Group #14548211 — subgroups

Subgroups — #14548211

Order 2187.

1454 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 1454 conjugacy classes of subgroups — too many to draw. Only its 71 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	C₃³×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
2	#10624811	C₃	1	6	yes	maximal, complemented
3	#10624768	C₃	1	2	yes	maximal, complemented
4	#10624732	C₃	1	1	yes	maximal, characteristic, complemented
5	#10624730	C₃	1	3	yes	maximal, complemented
6	C₃⁵	C₃²	1	1	yes	characteristic
7	C₃²×3₋¹⁺²	—	3	3	no	—
8	C₃²×3₋¹⁺²	—	3	27	no	—
9	C₃²×3₋¹⁺²	—	3	6	no	—
10	C₃²×3₋¹⁺²	—	3	3	no	—
11	C₃³×C₉	C₃²	1	1	yes	characteristic
12	C₃³×C₉	C₃²	1	2	yes	—
13	C₃×#732	—	3	3	no	—
14	C₃×AΣL(1,𝔽₂₇)	—	3	9	no	—
15	#6891	C₃²	1	6	yes	—
16	C₃²⋊H(3,𝔽₃)	C₃²	1	3	yes	—
17	H(3,𝔽₃)⋊C₉	—	3	18	no	—
18	#6868	—	3	6	no	—
19	C₃⁴	—	3	9	no	—
20	C₃⁴	—	3	2	no	—
21	C₃⁴	—	3	1	no	—
22	C₃⁴	—	9	3	no	—
23	C₃⁴	—	9	6	no	—
24	C₃⁴	H(3,𝔽₃)	1	3	yes	—
25	C₃⁴	C₃³	1	1	yes	characteristic
26	C₃×3₋¹⁺²	—	3	81	no	—
27	C₃×3₋¹⁺²	—	3	162	no	—
28	C₃×3₋¹⁺²	—	3	27	no	—
29	C₃×3₋¹⁺²	—	3	54	no	—
30	C₃×3₋¹⁺²	—	3	27	no	—
31	C₃×3₋¹⁺²	—	3	6	no	—
32	C₃×H(3,𝔽₃)	—	3	6	no	—
33	C₃×H(3,𝔽₃)	—	3	3	no	—
34	C₃×H(3,𝔽₃)	—	3	3	no	—
35	C₃²×C₉	—	3	2	no	—
36	C₃²×C₉	—	3	2	no	—
37	C₃²×C₉	—	3	1	no	—
38	C₃²×C₉	—	3	2	no	—
39	C₃²×C₉	—	3	2	no	—
40	C₃²×C₉	—	3	9	no	—
41	C₃²×C₉	—	3	18	no	—
42	C₃²×C₉	H(3,𝔽₃)	1	6	yes	—
43	C₃²×C₉	H(3,𝔽₃)	1	3	yes	—
44	#732	—	9	9	no	—
45	AΣL(1,𝔽₂₇)	—	9	27	no	—
46	C₃²⋊C₉	—	3	6	no	—
47	C₃²⋊C₉	—	3	12	no	—
48	C₃³	—	3	6	no	—
49	C₃³	—	3	3	no	—
50	C₃³	—	3	3	no	—
51	C₃³	—	3	18	no	—
52	C₃³	—	3	9	no	—
53	C₃³	—	3	3	no	—
54	C₃³	—	3	9	no	—
55	C₃³	—	3	9	no	—
56	C₃³	—	9	6	no	—
57	C₃³	—	9	81	no	—
58	C₃³	—	9	6	no	—
59	C₃³	—	9	3	no	—
60	C₃³	—	9	6	no	—
61	C₃³	—	9	6	no	—
62	C₃³	—	9	6	no	—
63	C₃³	—	9	3	no	—
64	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
65	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
66	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
67	3₋¹⁺²	—	3	243	no	—
68	3₋¹⁺²	—	9	18	no	—
69	H(3,𝔽₃)	—	9	18	no	—
70	H(3,𝔽₃)	—	9	9	no	—
71	H(3,𝔽₃)	—	9	9	no	—
72	C₃×C₉	—	3	18	no	—
73	C₃×C₉	—	3	9	no	—
74	C₃×C₉	—	3	18	no	—
75	C₃×C₉	—	3	18	no	—
76	C₃×C₉	—	3	3	no	—
77	C₃×C₉	—	3	6	no	—
78	C₃×C₉	—	3	6	no	—
79	C₃×C₉	—	3	18	no	—
80	C₃×C₉	—	3	18	no	—
81	C₃×C₉	—	9	6	no	—
82	C₃×C₉	AΣL(1,𝔽₂₇)	1	9	yes	complemented
83	C₃²	—	3	27	no	—
84	C₃²	—	3	2	no	—
85	C₃²	—	3	1	no	—
86	C₃²	—	3	2	no	—
87	C₃²	—	3	1	no	—
88	C₃²	—	3	2	no	—
89	C₃²	—	3	2	no	—
90	C₃²	—	3	2	no	—
91	C₃²	—	3	9	no	—
92	C₃²	—	9	27	no	—
93	C₃²	—	9	54	no	—
94	C₃²	—	9	27	no	—
95	C₃²	—	9	3	no	—
96	C₃²	—	9	6	no	—
97	C₃²	—	27	3	no	—
98	C₃²	—	27	3	no	—
99	C₃²	—	27	6	no	—
100	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
101	C₃²	C₃×AΣL(1,𝔽₂₇)	1	3	yes	—
102	C₃²	#6911	1	3	yes	—
103	C₃²	#6913	1	6	yes	—
104	C₉	—	3	9	no	—
105	C₉	—	3	18	no	—
106	C₉	—	27	6	no	—
107	C₃	—	3	6	no	—
108	C₃	—	3	3	no	—
109	C₃	—	3	3	no	—
110	C₃	—	9	9	no	—
111	C₃	—	81	3	no	—
112	C₃	#10624730	1	1	yes	characteristic
113	C₃	#10624737	1	1	yes	characteristic
114	C₃	#10624739	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃³×3₋¹⁺²

C₃

yes

maximal, characteristic, complemented

#10624811

C₃

yes

maximal, complemented

#10624768

C₃

yes

maximal, complemented

#10624732

C₃

yes

maximal, characteristic, complemented

#10624730

C₃

yes

maximal, complemented

C₃⁵

C₃²

yes

characteristic

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

—

C₃³×C₉

C₃²

yes

characteristic

C₃³×C₉

C₃²

yes

—

C₃×#732

—

C₃×AΣL(1,𝔽₂₇)

—

#6891

C₃²

yes

—

C₃²⋊H(3,𝔽₃)

C₃²

yes

—

H(3,𝔽₃)⋊C₉

—

#6868

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

H(3,𝔽₃)

yes

—

C₃⁴

C₃³

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

162

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

H(3,𝔽₃)

yes

—

C₃²×C₉

H(3,𝔽₃)

yes

—

#732

—

AΣL(1,𝔽₂₇)

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃×H(3,𝔽₃)

yes

—

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

3₋¹⁺²

—

243

—

3₋¹⁺²

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

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

AΣL(1,𝔽₂₇)

yes

complemented

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

100

C₃²

C₃²⋊H(3,𝔽₃)

yes

characteristic

101

C₃²

C₃×AΣL(1,𝔽₂₇)

yes

—

102

C₃²

#6911

yes

—

103

C₃²

#6913

yes

—

104

C₉

—

105

C₉

—

106

C₉

—

107

C₃

—

108

C₃

—

109

C₃

—

110

C₃

—

111

C₃

—

112

C₃

#10624730

yes

characteristic

113

C₃

#10624737

yes

characteristic

114

C₃

#10624739

yes

—

	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. C₃³×3₋¹⁺²	C₃	semidirect
2. #10624811	C₃	semidirect
3. #10624768	C₃	semidirect
4. #10624732	C₃	semidirect
5. #10624730	C₃	semidirect
82. C₃×C₉	AΣL(1,𝔽₂₇)	semidirect

H ≅ G/N

type

1. C₃³×3₋¹⁺²

C₃

semidirect

2. #10624811

C₃

semidirect

3. #10624768

C₃

semidirect

4. #10624732

C₃

semidirect

5. #10624730

C₃

semidirect

82. C₃×C₉

AΣL(1,𝔽₂₇)

semidirect

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

subgroup	index
C₃³×3₋¹⁺²	3
#10624811	3
#10624768	3
#10624732	3
#10624730	3

subgroup

index

C₃³×3₋¹⁺²

Subgroups — #14548211

Conjugacy classes of subgroups

Complements

Maximal subgroups