	C₄³	C₂×D₁₁	1	1	yes	characteristic
96	#336	—	22	2	no	—
97	#336	—	22	1	no	—
98	#333	—	22	8	no	—
99	C₂²×C₁₁	#390	1	1	yes	characteristic
100	C₂²×C₁₁	C₂×#100	1	2	yes	—
101	C₂²×C₁₁	C₂×#103	1	1	yes	characteristic
102	C₂²×C₁₁	#419	1	2	yes	—
103	C₂²×C₁₁	#419	1	1	yes	characteristic
104	C₄×C₁₁	—	2	4	no	—
105	C₄×C₁₁	—	2	4	no	—
106	C₄×C₁₁	—	2	2	no	—
107	C₄×C₁₁	—	8	4	no	—
108	C₄×C₁₁	#461	1	2	yes	—
109	C₄×C₁₁	#462	1	2	yes	—
110	C₄×C₁₁	#491	1	2	yes	—
111	C₄×C₁₁	#497	1	2	yes	—
112	C₂²×Q₈	C₂×Dic₁₁	1	4	yes	—
113	C₂²×C₈	—	22	2	no	—
114	C₂²×C₈	—	22	2	no	—
115	C₄⋊Q₈	—	2	4	no	—
116	C₄⋊Q₈	—	2	4	no	—
117	C₄⋊Q₈	—	4	8	no	—
118	C₂×C₄⋊C₄	—	2	4	no	—
119	C₂×C₄⋊C₄	—	2	4	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	—	2	2	no	—
122	C₂×C₄²	C₂×Dic₁₁	1	2	yes	—
123	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
124	C₂×C₄²	C₂²⋊D₁₁	1	2	yes	—
125	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
126	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
127	C₄⋊C₈	—	11	4	no	—
128	C₄⋊C₈	—	22	4	no	—
129	C₄⋊C₈	—	22	2	no	—
130	#97	—	44	2	no	—
131	C₂×C₁₁	#1548	1	2	yes	—
132	C₂×C₁₁	#1558	1	2	yes	—
133	C₂×C₁₁	#1960	1	1	yes	characteristic
134	C₂×C₁₁	#1965	1	1	yes	characteristic
135	C₂×C₁₁	#1981	1	1	yes	characteristic
136	C₂×Q₈	—	2	8	no	—
137	C₂×Q₈	—	4	8	no	—
138	C₂²×C₄	—	2	4	no	—
139	C₂²×C₄	C₂²⋊Dic₁₁	1	2	yes	—
140	C₂²×C₄	C₂²⋊Dic₁₁	1	2	yes	—
141	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
142	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
143	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
144	C₂×C₈	—	22	4	no	—
145	C₂×C₈	—	44	2	no	—
146	C₂×C₈	—	44	2	no	—
147	C₂×C₈	—	44	2	no	—
148	C₂×C₈	—	44	4	no	—
149	C₄⋊C₄	—	4	8	no	—
150	C₄⋊C₄	—	4	8	no	—
151	C₄⋊C₄	—	4	2	no	—
152	C₄⋊C₄	—	4	4	no	—
153	C₄⋊C₄	—	4	2	no	—
154	C₄²	—	2	4	no	—
155	C₄²	—	2	4	no	—
156	C₄²	—	2	2	no	—
157	C₄²	#4325	1	2	yes	—
158	C₄²	#4326	1	2	yes	—
159	C₄²	D₄×D₁₁	1	2	yes	—
160	C₄²	#4341	1	2	yes	—
161	C₁₁	#9702	1	1	yes	characteristic, complemented, Sylow
162	C₂³	C₂×C₂²⋊Dic₁₁	1	1	yes	characteristic
163	Q₈	—	8	8	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	4	8	no	—
170	C₂×C₄	—	4	4	no	—
171	C₂×C₄	#66897	1	4	yes	—
172	C₂×C₄	#66899	1	4	yes	—
173	C₂×C₄	D₄×Dic₁₁	1	2	yes	—
174	C₂×C₄	#66989	1	1	yes	characteristic
175	C₂×C₄	#66991	1	2	yes	—
176	C₂×C₄	#66993	1	1	yes	characteristic
177	C₂×C₄	C₂×#4325	1	1	yes	characteristic
178	C₂×C₄	#66995	1	4	yes	—
179	C₂×C₄	#66995	1	2	yes	—
180	C₂×C₄	C₂×#4326	1	1	yes	characteristic
181	C₈	—	44	4	no	—
182	C₈	—	88	2	no	—
183	C₂²	C₂×#66897	1	2	yes	—
184	C₂²	#10622834	1	1	yes	characteristic
185	C₂²	C₂×#66899	1	1	yes	characteristic
186	C₂²	#10622838	1	2	yes	—
187	C₂²	#10622838	1	1	yes	characteristic
188	C₄	—	2	4	no	—
189	C₄	—	2	4	no	—
190	C₄	—	2	2	no	—
191	C₄	—	8	4	no	—
192	C₄	#10622695	1	2	yes	—
193	C₄	#10622696	1	2	yes	—
194	C₄	#10622699	1	2	yes	—
195	C₄	#10622703	1	2	yes	—
196	C₂	#13067297	1	2	yes	—
197	C₂	#13067342	1	2	yes	—
198	C₂	#13069437	1	1	yes	characteristic
199	C₂	#13069457	1	1	yes	characteristic
200	C₂	#13069540	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13068733

C₂

yes

maximal, characteristic

#13068469

C₂

yes

maximal, characteristic

C₂×#10622184

C₂

yes

maximal

C₁₁×#2868

C₂

yes

maximal, characteristic

C₂×C₁₁×C₄⋊Q₈

C₄

yes

—

C₂×C₁₁×C₄⋊Q₈

C₂²

yes

—

C₁₁×#384

—

C₁₁×#384

—

C₁₁×#384

—

C₄³×C₁₁

C₂²

yes

characteristic

C₂×C₄⋊(C₁₁⋊C₈)

C₂²

yes

—

C₂×C₄⋊(C₁₁⋊C₈)

C₂²

yes

—

—

—

—

—

C₂²×Q₈×C₁₁

C₂×C₄

yes

—

C₁₁×C₄⋊Q₈

—

C₁₁×C₄⋊Q₈

—

C₁₁×C₄⋊Q₈

—

C₂×C₁₁×C₄⋊C₄

—

C₂×C₁₁×C₄⋊C₄

—

C₂×C₁₁×C₄⋊C₄

—

C₂×C₁₁×C₄⋊C₄

—

C₂×C₄²×C₁₁

C₂×C₄

yes

—

C₂×C₄²×C₁₁

D₄

yes

characteristic

C₂×C₄²×C₁₁

D₄

yes

—

C₂×C₄²×C₁₁

D₄

yes

characteristic

C₂×C₄²×C₁₁

C₂³

yes

characteristic

C₂²×C₁₁⋊C₈

—

C₂²×C₁₁⋊C₈

—

C₄⋊(C₁₁⋊C₈)

—

C₄⋊(C₁₁⋊C₈)

—

C₄⋊(C₁₁⋊C₈)

D₄

yes

—

#66867

—

#9702

—

maximal, Sylow

C₂×Q₈×C₁₁

—

C₂×Q₈×C₁₁

—

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

SD₁₆

yes

—

C₄²×C₁₁

Q₁₆

yes

—

C₄²×C₁₁

C₂×D₄

yes

—

C₄²×C₁₁

#41

yes

—

C₂×C₁₁⋊C₈

—

C₂×C₁₁⋊C₈

—

C₂×C₁₁⋊C₈

—

C₂×C₁₁⋊C₈

—

C₂×C₁₁⋊C₈

—

#2868

D₁₁

yes

—

—

—

C₂³×C₁₁

C₂×C₂²⋊C₄

yes

characteristic

Q₈×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

#100

yes

—

C₂×C₄×C₁₁

#103

yes

—

C₂×C₄×C₁₁

C₄×D₄

yes

—

C₂×C₄×C₁₁

#121

yes

—

C₂×C₄×C₁₁

yes

C₂×C₄×C₁₁

yes

C₂×C₄×C₁₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₁₁

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₁₁

#137

yes

—

C₂×C₄×C₁₁

#137

yes

—

C₁₁⋊C₈

—

C₁₁⋊C₈

—

C₂×C₄⋊Q₈

Dic₁₁

yes

—

C₂×C₄⋊Q₈

C₂×D₁₁

yes

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×#97

—

#384

—

#384

—

#384

—

C₄³

C₂×D₁₁

yes

—

—

—

C₂²×C₁₁

#390

yes

characteristic

100

C₂²×C₁₁

C₂×#100

yes

—

101

C₂²×C₁₁

C₂×#103

yes

characteristic

102

C₂²×C₁₁

#419

yes

—

103

C₂²×C₁₁

#419

yes

characteristic

104

C₄×C₁₁

—

105

C₄×C₁₁

—

106

C₄×C₁₁

—

107

C₄×C₁₁

—

108

C₄×C₁₁

#461

yes

—

109

C₄×C₁₁

#462

yes

—

110

C₄×C₁₁

#491

yes

—

111

C₄×C₁₁

#497

yes

—

112

C₂²×Q₈

C₂×Dic₁₁

yes

—

113

C₂²×C₈

—

114

C₂²×C₈

—

115

C₄⋊Q₈

—

116

C₄⋊Q₈

—

117

C₄⋊Q₈

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄²

C₂×Dic₁₁

yes

—

123

C₂×C₄²

C₂²⋊D₁₁

yes

characteristic

124

C₂×C₄²

C₂²⋊D₁₁

yes

—

125

C₂×C₄²

C₂²⋊D₁₁

yes

characteristic

126

C₂×C₄²

C₂²×D₁₁

yes

characteristic

127

C₄⋊C₈

—

128

C₄⋊C₈

—

129

C₄⋊C₈

—

130

#97

—

131

C₂×C₁₁

#1548

yes

—

132

C₂×C₁₁

#1558

yes

—

133

C₂×C₁₁

#1960

yes

characteristic

134

C₂×C₁₁

#1965

yes

characteristic

135

C₂×C₁₁

#1981

yes

characteristic

136

C₂×Q₈

—

137

C₂×Q₈

—

138

C₂²×C₄

—

139

C₂²×C₄

C₂²⋊Dic₁₁

yes

—

140

C₂²×C₄

C₂²⋊Dic₁₁

yes

—

141

C₂²×C₄

C₂²×Dic₁₁

yes

characteristic

142

C₂²×C₄

C₂×C₂²⋊D₁₁

yes

characteristic

143

C₂²×C₄

C₂×C₂²⋊D₁₁

yes

characteristic

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄²

—

155

C₄²

—

156

C₄²

—

157

C₄²

#4325

yes

—

158

C₄²

#4326

yes

—

159

C₄²

D₄×D₁₁

yes

—

160

C₄²

#4341

yes

—

161

C₁₁

#9702

yes

characteristic, complemented, Sylow

162

C₂³

C₂×C₂²⋊Dic₁₁

yes

characteristic

163

Q₈

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

#66897

yes

—

172

C₂×C₄

#66899

yes

—

173

C₂×C₄

D₄×Dic₁₁

yes

—

174

C₂×C₄

#66989

yes

characteristic

175

C₂×C₄

#66991

yes

—

176

C₂×C₄

#66993

yes

characteristic

177

C₂×C₄

C₂×#4325

yes

characteristic

178

C₂×C₄

#66995

yes

—

179

C₂×C₄

#66995

yes

—

180

C₂×C₄

C₂×#4326

yes

characteristic

181

C₈

—

182

C₈

—

183

C₂²

C₂×#66897

yes

—

184

C₂²

#10622834

yes

characteristic

185

C₂²

C₂×#66899

yes

characteristic

186

C₂²

#10622838

yes

—

187

C₂²

#10622838

yes

characteristic

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

#10622695

yes

—

193

C₄

#10622696

yes

—

194

C₄

#10622699

yes

—

195

C₄

#10622703

yes

—

196

C₂

#13067297

yes

—

197

C₂

#13067342

yes

—

198

C₂

#13069437

yes

characteristic

199

C₂

#13069457

yes

characteristic

200

C₂

#13069540

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₁
commutator	C₂²×C₄×C₁₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₁
Fitting	C₁₁×#2868	C₂
socle, prime radical	C₂³×C₁₁	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₁	C₂³
hypercenter	#2868	D₁₁
hypocenter	C₁₁	#9702
LCS	G ▷ C₂²×C₄×C₁₁ ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2868

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₁₁

commutator

C₂²×C₄×C₁₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₁

Fitting

C₁₁×#2868

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 ◁ #2868

N	H ≅ G/N	type
161. C₁₁	#9702	semidirect

H ≅ G/N

type

161. C₁₁

#9702

semidirect

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

subgroup	index
#13068733	2
#13068469	2
C₂×#10622184	2
C₁₁×#2868	2
#9702	11

subgroup

index

C₁₁×#2868

Subgroups — #14652855

Conjugacy classes of subgroups

Complements

Maximal subgroups