Group #14547563 — subgroups

Subgroups — #14547563

Order 2187.

1370 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 1370 conjugacy classes of subgroups — too many to draw. Only its 62 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	#10624779	C₃	1	3	yes	maximal, complemented
3	#10624773	C₃	1	2	yes	maximal, complemented
4	#10624590	C₃	1	6	yes	maximal, complemented
5	#10624562	C₃	1	1	yes	maximal, characteristic
6	3₋¹⁺⁴	—	3	27	no	—
7	C₃×#737	—	3	6	no	—
8	C₃×#737	—	3	3	no	—
9	C₃²×3₋¹⁺²	—	3	3	no	—
10	C₃²×3₋¹⁺²	C₃²	1	2	yes	—
11	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
12	C₃²×H(3,𝔽₃)	—	3	3	no	—
13	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
14	C₃²×H(3,𝔽₃)	C₃²	1	3	yes	—
15	#6891	C₃²	1	6	yes	—
16	H(3,𝔽₃)⋊C₉	—	3	9	no	—
17	#6866	—	3	6	no	—
18	#6857	—	3	18	no	—
19	C₃⁴	—	3	3	no	—
20	C₃⁴	H(3,𝔽₃)	1	3	yes	—
21	C₃⁴	C₃³	1	1	yes	characteristic
22	#737	—	3	162	no	—
23	#737	—	3	81	no	—
24	C₃×3₋¹⁺²	—	3	18	no	—
25	C₃×3₋¹⁺²	—	3	54	no	—
26	C₃×3₋¹⁺²	—	3	27	no	—
27	C₃×3₋¹⁺²	—	3	2	no	—
28	C₃×3₋¹⁺²	—	3	1	no	—
29	C₃×3₋¹⁺²	—	3	2	no	—
30	C₃×3₋¹⁺²	—	3	2	no	—
31	C₃×3₋¹⁺²	—	3	18	no	—
32	C₃×3₋¹⁺²	—	3	9	no	—
33	C₃×3₋¹⁺²	—	3	6	no	—
34	C₃×3₋¹⁺²	H(3,𝔽₃)	1	6	yes	complemented
35	C₃×H(3,𝔽₃)	—	3	9	no	—
36	C₃×H(3,𝔽₃)	—	3	1	no	—
37	C₃×H(3,𝔽₃)	—	3	2	no	—
38	C₃×H(3,𝔽₃)	—	3	3	no	—
39	C₃×H(3,𝔽₃)	—	3	6	no	—
40	C₃×H(3,𝔽₃)	—	9	9	no	—
41	C₃×H(3,𝔽₃)	—	9	27	no	—
42	C₃×H(3,𝔽₃)	—	9	9	no	—
43	C₃²×C₉	—	3	9	no	—
44	C₃²×C₉	—	3	2	no	—
45	C₃²×C₉	H(3,𝔽₃)	1	3	yes	complemented
46	C₃²⋊C₉	—	3	6	no	—
47	C₃²⋊C₉	—	3	6	no	—
48	C₃²⋊C₉	—	3	6	no	—
49	C₃³	—	3	3	no	—
50	C₃³	—	3	6	no	—
51	C₃³	—	3	3	no	—
52	C₃³	—	3	9	no	—
53	C₃³	—	9	3	no	—
54	C₃³	—	9	9	no	—
55	C₃³	—	9	18	no	—
56	C₃³	—	9	6	no	—
57	C₃³	—	9	3	no	—
58	C₃³	—	9	9	no	—
59	C₃³	AΣL(1,𝔽₂₇)	1	9	yes	—
60	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
61	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
62	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
63	3₋¹⁺²	—	3	18	no	—
64	3₋¹⁺²	—	3	162	no	—
65	3₋¹⁺²	—	3	18	no	—
66	3₋¹⁺²	—	3	18	no	—
67	3₋¹⁺²	—	3	9	no	—
68	3₋¹⁺²	—	3	18	no	—
69	3₋¹⁺²	—	27	6	no	—
70	H(3,𝔽₃)	—	3	18	no	—
71	H(3,𝔽₃)	—	3	9	no	—
72	H(3,𝔽₃)	—	3	9	no	—
73	H(3,𝔽₃)	—	9	18	no	—
74	H(3,𝔽₃)	—	9	6	no	—
75	H(3,𝔽₃)	—	9	81	no	—
76	C₃×C₉	—	3	18	no	—
77	C₃×C₉	—	3	81	no	—
78	C₃×C₉	—	3	6	no	—
79	C₃×C₉	—	3	3	no	—
80	C₃×C₉	—	3	9	no	—
81	C₃×C₉	—	9	6	no	—
82	C₃²	—	3	3	no	—
83	C₃²	—	3	9	no	—
84	C₃²	—	3	2	no	—
85	C₃²	—	3	1	no	—
86	C₃²	—	3	3	no	—
87	C₃²	—	3	6	no	—
88	C₃²	—	9	9	no	—
89	C₃²	—	9	27	no	—
90	C₃²	—	9	9	no	—
91	C₃²	—	9	9	no	—
92	C₃²	—	27	3	no	—
93	C₃²	—	27	6	no	—
94	C₃²	—	27	18	no	—
95	C₃²	—	27	3	no	—
96	C₃²	—	27	6	no	—
97	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
98	C₃²	C₃×AΣL(1,𝔽₂₇)	1	3	yes	—
99	C₉	—	3	18	no	—
100	C₉	—	3	9	no	—
101	C₉	—	27	6	no	—
102	C₃	—	3	3	no	—
103	C₃	—	9	1	no	—
104	C₃	—	9	2	no	—
105	C₃	—	9	9	no	—
106	C₃	—	27	9	no	—
107	C₃	#10624730	1	1	yes	characteristic
108	C₃	#10624749	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×3₋¹⁺⁴

C₃

yes

maximal, characteristic, complemented

#10624779

C₃

yes

maximal, complemented

#10624773

C₃

yes

maximal, complemented

#10624590

C₃

yes

maximal, complemented

#10624562

C₃

yes

maximal, characteristic

3₋¹⁺⁴

—

C₃×#737

—

C₃×#737

—

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

C₃²

yes

—

C₃²×3₋¹⁺²

C₃²

yes

characteristic

C₃²×H(3,𝔽₃)

—

C₃²×H(3,𝔽₃)

C₃²

yes

characteristic

C₃²×H(3,𝔽₃)

C₃²

yes

—

#6891

C₃²

yes

—

H(3,𝔽₃)⋊C₉

—

#6866

—

#6857

—

C₃⁴

—

C₃⁴

H(3,𝔽₃)

yes

—

C₃⁴

C₃³

yes

characteristic

#737

—

162

—

#737

—

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

—

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)

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

H(3,𝔽₃)

yes

complemented

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

AΣL(1,𝔽₂₇)

yes

—

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

—

162

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

—

—

—

—

—

—

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⋊H(3,𝔽₃)

yes

characteristic

C₃²

C₃×AΣL(1,𝔽₂₇)

yes

—

C₉

—

100

C₉

—

101

C₉

—

102

C₃

—

103

C₃

—

104

C₃

—

105

C₃

—

106

C₃

—

107

C₃

#10624730

yes

characteristic

108

C₃

#10624749

yes

—

	subgroup	G/H
center, socle	C₃²	C₃²⋊H(3,𝔽₃)
commutator, Frattini, cosocle	C₃⁴	C₃³
prime radical	#10624562	C₃
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

C₃

G ▷ C₃⁴ ▷ C₃² ▷ C₁

C₁ ◁ C₃² ◁ C₃⁴ ◁ G

G ▷ C₃⁴ ▷ C₃² ▷ C₁

N	H ≅ G/N	type
1. C₃×3₋¹⁺⁴	C₃	semidirect
2. #10624779	C₃	semidirect
3. #10624773	C₃	semidirect
4. #10624590	C₃	semidirect
34. C₃×3₋¹⁺²	H(3,𝔽₃)	semidirect
45. C₃²×C₉	H(3,𝔽₃)	semidirect

H ≅ G/N

type

1. C₃×3₋¹⁺⁴

C₃

semidirect

2. #10624779

C₃

semidirect

3. #10624773

C₃

semidirect

4. #10624590

C₃

semidirect

34. C₃×3₋¹⁺²

H(3,𝔽₃)

semidirect

45. C₃²×C₉

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
#10624779	3
#10624773	3
#10624590	3
#10624562	3

subgroup

index

C₃×3₋¹⁺⁴

Subgroups — #14547563

Conjugacy classes of subgroups

Complements

Maximal subgroups