Group #14554274 — subgroups

Subgroups — #14554274

Order 2187.

2024 conjugacy classes of subgroups; only the 314 normal subgroups are shown.

normal subgroups only

This group has 2024 conjugacy classes of subgroups — too many to draw. Only its 314 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	C₃²×#737	C₃	1	1	yes	maximal, characteristic
2	C₃×#6912	C₃	1	6	yes	maximal, complemented
3	C₃²×#733	C₃	1	3	yes	maximal, complemented
4	#10624881	C₃	1	3	yes	maximal
5	#10624738	C₃	1	18	yes	maximal, complemented
6	#10624733	C₃	1	9	yes	maximal, complemented
7	C₃×#737	—	3	18	no	—
8	C₃×#737	—	3	9	no	—
9	C₃×#737	—	3	9	no	—
10	C₃²×3₋¹⁺²	C₃²	1	9	yes	—
11	C₃²×3₋¹⁺²	C₃²	1	6	yes	—
12	C₃²×3₋¹⁺²	C₃²	1	2	yes	—
13	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
14	C₃³×C₉	C₃²	1	3	yes	—
15	C₃³×C₉	C₃²	1	1	yes	characteristic
16	#6912	—	3	54	no	—
17	C₃×#733	—	3	27	no	—
18	C₃×#733	C₃²	1	9	yes	—
19	C₃×#733	C₃²	1	18	yes	—
20	C₃×#733	C₃²	1	54	yes	—
21	#6893	—	3	27	no	—
22	#6891	C₃²	1	27	yes	—
23	C₃⁴	C₃³	1	3	yes	—
24	C₃⁴	C₃³	1	1	yes	characteristic
25	#737	—	3	243	no	—
26	C₃×3₋¹⁺²	—	3	6	no	—
27	C₃×3₋¹⁺²	—	3	6	no	—
28	C₃×3₋¹⁺²	—	3	18	no	—
29	C₃×3₋¹⁺²	—	3	6	no	—
30	C₃×3₋¹⁺²	—	3	18	no	—
31	C₃×3₋¹⁺²	—	3	36	no	—
32	C₃×3₋¹⁺²	—	3	18	no	—
33	C₃×3₋¹⁺²	—	3	54	no	—
34	C₃×3₋¹⁺²	—	3	27	no	—
35	C₃×3₋¹⁺²	H(3,𝔽₃)	1	18	yes	—
36	C₃×3₋¹⁺²	C₃³	1	27	yes	—
37	C₃×H(3,𝔽₃)	—	3	6	no	—
38	C₃×H(3,𝔽₃)	—	3	3	no	—
39	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	9	yes	—
40	C₃²×C₉	—	3	9	no	—
41	C₃²×C₉	—	3	6	no	—
42	C₃²×C₉	—	3	3	no	—
43	C₃²×C₉	—	3	18	no	—
44	C₃²×C₉	—	3	9	no	—
45	C₃²×C₉	—	3	9	no	—
46	C₃²×C₉	H(3,𝔽₃)	1	9	yes	—
47	C₃²×C₉	C₃³	1	1	yes	characteristic
48	C₃²×C₉	C₃³	1	6	yes	—
49	C₃²×C₉	C₃³	1	2	yes	—
50	#733	—	3	162	no	—
51	#733	—	3	27	no	—
52	#733	—	3	54	no	—
53	C₃²⋊C₉	—	3	81	no	—
54	C₃³	—	3	6	no	—
55	C₃³	—	3	3	no	—
56	C₃³	—	9	3	no	—
57	C₃³	—	9	9	no	—
58	C₃³	C₃×H(3,𝔽₃)	1	9	yes	—
59	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
60	C₃³	C₃×H(3,𝔽₃)	1	6	yes	—
61	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
62	C₃³	C₃⁴	1	1	yes	characteristic
63	3₋¹⁺²	—	3	162	no	—
64	3₋¹⁺²	—	3	54	no	—
65	3₋¹⁺²	—	3	54	no	—
66	3₋¹⁺²	—	3	27	no	—
67	3₋¹⁺²	—	9	54	no	—
68	H(3,𝔽₃)	—	3	27	no	—
69	C₃×C₉	—	3	81	no	—
70	C₃×C₉	—	3	2	no	—
71	C₃×C₉	—	3	4	no	—
72	C₃×C₉	—	3	6	no	—
73	C₃×C₉	—	3	6	no	—
74	C₃×C₉	—	3	1	no	—
75	C₃×C₉	—	3	6	no	—
76	C₃×C₉	—	3	2	no	—
77	C₃×C₉	—	3	27	no	—
78	C₃×C₉	—	3	27	no	—
79	C₃×C₉	—	9	27	no	—
80	C₃×C₉	C₃×H(3,𝔽₃)	1	3	yes	—
81	C₃×C₉	C₃×H(3,𝔽₃)	1	18	yes	—
82	C₃×C₉	C₃×H(3,𝔽₃)	1	6	yes	—
83	C₃²	—	3	3	no	—
84	C₃²	—	3	9	no	—
85	C₃²	—	3	9	no	—
86	C₃²	—	3	27	no	—
87	C₃²	—	9	6	no	—
88	C₃²	—	9	3	no	—
89	C₃²	—	9	18	no	—
90	C₃²	—	9	9	no	—
91	C₃²	C₃²⋊H(3,𝔽₃)	1	9	yes	—
92	C₃²	C₃²×H(3,𝔽₃)	1	2	yes	—
93	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
94	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
95	C₉	—	3	3	no	—
96	C₉	—	3	18	no	—
97	C₉	—	3	6	no	—
98	C₉	—	9	27	no	—
99	C₃	—	3	2	no	—
100	C₃	—	3	1	no	—
101	C₃	—	3	9	no	—
102	C₃	—	9	9	no	—
103	C₃	C₃×C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
104	C₃	#10624933	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#737

C₃

yes

maximal, characteristic

C₃×#6912

C₃

yes

maximal, complemented

C₃²×#733

C₃

yes

maximal, complemented

#10624881

C₃

yes

maximal

#10624738

C₃

yes

maximal, complemented

#10624733

C₃

yes

maximal, complemented

C₃×#737

—

C₃×#737

—

C₃×#737

—

C₃²×3₋¹⁺²

C₃²

yes

—

C₃²×3₋¹⁺²

C₃²

yes

—

C₃²×3₋¹⁺²

C₃²

yes

—

C₃²×H(3,𝔽₃)

C₃²

yes

characteristic

C₃³×C₉

C₃²

yes

—

C₃³×C₉

C₃²

yes

—

—

C₃²

yes

—

C₃×#733

C₃²

yes

—

C₃×#733

C₃²

yes

—

#6893

—

#6891

C₃²

yes

—

C₃⁴

C₃³

yes

—

C₃⁴

C₃³

yes

characteristic

#737

—

243

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

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

—

C₃×3₋¹⁺²

C₃³

yes

—

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

—

C₃²×C₉

—

C₃²×C₉

H(3,𝔽₃)

yes

—

C₃²×C₉

C₃³

yes

characteristic

C₃²×C₉

C₃³

yes

—

C₃²×C₉

C₃³

yes

—

#733

—

162

—

#733

—

#733

—

C₃²⋊C₉

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃×H(3,𝔽₃)

yes

—

C₃³

C₃×H(3,𝔽₃)

yes

—

C₃³

C₃×H(3,𝔽₃)

yes

—

C₃³

C₃×H(3,𝔽₃)

yes

—

C₃³

C₃⁴

yes

characteristic

3₋¹⁺²

—

162

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

H(3,𝔽₃)

—

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

—

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

—

C₉

—

C₉

—

C₉

—

C₃

—

100

C₃

—

101

C₃

—

102

C₃

—

103

C₃

C₃×C₃²⋊H(3,𝔽₃)

yes

characteristic

104

C₃

#10624933

yes

—

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

subgroup

G/H

center, socle

C₃²

C₃²×H(3,𝔽₃)

commutator, Frattini, cosocle

C₃³

C₃⁴

prime radical

C₃²×H(3,𝔽₃)

C₃²

LCS

G ▷ C₃³ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃² ◁ C₃²×H(3,𝔽₃) ◁ G

LECS

G ▷ C₃³ ▷ C₃ ▷ C₁

N	H ≅ G/N	type
2. C₃×#6912	C₃	semidirect
3. C₃²×#733	C₃	semidirect
5. #10624738	C₃	semidirect
6. #10624733	C₃	semidirect

H ≅ G/N

type

2. C₃×#6912

C₃

semidirect

3. C₃²×#733

C₃

semidirect

5. #10624738

C₃

semidirect

6. #10624733

C₃

semidirect

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

subgroup	index
C₃²×#737	3
C₃×#6912	3
C₃²×#733	3
#10624881	3
#10624738	3
#10624733	3

subgroup

index

C₃²×#737

C₃²×#733

Subgroups — #14554274

Conjugacy classes of subgroups

Complements

Maximal subgroups