Group #14554422 — subgroups

Subgroups — #14554422

Order 2187.

786 conjugacy classes of subgroups; only the 281 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 281 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×#6912	C₃	1	1	yes	maximal, characteristic
2	C₃×#6894	C₃	1	3	yes	maximal
3	C₃×C₉×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic
4	#10624791	C₃	1	18	yes	maximal, complemented
5	#10624787	C₃	1	9	yes	maximal, complemented
6	#10624699	C₃	1	6	yes	maximal
7	#10624691	C₃	1	2	yes	maximal
8	C₃²×3₋¹⁺²	C₃²	1	3	yes	—
9	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
10	C₃³×C₉	C₃²	1	1	yes	characteristic
11	#6912	—	3	9	no	—
12	C₃×#733	C₃²	1	9	yes	—
13	#6894	—	3	6	no	—
14	#6894	—	3	12	no	—
15	#6894	C₃²	1	9	yes	—
16	#6894	C₃²	1	18	yes	—
17	C₉×3₋¹⁺²	—	3	3	no	—
18	C₉×3₋¹⁺²	—	3	6	no	—
19	C₃×C₉⋊C₉	—	3	3	no	—
20	C₃×C₉⋊C₉	—	3	6	no	—
21	C₃×C₉⋊C₉	C₃²	1	6	yes	—
22	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
23	C₃×C₃²⋊C₉	C₃²	1	6	yes	—
24	C₃×C₉²	C₃²	1	3	yes	—
25	3₋¹⁺²⋊C₉	C₃²	1	54	yes	—
26	#6868	C₃²	1	18	yes	—
27	C₃⁴	C₃³	1	1	yes	characteristic
28	C₃×3₋¹⁺²	—	3	3	no	—
29	C₃×3₋¹⁺²	—	3	6	no	—
30	C₃×3₋¹⁺²	—	3	6	no	—
31	C₃×3₋¹⁺²	—	3	9	no	—
32	C₃×3₋¹⁺²	—	3	18	no	—
33	C₃×3₋¹⁺²	3₋¹⁺²	1	9	yes	—
34	C₃×3₋¹⁺²	C₃³	1	9	yes	—
35	C₃²×C₉	—	3	3	no	—
36	C₃²×C₉	—	3	6	no	—
37	C₃²×C₉	—	9	3	no	—
38	C₃²×C₉	H(3,𝔽₃)	1	3	yes	—
39	C₃²×C₉	H(3,𝔽₃)	1	6	yes	—
40	C₃²×C₉	C₃³	1	1	yes	characteristic
41	C₃²×C₉	C₃³	1	3	yes	—
42	C₃²×C₉	C₃³	1	6	yes	—
43	C₃²×C₉	C₃³	1	2	yes	—
44	#733	—	3	27	no	—
45	C₉⋊C₉	—	3	18	no	—
46	C₉⋊C₉	—	3	18	no	—
47	C₉⋊C₉	—	3	36	no	—
48	C₉⋊C₉	—	3	9	no	—
49	C₉⋊C₉	—	3	18	no	—
50	C₃²⋊C₉	—	3	6	no	—
51	C₃²⋊C₉	—	3	12	no	—
52	C₃²⋊C₉	C₃³	1	18	yes	—
53	C₉²	—	3	6	no	—
54	C₉²	—	3	3	no	—
55	C₃³	—	9	1	no	—
56	C₃³	—	9	2	no	—
57	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
58	C₃³	C₃×3₋¹⁺²	1	3	yes	—
59	C₃³	#737	1	6	yes	—
60	C₃³	C₃⁴	1	1	yes	characteristic
61	3₋¹⁺²	—	3	27	no	—
62	3₋¹⁺²	—	3	9	no	—
63	3₋¹⁺²	—	3	18	no	—
64	3₋¹⁺²	—	9	18	no	—
65	C₃×C₉	—	3	3	no	—
66	C₃×C₉	—	3	6	no	—
67	C₃×C₉	—	3	9	no	—
68	C₃×C₉	—	3	3	no	—
69	C₃×C₉	—	3	6	no	—
70	C₃×C₉	—	3	18	no	—
71	C₃×C₉	—	3	9	no	—
72	C₃×C₉	—	9	6	no	—
73	C₃×C₉	—	9	2	no	—
74	C₃×C₉	—	9	18	no	—
75	C₃×C₉	—	9	9	no	—
76	C₃×C₉	—	9	6	no	—
77	C₃×C₉	C₃×H(3,𝔽₃)	1	2	yes	—
78	C₃×C₉	C₃×H(3,𝔽₃)	1	6	yes	—
79	C₃×C₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
80	C₃×C₉	C₃×3₋¹⁺²	1	9	yes	—
81	C₃²	—	3	3	no	—
82	C₃²	—	3	1	no	—
83	C₃²	—	3	6	no	—
84	C₃²	—	3	2	no	—
85	C₃²	—	9	3	no	—
86	C₃²	—	9	6	no	—
87	C₃²	C₉⋊H(3,𝔽₃)	1	9	yes	—
88	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
89	C₃²	C₃²×3₋¹⁺²	1	1	yes	characteristic
90	C₃²	C₃×#737	1	2	yes	—
91	C₉	—	3	9	no	—
92	C₉	—	3	6	no	—
93	C₉	—	3	2	no	—
94	C₉	—	9	9	no	—
95	C₉	—	9	6	no	—
96	C₉	—	9	18	no	—
97	C₉	#6912	1	3	yes	complemented
98	C₃	—	3	1	no	—
99	C₃	—	3	2	no	—
100	C₃	—	3	9	no	—
101	C₃	C₃×C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
102	C₃	C₃×#6912	1	1	yes	characteristic
103	C₃	#10624929	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#6912

C₃

yes

maximal, characteristic

C₃×#6894

C₃

yes

maximal

C₃×C₉×3₋¹⁺²

C₃

yes

maximal, characteristic

#10624791

C₃

yes

maximal, complemented

#10624787

C₃

yes

maximal, complemented

#10624699

C₃

yes

maximal

#10624691

C₃

yes

maximal

C₃²×3₋¹⁺²

C₃²

yes

—

C₃²×3₋¹⁺²

C₃²

yes

characteristic

C₃³×C₉

C₃²

yes

characteristic

#6912

—

C₃×#733

C₃²

yes

—

#6894

—

#6894

—

#6894

C₃²

yes

—

#6894

C₃²

yes

—

C₉×3₋¹⁺²

—

C₉×3₋¹⁺²

—

C₃×C₉⋊C₉

—

C₃×C₉⋊C₉

—

C₃×C₉⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₉²

C₃²

yes

—

3₋¹⁺²⋊C₉

C₃²

yes

—

#6868

C₃²

yes

—

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

3₋¹⁺²

yes

—

C₃×3₋¹⁺²

C₃³

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

H(3,𝔽₃)

yes

—

C₃²×C₉

H(3,𝔽₃)

yes

—

C₃²×C₉

C₃³

yes

characteristic

C₃²×C₉

C₃³

yes

—

C₃²×C₉

C₃³

yes

—

C₃²×C₉

C₃³

yes

—

#733

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃³

yes

—

C₉²

—

C₉²

—

C₃³

—

C₃³

—

C₃³

C₃×H(3,𝔽₃)

yes

—

C₃³

C₃×3₋¹⁺²

yes

—

C₃³

#737

yes

—

C₃³

C₃⁴

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

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₃×H(3,𝔽₃)

yes

—

C₃×C₉

C₃×H(3,𝔽₃)

yes

—

C₃×C₉

C₃×H(3,𝔽₃)

yes

characteristic

C₃×C₉

C₃×3₋¹⁺²

yes

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₉⋊H(3,𝔽₃)

yes

—

C₃²

C₃²×H(3,𝔽₃)

yes

characteristic

C₃²

C₃²×3₋¹⁺²

yes

characteristic

C₃²

C₃×#737

yes

—

C₉

—

C₉

—

C₉

—

C₉

—

C₉

—

C₉

—

C₉

#6912

yes

complemented

C₃

—

C₃

—

100

C₃

—

101

C₃

C₃×C₉⋊H(3,𝔽₃)

yes

characteristic

102

C₃

C₃×#6912

yes

characteristic

103

C₃

#10624929

yes

—

	subgroup	G/H
center, socle	C₃²	C₃²×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃⁴
prime radical	C₃⁴	C₃³
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃³×C₉ ◁ G
LECS	G ▷ C₃³ ▷ C₃ ▷ C₁

subgroup

G/H

center, socle

C₃²

C₃²×H(3,𝔽₃)

commutator, Frattini, cosocle

C₃³

C₃⁴

prime radical

C₃⁴

C₃³

LCS

G ▷ C₃³ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃² ◁ C₃³×C₉ ◁ G

LECS

G ▷ C₃³ ▷ C₃ ▷ C₁

N	H ≅ G/N	type
4. #10624791	C₃	semidirect
5. #10624787	C₃	semidirect
97. C₉	#6912	semidirect

H ≅ G/N

type

4. #10624791

C₃

semidirect

5. #10624787

C₃

semidirect

97. C₉

#6912

semidirect

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

subgroup	index
C₃×#6912	3
C₃×#6894	3
C₃×C₉×3₋¹⁺²	3
#10624791	3
#10624787	3
#10624699	3
#10624691	3

subgroup

index

C₃×C₉×3₋¹⁺²

Subgroups — #14554422

Conjugacy classes of subgroups

Complements

Maximal subgroups