Group #13091226 — subgroups

Subgroups — #13091226

Order 1458.

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

normal subgroups only

This group has 160 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	#10624792	C₂	1	1	yes	maximal, characteristic, complemented, Sylow
2	C₉⋊#272	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃⋊#4178	—	3	1	no	maximal
4	C₃⋊(D₉⋊C₉)	—	3	1	no	maximal
5	#92178	—	3	2	no	maximal
6	C₃×#732	D₃	1	1	yes	characteristic
7	C₉⋊H(3,𝔽₃)	—	2	1	no	—
8	C₉×H(3,𝔽₃)	C₂×C₃	1	1	yes	characteristic
9	C₃×C₉⋊C₉	D₃	1	1	yes	characteristic, complemented
10	H(3,𝔽₃)⋊C₉	—	2	1	no	—
11	H(3,𝔽₃)⋊C₉	—	2	2	no	—
12	#6868	D₃	1	2	yes	complemented
13	C₃⋊#272	—	3	1	no	—
14	C₉×C₃⋊D₃	—	9	1	no	—
15	C₃×C₃⋊D₉	—	3	1	no	—
16	C₃×C₃⋊D₉	—	9	1	no	—
17	#4181	—	3	2	no	—
18	#4181	—	9	2	no	—
19	#4178	—	9	3	no	—
20	D₉⋊C₉	—	9	2	no	—
21	D₉⋊C₉	—	9	1	no	—
22	C₃×3₋¹⁺²	—	6	1	no	—
23	C₃×H(3,𝔽₃)	—	2	1	no	—
24	C₃×H(3,𝔽₃)	C₃×D₃	1	1	yes	characteristic
25	C₃²×C₉	—	2	1	no	—
26	C₃²×C₉	—	3	1	no	—
27	C₃²×C₉	C₃×D₃	1	1	yes	characteristic
28	C₃²×C₉	C₃⋊D₃	1	1	yes	characteristic
29	#732	—	3	3	no	—
30	C₉⋊C₉	—	3	1	no	—
31	C₉⋊C₉	—	3	2	no	—
32	C₃²⋊C₉	—	2	2	no	—
33	C₃²⋊C₉	—	3	2	no	—
34	C₃²⋊C₉	—	6	2	no	—
35	C₃²⋊C₉	C₃×D₃	1	2	yes	—
36	C₃×C₃⋊D₃	—	9	1	no	—
37	C₃×C₃⋊D₃	—	27	1	no	—
38	#272	—	9	3	no	—
39	#272	—	27	2	no	—
40	C₃⋊D₉	—	3	1	no	—
41	C₃⋊D₉	—	9	1	no	—
42	D₃×C₉	—	27	1	no	—
43	D₃×C₉	—	27	1	no	—
44	D₃×C₉	—	27	2	no	—
45	C₃×D₉	—	9	2	no	—
46	C₃×D₉	—	9	1	no	—
47	C₃×D₉	—	27	2	no	—
48	C₃³	—	3	1	no	—
49	C₃³	—	6	1	no	—
50	C₃³	C₃×C₃⋊D₃	1	1	yes	characteristic
51	3₋¹⁺²	—	18	2	no	—
52	3₋¹⁺²	—	18	1	no	—
53	H(3,𝔽₃)	—	3	2	no	—
54	H(3,𝔽₃)	—	6	1	no	—
55	H(3,𝔽₃)	—	6	2	no	—
56	H(3,𝔽₃)	AGL(1,ℤ/9)	1	3	yes	—
57	C₃×C₉	—	3	1	no	—
58	C₃×C₉	—	3	2	no	—
59	C₃×C₉	—	3	1	no	—
60	C₃×C₉	—	3	1	no	—
61	C₃×C₉	—	3	2	no	—
62	C₃×C₉	—	6	1	no	—
63	C₃×C₉	—	6	2	no	—
64	C₃×C₉	—	6	1	no	—
65	C₃×C₉	—	9	1	no	—
66	C₃×C₉	—	18	1	no	—
67	C₃×C₉	—	18	1	no	—
68	C₃×C₉	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
69	C₃×C₉	(C₃⋊D₃)⋊C₃	1	2	yes	—
70	C₃⋊D₃	—	9	1	no	—
71	C₃⋊D₃	—	27	1	no	—
72	C₃×D₃	—	27	1	no	—
73	C₃×D₃	—	27	1	no	—
74	C₃×D₃	—	27	2	no	—
75	C₃×D₃	—	81	3	no	—
76	C₂×C₉	—	81	1	no	—
77	D₉	—	9	2	no	—
78	D₉	—	9	1	no	—
79	D₉	—	27	2	no	—
80	C₃²	—	3	1	no	—
81	C₃²	—	3	3	no	—
82	C₃²	—	6	1	no	—
83	C₃²	—	9	1	no	—
84	C₃²	—	18	1	no	—
85	C₃²	—	18	2	no	—
86	C₃²	—	18	1	no	—
87	C₃²	—	18	1	no	—
88	C₃²	C₃⋊((C₃⋊D₃)⋊C₃)	1	1	yes	characteristic
89	C₃²	C₃⋊AGL(1,ℤ/9)	1	1	yes	characteristic
90	C₃²	#4207	1	2	yes	—
91	C₉	—	2	1	no	—
92	C₉	—	3	2	no	—
93	C₉	—	9	1	no	—
94	C₉	—	9	2	no	—
95	C₉	—	18	1	no	—
96	C₉	—	18	2	no	—
97	C₉	#4178	1	1	yes	characteristic, complemented
98	C₂×C₃	—	81	1	no	—
99	D₃	—	27	1	no	—
100	D₃	—	27	2	no	—
101	D₃	—	27	1	no	—
102	D₃	—	81	3	no	—
103	C₃	—	2	1	no	—
104	C₃	—	3	1	no	—
105	C₃	—	3	2	no	—
106	C₃	—	9	3	no	—
107	C₃	—	54	1	no	—
108	C₃	#92303	1	1	yes	characteristic
109	C₃	C₃⋊#4178	1	1	yes	characteristic
110	C₂	—	81	1	no	Sylow

subgroup

G/H

class size

multiplicity

normal?

properties

#10624792

C₂

yes

maximal, characteristic, complemented, Sylow

C₉⋊#272

C₃

yes

maximal, characteristic, complemented

C₃⋊#4178

—

maximal

C₃⋊(D₉⋊C₉)

—

—

D₃

yes

characteristic

C₉⋊H(3,𝔽₃)

—

C₉×H(3,𝔽₃)

C₂×C₃

yes

characteristic

C₃×C₉⋊C₉

D₃

yes

characteristic, complemented

H(3,𝔽₃)⋊C₉

—

H(3,𝔽₃)⋊C₉

—

#6868

D₃

yes

complemented

C₃⋊#272

—

C₉×C₃⋊D₃

—

C₃×C₃⋊D₉

—

C₃×C₃⋊D₉

—

#4181

—

#4181

—

#4178

—

D₉⋊C₉

—

D₉⋊C₉

—

C₃×3₋¹⁺²

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₃×D₃

yes

characteristic

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃×D₃

yes

characteristic

C₃²×C₉

C₃⋊D₃

yes

characteristic

#732

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×D₃

yes

—

C₃×C₃⋊D₃

—

C₃×C₃⋊D₃

—

#272

—

#272

—

C₃⋊D₉

—

C₃⋊D₉

—

D₃×C₉

—

D₃×C₉

—

D₃×C₉

—

C₃×D₉

—

C₃×D₉

—

C₃×D₉

—

C₃³

—

C₃³

—

C₃³

C₃×C₃⋊D₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

—

—

—

yes

—

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⋊D₃)⋊C₃

yes

characteristic

C₃×C₉

(C₃⋊D₃)⋊C₃

yes

—

C₃⋊D₃

—

C₃⋊D₃

—

C₃×D₃

—

C₃×D₃

—

C₃×D₃

—

C₃×D₃

—

C₂×C₉

—

D₉

—

D₉

—

D₉

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₃⋊((C₃⋊D₃)⋊C₃)

yes

C₃²

yes

C₃²

yes

—

C₉

—

C₉

—

C₉

—

C₉

—

C₉

—

C₉

—

C₉

#4178

yes

characteristic, complemented

C₂×C₃

—

D₃

—

100

D₃

—

101

D₃

—

102

D₃

—

103

C₃

—

104

C₃

—

105

C₃

—

106

C₃

—

107

C₃

—

108

C₃

#92303

yes

characteristic

109

C₃

C₃⋊#4178

yes

characteristic

110

C₂

—

Sylow

	subgroup	G/H
center, hypercenter	C₃	#92303
commutator, cosocle, hypocenter	C₉×H(3,𝔽₃)	C₂×C₃
Frattini	C₃³	C₃×C₃⋊D₃
Fitting	#10624792	C₂
socle	C₃²	C₃⋊((C₃⋊D₃)⋊C₃)
LCS	G ▷ C₉×H(3,𝔽₃)
UCS	C₁ ◁ C₃

subgroup

G/H

center, hypercenter

C₃

#92303

commutator, cosocle, hypocenter

C₉×H(3,𝔽₃)

C₂×C₃

Frattini

C₃³

C₃×C₃⋊D₃

Fitting

#10624792

C₂

socle

C₃²

C₃⋊((C₃⋊D₃)⋊C₃)

LCS

G ▷ C₉×H(3,𝔽₃)

UCS

C₁ ◁ C₃

N	H ≅ G/N	type
1. #10624792	C₂	semidirect
2. C₉⋊#272	C₃	semidirect
9. C₃×C₉⋊C₉	D₃	semidirect
12. #6868	D₃	semidirect
97. C₉	#4178	semidirect

H ≅ G/N

type

1. #10624792

C₂

semidirect

2. C₉⋊#272

C₃

semidirect

9. C₃×C₉⋊C₉

D₃

semidirect

12. #6868

D₃

semidirect

97. C₉

#4178

semidirect

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

subgroup	index
#10624792	2
C₉⋊#272	3
C₃⋊#4178	3
C₃⋊(D₉⋊C₉)	3
#92178	3

subgroup

index

C₃⋊(D₉⋊C₉)

Subgroups — #13091226

Conjugacy classes of subgroups

Complements

Maximal subgroups