Group #14641008 — subgroups

Subgroups — #14641008

Order 2816.

198 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 91 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	#13070419	C₂	1	1	yes	maximal, characteristic
2	C₂×#10622096	C₂	1	1	yes	maximal, characteristic
3	C₁₁×#1566	C₂	1	1	yes	maximal, characteristic
4	C₁₁×#423	—	2	1	no	—
5	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
6	C₂×C₄⋊(C₁₁⋊C₈)	—	2	1	no	—
7	C₁₁×#334	—	2	1	no	—
8	C₁₁×#334	C₄	1	2	yes	—
9	C₂²⋊(C₁₁⋊C₁₆)	C₄	1	2	yes	—
10	C₄⋊(C₁₁⋊C₁₆)	—	2	1	no	—
11	#10622099	—	2	1	no	—
12	#10622096	—	2	1	no	—
13	#10622096	C₄	1	2	yes	—
14	C₂×C₁₁×M₁₆	—	4	1	no	—
15	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₁	Q₈	1	1	yes	characteristic
17	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
18	C₂²×C₁₁⋊C₈	—	4	1	no	—
19	C₁₁×C₄⋊C₈	—	2	2	no	—
20	C₁₁×C₄⋊C₈	—	4	1	no	—
21	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
24	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
25	C₂×C₁₁⋊C₁₆	—	2	2	no	—
26	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
27	C₄⋊(C₁₁⋊C₈)	—	4	1	no	—
28	#7292	—	11	1	no	maximal, Sylow
29	C₂²×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₁	SD₁₆	1	1	yes	characteristic
31	C₂²×C₄×C₁₁	SD₁₆	1	1	yes	characteristic
32	C₁₁×M₁₆	—	8	2	no	—
33	C₂×C₈×C₁₁	—	2	2	no	—
34	C₂×C₈×C₁₁	—	4	1	no	—
35	C₂×C₈×C₁₁	—	4	2	no	—
36	C₂×C₈×C₁₁	C₄²	1	1	yes	characteristic
37	C₂×C₈×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₁₁	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₁₁	C₄⋊C₄	1	1	yes	characteristic
40	C₄²×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
41	C₄²×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
42	C₄²×C₁₁	D₈	1	1	yes	characteristic
43	C₄²×C₁₁	Q₁₆	1	1	yes	characteristic
44	C₂×C₁₁⋊C₈	—	4	2	no	—
45	C₂×C₁₁⋊C₈	—	8	2	no	—
46	C₁₁⋊C₁₆	—	4	2	no	—
47	#2163	—	11	1	no	—
48	#1566	D₁₁	1	1	yes	characteristic
49	C₂×#335	—	11	1	no	—
50	C₂³×C₁₁	#106	1	1	yes	characteristic
51	C₂×C₄×C₁₁	—	2	2	no	—
52	C₂×C₄×C₁₁	—	2	1	no	—
53	C₂×C₄×C₁₁	—	2	1	no	—
54	C₂×C₄×C₁₁	—	2	1	no	—
55	C₂×C₄×C₁₁	—	2	1	no	—
56	C₂×C₄×C₁₁	#95	1	1	yes	characteristic
57	C₂×C₄×C₁₁	#102	1	1	yes	characteristic
58	C₂×C₄×C₁₁	#102	1	1	yes	characteristic
59	C₂×C₄×C₁₁	#103	1	1	yes	characteristic
60	C₂×C₄×C₁₁	#103	1	1	yes	characteristic
61	C₂×C₄×C₁₁	#107	1	1	yes	characteristic
62	C₈×C₁₁	—	2	1	no	—
63	C₈×C₁₁	—	2	1	no	—
64	C₈×C₁₁	—	2	1	no	—
65	C₈×C₁₁	—	8	2	no	—
66	C₈×C₁₁	#108	1	2	yes	—
67	C₁₁⋊C₈	—	8	2	no	—
68	#423	—	2	1	no	—
69	C₂×C₄⋊C₈	—	22	1	no	—
70	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
71	C₄⋊C₁₆	—	22	1	no	—
72	C₂²⋊C₁₆	—	11	2	no	—
73	#346	—	22	1	no	—
74	#335	—	11	2	no	—
75	#335	—	22	1	no	—
76	#334	—	2	1	no	—
77	#334	Dic₁₁	1	2	yes	—
78	C₂²×C₁₁	—	2	1	no	—
79	C₂²×C₁₁	—	2	1	no	—
80	C₂²×C₁₁	#340	1	1	yes	characteristic
81	C₂²×C₁₁	#365	1	2	yes	—
82	C₄×C₁₁	—	2	1	no	—
83	C₄×C₁₁	—	2	1	no	—
84	C₄×C₁₁	—	2	1	no	—
85	C₄×C₁₁	—	2	1	no	—
86	C₄×C₁₁	#341	1	1	yes	characteristic
87	C₄×C₁₁	#344	1	1	yes	characteristic
88	C₄×C₁₁	#361	1	1	yes	characteristic
89	C₄×C₁₁	#362	1	1	yes	characteristic
90	C₂×M₁₆	—	4	1	no	—
91	C₂²×C₈	—	44	1	no	—
92	C₂²×C₈	Dic₂₂	1	1	yes	characteristic
93	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
94	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
95	C₂×C₁₆	—	22	2	no	—
96	C₄⋊C₈	—	2	2	no	—
97	C₄⋊C₈	—	4	1	no	—
98	C₄⋊C₈	—	22	2	no	—
99	C₄⋊C₈	—	44	1	no	—
100	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
101	C₄×C₈	D₄₄	1	1	yes	characteristic
102	C₄×C₈	C₂×Dic₁₁	1	1	yes	characteristic
103	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
104	C₂×C₁₁	—	2	2	no	—
105	C₂×C₁₁	#1301	1	1	yes	characteristic
106	C₂×C₁₁	#1383	1	1	yes	characteristic
107	C₂×C₁₁	#1387	1	1	yes	characteristic
108	C₂²×C₄	Dic₁₁⋊C₄	1	1	yes	characteristic
109	C₂²×C₄	#4324	1	1	yes	characteristic
110	C₂²×C₄	#4325	1	1	yes	characteristic
111	M₁₆	—	8	2	no	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	4	1	no	—
114	C₂×C₈	—	4	2	no	—
115	C₂×C₈	—	44	2	no	—
116	C₂×C₈	—	88	2	no	—
117	C₂×C₈	C₄×Dic₁₁	1	1	yes	characteristic
118	C₂×C₈	Dic₁₁⋊C₄	1	1	yes	characteristic
119	C₂×C₈	C₄⋊Dic₁₁	1	1	yes	characteristic
120	C₂×C₈	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
121	C₄²	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
122	C₄²	#4323	1	1	yes	characteristic
123	C₄²	#4326	1	1	yes	characteristic
124	C₄²	C₂²⋊Dic₁₁	1	1	yes	characteristic
125	C₁₆	—	44	2	no	—
126	C₁₁	#7292	1	1	yes	characteristic, complemented, Sylow
127	C₂³	#66872	1	1	yes	characteristic
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	#66871	1	1	yes	characteristic
134	C₂×C₄	#66873	1	1	yes	characteristic
135	C₂×C₄	#66874	1	1	yes	characteristic
136	C₂×C₄	C₁₁⋊#95	1	1	yes	characteristic
137	C₂×C₄	#66896	1	1	yes	characteristic
138	C₂×C₄	#66899	1	1	yes	characteristic
139	C₈	—	2	1	no	—
140	C₈	—	2	1	no	—
141	C₈	—	2	1	no	—
142	C₈	—	8	2	no	—
143	C₈	—	88	2	no	—
144	C₈	#66886	1	2	yes	—
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	#10622125	1	2	yes	—
148	C₂²	#10622167	1	1	yes	characteristic
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	#10622133	1	1	yes	characteristic
154	C₄	#10622134	1	1	yes	characteristic
155	C₄	#10622194	1	1	yes	characteristic
156	C₄	#10622196	1	1	yes	characteristic
157	C₂	—	2	2	no	—
158	C₂	#13066511	1	1	yes	characteristic
159	C₂	#13066714	1	1	yes	characteristic
160	C₂	#13066726	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13070419

C₂

yes

maximal, characteristic

C₂×#10622096

C₂

yes

maximal, characteristic

C₁₁×#1566

C₂

yes

maximal, characteristic

C₁₁×#423

—

C₂×C₄×C₈×C₁₁

C₂²

yes

characteristic

C₂×C₄⋊(C₁₁⋊C₈)

—

C₁₁×#334

—

C₁₁×#334

C₄

yes

—

C₂²⋊(C₁₁⋊C₁₆)

C₄

yes

—

C₄⋊(C₁₁⋊C₁₆)

—

#10622099

—

#10622096

—

#10622096

C₄

yes

—

C₂×C₁₁×M₁₆

—

C₂²×C₈×C₁₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₁

Q₈

yes

characteristic

C₂×C₄²×C₁₁

D₄

yes

characteristic

C₂²×C₁₁⋊C₈

—

C₁₁×C₄⋊C₈

—

C₁₁×C₄⋊C₈

—

C₄×C₈×C₁₁

C₂×C₄

yes

characteristic

C₄×C₈×C₁₁

C₂×C₄

yes

characteristic

C₄×C₈×C₁₁

D₄

yes

characteristic

C₄×C₈×C₁₁

D₄

yes

characteristic

C₂×C₁₁⋊C₁₆

—

C₄⋊(C₁₁⋊C₈)

—

C₄⋊(C₁₁⋊C₈)

—

#7292

—

maximal, Sylow

C₂²×C₄×C₁₁

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₁₁

SD₁₆

yes

characteristic

C₂²×C₄×C₁₁

SD₁₆

yes

characteristic

C₁₁×M₁₆

—

C₂×C₈×C₁₁

—

C₂×C₈×C₁₁

—

C₂×C₈×C₁₁

—

C₂×C₈×C₁₁

C₄²

yes

characteristic

C₂×C₈×C₁₁

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₁₁

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₁

C₄⋊C₄

yes

characteristic

C₄²×C₁₁

C₂²⋊C₄

yes

characteristic

C₄²×C₁₁

C₂²⋊C₄

yes

characteristic

C₄²×C₁₁

D₈

yes

characteristic

C₄²×C₁₁

Q₁₆

yes

characteristic

C₂×C₁₁⋊C₈

—

C₂×C₁₁⋊C₈

—

C₁₁⋊C₁₆

—

#2163

—

#1566

D₁₁

yes

characteristic

C₂×#335

—

C₂³×C₁₁

#106

yes

characteristic

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

yes

C₂×C₄×C₁₁

yes

C₂×C₄×C₁₁

yes

C₂×C₄×C₁₁

yes

C₂×C₄×C₁₁

yes

C₂×C₄×C₁₁

yes

C₈×C₁₁

—

C₈×C₁₁

—

C₈×C₁₁

—

C₈×C₁₁

—

C₈×C₁₁

#108

yes

—

C₁₁⋊C₈

—

#423

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂×D₁₁

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

—

—

—

—

Dic₁₁

yes

—

C₂²×C₁₁

—

C₂²×C₁₁

—

C₂²×C₁₁

#340

yes

characteristic

C₂²×C₁₁

#365

yes

—

C₄×C₁₁

—

C₄×C₁₁

—

C₄×C₁₁

—

C₄×C₁₁

—

C₄×C₁₁

yes

C₄×C₁₁

yes

C₄×C₁₁

yes

C₄×C₁₁

yes

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

Dic₂₂

yes

characteristic

C₂²×C₈

C₄×D₁₁

yes

characteristic

C₂×C₄²

C₂²⋊D₁₁

yes

characteristic

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

100

C₄×C₈

C₄×D₁₁

yes

characteristic

101

C₄×C₈

D₄₄

yes

characteristic

102

C₄×C₈

C₂×Dic₁₁

yes

characteristic

103

C₄×C₈

C₂²⋊D₁₁

yes

characteristic

104

C₂×C₁₁

—

105

C₂×C₁₁

#1301

yes

characteristic

106

C₂×C₁₁

#1383

yes

characteristic

107

C₂×C₁₁

#1387

yes

characteristic

108

C₂²×C₄

Dic₁₁⋊C₄

yes

characteristic

109

C₂²×C₄

#4324

yes

characteristic

110

C₂²×C₄

#4325

yes

characteristic

111

M₁₆

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

C₄×Dic₁₁

yes

characteristic

118

C₂×C₈

Dic₁₁⋊C₄

yes

characteristic

119

C₂×C₈

C₄⋊Dic₁₁

yes

characteristic

120

C₂×C₈

(C₂×D₁₁)⋊C₄

yes

characteristic

121

C₄²

(C₂×D₁₁)⋊C₄

yes

characteristic

122

C₄²

#4323

yes

characteristic

123

C₄²

#4326

yes

characteristic

124

C₄²

C₂²⋊Dic₁₁

yes

characteristic

125

C₁₆

—

126

C₁₁

#7292

yes

characteristic, complemented, Sylow

127

C₂³

#66872

yes

characteristic

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

#66871

yes

characteristic

134

C₂×C₄

#66873

yes

characteristic

135

C₂×C₄

#66874

yes

characteristic

136

C₂×C₄

C₁₁⋊#95

yes

characteristic

137

C₂×C₄

#66896

yes

characteristic

138

C₂×C₄

#66899

yes

characteristic

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

#66886

yes

—

145

C₂²

—

146

C₂²

—

147

C₂²

#10622125

yes

—

148

C₂²

#10622167

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

#10622133

yes

characteristic

154

C₄

#10622134

yes

characteristic

155

C₄

#10622194

yes

characteristic

156

C₄

#10622196

yes

characteristic

157

C₂

—

158

C₂

#13066511

yes

characteristic

159

C₂

#13066714

yes

characteristic

160

C₂

#13066726

yes

characteristic

	subgroup	G/H
center	C₂²	#10622167
commutator	C₂×C₈×C₁₁	C₄²
Frattini	C₂×C₄×C₈	C₂×D₁₁
Fitting	C₁₁×#1566	C₂
socle	C₂²×C₁₁	#340
prime radical	C₂³×C₁₁	#106
cosocle	C₂×C₄×C₈×C₁₁	C₂²
hypercenter	#1566	D₁₁
hypocenter	C₁₁	#7292
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1566

subgroup

G/H

center

C₂²

#10622167

commutator

C₂×C₈×C₁₁

C₄²

Frattini

C₂×C₄×C₈

C₂×D₁₁

Fitting

C₁₁×#1566

C₂

C₂²×C₁₁

C₂³×C₁₁

C₂×C₄×C₈×C₁₁

C₂²

D₁₁

C₁₁

G ▷ C₂×C₈×C₁₁ ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1566

N	H ≅ G/N	type
126. C₁₁	#7292	semidirect

H ≅ G/N

type

126. C₁₁

#7292

semidirect

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

subgroup	index
#13070419	2
C₂×#10622096	2
C₁₁×#1566	2
#7292	11

subgroup

index

#13070419

C₂×#10622096

C₁₁×#1566

#7292

Subgroups — #14641008

Conjugacy classes of subgroups

Complements

Maximal subgroups