Group #14548129 — subgroups

Subgroups — #14548129

Order 2187.

395 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 395 conjugacy classes of subgroups — too many to draw. Only its 53 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×#6893	C₃	1	1	yes	maximal, characteristic, complemented
2	#10624768	C₃	1	2	yes	maximal, complemented
3	#10624732	C₃	1	1	yes	maximal, characteristic, complemented
4	#10624629	C₃	1	6	yes	maximal, complemented
5	#10624628	C₃	1	3	yes	maximal, complemented
6	C₃²×3₋¹⁺²	C₃²	1	2	yes	—
7	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
8	C₃³×C₉	C₃²	1	1	yes	characteristic
9	C₃×#732	—	3	3	no	—
10	#6893	—	3	6	no	—
11	#6893	—	3	3	no	—
12	#6891	C₃²	1	6	yes	—
13	C₃²⋊H(3,𝔽₃)	C₃²	1	3	yes	—
14	C₃×C₃²⋊C₉	—	3	3	no	—
15	#6868	—	3	6	no	—
16	#6857	—	3	18	no	—
17	#6856	—	3	9	no	—
18	C₃⁴	—	3	1	no	—
19	C₃⁴	C₃³	1	1	yes	characteristic
20	C₃×3₋¹⁺²	—	3	2	no	—
21	C₃×3₋¹⁺²	—	3	2	no	—
22	C₃×3₋¹⁺²	—	3	6	no	—
23	C₃×3₋¹⁺²	—	9	6	no	—
24	C₃×3₋¹⁺²	H(3,𝔽₃)	1	6	yes	—
25	C₃×H(3,𝔽₃)	—	3	2	no	—
26	C₃×H(3,𝔽₃)	—	3	3	no	—
27	C₃×H(3,𝔽₃)	—	3	3	no	—
28	C₃×H(3,𝔽₃)	—	3	6	no	—
29	C₃×H(3,𝔽₃)	—	9	3	no	—
30	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	3	yes	—
31	C₃²×C₉	—	3	1	no	—
32	C₃²×C₉	—	3	2	no	—
33	C₃²×C₉	—	3	2	no	—
34	C₃²×C₉	—	9	3	no	—
35	C₃²×C₉	H(3,𝔽₃)	1	3	yes	—
36	#732	—	9	9	no	—
37	C₃²⋊C₉	—	3	27	no	—
38	C₃²⋊C₉	—	3	6	no	—
39	C₃²⋊C₉	—	3	6	no	—
40	C₃²⋊C₉	—	3	6	no	—
41	C₃³	—	3	3	no	—
42	C₃³	—	3	3	no	—
43	C₃³	—	9	2	no	—
44	C₃³	—	9	1	no	—
45	C₃³	—	9	6	no	—
46	C₃³	—	9	3	no	—
47	C₃³	—	9	3	no	—
48	C₃³	—	9	3	no	—
49	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
50	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
51	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
52	3₋¹⁺²	—	9	6	no	—
53	3₋¹⁺²	—	9	6	no	—
54	3₋¹⁺²	—	9	6	no	—
55	3₋¹⁺²	—	27	6	no	—
56	H(3,𝔽₃)	—	9	3	no	—
57	H(3,𝔽₃)	—	9	6	no	—
58	H(3,𝔽₃)	—	9	9	no	—
59	H(3,𝔽₃)	—	27	6	no	—
60	H(3,𝔽₃)	—	27	3	no	—
61	C₃×C₉	—	3	3	no	—
62	C₃×C₉	—	3	6	no	—
63	C₃×C₉	—	3	3	no	—
64	C₃×C₉	—	3	6	no	—
65	C₃×C₉	—	9	3	no	—
66	C₃×C₉	—	9	6	no	—
67	C₃×C₉	—	9	6	no	—
68	C₃×C₉	—	9	6	no	—
69	C₃²	—	3	2	no	—
70	C₃²	—	3	1	no	—
71	C₃²	—	3	1	no	—
72	C₃²	—	3	2	no	—
73	C₃²	—	3	2	no	—
74	C₃²	—	3	2	no	—
75	C₃²	—	3	2	no	—
76	C₃²	—	3	1	no	—
77	C₃²	—	3	2	no	—
78	C₃²	—	9	9	no	—
79	C₃²	—	9	3	no	—
80	C₃²	—	9	6	no	—
81	C₃²	—	9	3	no	—
82	C₃²	—	27	9	no	—
83	C₃²	—	27	3	no	—
84	C₃²	—	27	6	no	—
85	C₃²	—	27	3	no	—
86	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
87	C₃²	#6909	1	3	yes	—
88	C₉	—	9	3	no	—
89	C₉	—	9	6	no	—
90	C₉	—	27	6	no	—
91	C₃	—	3	3	no	—
92	C₃	—	9	2	no	—
93	C₃	—	9	1	no	—
94	C₃	—	27	3	no	—
95	C₃	—	81	3	no	—
96	C₃	#10624735	1	1	yes	characteristic
97	C₃	#10624757	1	2	yes	—
98	C₃	#10624759	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#6893

C₃

yes

maximal, characteristic, complemented

#10624768

C₃

yes

maximal, complemented

#10624732

C₃

yes

maximal, characteristic, complemented

#10624629

C₃

yes

maximal, complemented

#10624628

C₃

yes

maximal, complemented

C₃²×3₋¹⁺²

C₃²

yes

—

C₃²×H(3,𝔽₃)

C₃²

yes

characteristic

C₃³×C₉

C₃²

yes

—

—

—

C₃²

yes

—

C₃²⋊H(3,𝔽₃)

C₃²

yes

—

C₃×C₃²⋊C₉

—

#6868

—

#6857

—

#6856

—

C₃⁴

—

C₃⁴

C₃³

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

H(3,𝔽₃)

yes

—

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

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

H(3,𝔽₃)

yes

—

#732

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

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

—

—

—

—

—

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

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⋊H(3,𝔽₃)

yes

characteristic

C₃²

#6909

yes

—

C₉

—

C₉

—

C₉

—

C₃

—

C₃

—

C₃

—

C₃

—

C₃

—

C₃

#10624735

yes

characteristic

C₃

#10624757

yes

—

C₃

#10624759

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. C₃×#6893	C₃	semidirect
2. #10624768	C₃	semidirect
3. #10624732	C₃	semidirect
4. #10624629	C₃	semidirect
5. #10624628	C₃	semidirect

H ≅ G/N

type

1. C₃×#6893

C₃

semidirect

2. #10624768

C₃

semidirect

3. #10624732

C₃

semidirect

4. #10624629

C₃

semidirect

5. #10624628

C₃

semidirect

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

subgroup	index
C₃×#6893	3
#10624768	3
#10624732	3
#10624629	3
#10624628	3

subgroup

index

Subgroups — #14548129

Conjugacy classes of subgroups

Complements

Maximal subgroups