Group #13091238 — subgroups

Subgroups — #10624800⋊C₂ #13091238

Order 1458.

122 conjugacy classes of subgroups; only the 28 normal subgroups are shown.

normal subgroups only

This group has 122 conjugacy classes of subgroups — too many to draw. Only its 28 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	#10624800	C₂	1	1	yes	maximal, characteristic, complemented, Sylow
2	D₃×#731	—	3	1	no	maximal
3	#6894⋊C₂	C₃	1	1	yes	maximal, characteristic, complemented
4	D₉⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
5	D₉×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃×D₉⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃×#731	D₃	1	1	yes	characteristic
8	#6894	C₂×C₃	1	1	yes	characteristic, complemented
9	C₉⋊H(3,𝔽₃)	C₂×C₃	1	1	yes	characteristic
10	C₉×3₋¹⁺²	C₂×C₃	1	1	yes	characteristic, complemented
11	C₃×C₉⋊C₉	C₂×C₃	1	1	yes	characteristic, complemented
12	3₋¹⁺²⋊C₉	—	2	1	no	—
13	3₋¹⁺²⋊C₉	—	2	1	no	—
14	H(3,𝔽₃)⋊C₉	—	2	1	no	—
15	#6867	—	2	1	no	—
16	D₃×3₋¹⁺²	—	3	1	no	—
17	D₃×3₋¹⁺²	—	3	1	no	—
18	C₃×AGL(1,ℤ/9)	—	3	1	no	—
19	D₃×H(3,𝔽₃)	—	3	1	no	—
20	C₃×D₃×C₉	—	3	1	no	—
21	C₃²×D₉	C₃²	1	1	yes	characteristic
22	C₂×#731	—	9	1	no	—
23	D₉⋊C₉	—	3	1	no	—
24	D₉⋊C₉	—	3	1	no	—
25	C₉×D₉	—	3	1	no	—
26	C₃×3₋¹⁺²	—	3	1	no	—
27	C₃×3₋¹⁺²	C₃×D₃	1	1	yes	characteristic
28	C₃×3₋¹⁺²	C₃×D₃	1	1	yes	characteristic
29	C₃×H(3,𝔽₃)	C₃×D₃	1	1	yes	characteristic
30	C₃²×C₉	—	2	1	no	—
31	C₃²×C₉	C₃×D₃	1	1	yes	characteristic
32	C₃²×C₉	C₂×C₃²	1	1	yes	characteristic
33	#731	—	3	1	no	—
34	#731	—	6	1	no	—
35	C₉⋊C₉	—	3	1	no	—
36	C₉⋊C₉	—	3	1	no	—
37	C₉⋊C₉	—	6	1	no	—
38	C₉⋊C₉	—	6	1	no	—
39	C₉⋊C₉	—	6	1	no	—
40	C₃²⋊C₉	—	2	1	no	—
41	C₃²⋊C₉	—	2	1	no	—
42	C₃²⋊C₉	—	2	1	no	—
43	C₃²⋊C₉	—	6	1	no	—
44	C₉²	—	3	1	no	—
45	C₃²×D₃	—	3	1	no	—
46	C₃²×D₃	—	9	1	no	—
47	C₂×3₋¹⁺²	—	9	1	no	—
48	C₂×3₋¹⁺²	—	9	1	no	—
49	C₂×H(3,𝔽₃)	—	9	1	no	—
50	C₂×C₃×C₉	—	9	1	no	—
51	AGL(1,ℤ/9)	—	9	1	no	—
52	D₃×C₉	—	9	1	no	—
53	D₃×C₉	—	9	1	no	—
54	D₃×C₉	—	9	1	no	—
55	C₃×D₉	—	3	1	no	—
56	C₃×D₉	H(3,𝔽₃)	1	1	yes	characteristic
57	C₃³	—	3	1	no	—
58	C₃³	C₃²×D₃	1	1	yes	characteristic
59	3₋¹⁺²	—	2	1	no	—
60	3₋¹⁺²	—	3	1	no	—
61	3₋¹⁺²	—	6	1	no	—
62	3₋¹⁺²	—	6	1	no	—
63	3₋¹⁺²	—	9	1	no	—
64	3₋¹⁺²	—	18	1	no	—
65	3₋¹⁺²	AGL(1,ℤ/9)	1	1	yes	characteristic, complemented
66	H(3,𝔽₃)	—	3	1	no	—
67	H(3,𝔽₃)	—	6	1	no	—
68	C₃×C₉	—	2	1	no	—
69	C₃×C₉	—	3	1	no	—
70	C₃×C₉	—	3	1	no	—
71	C₃×C₉	—	3	1	no	—
72	C₃×C₉	—	3	1	no	—
73	C₃×C₉	—	3	1	no	—
74	C₃×C₉	—	6	1	no	—
75	C₃×C₉	—	6	1	no	—
76	C₃×C₉	—	6	1	no	—
77	C₃×C₉	—	6	1	no	—
78	C₃×C₉	—	6	1	no	—
79	C₃×C₉	—	6	1	no	—
80	C₃×C₉	—	18	1	no	—
81	C₃×C₉	C₂×H(3,𝔽₃)	1	1	yes	characteristic
82	C₂×C₃²	—	9	1	no	—
83	C₂×C₃²	—	27	1	no	—
84	C₃×D₃	—	3	1	no	—
85	C₃×D₃	—	9	1	no	—
86	C₃×D₃	—	27	1	no	—
87	C₂×C₉	—	27	1	no	—
88	C₂×C₉	—	27	1	no	—
89	C₂×C₉	—	27	1	no	—
90	D₉	#731	1	1	yes	characteristic, complemented
91	C₃²	—	2	1	no	—
92	C₃²	—	3	1	no	—
93	C₃²	—	6	1	no	—
94	C₃²	—	9	1	no	—
95	C₃²	—	9	1	no	—
96	C₃²	—	18	1	no	—
97	C₃²	D₃×H(3,𝔽₃)	1	1	yes	characteristic
98	C₃²	C₃×AGL(1,ℤ/9)	1	1	yes	characteristic
99	C₉	—	2	1	no	—
100	C₉	—	3	1	no	—
101	C₉	—	6	1	no	—
102	C₉	—	6	1	no	—
103	C₉	—	9	1	no	—
104	C₉	—	9	1	no	—
105	C₉	—	18	1	no	—
106	C₉	—	18	1	no	—
107	C₉	—	18	1	no	—
108	C₉	—	18	1	no	—
109	C₉	C₂×#731	1	1	yes	characteristic, complemented
110	C₂×C₃	—	9	1	no	—
111	C₂×C₃	—	27	1	no	—
112	C₂×C₃	—	81	1	no	—
113	D₃	—	3	1	no	—
114	C₃	—	2	1	no	—
115	C₃	—	3	1	no	—
116	C₃	—	6	1	no	—
117	C₃	—	27	1	no	—
118	C₃	D₉⋊H(3,𝔽₃)	1	1	yes	characteristic
119	C₃	D₃×#731	1	1	yes	characteristic
120	C₂	—	9	1	no	Sylow

subgroup

G/H

class size

multiplicity

normal?

properties

#10624800

C₂

yes

maximal, characteristic, complemented, Sylow

D₃×#731

—

maximal

#6894⋊C₂

C₃

yes

maximal, characteristic, complemented

D₉⋊H(3,𝔽₃)

C₃

yes

maximal, characteristic

D₉×3₋¹⁺²

C₃

yes

maximal, characteristic, complemented

C₃×D₉⋊C₉

C₃

yes

maximal, characteristic, complemented

C₃×#731

D₃

yes

characteristic

#6894

C₂×C₃

yes

characteristic, complemented

C₉⋊H(3,𝔽₃)

C₂×C₃

yes

characteristic

C₉×3₋¹⁺²

C₂×C₃

yes

characteristic, complemented

C₃×C₉⋊C₉

C₂×C₃

yes

characteristic, complemented

3₋¹⁺²⋊C₉

—

3₋¹⁺²⋊C₉

—

H(3,𝔽₃)⋊C₉

—

#6867

—

D₃×3₋¹⁺²

—

D₃×3₋¹⁺²

—

C₃×AGL(1,ℤ/9)

—

D₃×H(3,𝔽₃)

—

C₃×D₃×C₉

—

C₃²×D₉

C₃²

yes

characteristic

C₂×#731

—

D₉⋊C₉

—

D₉⋊C₉

—

C₉×D₉

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃×D₃

yes

characteristic

C₃×3₋¹⁺²

C₃×D₃

yes

characteristic

C₃×H(3,𝔽₃)

C₃×D₃

yes

characteristic

C₃²×C₉

—

C₃²×C₉

C₃×D₃

yes

characteristic

C₃²×C₉

C₂×C₃²

yes

characteristic

#731

—

#731

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₉²

—

C₃²×D₃

—

C₃²×D₃

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×H(3,𝔽₃)

—

C₂×C₃×C₉

—

AGL(1,ℤ/9)

—

D₃×C₉

—

D₃×C₉

—

D₃×C₉

—

C₃×D₉

—

C₃×D₉

H(3,𝔽₃)

yes

characteristic

C₃³

—

C₃³

C₃²×D₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

AGL(1,ℤ/9)

yes

characteristic, complemented

H(3,𝔽₃)

—

H(3,𝔽₃)

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₂×H(3,𝔽₃)

yes

characteristic

C₂×C₃²

—

C₂×C₃²

—

C₃×D₃

—

C₃×D₃

—

C₃×D₃

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

D₉

#731

yes

characteristic, complemented

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

D₃×H(3,𝔽₃)

yes

characteristic

C₃²

C₃×AGL(1,ℤ/9)

yes

characteristic

C₉

—

100

C₉

—

101

C₉

—

102

C₉

—

103

C₉

—

104

C₉

—

105

C₉

—

106

C₉

—

107

C₉

—

108

C₉

—

109

C₉

C₂×#731

yes

characteristic, complemented

110

C₂×C₃

—

111

C₂×C₃

—

112

C₂×C₃

—

113

D₃

—

114

C₃

—

115

C₃

—

116

C₃

—

117

C₃

—

118

C₃

D₉⋊H(3,𝔽₃)

yes

characteristic

119

C₃

D₃×#731

yes

characteristic

120

C₂

—

Sylow

	subgroup	G/H
center	C₃	D₉⋊H(3,𝔽₃)
commutator, cosocle	C₃²×C₉	C₂×C₃²
Frattini	C₃³	C₃²×D₃
Fitting	#10624800	C₂
socle	C₃²	D₃×H(3,𝔽₃)
prime radical	D₉⋊H(3,𝔽₃)	C₃
hypercenter	3₋¹⁺²	AGL(1,ℤ/9)
hypocenter	C₉	C₂×#731
LCS	G ▷ C₃²×C₉ ▷ C₃×C₉ ▷ C₉
UCS	C₁ ◁ C₃ ◁ C₃² ◁ 3₋¹⁺²

subgroup

G/H

center

C₃

D₉⋊H(3,𝔽₃)

commutator, cosocle

C₃²×C₉

C₂×C₃²

Frattini

C₃³

C₃²×D₃

Fitting

#10624800

C₂

socle

C₃²

D₃×H(3,𝔽₃)

prime radical

D₉⋊H(3,𝔽₃)

C₃

3₋¹⁺²

C₉

G ▷ C₃²×C₉ ▷ C₃×C₉ ▷ C₉

UCS

C₁ ◁ C₃ ◁ C₃² ◁ 3₋¹⁺²

N	H ≅ G/N	type
1. #10624800	C₂	semidirect
3. #6894⋊C₂	C₃	semidirect
5. D₉×3₋¹⁺²	C₃	semidirect
6. C₃×D₉⋊C₉	C₃	semidirect
8. #6894	C₂×C₃	semidirect
10. C₉×3₋¹⁺²	C₂×C₃	semidirect
11. C₃×C₉⋊C₉	C₂×C₃	semidirect
65. 3₋¹⁺²	AGL(1,ℤ/9)	semidirect
90. D₉	#731	semidirect
109. C₉	C₂×#731	semidirect

H ≅ G/N

type

1. #10624800

C₂

semidirect

3. #6894⋊C₂

C₃

semidirect

5. D₉×3₋¹⁺²

C₃

semidirect

6. C₃×D₉⋊C₉

C₃

semidirect

8. #6894

C₂×C₃

semidirect

10. C₉×3₋¹⁺²

C₂×C₃

semidirect

11. C₃×C₉⋊C₉

C₂×C₃

semidirect

65. 3₋¹⁺²

AGL(1,ℤ/9)

semidirect

90. D₉

#731

semidirect

109. C₉

C₂×#731

semidirect

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

subgroup	index
#10624800	2
D₃×#731	3
#6894⋊C₂	3
D₉⋊H(3,𝔽₃)	3
D₉×3₋¹⁺²	3
C₃×D₉⋊C₉	3

subgroup

index

#10624800

D₃×#731

#6894⋊C₂

D₉⋊H(3,𝔽₃)

D₉×3₋¹⁺²

C₃×D₉⋊C₉

Subgroups — #10624800⋊C₂ #13091238

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10624800⋊C2 #13091238

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10624800⋊C₂ #13091238