Group #14550603 — subgroups

Subgroups — #14550603

Order 2187.

1337 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 1337 conjugacy classes of subgroups — too many to draw. Only its 95 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	C₃×#6909	C₃	1	9	yes	maximal, complemented
2	#10624881	C₃	1	2	yes	maximal, complemented
3	#10624880	C₃	1	1	yes	maximal, characteristic, complemented
4	#10624880	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃²×3₋¹⁺²	C₃²	1	6	yes	—
6	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
7	C₃²×H(3,𝔽₃)	C₃²	1	3	yes	—
8	C₃²×H(3,𝔽₃)	C₃²	1	3	yes	—
9	#6909	—	3	54	no	—
10	#6909	—	3	27	no	—
11	C₃×AΣL(1,𝔽₂₇)	—	3	27	no	—
12	#6893	—	3	18	no	—
13	#6891	—	3	6	no	—
14	C₃²⋊H(3,𝔽₃)	—	3	3	no	—
15	C₃²⋊H(3,𝔽₃)	—	3	3	no	—
16	C₃²⋊H(3,𝔽₃)	—	3	9	no	—
17	C₃²⋊H(3,𝔽₃)	—	3	9	no	—
18	C₃⁴	—	3	3	no	—
19	C₃⁴	H(3,𝔽₃)	1	3	yes	—
20	C₃⁴	H(3,𝔽₃)	1	9	yes	complemented
21	C₃⁴	C₃³	1	1	yes	characteristic
22	C₃×3₋¹⁺²	—	3	6	no	—
23	C₃×3₋¹⁺²	—	3	6	no	—
24	C₃×3₋¹⁺²	—	3	18	no	—
25	C₃×3₋¹⁺²	—	3	18	no	—
26	C₃×3₋¹⁺²	—	3	18	no	—
27	C₃×3₋¹⁺²	H(3,𝔽₃)	1	18	yes	complemented
28	C₃×H(3,𝔽₃)	—	3	9	no	—
29	C₃×H(3,𝔽₃)	—	3	18	no	—
30	C₃×H(3,𝔽₃)	—	3	3	no	—
31	C₃×H(3,𝔽₃)	—	3	3	no	—
32	C₃×H(3,𝔽₃)	—	3	3	no	—
33	C₃×H(3,𝔽₃)	—	3	6	no	—
34	C₃×H(3,𝔽₃)	—	3	3	no	—
35	C₃×H(3,𝔽₃)	—	3	3	no	—
36	C₃×H(3,𝔽₃)	—	3	6	no	—
37	C₃×H(3,𝔽₃)	—	3	18	no	—
38	C₃×H(3,𝔽₃)	—	3	9	no	—
39	C₃×H(3,𝔽₃)	—	3	27	no	—
40	C₃×H(3,𝔽₃)	—	3	27	no	—
41	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	9	yes	complemented
42	C₃²×C₉	—	3	6	no	—
43	AΣL(1,𝔽₂₇)	—	3	243	no	—
44	C₃²⋊C₉	—	3	54	no	—
45	C₃³	—	3	6	no	—
46	C₃³	—	3	3	no	—
47	C₃³	—	3	3	no	—
48	C₃³	—	3	9	no	—
49	C₃³	—	3	9	no	—
50	C₃³	—	3	27	no	—
51	C₃³	—	9	9	no	—
52	C₃³	—	27	3	no	—
53	C₃³	—	27	1	no	—
54	C₃³	—	27	9	no	—
55	C₃³	—	27	9	no	—
56	C₃³	C₃×H(3,𝔽₃)	1	6	yes	—
57	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
58	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
59	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
60	3₋¹⁺²	—	3	54	no	—
61	3₋¹⁺²	—	9	6	no	—
62	3₋¹⁺²	—	9	18	no	—
63	3₋¹⁺²	—	9	6	no	—
64	3₋¹⁺²	—	9	6	no	—
65	H(3,𝔽₃)	—	3	27	no	—
66	H(3,𝔽₃)	—	9	6	no	—
67	H(3,𝔽₃)	—	9	9	no	—
68	H(3,𝔽₃)	—	9	3	no	—
69	H(3,𝔽₃)	—	9	6	no	—
70	H(3,𝔽₃)	—	9	9	no	—
71	H(3,𝔽₃)	—	9	18	no	—
72	H(3,𝔽₃)	—	9	3	no	—
73	H(3,𝔽₃)	—	27	27	no	—
74	H(3,𝔽₃)	—	27	27	no	—
75	C₃×C₉	—	3	18	no	—
76	C₃×C₉	—	9	18	no	—
77	C₃²	—	9	1	no	—
78	C₃²	—	9	3	no	—
79	C₃²	—	9	3	no	—
80	C₃²	—	9	6	no	—
81	C₃²	—	9	9	no	—
82	C₃²	—	9	9	no	—
83	C₃²	—	9	27	no	—
84	C₃²	—	9	27	no	—
85	C₃²	—	9	18	no	—
86	C₃²	—	9	9	no	—
87	C₃²	—	9	3	no	—
88	C₃²	—	9	9	no	—
89	C₃²	—	27	27	no	—
90	C₃²	—	27	27	no	—
91	C₃²	C₃²⋊H(3,𝔽₃)	1	2	yes	—
92	C₃²	C₃²⋊H(3,𝔽₃)	1	9	yes	—
93	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
94	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
95	C₉	—	9	18	no	—
96	C₃	—	3	1	no	—
97	C₃	—	3	1	no	—
98	C₃	—	3	2	no	—
99	C₃	—	3	9	no	—
100	C₃	—	27	3	no	—
101	C₃	—	27	9	no	—
102	C₃	—	27	9	no	—
103	C₃	#10624562	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#6909

C₃

yes

maximal, complemented

#10624881

C₃

yes

maximal, complemented

#10624880

C₃

yes

maximal, characteristic, complemented

#10624880

C₃

yes

maximal, characteristic, complemented

C₃²×3₋¹⁺²

C₃²

yes

—

C₃²×H(3,𝔽₃)

C₃²

yes

characteristic

C₃²×H(3,𝔽₃)

C₃²

yes

—

C₃²×H(3,𝔽₃)

C₃²

yes

—

#6909

—

#6909

—

C₃×AΣL(1,𝔽₂₇)

—

#6893

—

#6891

—

C₃²⋊H(3,𝔽₃)

—

C₃²⋊H(3,𝔽₃)

—

C₃²⋊H(3,𝔽₃)

—

C₃²⋊H(3,𝔽₃)

—

C₃⁴

—

C₃⁴

H(3,𝔽₃)

yes

—

C₃⁴

H(3,𝔽₃)

yes

complemented

C₃⁴

C₃³

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

H(3,𝔽₃)

yes

complemented

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)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

H(3,𝔽₃)

yes

complemented

C₃²×C₉

—

AΣL(1,𝔽₂₇)

—

243

—

C₃²⋊C₉

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃×H(3,𝔽₃)

yes

—

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

C₃³

C₃×H(3,𝔽₃)

yes

—

C₃³

C₃×H(3,𝔽₃)

yes

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

—

—

—

—

—

—

—

—

—

—

C₃×C₉

—

C₃×C₉

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₃²⋊H(3,𝔽₃)

yes

—

C₃²

C₃²⋊H(3,𝔽₃)

yes

—

C₃²

C₃²⋊H(3,𝔽₃)

yes

characteristic

C₃²

C₃²⋊H(3,𝔽₃)

yes

characteristic

C₉

—

C₃

—

C₃

—

C₃

—

C₃

—

100

C₃

—

101

C₃

—

102

C₃

—

103

C₃

#10624562

yes

characteristic

	subgroup	G/H
center, socle	C₃	#10624562
commutator, Frattini, cosocle	C₃⁴	C₃³
LCS	G ▷ C₃⁴ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃ ◁ C₃⁴ ◁ G
LECS	G ▷ C₃⁴ ▷ C₃ ▷ C₁

subgroup

G/H

center, socle

C₃

#10624562

commutator, Frattini, cosocle

C₃⁴

C₃³

LCS

G ▷ C₃⁴ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃ ◁ C₃⁴ ◁ G

LECS

G ▷ C₃⁴ ▷ C₃ ▷ C₁

N	H ≅ G/N	type
1. C₃×#6909	C₃	semidirect
2. #10624881	C₃	semidirect
3. #10624880	C₃	semidirect
4. #10624880	C₃	semidirect
20. C₃⁴	H(3,𝔽₃)	semidirect
27. C₃×3₋¹⁺²	H(3,𝔽₃)	semidirect
41. C₃×H(3,𝔽₃)	H(3,𝔽₃)	semidirect

H ≅ G/N

type

1. C₃×#6909

C₃

semidirect

2. #10624881

C₃

semidirect

3. #10624880

C₃

semidirect

4. #10624880

C₃

semidirect

20. C₃⁴

H(3,𝔽₃)

semidirect

27. C₃×3₋¹⁺²

H(3,𝔽₃)

semidirect

41. C₃×H(3,𝔽₃)

H(3,𝔽₃)

semidirect

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

subgroup	index
C₃×#6909	3
#10624881	3
#10624880	3

subgroup

index

C₃×#6909

#10624881

#10624880

Subgroups — #14550603

Conjugacy classes of subgroups

Complements

Maximal subgroups