Group #14554302 — subgroups

Subgroups — #14554302

Order 2187.

2105 conjugacy classes of subgroups; only the 314 normal subgroups are shown.

normal subgroups only

This group has 2105 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	C₃×3₋¹⁺⁴	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×#6910	C₃	1	6	yes	maximal, complemented
3	C₃²×#731	C₃	1	3	yes	maximal, complemented
4	#10624881	C₃	1	1	yes	maximal, characteristic, complemented
5	#10624880	C₃	1	1	yes	maximal, characteristic
6	C₃×#6893	C₃	1	1	yes	maximal, characteristic, complemented
7	#10624745	C₃	1	27	yes	maximal, complemented
8	3₋¹⁺⁴	—	3	27	no	—
9	C₃×#737	—	3	9	no	—
10	C₃²×3₋¹⁺²	C₃²	1	3	yes	—
11	C₃²×3₋¹⁺²	C₃²	1	3	yes	—
12	C₃²×3₋¹⁺²	C₃²	1	2	yes	—
13	C₃²×3₋¹⁺²	C₃²	1	9	yes	complemented
14	C₃²×H(3,𝔽₃)	C₃²	1	3	yes	—
15	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
16	C₃³×C₉	C₃²	1	1	yes	characteristic
17	#6910	—	3	54	no	—
18	C₃×#731	—	3	27	no	—
19	C₃×#731	C₃²	1	27	yes	complemented
20	C₃×#731	C₃²	1	54	yes	complemented
21	#6893	—	3	9	no	—
22	#6893	—	3	9	no	—
23	#6891	C₃²	1	9	yes	complemented
24	C₃²⋊H(3,𝔽₃)	—	3	9	no	—
25	C₃²⋊H(3,𝔽₃)	C₃²	1	9	yes	—
26	C₃×C₃²⋊C₉	C₃²	1	9	yes	complemented
27	C₃⁴	—	3	3	no	—
28	C₃⁴	C₃³	1	3	yes	—
29	C₃⁴	C₃³	1	1	yes	characteristic
30	#737	—	3	243	no	—
31	C₃×3₋¹⁺²	—	3	6	no	—
32	C₃×3₋¹⁺²	—	3	18	no	—
33	C₃×3₋¹⁺²	—	3	18	no	—
34	C₃×3₋¹⁺²	—	3	81	no	—
35	C₃×3₋¹⁺²	—	3	6	no	—
36	C₃×3₋¹⁺²	—	3	6	no	—
37	C₃×3₋¹⁺²	—	3	18	no	—
38	C₃×3₋¹⁺²	—	3	9	no	—
39	C₃×3₋¹⁺²	—	3	18	no	—
40	C₃×3₋¹⁺²	H(3,𝔽₃)	1	27	yes	complemented
41	C₃×3₋¹⁺²	C₃³	1	9	yes	—
42	C₃×3₋¹⁺²	C₃³	1	9	yes	—
43	C₃×H(3,𝔽₃)	—	3	9	no	—
44	C₃×H(3,𝔽₃)	—	3	9	no	—
45	C₃×H(3,𝔽₃)	—	3	18	no	—
46	C₃×H(3,𝔽₃)	—	3	6	no	—
47	C₃×H(3,𝔽₃)	—	3	27	no	—
48	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	9	yes	—
49	C₃×H(3,𝔽₃)	C₃³	1	9	yes	—
50	C₃²×C₉	—	3	3	no	—
51	C₃²×C₉	—	3	9	no	—
52	C₃²×C₉	—	3	3	no	—
53	C₃²×C₉	—	3	9	no	—
54	C₃²×C₉	—	3	3	no	—
55	C₃²×C₉	C₃³	1	3	yes	—
56	C₃²×C₉	C₃³	1	6	yes	—
57	#731	—	3	81	no	—
58	#731	—	3	162	no	—
59	C₃²⋊C₉	—	3	27	no	—
60	C₃²⋊C₉	—	3	27	no	—
61	C₃³	—	3	9	no	—
62	C₃³	—	3	9	no	—
63	C₃³	—	9	3	no	—
64	C₃³	—	9	9	no	—
65	C₃³	—	9	9	no	—
66	C₃³	—	27	9	no	—
67	C₃³	C₃×H(3,𝔽₃)	1	9	yes	—
68	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
69	C₃³	C₃×H(3,𝔽₃)	1	9	yes	—
70	C₃³	C₃⁴	1	1	yes	characteristic
71	3₋¹⁺²	—	3	162	no	—
72	3₋¹⁺²	—	3	18	no	—
73	3₋¹⁺²	—	3	54	no	—
74	3₋¹⁺²	—	3	9	no	—
75	3₋¹⁺²	—	3	18	no	—
76	3₋¹⁺²	—	9	18	no	—
77	3₋¹⁺²	—	9	18	no	—
78	3₋¹⁺²	C₃×H(3,𝔽₃)	1	27	yes	—
79	H(3,𝔽₃)	—	3	27	no	—
80	H(3,𝔽₃)	—	3	9	no	—
81	H(3,𝔽₃)	—	3	18	no	—
82	H(3,𝔽₃)	—	9	18	no	—
83	H(3,𝔽₃)	—	27	27	no	—
84	C₃×C₉	—	3	81	no	—
85	C₃×C₉	—	3	9	no	—
86	C₃×C₉	—	3	27	no	—
87	C₃×C₉	—	3	9	no	—
88	C₃×C₉	—	3	18	no	—
89	C₃×C₉	—	3	3	no	—
90	C₃×C₉	—	3	9	no	—
91	C₃×C₉	—	3	6	no	—
92	C₃×C₉	—	9	9	no	—
93	C₃×C₉	—	9	9	no	—
94	C₃²	—	3	3	no	—
95	C₃²	—	3	9	no	—
96	C₃²	—	3	9	no	—
97	C₃²	—	3	27	no	—
98	C₃²	—	9	9	no	—
99	C₃²	—	9	9	no	—
100	C₃²	—	9	27	no	—
101	C₃²	—	27	9	no	—
102	C₃²	—	27	27	no	—
103	C₃²	C₃²⋊H(3,𝔽₃)	1	9	yes	—
104	C₃²	C₃²×H(3,𝔽₃)	1	3	yes	—
105	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
106	C₉	—	3	27	no	—
107	C₉	—	9	3	no	—
108	C₉	—	9	6	no	—
109	C₉	—	9	9	no	—
110	C₃	—	3	3	no	—
111	C₃	—	3	9	no	—
112	C₃	—	9	9	no	—
113	C₃	—	27	9	no	—
114	C₃	C₃×C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
115	C₃	#10624931	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×3₋¹⁺⁴

C₃

yes

maximal, characteristic, complemented

C₃×#6910

C₃

yes

maximal, complemented

C₃²×#731

C₃

yes

maximal, complemented

#10624881

C₃

yes

maximal, characteristic, complemented

#10624880

C₃

yes

maximal, characteristic

C₃×#6893

C₃

yes

maximal, characteristic, complemented

#10624745

C₃

yes

maximal, complemented

3₋¹⁺⁴

—

C₃×#737

—

C₃²×3₋¹⁺²

C₃²

yes

—

C₃²×3₋¹⁺²

C₃²

yes

—

C₃²×3₋¹⁺²

C₃²

yes

—

C₃²×3₋¹⁺²

C₃²

yes

complemented

C₃²×H(3,𝔽₃)

C₃²

yes

—

C₃²×H(3,𝔽₃)

C₃²

yes

characteristic

C₃³×C₉

C₃²

yes

—

—

C₃²

yes

complemented

C₃×#731

C₃²

yes

—

—

C₃²

yes

complemented

C₃²⋊H(3,𝔽₃)

—

C₃²⋊H(3,𝔽₃)

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

complemented

C₃⁴

—

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

complemented

C₃×3₋¹⁺²

C₃³

yes

—

C₃×3₋¹⁺²

C₃³

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

—

C₃×H(3,𝔽₃)

C₃³

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃³

yes

—

C₃²×C₉

C₃³

yes

—

#731

—

#731

—

162

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

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

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

C₃×H(3,𝔽₃)

yes

—

—

—

—

—

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

100

C₃²

—

101

C₃²

—

102

C₃²

—

103

C₃²

C₃²⋊H(3,𝔽₃)

yes

—

104

C₃²

C₃²×H(3,𝔽₃)

yes

—

105

C₃²

C₃²×H(3,𝔽₃)

yes

characteristic

106

C₉

—

107

C₉

—

108

C₉

—

109

C₉

—

110

C₃

—

111

C₃

—

112

C₃

—

113

C₃

—

114

C₃

C₃×C₃²⋊H(3,𝔽₃)

yes

characteristic

115

C₃

#10624931

yes

—

	subgroup	G/H
center, socle	C₃²	C₃²×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃⁴
prime radical	#10624880	C₃
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

C₃

G ▷ C₃³ ▷ C₃ ▷ C₁

C₁ ◁ C₃² ◁ C₃²×H(3,𝔽₃) ◁ G

LECS

G ▷ C₃³ ▷ C₃ ▷ C₁

N	H ≅ G/N	type
1. C₃×3₋¹⁺⁴	C₃	semidirect
2. C₃×#6910	C₃	semidirect
3. C₃²×#731	C₃	semidirect
4. #10624881	C₃	semidirect
6. C₃×#6893	C₃	semidirect
7. #10624745	C₃	semidirect
13. C₃²×3₋¹⁺²	C₃²	semidirect
19. C₃×#731	C₃²	semidirect
20. C₃×#731	C₃²	semidirect
23. #6891	C₃²	semidirect
26. C₃×C₃²⋊C₉	C₃²	semidirect
40. C₃×3₋¹⁺²	H(3,𝔽₃)	semidirect

H ≅ G/N

type

1. C₃×3₋¹⁺⁴

C₃

semidirect

2. C₃×#6910

C₃

semidirect

3. C₃²×#731

C₃

semidirect

4. #10624881

C₃

semidirect

6. C₃×#6893

C₃

semidirect

7. #10624745

C₃

semidirect

13. C₃²×3₋¹⁺²

C₃²

semidirect

19. C₃×#731

C₃²

semidirect

20. C₃×#731

C₃²

semidirect

23. #6891

C₃²

semidirect

26. C₃×C₃²⋊C₉

C₃²

semidirect

40. C₃×3₋¹⁺²

H(3,𝔽₃)

semidirect

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

subgroup	index
C₃×3₋¹⁺⁴	3
C₃×#6910	3
C₃²×#731	3
#10624881	3
#10624880	3
C₃×#6893	3
#10624745	3

subgroup

index

C₃×3₋¹⁺⁴

C₃²×#731

Subgroups — #14554302

Conjugacy classes of subgroups

Complements

Maximal subgroups