Group #13091228 — subgroups

Subgroups — #13091228

Order 1458.

138 conjugacy classes of subgroups; only the 31 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 31 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	#10624793	C₂	1	1	yes	maximal, characteristic, complemented, Sylow
2	C₉×#272	C₃	1	1	yes	maximal, characteristic
3	C₃×#4176	C₃	1	1	yes	maximal, characteristic
4	D₃×C₉⋊C₉	—	3	1	no	maximal
5	#272⋊C₉	C₃	1	2	yes	maximal
6	C₃×#731	C₂×C₃	1	1	yes	characteristic
7	#6894	—	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	3₋¹⁺²⋊C₉	—	2	2	no	—
11	H(3,𝔽₃)⋊C₉	C₂×C₃	1	2	yes	—
12	#6867	—	2	1	no	—
13	C₃×#272	C₃²	1	1	yes	characteristic
14	C₃×D₃×C₉	—	3	1	no	—
15	C₃×D₃×C₉	—	3	2	no	—
16	C₃×D₃×C₉	—	3	1	no	—
17	C₃×D₃×C₉	—	9	1	no	—
18	C₂×C₉⋊C₉	—	9	1	no	—
19	#4176	—	3	1	no	—
20	C₃×3₋¹⁺²	—	2	1	no	—
21	C₃×H(3,𝔽₃)	C₂×C₃²	1	1	yes	characteristic
22	C₃²×C₉	—	3	1	no	—
23	C₃²×C₉	C₃×D₃	1	1	yes	characteristic
24	C₃²×C₉	C₃×D₃	1	1	yes	characteristic
25	C₃²×C₉	C₃×D₃	1	2	yes	—
26	#731	—	3	1	no	—
27	#731	—	6	1	no	—
28	C₉⋊C₉	—	3	1	no	—
29	C₉⋊C₉	—	6	1	no	—
30	C₉⋊C₉	—	6	2	no	—
31	C₉⋊C₉	—	6	1	no	—
32	C₃²⋊C₉	—	2	2	no	—
33	C₃²⋊C₉	—	2	1	no	—
34	C₃²⋊C₉	—	6	1	no	—
35	C₃²×D₃	—	3	1	no	—
36	C₃²×D₃	—	9	1	no	—
37	C₂×C₃×C₉	—	9	1	no	—
38	C₂×C₃×C₉	—	9	2	no	—
39	C₂×C₃×C₉	—	9	1	no	—
40	#272	3₋¹⁺²	1	1	yes	characteristic
41	D₃×C₉	—	3	3	no	—
42	D₃×C₉	—	9	3	no	—
43	D₃×C₉	—	9	1	no	—
44	D₃×C₉	—	9	2	no	—
45	C₃³	—	3	1	no	—
46	C₃³	C₃²×D₃	1	1	yes	characteristic
47	3₋¹⁺²	—	6	2	no	—
48	3₋¹⁺²	—	6	1	no	—
49	H(3,𝔽₃)	—	2	1	no	—
50	H(3,𝔽₃)	—	6	1	no	—
51	H(3,𝔽₃)	C₂×3₋¹⁺²	1	1	yes	characteristic
52	C₃×C₉	—	2	1	no	—
53	C₃×C₉	—	3	3	no	—
54	C₃×C₉	—	3	1	no	—
55	C₃×C₉	—	3	2	no	—
56	C₃×C₉	—	3	1	no	—
57	C₃×C₉	—	6	1	no	—
58	C₃×C₉	—	6	1	no	—
59	C₃×C₉	—	6	2	no	—
60	C₃×C₉	—	6	1	no	—
61	C₃×C₉	—	9	2	no	—
62	C₃×C₉	—	9	3	no	—
63	C₃×C₉	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
64	C₂×C₃²	—	9	1	no	—
65	C₃×D₃	—	3	1	no	—
66	C₃×D₃	—	3	2	no	—
67	C₃×D₃	—	3	1	no	—
68	C₃×D₃	—	9	1	no	—
69	C₃×D₃	—	9	1	no	—
70	C₃×D₃	—	9	2	no	—
71	C₂×C₉	—	9	3	no	—
72	C₂×C₉	—	27	1	no	—
73	C₂×C₉	—	27	2	no	—
74	C₃²	—	2	1	no	—
75	C₃²	—	3	1	no	—
76	C₃²	—	3	2	no	—
77	C₃²	—	3	1	no	—
78	C₃²	—	6	1	no	—
79	C₃²	—	9	1	no	—
80	C₃²	—	9	2	no	—
81	C₃²	C₃×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
82	C₃²	D₃×3₋¹⁺²	1	1	yes	characteristic
83	C₉	—	6	1	no	—
84	C₉	—	9	1	no	—
85	C₉	—	9	2	no	—
86	C₉	—	18	1	no	—
87	C₉	—	18	1	no	—
88	C₉	—	18	2	no	—
89	C₉	#4176	1	3	yes	complemented
90	C₂×C₃	—	9	1	no	—
91	C₂×C₃	—	9	1	no	—
92	C₂×C₃	—	9	2	no	—
93	D₃	—	3	1	no	—
94	D₃	—	9	1	no	—
95	C₃	—	3	1	no	—
96	C₃	—	6	1	no	—
97	C₃	—	9	1	no	—
98	C₃	—	18	1	no	—
99	C₃	#92262	1	1	yes	characteristic
100	C₃	C₃×#4176	1	1	yes	characteristic
101	C₃	#6913⋊C₂	1	2	yes	—
102	C₂	—	9	1	no	Sylow

subgroup

G/H

class size

multiplicity

normal?

properties

#10624793

C₂

yes

maximal, characteristic, complemented, Sylow

C₉×#272

C₃

yes

maximal, characteristic

C₃×#4176

C₃

yes

maximal, characteristic

D₃×C₉⋊C₉

—

maximal

#272⋊C₉

C₃

yes

maximal

C₃×#731

C₂×C₃

yes

characteristic

#6894

—

C₉×H(3,𝔽₃)

C₂×C₃

yes

characteristic

C₃×C₉⋊C₉

D₃

yes

characteristic, complemented

3₋¹⁺²⋊C₉

—

H(3,𝔽₃)⋊C₉

C₂×C₃

yes

—

#6867

—

C₃×#272

C₃²

yes

characteristic

C₃×D₃×C₉

—

C₃×D₃×C₉

—

C₃×D₃×C₉

—

C₃×D₃×C₉

—

C₂×C₉⋊C₉

—

#4176

—

C₃×3₋¹⁺²

—

C₃×H(3,𝔽₃)

C₂×C₃²

yes

characteristic

C₃²×C₉

—

C₃²×C₉

C₃×D₃

yes

characteristic

C₃²×C₉

C₃×D₃

yes

characteristic

C₃²×C₉

C₃×D₃

yes

—

#731

—

#731

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²×D₃

—

C₃²×D₃

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

#272

3₋¹⁺²

yes

characteristic

D₃×C₉

—

D₃×C₉

—

D₃×C₉

—

D₃×C₉

—

C₃³

—

C₃³

C₃²×D₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

H(3,𝔽₃)

C₂×3₋¹⁺²

yes

characteristic

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

C₂×C₃²

—

C₃×D₃

—

C₃×D₃

—

C₃×D₃

—

C₃×D₃

—

C₃×D₃

—

C₃×D₃

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

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

—

C₉

—

C₉

#4176

yes

complemented

C₂×C₃

—

C₂×C₃

—

C₂×C₃

—

D₃

—

D₃

—

C₃

—

C₃

—

C₃

—

C₃

—

C₃

#92262

yes

characteristic

100

C₃

C₃×#4176

yes

characteristic

101

C₃

#6913⋊C₂

yes

—

102

C₂

—

Sylow

	subgroup	G/H
center, socle	C₃²	C₃×(C₃⋊D₃)⋊C₃
commutator, cosocle	C₃×H(3,𝔽₃)	C₂×C₃²
Frattini	C₃³	C₃²×D₃
Fitting	#10624793	C₂
prime radical	C₃×#272	C₃²
hypercenter	C₃×C₉	(C₃⋊D₃)⋊C₃
hypocenter	H(3,𝔽₃)	C₂×3₋¹⁺²
LCS	G ▷ C₃×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₃² ◁ C₃×C₉

subgroup

G/H

center, socle

C₃²

C₃×(C₃⋊D₃)⋊C₃

commutator, cosocle

C₃×H(3,𝔽₃)

C₂×C₃²

Frattini

C₃³

C₃²×D₃

C₂

C₃²

C₃×C₉

(C₃⋊D₃)⋊C₃

hypocenter

H(3,𝔽₃)

C₂×3₋¹⁺²

LCS

G ▷ C₃×H(3,𝔽₃) ▷ H(3,𝔽₃)

UCS

C₁ ◁ C₃² ◁ C₃×C₉

N	H ≅ G/N	type
1. #10624793	C₂	semidirect
9. C₃×C₉⋊C₉	D₃	semidirect
89. C₉	#4176	semidirect

H ≅ G/N

type

1. #10624793

C₂

semidirect

9. C₃×C₉⋊C₉

D₃

semidirect

89. C₉

#4176

semidirect

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

subgroup	index
#10624793	2
C₉×#272	3
C₃×#4176	3
D₃×C₉⋊C₉	3
#272⋊C₉	3

subgroup

index

D₃×C₉⋊C₉

Subgroups — #13091228

Conjugacy classes of subgroups

Complements

Maximal subgroups