Group #13091195 — subgroups

Subgroups — #13091195

Order 1458.

155 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 155 conjugacy classes of subgroups — too many to draw. Only its 26 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	#10624776	C₂	1	1	yes	maximal, characteristic, complemented, Sylow
2	C₃×C₃⋊AGL(1,ℤ/9)	C₃	1	1	yes	maximal, characteristic
3	#92259	—	3	1	no	maximal
4	#92177	C₃	1	3	yes	maximal, complemented
5	C₃²×3₋¹⁺²	C₂×C₃	1	1	yes	characteristic
6	#6891	—	2	1	no	—
7	#6891	D₃	1	1	yes	characteristic
8	#6868	—	2	3	no	—
9	#6868	C₂×C₃	1	3	yes	complemented
10	C₃²×C₃⋊D₃	—	3	1	no	—
11	C₃⋊AGL(1,ℤ/9)	—	3	1	no	—
12	C₃×C₃⋊D₉	C₃²	1	1	yes	characteristic
13	D₃×3₋¹⁺²	—	9	1	no	—
14	C₃×AGL(1,ℤ/9)	—	9	1	no	—
15	(C₃⋊D₃)⋊C₉	—	3	3	no	—
16	C₃⁴	C₃×D₃	1	1	yes	characteristic
17	C₃×3₋¹⁺²	—	3	1	no	—
18	C₃×3₋¹⁺²	—	3	1	no	—
19	C₃×3₋¹⁺²	—	3	1	no	—
20	C₃×3₋¹⁺²	—	6	3	no	—
21	C₃×3₋¹⁺²	—	6	1	no	—
22	C₃×3₋¹⁺²	—	6	1	no	—
23	C₃×3₋¹⁺²	—	6	1	no	—
24	C₃²×C₉	—	2	1	no	—
25	C₃²×C₉	C₂×C₃²	1	1	yes	characteristic
26	C₃²⋊C₉	—	2	3	no	—
27	C₃²⋊C₉	—	6	1	no	—
28	C₃²⋊C₉	—	6	1	no	—
29	C₃²⋊C₉	—	6	1	no	—
30	C₃²⋊C₉	C₃×D₃	1	3	yes	—
31	C₃×C₃⋊D₃	—	3	1	no	—
32	C₃×C₃⋊D₃	—	9	1	no	—
33	C₃²×D₃	—	9	1	no	—
34	C₃²×D₃	—	27	1	no	—
35	C₂×3₋¹⁺²	—	27	1	no	—
36	C₃⋊D₉	3₋¹⁺²	1	1	yes	characteristic, complemented
37	AGL(1,ℤ/9)	—	9	3	no	—
38	D₃×C₉	—	9	3	no	—
39	C₃×D₉	—	9	1	no	—
40	C₃³	—	2	1	no	—
41	C₃³	—	3	1	no	—
42	C₃³	—	6	1	no	—
43	C₃³	—	9	1	no	—
44	C₃³	—	18	1	no	—
45	C₃³	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
46	C₃³	C₃²×D₃	1	1	yes	characteristic
47	3₋¹⁺²	—	3	3	no	—
48	3₋¹⁺²	—	6	3	no	—
49	3₋¹⁺²	—	6	3	no	—
50	3₋¹⁺²	—	6	3	no	—
51	3₋¹⁺²	—	6	3	no	—
52	3₋¹⁺²	—	9	1	no	—
53	3₋¹⁺²	—	18	1	no	—
54	3₋¹⁺²	—	18	3	no	—
55	C₃×C₉	—	2	3	no	—
56	C₃×C₉	—	2	1	no	—
57	C₃×C₉	—	3	1	no	—
58	C₃×C₉	—	3	3	no	—
59	C₃×C₉	—	6	1	no	—
60	C₃×C₉	—	6	1	no	—
61	C₃×C₉	—	6	1	no	—
62	C₃×C₉	—	6	1	no	—
63	C₃×C₉	—	6	3	no	—
64	C₃×C₉	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
65	C₂×C₃²	—	27	1	no	—
66	C₃⋊D₃	—	3	1	no	—
67	C₃×D₃	—	9	1	no	—
68	C₃×D₃	—	27	1	no	—
69	C₃×D₃	—	27	1	no	—
70	C₃×D₃	—	27	3	no	—
71	C₂×C₉	—	27	3	no	—
72	D₉	—	9	1	no	—
73	C₃²	—	2	1	no	—
74	C₃²	—	3	1	no	—
75	C₃²	—	3	1	no	—
76	C₃²	—	3	1	no	—
77	C₃²	—	6	1	no	—
78	C₃²	—	6	1	no	—
79	C₃²	—	6	2	no	—
80	C₃²	—	6	1	no	—
81	C₃²	—	6	1	no	—
82	C₃²	—	9	3	no	—
83	C₃²	—	18	3	no	—
84	C₃²	C₃×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
85	C₃²	D₃×3₋¹⁺²	1	1	yes	characteristic
86	C₉	—	3	1	no	—
87	C₉	—	6	1	no	—
88	C₉	—	6	3	no	—
89	C₉	—	9	3	no	—
90	C₉	—	18	3	no	—
91	C₂×C₃	—	27	1	no	—
92	C₂×C₃	—	81	1	no	—
93	D₃	—	9	1	no	—
94	D₃	—	27	1	no	—
95	C₃	—	2	1	no	—
96	C₃	—	3	1	no	—
97	C₃	—	6	1	no	—
98	C₃	—	9	1	no	—
99	C₃	—	18	1	no	—
100	C₃	#92259	1	1	yes	characteristic
101	C₃	#4179⋊C₃	1	1	yes	characteristic
102	C₂	—	27	1	no	Sylow

subgroup

G/H

class size

multiplicity

normal?

properties

#10624776

C₂

yes

maximal, characteristic, complemented, Sylow

C₃×C₃⋊AGL(1,ℤ/9)

C₃

yes

maximal, characteristic

#92259

—

maximal

#92177

C₃

yes

maximal, complemented

C₃²×3₋¹⁺²

C₂×C₃

yes

characteristic

#6891

—

#6891

D₃

yes

characteristic

#6868

—

#6868

C₂×C₃

yes

complemented

C₃²×C₃⋊D₃

—

C₃⋊AGL(1,ℤ/9)

—

C₃×C₃⋊D₉

C₃²

yes

characteristic

D₃×3₋¹⁺²

—

C₃×AGL(1,ℤ/9)

—

(C₃⋊D₃)⋊C₉

—

C₃⁴

C₃×D₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃²×C₉

—

C₃²×C₉

C₂×C₃²

yes

characteristic

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×D₃

yes

—

C₃×C₃⋊D₃

—

C₃×C₃⋊D₃

—

C₃²×D₃

—

C₃²×D₃

—

C₂×3₋¹⁺²

—

C₃⋊D₉

3₋¹⁺²

yes

characteristic, complemented

AGL(1,ℤ/9)

—

D₃×C₉

—

C₃×D₉

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

(C₃⋊D₃)⋊C₃

yes

characteristic

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

—

3₋¹⁺²

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₂×3₋¹⁺²

yes

characteristic, complemented

C₂×C₃²

—

C₃⋊D₃

—

C₃×D₃

—

C₃×D₃

—

C₃×D₃

—

C₃×D₃

—

C₂×C₉

—

D₉

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₃×(C₃⋊D₃)⋊C₃

yes

characteristic

C₃²

D₃×3₋¹⁺²

yes

characteristic

C₉

—

C₉

—

C₉

—

C₉

—

C₉

—

C₂×C₃

—

C₂×C₃

—

D₃

—

D₃

—

C₃

—

C₃

—

C₃

—

C₃

—

C₃

—

100

C₃

#92259

yes

characteristic

101

C₃

#4179⋊C₃

yes

characteristic

102

C₂

—

Sylow

	subgroup	G/H
center, hypercenter	C₃	#4179⋊C₃
commutator, cosocle	C₃²×C₉	C₂×C₃²
Frattini	C₃³	C₃²×D₃
Fitting	#10624776	C₂
socle	C₃²	C₃×(C₃⋊D₃)⋊C₃
prime radical	C₃×C₃⋊AGL(1,ℤ/9)	C₃
hypocenter	C₃×C₉	C₂×3₋¹⁺²
LCS	G ▷ C₃²×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₃

subgroup

G/H

center, hypercenter

C₃

#4179⋊C₃

commutator, cosocle

C₃²×C₉

C₂×C₃²

Frattini

C₃³

C₃²×D₃

Fitting

#10624776

C₂

socle

C₃²

C₃×(C₃⋊D₃)⋊C₃

prime radical

C₃×C₃⋊AGL(1,ℤ/9)

C₃

hypocenter

C₃×C₉

C₂×3₋¹⁺²

LCS

G ▷ C₃²×C₉ ▷ C₃×C₉

UCS

C₁ ◁ C₃

N	H ≅ G/N	type
1. #10624776	C₂	semidirect
4. #92177	C₃	semidirect
9. #6868	C₂×C₃	semidirect
36. C₃⋊D₉	3₋¹⁺²	semidirect
64. C₃×C₉	C₂×3₋¹⁺²	semidirect

H ≅ G/N

type

1. #10624776

C₂

semidirect

4. #92177

C₃

semidirect

9. #6868

C₂×C₃

semidirect

36. C₃⋊D₉

3₋¹⁺²

semidirect

64. C₃×C₉

C₂×3₋¹⁺²

semidirect

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

subgroup	index
#10624776	2
C₃×C₃⋊AGL(1,ℤ/9)	3
#92259	3
#92177	3

subgroup

index

Subgroups — #13091195

Conjugacy classes of subgroups

Complements

Maximal subgroups