Group #13077715 — subgroups

Subgroups — #13077715

Order 1408.

330 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 161 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	#10623048	C₂	1	1	yes	maximal, characteristic
2	C₁₁×#433	C₂	1	1	yes	maximal, characteristic
3	#10622613	C₂	1	1	yes	maximal, characteristic
4	#10622443	C₂	1	2	yes	maximal, complemented
5	#10622441	C₂	1	2	yes	maximal, complemented
6	#10622333	C₂	1	2	yes	maximal
7	#10622331	C₂	1	2	yes	maximal
8	#10622314	C₂	1	2	yes	maximal, complemented
9	C₈×Dic₂₂	C₂	1	2	yes	maximal, complemented
10	C₁₁×#117	C₂²	1	1	yes	characteristic
11	#66979	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₁₁	C₂²	1	1	yes	characteristic
13	#66943	C₄	1	2	yes	—
14	#66941	C₄	1	2	yes	—
15	#66931	C₄	1	2	yes	—
16	#66931	C₄	1	2	yes	—
17	C₄×Dic₂₂	C₂²	1	2	yes	—
18	C₄×Dic₂₂	C₂²	1	2	yes	—
19	C₁₁×C₄⋊C₈	C₂²	1	2	yes	—
20	C₁₁×C₂²⋊C₈	C₂²	1	2	yes	—
21	C₁₁×#97	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
23	C₂²⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
24	#66879	C₂²	1	2	yes	—
25	Dic₁₁⋊C₈	C₂²	1	2	yes	—
26	Dic₁₁⋊C₈	C₂²	1	2	yes	—
27	Dic₁₁⋊C₈	C₂²	1	2	yes	—
28	Dic₁₁⋊C₈	C₂²	1	2	yes	—
29	C₈×Dic₁₁	C₂²	1	2	yes	—
30	C₈×Dic₁₁	C₂²	1	2	yes	—
31	C₈×Dic₁₁	C₂²	1	2	yes	—
32	C₄⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
33	#66867	C₂²	1	1	yes	characteristic
34	C₄×C₁₁⋊C₈	C₂²	1	1	yes	characteristic
35	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
36	C₂²×Dic₁₁	C₂×C₄	1	2	yes	—
37	C₂×Dic₂₂	C₂×C₄	1	2	yes	—
38	C₂×Dic₂₂	C₂×C₄	1	2	yes	—
39	C₂×C₈×C₁₁	C₂³	1	2	yes	—
40	C₂×C₈×C₁₁	C₂³	1	2	yes	—
41	C₁₁×C₄⋊C₄	C₂×C₄	1	2	yes	—
42	C₁₁×C₂²⋊C₄	C₂×C₄	1	2	yes	—
43	C₄²×C₁₁	C₂³	1	1	yes	characteristic
44	C₄²×C₁₁	C₂³	1	1	yes	characteristic
45	C₂²⋊Dic₁₁	C₂×C₄	1	1	yes	characteristic
46	C₂²⋊Dic₁₁	C₂×C₄	1	1	yes	characteristic
47	C₄⋊Dic₁₁	C₂×C₄	1	2	yes	—
48	C₄⋊Dic₁₁	C₂×C₄	1	2	yes	—
49	C₄⋊Dic₁₁	C₂×C₄	1	2	yes	—
50	Dic₁₁⋊C₄	C₂×C₄	1	2	yes	—
51	Dic₁₁⋊C₄	C₂×C₄	1	2	yes	—
52	Dic₁₁⋊C₄	C₂×C₄	1	2	yes	—
53	Dic₁₁⋊C₄	C₂×C₄	1	2	yes	—
54	C₄×Dic₁₁	C₂×C₄	1	1	yes	characteristic
55	C₄×Dic₁₁	C₂×C₄	1	1	yes	characteristic
56	C₄×Dic₁₁	C₂³	1	2	yes	—
57	C₄×Dic₁₁	C₂³	1	2	yes	—
58	C₂×C₁₁⋊C₈	C₂³	1	2	yes	—
59	C₂×C₁₁⋊C₈	C₂³	1	2	yes	—
60	#2986	—	11	1	no	maximal, Sylow
61	C₂³×C₁₁	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₁₁	—	2	2	no	—
63	C₂×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₁₁	C₂²×C₄	1	2	yes	—
65	C₂×C₄×C₁₁	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
67	C₂×Dic₁₁	—	2	4	no	—
68	C₂×Dic₁₁	C₂²×C₄	1	2	yes	—
69	C₂×Dic₁₁	C₂²×C₄	1	2	yes	—
70	C₂×Dic₁₁	C₂²×C₄	1	2	yes	—
71	C₂×Dic₁₁	C₂²×C₄	1	2	yes	—
72	Dic₂₂	—	2	4	no	—
73	Dic₂₂	—	2	4	no	—
74	C₈×C₁₁	—	2	2	no	—
75	C₈×C₁₁	—	2	2	no	—
76	C₁₁⋊C₈	—	2	2	no	—
77	C₁₁⋊C₈	—	2	2	no	—
78	#533	—	11	1	no	—
79	#446	—	11	2	no	—
80	#446	—	11	2	no	—
81	C₈×Q₈	—	11	2	no	—
82	C₈×Q₈	—	11	2	no	—
83	#433	—	11	1	no	—
84	#433	—	11	2	no	—
85	#433	D₁₁	1	1	yes	characteristic
86	#431	—	11	2	no	—
87	C₂²×C₁₁	—	2	1	no	—
88	C₂²×C₁₁	—	2	2	no	—
89	C₂²×C₁₁	C₂³×C₄	1	1	yes	characteristic
90	C₄×C₁₁	—	2	1	no	—
91	C₄×C₁₁	—	2	2	no	—
92	C₄×C₁₁	—	2	2	no	—
93	C₄×C₁₁	2₋¹⁺⁴	1	2	yes	—
94	Dic₁₁	—	2	2	no	—
95	Dic₁₁	—	2	2	no	—
96	Dic₁₁	—	2	2	no	—
97	Dic₁₁	#131	1	4	yes	—
98	C₄⋊Q₈	—	11	2	no	—
99	#125	—	11	2	no	—
100	C₂²⋊Q₈	—	11	2	no	—
101	C₂²⋊Q₈	—	11	2	no	—
102	C₄×Q₈	—	11	2	no	—
103	C₄×Q₈	—	11	2	no	—
104	#117	—	11	1	no	—
105	#117	C₂×D₁₁	1	1	yes	characteristic
106	C₂×C₄²	—	11	1	no	—
107	C₄⋊C₈	—	11	2	no	—
108	C₄⋊C₈	—	11	2	no	—
109	C₄⋊C₈	—	11	2	no	—
110	C₄⋊C₈	—	11	2	no	—
111	C₄⋊C₈	—	11	2	no	—
112	C₄⋊C₈	C₂×D₁₁	1	2	yes	—
113	C₂²⋊C₈	—	11	2	no	—
114	C₂²⋊C₈	C₂×D₁₁	1	2	yes	—
115	#97	—	11	2	no	—
116	#97	—	11	1	no	—
117	#97	C₂×D₁₁	1	1	yes	characteristic
118	C₄×C₈	—	11	2	no	—
119	C₄×C₈	—	11	2	no	—
120	C₄×C₈	—	11	2	no	—
121	C₄×C₈	—	11	1	no	—
122	C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
123	C₂×C₁₁	—	4	1	no	—
124	C₂×C₁₁	#519	1	1	yes	characteristic
125	C₂×C₁₁	C₂×#131	1	2	yes	—
126	C₂×Q₈	—	11	2	no	—
127	C₂×Q₈	—	11	2	no	—
128	C₂²×C₄	—	11	2	no	—
129	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
130	C₂×C₈	—	11	2	no	—
131	C₂×C₈	—	11	2	no	—
132	C₂×C₈	C₂²×D₁₁	1	2	yes	—
133	C₂×C₈	C₂²×D₁₁	1	2	yes	—
134	C₄⋊C₄	—	11	2	no	—
135	C₄⋊C₄	—	11	2	no	—
136	C₄⋊C₄	—	11	2	no	—
137	C₄⋊C₄	—	11	2	no	—
138	C₄⋊C₄	—	11	2	no	—
139	C₄⋊C₄	—	11	2	no	—
140	C₄⋊C₄	—	11	2	no	—
141	C₄⋊C₄	C₄×D₁₁	1	2	yes	—
142	C₂²⋊C₄	—	11	1	no	—
143	C₂²⋊C₄	—	11	1	no	—
144	C₂²⋊C₄	C₄×D₁₁	1	2	yes	—
145	C₄²	—	11	1	no	—
146	C₄²	—	11	1	no	—
147	C₄²	—	11	2	no	—
148	C₄²	—	11	2	no	—
149	C₄²	C₂²×D₁₁	1	1	yes	characteristic
150	C₄²	C₂²×D₁₁	1	1	yes	characteristic
151	C₁₁	#2986	1	1	yes	characteristic, complemented, Sylow
152	C₂³	C₂×C₄×D₁₁	1	1	yes	characteristic
153	Q₈	—	22	4	no	—
154	Q₈	—	22	4	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	11	2	no	—
157	C₂×C₄	—	11	2	no	—
158	C₂×C₄	—	11	2	no	—
159	C₂×C₄	—	11	2	no	—
160	C₂×C₄	—	22	4	no	—
161	C₂×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
162	C₂×C₄	C₂×C₄×D₁₁	1	2	yes	—
163	C₂×C₄	C₂×C₄×D₁₁	1	2	yes	—
164	C₂×C₄	C₂³×D₁₁	1	1	yes	characteristic
165	C₈	—	2	2	no	—
166	C₈	—	2	2	no	—
167	C₈	—	22	2	no	—
168	C₈	—	22	2	no	—
169	C₂²	—	2	2	no	—
170	C₂²	—	2	1	no	—
171	C₂²	C₂²×C₄×D₁₁	1	1	yes	characteristic
172	C₄	—	2	2	no	—
173	C₄	—	2	2	no	—
174	C₄	—	2	1	no	—
175	C₄	—	11	4	no	—
176	C₄	—	22	2	no	—
177	C₄	—	22	2	no	—
178	C₄	—	22	2	no	—
179	C₄	#67043	1	2	yes	—
180	C₂	—	4	1	no	—
181	C₂	#10623047	1	1	yes	characteristic
182	C₂	D₁₁×#131	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10623048

C₂

yes

maximal, characteristic

C₁₁×#433

C₂

yes

maximal, characteristic

#10622613

C₂

yes

maximal, characteristic

#10622443

C₂

yes

maximal, complemented

#10622441

C₂

yes

maximal, complemented

#10622333

C₂

yes

maximal

#10622331

C₂

yes

maximal

#10622314

C₂

yes

maximal, complemented

C₈×Dic₂₂

C₂

yes

maximal, complemented

C₁₁×#117

C₂²

yes

characteristic

#66979

C₂²

yes

characteristic

C₂×C₄×Dic₁₁

C₂²

yes

characteristic

#66943

C₄

yes

—

#66941

C₄

yes

—

#66931

C₄

yes

—

#66931

C₄

yes

—

C₄×Dic₂₂

C₂²

yes

—

C₄×Dic₂₂

C₂²

yes

—

C₁₁×C₄⋊C₈

C₂²

yes

—

C₁₁×C₂²⋊C₈

C₂²

yes

—

C₁₁×#97

C₂²

yes

characteristic

C₄×C₈×C₁₁

C₂²

yes

characteristic

C₂²⋊(C₁₁⋊C₈)

C₂²

yes

—

#66879

C₂²

yes

—

Dic₁₁⋊C₈

C₂²

yes

—

Dic₁₁⋊C₈

C₂²

yes

—

Dic₁₁⋊C₈

C₂²

yes

—

Dic₁₁⋊C₈

C₂²

yes

—

C₈×Dic₁₁

C₂²

yes

—

C₈×Dic₁₁

C₂²

yes

—

C₈×Dic₁₁

C₂²

yes

—

C₄⋊(C₁₁⋊C₈)

C₂²

yes

—

#66867

C₂²

yes

characteristic

C₄×C₁₁⋊C₈

C₂²

yes

characteristic

C₂²×C₄×C₁₁

C₂³

yes

characteristic

C₂²×Dic₁₁

C₂×C₄

yes

—

C₂×Dic₂₂

C₂×C₄

yes

—

C₂×Dic₂₂

C₂×C₄

yes

—

C₂×C₈×C₁₁

C₂³

yes

—

C₂×C₈×C₁₁

C₂³

yes

—

C₁₁×C₄⋊C₄

C₂×C₄

yes

—

C₁₁×C₂²⋊C₄

C₂×C₄

yes

—

C₄²×C₁₁

C₂³

yes

characteristic

C₄²×C₁₁

C₂³

yes

characteristic

C₂²⋊Dic₁₁

C₂×C₄

yes

characteristic

C₂²⋊Dic₁₁

C₂×C₄

yes

characteristic

C₄⋊Dic₁₁

C₂×C₄

yes

—

C₄⋊Dic₁₁

C₂×C₄

yes

—

C₄⋊Dic₁₁

C₂×C₄

yes

—

Dic₁₁⋊C₄

C₂×C₄

yes

—

Dic₁₁⋊C₄

C₂×C₄

yes

—

Dic₁₁⋊C₄

C₂×C₄

yes

—

Dic₁₁⋊C₄

C₂×C₄

yes

—

C₄×Dic₁₁

C₂×C₄

yes

characteristic

C₄×Dic₁₁

C₂×C₄

yes

characteristic

C₄×Dic₁₁

C₂³

yes

—

C₄×Dic₁₁

C₂³

yes

—

C₂×C₁₁⋊C₈

C₂³

yes

—

C₂×C₁₁⋊C₈

C₂³

yes

—

#2986

—

maximal, Sylow

C₂³×C₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₁

C₂²×C₄

yes

—

C₂×C₄×C₁₁

C₂²×C₄

yes

—

C₂×C₄×C₁₁

C₂⁴

yes

characteristic

C₂×Dic₁₁

—

C₂×Dic₁₁

C₂²×C₄

yes

—

C₂×Dic₁₁

C₂²×C₄

yes

—

C₂×Dic₁₁

C₂²×C₄

yes

—

C₂×Dic₁₁

C₂²×C₄

yes

—

Dic₂₂

—

Dic₂₂

—

C₈×C₁₁

—

C₈×C₁₁

—

C₁₁⋊C₈

—

C₁₁⋊C₈

—

#533

—

#446

—

#446

—

C₈×Q₈

—

C₈×Q₈

—

#433

—

#433

—

#433

D₁₁

yes

characteristic

#431

—

C₂²×C₁₁

—

C₂²×C₁₁

—

C₂²×C₁₁

C₂³×C₄

yes

characteristic

C₄×C₁₁

—

C₄×C₁₁

—

C₄×C₁₁

—

C₄×C₁₁

2₋¹⁺⁴

yes

—

Dic₁₁

—

Dic₁₁

—

Dic₁₁

—

Dic₁₁

#131

yes

—

C₄⋊Q₈

—

#125

—

100

C₂²⋊Q₈

—

101

C₂²⋊Q₈

—

102

C₄×Q₈

—

103

C₄×Q₈

—

104

#117

—

105

#117

C₂×D₁₁

yes

characteristic

106

C₂×C₄²

—

107

C₄⋊C₈

—

108

C₄⋊C₈

—

109

C₄⋊C₈

—

110

C₄⋊C₈

—

111

C₄⋊C₈

—

112

C₄⋊C₈

C₂×D₁₁

yes

—

113

C₂²⋊C₈

—

114

C₂²⋊C₈

C₂×D₁₁

yes

—

115

#97

—

116

#97

—

117

#97

C₂×D₁₁

yes

characteristic

118

C₄×C₈

—

119

C₄×C₈

—

120

C₄×C₈

—

121

C₄×C₈

—

122

C₄×C₈

C₂×D₁₁

yes

characteristic

123

C₂×C₁₁

—

124

C₂×C₁₁

#519

yes

characteristic

125

C₂×C₁₁

C₂×#131

yes

—

126

C₂×Q₈

—

127

C₂×Q₈

—

128

C₂²×C₄

—

129

C₂²×C₄

C₂²×D₁₁

yes

characteristic

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

C₂²×D₁₁

yes

—

133

C₂×C₈

C₂²×D₁₁

yes

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄⋊C₄

C₄×D₁₁

yes

—

142

C₂²⋊C₄

—

143

C₂²⋊C₄

—

144

C₂²⋊C₄

C₄×D₁₁

yes

—

145

C₄²

—

146

C₄²

—

147

C₄²

—

148

C₄²

—

149

C₄²

C₂²×D₁₁

yes

characteristic

150

C₄²

C₂²×D₁₁

yes

characteristic

151

C₁₁

#2986

yes

characteristic, complemented, Sylow

152

C₂³

C₂×C₄×D₁₁

yes

characteristic

153

Q₈

—

154

Q₈

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

C₂×C₄×D₁₁

yes

characteristic

162

C₂×C₄

C₂×C₄×D₁₁

yes

—

163

C₂×C₄

C₂×C₄×D₁₁

yes

—

164

C₂×C₄

C₂³×D₁₁

yes

characteristic

165

C₈

—

166

C₈

—

167

C₈

—

168

C₈

—

169

C₂²

—

170

C₂²

—

171

C₂²

C₂²×C₄×D₁₁

yes

characteristic

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

#67043

yes

—

180

C₂

—

181

C₂

#10623047

yes

characteristic

182

C₂

D₁₁×#131

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₁
commutator, socle	C₂²×C₁₁	C₂³×C₄
Fitting	C₁₁×#433	C₂
prime radical	C₂³×C₁₁	C₂²×C₄
cosocle	C₂×C₄×C₁₁	C₂⁴
hypercenter	#433	D₁₁
hypocenter	C₁₁	#2986
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ #433

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₁

commutator, socle

C₂²×C₁₁

C₂³×C₄

Fitting

C₁₁×#433

C₂

prime radical

C₂³×C₁₁

C₂²×C₄

cosocle

C₂×C₄×C₁₁

C₂⁴

D₁₁

C₁₁

G ▷ C₂²×C₁₁ ▷ C₁₁

C₁ ◁ C₂×C₄ ◁ #433

N	H ≅ G/N	type
4. #10622443	C₂	semidirect
5. #10622441	C₂	semidirect
8. #10622314	C₂	semidirect
9. C₈×Dic₂₂	C₂	semidirect
151. C₁₁	#2986	semidirect

H ≅ G/N

type

4. #10622443

C₂

semidirect

5. #10622441

C₂

semidirect

8. #10622314

C₂

semidirect

9. C₈×Dic₂₂

C₂

semidirect

151. C₁₁

#2986

semidirect

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

subgroup	index
#10623048	2
C₁₁×#433	2
#10622613	2
#10622443	2
#10622441	2
#10622333	2
#10622331	2
#10622314	2
C₈×Dic₂₂	2
#2986	11

subgroup

index

C₁₁×#433

C₈×Dic₂₂

Subgroups — #13077715

Conjugacy classes of subgroups

Complements

Maximal subgroups