Group #13067351 — subgroups

Subgroups — #13067351

Order 1408.

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

normal subgroups only

This group has 254 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	C₁₁×#532	C₂	1	1	yes	maximal, characteristic
2	#10622537	C₂	1	2	yes	maximal, complemented
3	#10622184	C₂	1	2	yes	maximal, complemented
4	#10622182	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₁×#41	C₄	1	2	yes	—
6	C₁₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
7	C₁₁×#127	C₂²	1	1	yes	characteristic
8	C₁₁×#124	C₄	1	2	yes	—
9	C₁₁×#121	—	2	2	no	—
10	C₄×D₄×C₁₁	—	2	2	no	—
11	C₂×C₄²×C₁₁	C₂²	1	1	yes	characteristic
12	C₂×#4318	—	2	2	no	—
13	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
14	C₄⋊(C₁₁⋊C₈)	C₂²	1	4	yes	—
15	C₁₁×#41	—	4	4	no	—
16	C₂×Q₈×C₁₁	C₂×C₄	1	2	yes	—
17	C₂×D₄×C₁₁	—	2	2	no	—
18	C₂×D₄×C₁₁	—	2	2	no	—
19	C₂×D₄×C₁₁	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₁₁	—	2	2	no	—
21	C₂²×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₁	D₄	1	2	yes	—
23	C₁₁×C₄⋊C₄	—	2	2	no	—
24	C₁₁×C₂²⋊C₄	—	2	4	no	—
25	C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₁₁	D₄	1	2	yes	—
27	C₄²×C₁₁	C₂³	1	1	yes	characteristic
28	#4318	—	4	4	no	—
29	C₂×C₁₁⋊C₈	—	2	4	no	—
30	#1560	—	11	1	no	maximal, Sylow
31	C₂³×C₁₁	—	2	2	no	—
32	C₂³×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
33	Q₈×C₁₁	—	4	2	no	—
34	D₄×C₁₁	—	4	2	no	—
35	D₄×C₁₁	—	4	4	no	—
36	D₄×C₁₁	—	4	2	no	—
37	D₄×C₁₁	—	4	2	no	—
38	C₂×C₄×C₁₁	—	2	2	no	—
39	C₂×C₄×C₁₁	—	2	2	no	—
40	C₂×C₄×C₁₁	—	2	2	no	—
41	C₂×C₄×C₁₁	—	2	2	no	—
42	C₂×C₄×C₁₁	—	4	4	no	—
43	C₂×C₄×C₁₁	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₁	C₂×D₄	1	2	yes	—
47	C₁₁⋊C₈	—	4	4	no	—
48	#532	D₁₁	1	1	yes	characteristic
49	#423	—	11	2	no	—
50	#333	—	11	2	no	—
51	#331	—	11	2	no	—
52	C₂²×C₁₁	—	2	1	no	—
53	C₂²×C₁₁	—	2	2	no	—
54	C₂²×C₁₁	—	4	2	no	—
55	C₂²×C₁₁	—	4	4	no	—
56	C₂²×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₄×C₁₁	—	2	1	no	—
58	C₄×C₁₁	—	2	2	no	—
59	C₄×C₁₁	—	4	2	no	—
60	C₄×C₁₁	#100	1	2	yes	—
61	C₄×C₁₁	AGL(1,ℤ/8)	1	2	yes	—
62	C₄×C₁₁	#137	1	2	yes	—
63	C₂×#41	Dic₁₁	1	2	yes	—
64	C₂×M₁₆	—	22	2	no	—
65	C₄⋊Q₈	C₂×D₁₁	1	1	yes	characteristic
66	#127	C₂×D₁₁	1	1	yes	characteristic
67	#124	Dic₁₁	1	2	yes	—
68	#121	—	2	2	no	—
69	C₄×D₄	—	2	2	no	—
70	C₂×C₄²	C₂×D₁₁	1	1	yes	characteristic
71	C₄⋊C₈	—	11	4	no	—
72	C₄⋊C₈	—	22	2	no	—
73	C₂×C₁₁	—	4	1	no	—
74	C₂×C₁₁	—	8	2	no	—
75	C₂×C₁₁	C₂×#100	1	1	yes	characteristic
76	C₂×C₁₁	#417	1	2	yes	—
77	#41	—	4	4	no	—
78	C₂×Q₈	C₂×Dic₁₁	1	2	yes	—
79	C₂×D₄	—	2	2	no	—
80	C₂×D₄	—	2	2	no	—
81	C₂×D₄	C₂×Dic₁₁	1	2	yes	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	C₂×Dic₁₁	1	1	yes	characteristic
84	C₂²×C₄	C₂²⋊D₁₁	1	2	yes	—
85	M₁₆	—	44	4	no	—
86	C₂×C₈	—	22	4	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₂²⋊C₄	—	2	4	no	—
89	C₄²	C₂×Dic₁₁	1	1	yes	characteristic
90	C₄²	C₂²⋊D₁₁	1	2	yes	—
91	C₄²	C₂²×D₁₁	1	1	yes	characteristic
92	C₁₁	#1560	1	1	yes	characteristic, complemented, Sylow
93	C₂³	—	2	2	no	—
94	C₂³	C₂²⋊Dic₁₁	1	1	yes	characteristic
95	Q₈	—	4	2	no	—
96	D₄	—	4	2	no	—
97	D₄	—	4	4	no	—
98	D₄	—	4	2	no	—
99	D₄	—	4	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	4	4	no	—
105	C₂×C₄	C₂²⋊Dic₁₁	1	2	yes	—
106	C₂×C₄	C₂²⋊Dic₁₁	1	1	yes	characteristic
107	C₂×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
108	C₂×C₄	C₂×C₂²⋊D₁₁	1	2	yes	—
109	C₈	—	44	4	no	—
110	C₂²	—	2	1	no	—
111	C₂²	—	2	2	no	—
112	C₂²	—	4	2	no	—
113	C₂²	—	4	4	no	—
114	C₂²	C₂×C₂²⋊Dic₁₁	1	1	yes	characteristic
115	C₄	—	2	1	no	—
116	C₄	—	2	2	no	—
117	C₄	—	4	2	no	—
118	C₄	#66897	1	2	yes	—
119	C₄	#67002	1	2	yes	—
120	C₄	#67004	1	2	yes	—
121	C₂	—	4	1	no	—
122	C₂	—	8	2	no	—
123	C₂	C₂×#66897	1	1	yes	characteristic
124	C₂	#10622845	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₁×#532

C₂

yes

maximal, characteristic

#10622537

C₂

yes

maximal, complemented

#10622184

C₂

yes

maximal, complemented

#10622182

C₂

yes

maximal, complemented

C₂×C₁₁×#41

C₄

yes

—

C₁₁×C₄⋊Q₈

C₂²

yes

characteristic

C₁₁×#127

C₂²

yes

characteristic

C₁₁×#124

C₄

yes

—

C₁₁×#121

—

C₄×D₄×C₁₁

—

C₂×C₄²×C₁₁

C₂²

yes

characteristic

C₂×#4318

—

C₄⋊(C₁₁⋊C₈)

—

C₄⋊(C₁₁⋊C₈)

C₂²

yes

—

C₁₁×#41

—

C₂×Q₈×C₁₁

C₂×C₄

yes

—

C₂×D₄×C₁₁

—

C₂×D₄×C₁₁

—

C₂×D₄×C₁₁

C₂×C₄

yes

—

C₂²×C₄×C₁₁

—

C₂²×C₄×C₁₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₁

D₄

yes

—

C₁₁×C₄⋊C₄

—

C₁₁×C₂²⋊C₄

—

C₄²×C₁₁

C₂×C₄

yes

characteristic

C₄²×C₁₁

D₄

yes

—

C₄²×C₁₁

C₂³

yes

characteristic

#4318

—

C₂×C₁₁⋊C₈

—

#1560

—

maximal, Sylow

C₂³×C₁₁

—

C₂³×C₁₁

C₂²⋊C₄

yes

characteristic

Q₈×C₁₁

—

D₄×C₁₁

—

D₄×C₁₁

—

D₄×C₁₁

—

D₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₁

C₂×D₄

yes

—

C₁₁⋊C₈

—

#532

D₁₁

yes

—

—

—

C₂²×C₁₁

—

C₂²×C₁₁

—

C₂²×C₁₁

—

C₂²×C₁₁

—

C₂²×C₁₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₁

—

C₄×C₁₁

—

C₄×C₁₁

—

C₄×C₁₁

#100

yes

—

C₄×C₁₁

AGL(1,ℤ/8)

yes

—

C₄×C₁₁

#137

yes

—

C₂×#41

Dic₁₁

yes

—

C₂×M₁₆

—

C₄⋊Q₈

C₂×D₁₁

yes

characteristic

#127

C₂×D₁₁

yes

characteristic

#124

Dic₁₁

yes

—

#121

—

C₄×D₄

—

C₂×C₄²

C₂×D₁₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂×C₁₁

—

C₂×C₁₁

—

C₂×C₁₁

C₂×#100

yes

characteristic

C₂×C₁₁

#417

yes

—

#41

—

C₂×Q₈

C₂×Dic₁₁

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₁₁

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₁₁

yes

characteristic

C₂²×C₄

C₂²⋊D₁₁

yes

—

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂×Dic₁₁

yes

characteristic

C₄²

C₂²⋊D₁₁

yes

—

C₄²

C₂²×D₁₁

yes

characteristic

C₁₁

#1560

yes

characteristic, complemented, Sylow

C₂³

—

C₂³

C₂²⋊Dic₁₁

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂²⋊Dic₁₁

yes

—

106

C₂×C₄

C₂²⋊Dic₁₁

yes

characteristic

107

C₂×C₄

C₂²×Dic₁₁

yes

characteristic

108

C₂×C₄

C₂×C₂²⋊D₁₁

yes

—

109

C₈

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

C₂×C₂²⋊Dic₁₁

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

#66897

yes

—

119

C₄

#67002

yes

—

120

C₄

#67004

yes

—

121

C₂

—

122

C₂

—

123

C₂

C₂×#66897

yes

characteristic

124

C₂

#10622845

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₁₁
commutator	C₂×C₄×C₁₁	C₂²×C₄
Frattini	C₄²	C₂²×D₁₁
Fitting, prime radical	C₁₁×#532	C₂
socle	C₂²×C₁₁	C₂×C₂²⋊C₄
cosocle	C₄²×C₁₁	C₂³
hypercenter	#532	D₁₁
hypocenter	C₁₁	#1560
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₁₁

commutator

C₂×C₄×C₁₁

C₂²×C₄

Frattini

C₄²

C₂²×D₁₁

Fitting, prime radical

C₁₁×#532

C₂

socle

C₂²×C₁₁

C₂×C₂²⋊C₄

cosocle

C₄²×C₁₁

C₂³

D₁₁

C₁₁

G ▷ C₂×C₄×C₁₁ ▷ C₂²×C₁₁ ▷ C₁₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
2. #10622537	C₂	semidirect
3. #10622184	C₂	semidirect
4. #10622182	C₂	semidirect
92. C₁₁	#1560	semidirect

H ≅ G/N

type

2. #10622537

C₂

semidirect

3. #10622184

C₂

semidirect

4. #10622182

C₂

semidirect

92. C₁₁

#1560

semidirect

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

subgroup	index
C₁₁×#532	2
#10622537	2
#10622184	2
#10622182	2
#1560	11

subgroup

index

C₁₁×#532

Subgroups — #13067351

Conjugacy classes of subgroups

Complements

Maximal subgroups