Group #13069762 — subgroups

Subgroups — #13069762

Order 1408.

294 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 85 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	C₂×#66923	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₁×#390	C₂	1	1	yes	maximal, characteristic
3	C₂×#66896	C₂	1	2	yes	maximal
4	#10622167	C₂	1	2	yes	maximal, complemented
5	#10622162	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄×C₁₁	C₂²	1	1	yes	characteristic
7	C₂×C₁₁×C₄⋊C₄	C₂²	1	1	yes	characteristic
8	C₂×C₁₁×C₂²⋊C₄	—	2	2	no	—
9	C₂×C₄²×C₁₁	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₁₁	C₂²	1	2	yes	—
11	C₂×Dic₁₁⋊C₄	—	2	2	no	—
12	C₂²×C₁₁⋊C₈	C₂²	1	2	yes	—
13	#66923	—	2	4	no	—
14	#66896	—	2	4	no	—
15	C₂⁴×C₁₁	—	2	1	no	—
16	C₂×D₄×C₁₁	—	2	2	no	—
17	C₂×D₄×C₁₁	—	2	2	no	—
18	C₂×D₄×C₁₁	—	2	1	no	—
19	C₂×D₄×C₁₁	Q₈	1	2	yes	—
20	C₂²×C₄×C₁₁	—	2	1	no	—
21	C₂²×C₄×C₁₁	D₄	1	2	yes	—
22	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
23	C₂²×Dic₁₁	—	2	2	no	—
24	C₁₁×C₄⋊C₄	—	2	2	no	—
25	C₁₁×C₂²⋊C₄	—	4	4	no	—
26	C₄²×C₁₁	—	2	2	no	—
27	C₄⋊Dic₁₁	—	2	2	no	—
28	C₄⋊Dic₁₁	D₄	1	4	yes	—
29	Dic₁₁⋊C₄	—	2	8	no	—
30	C₂×C₁₁⋊C₈	—	2	2	no	—
31	C₂×C₁₁⋊C₈	—	2	2	no	—
32	C₂×C₁₁⋊C₈	—	2	2	no	—
33	#2025	—	11	1	no	maximal, Sylow
34	C₂³×C₁₁	—	2	2	no	—
35	C₂³×C₁₁	—	4	1	no	—
36	C₂³×C₁₁	—	4	1	no	—
37	C₂³×C₁₁	—	4	2	no	—
38	C₂³×C₁₁	—	4	2	no	—
39	C₂³×C₁₁	C₂×D₄	1	1	yes	characteristic
40	D₄×C₁₁	—	2	4	no	—
41	D₄×C₁₁	—	4	2	no	—
42	C₂×C₄×C₁₁	—	2	2	no	—
43	C₂×C₄×C₁₁	—	2	2	no	—
44	C₂×C₄×C₁₁	—	2	2	no	—
45	C₂×C₄×C₁₁	—	4	1	no	—
46	C₂×C₄×C₁₁	—	4	2	no	—
47	C₂×C₄×C₁₁	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₄×C₁₁	#41	1	1	yes	characteristic
50	C₂×C₄×C₁₁	#41	1	2	yes	—
51	C₂×Dic₁₁	—	2	4	no	—
52	C₂×Dic₁₁	—	4	2	no	—
53	C₂×Dic₁₁	—	4	2	no	—
54	C₁₁⋊C₈	—	4	2	no	—
55	C₂×#125	—	11	1	no	—
56	C₂×#102	—	11	2	no	—
57	#390	D₁₁	1	1	yes	characteristic
58	#340	—	11	2	no	—
59	#336	—	11	1	no	—
60	C₂²×C₁₁	—	4	2	no	—
61	C₂²×C₁₁	—	4	4	no	—
62	C₂²×C₁₁	—	8	2	no	—
63	C₂²×C₁₁	—	8	1	no	—
64	C₂²×C₁₁	—	8	1	no	—
65	C₂²×C₁₁	#120	1	1	yes	characteristic
66	C₂²×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
67	C₂²×C₁₁	#124	1	1	yes	characteristic
68	C₂²×C₁₁	#135	1	2	yes	—
69	C₂²×C₁₁	AGL(1,ℤ/8)	1	2	yes	—
70	C₄×C₁₁	—	4	2	no	—
71	C₄×C₁₁	—	8	1	no	—
72	C₄×C₁₁	C₂²⋊Q₈	1	2	yes	—
73	C₄×C₁₁	#124	1	2	yes	—
74	Dic₁₁	—	4	4	no	—
75	C₂²×D₄	C₂×D₁₁	1	1	yes	characteristic
76	C₂²×C₈	—	11	2	no	—
77	#125	—	22	4	no	—
78	C₂×C₄⋊C₄	—	11	2	no	—
79	C₂×C₄⋊C₄	—	22	2	no	—
80	C₂×C₄⋊C₄	C₂×D₁₁	1	1	yes	characteristic
81	C₂×C₂²⋊C₄	—	2	2	no	—
82	C₂×C₄²	C₂×D₁₁	1	1	yes	characteristic
83	#102	—	22	4	no	—
84	C₂×C₁₁	—	8	2	no	—
85	C₂×C₁₁	#393	1	1	yes	characteristic
86	C₂×C₁₁	#449	1	2	yes	—
87	C₂×C₁₁	#478	1	2	yes	—
88	C₂×C₁₁	#488	1	1	yes	characteristic
89	C₂×C₁₁	#490	1	1	yes	characteristic
90	C₂⁴	—	2	1	no	—
91	C₂×D₄	—	2	1	no	—
92	C₂×D₄	—	2	2	no	—
93	C₂×D₄	—	2	2	no	—
94	C₂×D₄	Dic₂₂	1	2	yes	—
95	C₂²×C₄	—	2	1	no	—
96	C₂²×C₄	—	22	2	no	—
97	C₂²×C₄	C₂²⋊D₁₁	1	2	yes	—
98	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
99	C₂×C₈	—	22	2	no	—
100	C₂×C₈	—	22	2	no	—
101	C₂×C₈	—	22	2	no	—
102	C₄⋊C₄	—	2	2	no	—
103	C₄⋊C₄	—	11	4	no	—
104	C₄⋊C₄	—	22	8	no	—
105	C₄⋊C₄	—	22	2	no	—
106	C₂²⋊C₄	—	4	4	no	—
107	C₄²	—	2	2	no	—
108	C₁₁	#2025	1	1	yes	characteristic, complemented, Sylow
109	C₂³	—	2	2	no	—
110	C₂³	—	4	2	no	—
111	C₂³	—	4	1	no	—
112	C₂³	—	4	1	no	—
113	C₂³	—	4	2	no	—
114	C₂³	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
115	D₄	—	2	4	no	—
116	D₄	—	4	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	22	4	no	—
123	C₂×C₄	—	44	2	no	—
124	C₂×C₄	—	44	2	no	—
125	C₂×C₄	C₂×Dic₂₂	1	1	yes	characteristic
126	C₂×C₄	#4339	1	1	yes	characteristic
127	C₂×C₄	D₄×D₁₁	1	2	yes	—
128	C₂×C₄	#4341	1	2	yes	—
129	C₈	—	44	2	no	—
130	C₂²	—	4	2	no	—
131	C₂²	—	4	4	no	—
132	C₂²	—	8	2	no	—
133	C₂²	—	8	1	no	—
134	C₂²	—	8	1	no	—
135	C₂²	#66977	1	1	yes	characteristic
136	C₂²	#66985	1	2	yes	—
137	C₂²	#66989	1	1	yes	characteristic
138	C₂²	#66990	1	1	yes	characteristic
139	C₂²	#67003	1	2	yes	—
140	C₄	—	4	2	no	—
141	C₄	—	8	1	no	—
142	C₄	—	44	4	no	—
143	C₄	#66931	1	2	yes	—
144	C₄	#66938	1	2	yes	—
145	C₂	—	8	2	no	—
146	C₂	#10622560	1	1	yes	characteristic
147	C₂	#10622622	1	2	yes	—
148	C₂	#10622650	1	2	yes	—
149	C₂	#10622669	1	1	yes	characteristic
150	C₂	#10622684	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#66923

C₂

yes

maximal, characteristic, complemented

C₁₁×#390

C₂

yes

maximal, characteristic

C₂×#66896

C₂

yes

maximal

#10622167

C₂

yes

maximal, complemented

#10622162

C₂

yes

maximal, characteristic, complemented

C₂²×D₄×C₁₁

C₂²

yes

characteristic

C₂×C₁₁×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₁₁×C₂²⋊C₄

—

C₂×C₄²×C₁₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₁

C₂²

yes

—

C₂×Dic₁₁⋊C₄

—

C₂²×C₁₁⋊C₈

C₂²

yes

—

#66923

—

#66896

—

C₂⁴×C₁₁

—

C₂×D₄×C₁₁

—

C₂×D₄×C₁₁

—

C₂×D₄×C₁₁

—

C₂×D₄×C₁₁

Q₈

yes

—

C₂²×C₄×C₁₁

—

C₂²×C₄×C₁₁

D₄

yes

—

C₂²×C₄×C₁₁

C₂³

yes

characteristic

C₂²×Dic₁₁

—

C₁₁×C₄⋊C₄

—

C₁₁×C₂²⋊C₄

—

C₄²×C₁₁

—

C₄⋊Dic₁₁

—

C₄⋊Dic₁₁

D₄

yes

—

Dic₁₁⋊C₄

—

C₂×C₁₁⋊C₈

—

C₂×C₁₁⋊C₈

—

C₂×C₁₁⋊C₈

—

#2025

—

maximal, Sylow

C₂³×C₁₁

—

C₂³×C₁₁

—

C₂³×C₁₁

—

C₂³×C₁₁

—

C₂³×C₁₁

—

C₂³×C₁₁

C₂×D₄

yes

characteristic

D₄×C₁₁

—

D₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

C₂×D₄

yes

—

C₂×C₄×C₁₁

C₂×Q₈

yes

C₂×C₄×C₁₁

yes

C₂×C₄×C₁₁

yes

—

C₂×Dic₁₁

—

C₂×Dic₁₁

—

C₂×Dic₁₁

—

C₁₁⋊C₈

—

C₂×#125

—

C₂×#102

—

#390

D₁₁

yes

characteristic

#340

—

#336

—

C₂²×C₁₁

—

C₂²×C₁₁

—

C₂²×C₁₁

—

C₂²×C₁₁

—

C₂²×C₁₁

—

C₂²×C₁₁

#120

yes

characteristic

C₂²×C₁₁

C₂²⋊Q₈

yes

C₂²×C₁₁

yes

C₂²×C₁₁

yes

—

C₂²×C₁₁

AGL(1,ℤ/8)

yes

—

C₄×C₁₁

—

C₄×C₁₁

—

C₄×C₁₁

C₂²⋊Q₈

yes

—

C₄×C₁₁

#124

yes

—

Dic₁₁

—

C₂²×D₄

C₂×D₁₁

yes

characteristic

C₂²×C₈

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₁

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₁₁

yes

characteristic

#102

—

C₂×C₁₁

—

C₂×C₁₁

#393

yes

characteristic

C₂×C₁₁

#449

yes

—

C₂×C₁₁

#478

yes

—

C₂×C₁₁

yes

C₂×C₁₁

yes

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

Dic₂₂

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₁

yes

—

C₂²×C₄

C₂²×D₁₁

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₂²⋊C₄

—

107

C₄²

—

108

C₁₁

#2025

yes

characteristic, complemented, Sylow

109

C₂³

—

110

C₂³

—

111

C₂³

—

112

C₂³

—

113

C₂³

—

114

C₂³

C₂×C₂²⋊D₁₁

yes

characteristic

115

D₄

—

116

D₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

C₂×Dic₂₂

yes

characteristic

126

C₂×C₄

#4339

yes

characteristic

127

C₂×C₄

D₄×D₁₁

yes

—

128

C₂×C₄

#4341

yes

—

129

C₈

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

#66977

yes

characteristic

136

C₂²

#66985

yes

—

137

C₂²

#66989

yes

characteristic

138

C₂²

#66990

yes

characteristic

139

C₂²

#67003

yes

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

#66931

yes

—

144

C₄

#66938

yes

—

145

C₂

—

146

C₂

#10622560

yes

characteristic

147

C₂

#10622622

yes

—

148

C₂

#10622650

yes

—

149

C₂

#10622669

yes

characteristic

150

C₂

#10622684

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₁
commutator, cosocle	C₂²×C₄×C₁₁	C₂³
Frattini	C₂²×C₄	C₂²×D₁₁
Fitting	C₁₁×#390	C₂
socle	C₂³×C₁₁	C₂×D₄
prime radical	C₂²×D₄×C₁₁	C₂²
hypercenter	#390	D₁₁
hypocenter	C₁₁	#2025
LCS	G ▷ C₂²×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₁

commutator, cosocle

C₂²×C₄×C₁₁

C₂³

Frattini

C₂²×C₄

C₂²×D₁₁

Fitting

C₁₁×#390

C₂

socle

C₂³×C₁₁

C₂×D₄

prime radical

C₂²×D₄×C₁₁

C₂²

D₁₁

C₁₁

G ▷ C₂²×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

N	H ≅ G/N	type
1. C₂×#66923	C₂	semidirect
4. #10622167	C₂	semidirect
5. #10622162	C₂	semidirect
108. C₁₁	#2025	semidirect

H ≅ G/N

type

1. C₂×#66923

C₂

semidirect

4. #10622167

C₂

semidirect

5. #10622162

C₂

semidirect

108. C₁₁

#2025

semidirect

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

subgroup	index
C₂×#66923	2
C₁₁×#390	2
C₂×#66896	2
#10622167	2
#10622162	2
#2025	11

subgroup

index

C₁₁×#390

Subgroups — #13069762

Conjugacy classes of subgroups

Complements

Maximal subgroups