Group #14548280 — subgroups

Subgroups — #14548280

Order 2187.

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

normal subgroups only

This group has 481 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	#10624862	C₃	1	1	yes	maximal, characteristic, complemented
2	#10624806	C₃	1	2	yes	maximal, complemented
3	#10624777	C₃	1	4	yes	maximal, complemented
4	#10624742	C₃	1	2	yes	maximal, complemented
5	#10624741	C₃	1	1	yes	maximal, characteristic, complemented
6	#10624637	C₃	1	2	yes	maximal, complemented
7	C₃×#6856	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
9	C₃²×H(3,𝔽₃)	C₃²	1	2	yes	—
10	C₃×#731	—	3	3	no	—
11	C₃×AΣL(1,𝔽₂₇)	—	3	6	no	—
12	#6891	C₃²	1	2	yes	—
13	#6891	C₃²	1	4	yes	—
14	C₉×H(3,𝔽₃)	—	3	3	no	—
15	C₉×H(3,𝔽₃)	—	3	6	no	—
16	C₃×C₃²⋊C₉	—	3	3	no	—
17	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
18	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
19	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
20	H(3,𝔽₃)⋊C₉	—	3	12	no	—
21	#6868	—	3	6	no	—
22	#6861	—	3	6	no	—
23	#6856	—	3	3	no	—
24	C₃⁴	—	3	1	no	—
25	C₃⁴	—	3	2	no	—
26	C₃⁴	C₃³	1	1	yes	characteristic
27	C₃×3₋¹⁺²	—	3	4	no	—
28	C₃×3₋¹⁺²	—	3	2	no	—
29	C₃×H(3,𝔽₃)	—	3	2	no	—
30	C₃×H(3,𝔽₃)	—	3	9	no	—
31	C₃×H(3,𝔽₃)	—	3	4	no	—
32	C₃×H(3,𝔽₃)	—	3	2	no	—
33	C₃×H(3,𝔽₃)	—	9	6	no	—
34	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	3	yes	—
35	C₃²×C₉	—	3	9	no	—
36	C₃²×C₉	—	3	1	no	—
37	C₃²×C₉	—	3	1	no	—
38	C₃²×C₉	H(3,𝔽₃)	1	6	yes	—
39	#731	—	9	9	no	—
40	AΣL(1,𝔽₂₇)	—	9	18	no	—
41	C₃²⋊C₉	—	3	18	no	—
42	C₃²⋊C₉	—	3	2	no	—
43	C₃²⋊C₉	—	3	2	no	—
44	C₃²⋊C₉	—	3	2	no	—
45	C₃²⋊C₉	—	3	2	no	—
46	C₃²⋊C₉	—	3	1	no	—
47	C₃²⋊C₉	—	3	2	no	—
48	C₃²⋊C₉	—	3	4	no	—
49	C₃²⋊C₉	—	3	2	no	—
50	C₃²⋊C₉	—	3	4	no	—
51	C₃²⋊C₉	—	3	4	no	—
52	C₃²⋊C₉	—	3	4	no	—
53	C₃²⋊C₉	H(3,𝔽₃)	1	3	yes	—
54	C₃³	—	3	9	no	—
55	C₃³	—	3	3	no	—
56	C₃³	—	3	3	no	—
57	C₃³	—	9	2	no	—
58	C₃³	—	9	1	no	—
59	C₃³	—	9	6	no	—
60	C₃³	—	9	2	no	—
61	C₃³	—	9	6	no	—
62	C₃³	—	27	1	no	—
63	C₃³	—	27	2	no	—
64	C₃³	—	27	2	no	—
65	C₃³	—	27	2	no	—
66	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
67	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
68	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
69	3₋¹⁺²	—	9	12	no	—
70	3₋¹⁺²	—	9	6	no	—
71	H(3,𝔽₃)	—	3	18	no	—
72	H(3,𝔽₃)	—	3	6	no	—
73	H(3,𝔽₃)	—	3	12	no	—
74	H(3,𝔽₃)	—	9	6	no	—
75	H(3,𝔽₃)	—	9	8	no	—
76	H(3,𝔽₃)	#732	1	9	yes	—
77	C₃×C₉	—	3	6	no	—
78	C₃×C₉	—	3	3	no	—
79	C₃×C₉	—	9	6	no	—
80	C₃×C₉	—	9	1	no	—
81	C₃×C₉	—	9	4	no	—
82	C₃×C₉	—	9	2	no	—
83	C₃×C₉	—	9	3	no	—
84	C₃×C₉	—	27	1	no	—
85	C₃×C₉	—	27	9	no	—
86	C₃²	—	3	1	no	—
87	C₃²	—	3	2	no	—
88	C₃²	—	3	2	no	—
89	C₃²	—	3	9	no	—
90	C₃²	—	3	1	no	—
91	C₃²	—	3	2	no	—
92	C₃²	—	3	1	no	—
93	C₃²	—	3	4	no	—
94	C₃²	—	3	2	no	—
95	C₃²	—	9	3	no	—
96	C₃²	—	9	6	no	—
97	C₃²	—	9	6	no	—
98	C₃²	—	27	3	no	—
99	C₃²	—	27	18	no	—
100	C₃²	—	27	9	no	—
101	C₃²	—	27	2	no	—
102	C₃²	—	27	2	no	—
103	C₃²	—	27	2	no	—
104	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
105	C₃²	C₃×#732	1	3	yes	—
106	C₉	—	9	6	no	—
107	C₉	—	9	3	no	—
108	C₉	—	27	1	no	—
109	C₉	—	27	4	no	—
110	C₉	—	27	2	no	—
111	C₃	—	3	3	no	—
112	C₃	—	9	2	no	—
113	C₃	—	9	1	no	—
114	C₃	—	9	9	no	—
115	C₃	—	27	6	no	—
116	C₃	#10624732	1	1	yes	characteristic
117	C₃	#10624748	1	1	yes	characteristic
118	C₃	#10624752	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10624862

C₃

yes

maximal, characteristic, complemented

#10624806

C₃

yes

maximal, complemented

#10624777

C₃

yes

maximal, complemented

#10624742

C₃

yes

maximal, complemented

#10624741

C₃

yes

maximal, characteristic, complemented

#10624637

C₃

yes

maximal, complemented

C₃×#6856

C₃

yes

maximal, characteristic, complemented

C₃²×H(3,𝔽₃)

C₃²

yes

characteristic

C₃²×H(3,𝔽₃)

C₃²

yes

—

C₃×#731

—

C₃×AΣL(1,𝔽₂₇)

—

#6891

C₃²

yes

—

#6891

C₃²

yes

—

C₉×H(3,𝔽₃)

—

C₉×H(3,𝔽₃)

—

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

C₃²

yes

characteristic

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

characteristic

H(3,𝔽₃)⋊C₉

—

#6868

—

#6861

—

#6856

—

C₃⁴

—

C₃⁴

—

C₃⁴

C₃³

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

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

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

H(3,𝔽₃)

yes

—

#731

—

AΣL(1,𝔽₂₇)

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

H(3,𝔽₃)

yes

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

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

—

—

—

—

—

—

yes

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

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

—

100

C₃²

—

101

C₃²

—

102

C₃²

—

103

C₃²

—

104

C₃²

C₃²⋊H(3,𝔽₃)

yes

characteristic

105

C₃²

C₃×#732

yes

—

106

C₉

—

107

C₉

—

108

C₉

—

109

C₉

—

110

C₉

—

111

C₃

—

112

C₃

—

113

C₃

—

114

C₃

—

115

C₃

—

116

C₃

#10624732

yes

characteristic

117

C₃

#10624748

yes

characteristic

118

C₃

#10624752

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. #10624862	C₃	semidirect
2. #10624806	C₃	semidirect
3. #10624777	C₃	semidirect
4. #10624742	C₃	semidirect
5. #10624741	C₃	semidirect
6. #10624637	C₃	semidirect
7. C₃×#6856	C₃	semidirect

H ≅ G/N

type

1. #10624862

C₃

semidirect

2. #10624806

C₃

semidirect

3. #10624777

C₃

semidirect

4. #10624742

C₃

semidirect

5. #10624741

C₃

semidirect

6. #10624637

C₃

semidirect

7. C₃×#6856

C₃

semidirect

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

subgroup	index
#10624862	3
#10624806	3
#10624777	3
#10624742	3
#10624741	3
#10624637	3
C₃×#6856	3

subgroup

index

Subgroups — #14548280

Conjugacy classes of subgroups

Complements

Maximal subgroups