Group #14547945 — subgroups

Subgroups — #14547945

Order 2187.

611 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 611 conjugacy classes of subgroups — too many to draw. Only its 50 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	#10624880	C₃	1	1	yes	maximal, characteristic, complemented
2	#10624743	C₃	1	3	yes	maximal, complemented
3	#10624737	C₃	1	3	yes	maximal, complemented
4	#10624665	C₃	1	3	yes	maximal, complemented
5	#10624608	C₃	1	3	yes	maximal, complemented
6	C₃²×3₋¹⁺²	C₃²	1	3	yes	complemented
7	C₃²×H(3,𝔽₃)	—	3	1	no	—
8	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
9	C₃×#732	—	3	9	no	—
10	C₃×AΣL(1,𝔽₂₇)	—	3	3	no	—
11	C₃×AΣL(1,𝔽₂₇)	—	3	6	no	—
12	#6891	C₃²	1	3	yes	complemented
13	C₃²⋊H(3,𝔽₃)	—	3	2	no	—
14	C₃²⋊H(3,𝔽₃)	—	3	9	no	—
15	C₃²⋊H(3,𝔽₃)	C₃²	1	3	yes	—
16	C₃²⋊H(3,𝔽₃)	C₃²	1	3	yes	complemented
17	#6857	—	3	9	no	—
18	#6856	—	3	9	no	—
19	C₃⁴	—	3	3	no	—
20	C₃⁴	H(3,𝔽₃)	1	1	yes	characteristic
21	C₃⁴	H(3,𝔽₃)	1	2	yes	—
22	C₃⁴	C₃³	1	1	yes	characteristic
23	C₃×3₋¹⁺²	—	3	3	no	—
24	C₃×3₋¹⁺²	—	3	6	no	—
25	C₃×3₋¹⁺²	—	3	3	no	—
26	C₃×3₋¹⁺²	—	9	9	no	—
27	C₃×H(3,𝔽₃)	—	3	6	no	—
28	C₃×H(3,𝔽₃)	—	3	3	no	—
29	C₃×H(3,𝔽₃)	—	3	3	no	—
30	C₃×H(3,𝔽₃)	—	3	3	no	—
31	C₃×H(3,𝔽₃)	—	3	6	no	—
32	C₃×H(3,𝔽₃)	—	3	18	no	—
33	C₃×H(3,𝔽₃)	—	3	6	no	—
34	C₃×H(3,𝔽₃)	—	3	3	no	—
35	C₃×H(3,𝔽₃)	—	9	9	no	—
36	C₃×H(3,𝔽₃)	—	9	3	no	—
37	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	9	yes	—
38	C₃²×C₉	—	3	3	no	—
39	#732	—	9	27	no	—
40	AΣL(1,𝔽₂₇)	—	9	9	no	—
41	AΣL(1,𝔽₂₇)	—	9	18	no	—
42	C₃²⋊C₉	—	3	6	no	—
43	C₃²⋊C₉	—	3	3	no	—
44	C₃³	—	3	1	no	—
45	C₃³	—	3	3	no	—
46	C₃³	—	3	2	no	—
47	C₃³	—	3	9	no	—
48	C₃³	—	9	1	no	—
49	C₃³	—	9	3	no	—
50	C₃³	—	9	9	no	—
51	C₃³	—	9	3	no	—
52	C₃³	—	9	6	no	—
53	C₃³	—	9	2	no	—
54	C₃³	—	9	3	no	—
55	C₃³	—	27	9	no	—
56	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
57	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
58	3₋¹⁺²	—	3	9	no	—
59	3₋¹⁺²	—	9	6	no	—
60	3₋¹⁺²	—	9	18	no	—
61	3₋¹⁺²	—	9	3	no	—
62	3₋¹⁺²	—	9	6	no	—
63	H(3,𝔽₃)	—	9	9	no	—
64	H(3,𝔽₃)	—	9	9	no	—
65	H(3,𝔽₃)	—	9	18	no	—
66	H(3,𝔽₃)	—	9	9	no	—
67	H(3,𝔽₃)	—	9	3	no	—
68	H(3,𝔽₃)	—	9	6	no	—
69	H(3,𝔽₃)	—	9	18	no	—
70	H(3,𝔽₃)	—	27	6	no	—
71	H(3,𝔽₃)	—	27	3	no	—
72	H(3,𝔽₃)	—	27	18	no	—
73	C₃×C₉	—	3	9	no	—
74	C₃×C₉	—	9	6	no	—
75	C₃×C₉	—	9	3	no	—
76	C₃×C₉	—	9	3	no	—
77	C₃²	—	3	3	no	—
78	C₃²	—	3	3	no	—
79	C₃²	—	3	6	no	—
80	C₃²	—	3	3	no	—
81	C₃²	—	3	1	no	—
82	C₃²	—	3	3	no	—
83	C₃²	—	9	3	no	—
84	C₃²	—	9	9	no	—
85	C₃²	—	9	6	no	—
86	C₃²	—	9	18	no	—
87	C₃²	—	9	2	no	—
88	C₃²	—	9	9	no	—
89	C₃²	—	27	3	no	—
90	C₃²	—	27	3	no	—
91	C₃²	—	27	6	no	—
92	C₃²	—	27	27	no	—
93	C₃²	—	27	3	no	—
94	C₃²	—	27	6	no	—
95	C₃²	—	27	3	no	—
96	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
97	C₉	—	9	9	no	—
98	C₉	—	27	3	no	—
99	C₃	—	9	3	no	—
100	C₃	—	9	1	no	—
101	C₃	—	9	3	no	—
102	C₃	—	27	2	no	—
103	C₃	—	27	9	no	—
104	C₃	—	81	3	no	—
105	C₃	UT(4,𝔽₃)	1	3	yes	—
106	C₃	#10624754	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10624880

C₃

yes

maximal, characteristic, complemented

#10624743

C₃

yes

maximal, complemented

#10624737

C₃

yes

maximal, complemented

#10624665

C₃

yes

maximal, complemented

#10624608

C₃

yes

maximal, complemented

C₃²×3₋¹⁺²

C₃²

yes

complemented

C₃²×H(3,𝔽₃)

—

C₃²×H(3,𝔽₃)

C₃²

yes

characteristic

C₃×#732

—

C₃×AΣL(1,𝔽₂₇)

—

C₃×AΣL(1,𝔽₂₇)

—

#6891

C₃²

yes

complemented

C₃²⋊H(3,𝔽₃)

—

C₃²⋊H(3,𝔽₃)

—

C₃²⋊H(3,𝔽₃)

C₃²

yes

—

C₃²⋊H(3,𝔽₃)

C₃²

yes

complemented

#6857

—

#6856

—

C₃⁴

—

C₃⁴

H(3,𝔽₃)

yes

characteristic

C₃⁴

H(3,𝔽₃)

yes

—

C₃⁴

C₃³

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

H(3,𝔽₃)

yes

—

C₃²×C₉

—

#732

—

AΣL(1,𝔽₂₇)

—

AΣL(1,𝔽₂₇)

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃×H(3,𝔽₃)

yes

—

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

—

—

—

—

—

—

—

—

—

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₃²⋊H(3,𝔽₃)

yes

characteristic

C₉

—

C₉

—

C₃

—

100

C₃

—

101

C₃

—

102

C₃

—

103

C₃

—

104

C₃

—

105

C₃

UT(4,𝔽₃)

yes

—

106

C₃

#10624754

yes

characteristic

	subgroup	G/H
center, socle	C₃²	C₃²⋊H(3,𝔽₃)
commutator, Frattini, cosocle	C₃⁴	C₃³
LCS	G ▷ C₃⁴ ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ G
LECS	G ▷ C₃⁴ ▷ C₃² ▷ C₁

subgroup

G/H

center, socle

C₃²

C₃²⋊H(3,𝔽₃)

commutator, Frattini, cosocle

C₃⁴

C₃³

LCS

G ▷ C₃⁴ ▷ C₃² ▷ C₁

UCS

C₁ ◁ C₃² ◁ C₃⁴ ◁ G

LECS

G ▷ C₃⁴ ▷ C₃² ▷ C₁

N	H ≅ G/N	type
1. #10624880	C₃	semidirect
2. #10624743	C₃	semidirect
3. #10624737	C₃	semidirect
4. #10624665	C₃	semidirect
5. #10624608	C₃	semidirect
6. C₃²×3₋¹⁺²	C₃²	semidirect
12. #6891	C₃²	semidirect
16. C₃²⋊H(3,𝔽₃)	C₃²	semidirect

H ≅ G/N

type

1. #10624880

C₃

semidirect

2. #10624743

C₃

semidirect

3. #10624737

C₃

semidirect

4. #10624665

C₃

semidirect

5. #10624608

C₃

semidirect

6. C₃²×3₋¹⁺²

C₃²

semidirect

12. #6891

C₃²

semidirect

16. C₃²⋊H(3,𝔽₃)

C₃²

semidirect

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

subgroup	index
#10624880	3
#10624743	3
#10624737	3
#10624665	3
#10624608	3

subgroup

index

Subgroups — #14547945

Conjugacy classes of subgroups

Complements

Maximal subgroups