Group #14547196 — subgroups

Subgroups — #14547196

Order 2187.

434 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 59 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×C₉⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
2	#10624768	C₃	1	2	yes	maximal, complemented
3	#10624741	C₃	1	3	yes	maximal
4	C₃×#6868	C₃	1	1	yes	maximal, characteristic, complemented
5	#10624597	C₃	1	6	yes	maximal, complemented
6	C₃²×3₋¹⁺²	—	3	1	no	—
7	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
8	C₃³×C₉	C₃²	1	1	yes	characteristic
9	C₃×#731	—	3	9	no	—
10	C₉⋊H(3,𝔽₃)	—	3	3	no	—
11	C₉⋊H(3,𝔽₃)	—	3	6	no	—
12	#6891	C₃²	1	6	yes	—
13	C₃×C₃²⋊C₉	—	3	2	no	—
14	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
15	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
16	#6868	—	3	4	no	—
17	#6868	—	3	2	no	—
18	#6868	—	3	2	no	—
19	#6868	—	3	1	no	—
20	#6861	—	3	18	no	—
21	C₃⁴	—	3	1	no	—
22	C₃⁴	C₃³	1	1	yes	characteristic
23	C₃×3₋¹⁺²	—	3	9	no	—
24	C₃×3₋¹⁺²	—	3	6	no	—
25	C₃×3₋¹⁺²	—	9	6	no	—
26	C₃×3₋¹⁺²	—	9	3	no	—
27	C₃×H(3,𝔽₃)	—	3	2	no	—
28	C₃×H(3,𝔽₃)	—	9	3	no	—
29	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	3	yes	—
30	C₃²×C₉	—	3	2	no	—
31	C₃²×C₉	—	3	1	no	—
32	C₃²×C₉	—	3	2	no	—
33	C₃²×C₉	—	3	3	no	—
34	C₃²×C₉	—	3	3	no	—
35	C₃²×C₉	—	9	2	no	—
36	C₃²×C₉	H(3,𝔽₃)	1	2	yes	—
37	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
38	#731	—	9	27	no	—
39	C₃²⋊C₉	—	3	18	no	—
40	C₃²⋊C₉	—	3	2	no	—
41	C₃²⋊C₉	—	3	2	no	—
42	C₃²⋊C₉	—	3	6	no	—
43	C₃²⋊C₉	—	3	12	no	—
44	C₃²⋊C₉	—	3	3	no	—
45	C₃²⋊C₉	—	3	6	no	—
46	C₃²⋊C₉	H(3,𝔽₃)	1	6	yes	—
47	C₃³	—	3	1	no	—
48	C₃³	—	3	2	no	—
49	C₃³	—	3	3	no	—
50	C₃³	—	3	3	no	—
51	C₃³	—	9	2	no	—
52	C₃³	—	9	3	no	—
53	C₃³	—	9	6	no	—
54	C₃³	—	9	1	no	—
55	C₃³	—	9	2	no	—
56	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
57	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
58	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
59	3₋¹⁺²	—	9	27	no	—
60	3₋¹⁺²	—	9	18	no	—
61	H(3,𝔽₃)	—	3	9	no	—
62	H(3,𝔽₃)	—	9	6	no	—
63	C₃×C₉	—	3	1	no	—
64	C₃×C₉	—	3	2	no	—
65	C₃×C₉	—	3	2	no	—
66	C₃×C₉	—	3	6	no	—
67	C₃×C₉	—	3	2	no	—
68	C₃×C₉	—	3	4	no	—
69	C₃×C₉	—	9	3	no	—
70	C₃×C₉	—	9	6	no	—
71	C₃×C₉	—	9	2	no	—
72	C₃×C₉	—	9	4	no	—
73	C₃×C₉	—	9	3	no	—
74	C₃×C₉	—	9	6	no	—
75	C₃×C₉	—	9	9	no	—
76	C₃×C₉	#731	1	3	yes	complemented
77	C₃²	—	3	1	no	—
78	C₃²	—	3	1	no	—
79	C₃²	—	3	2	no	—
80	C₃²	—	3	2	no	—
81	C₃²	—	3	2	no	—
82	C₃²	—	3	2	no	—
83	C₃²	—	3	4	no	—
84	C₃²	—	9	3	no	—
85	C₃²	—	9	6	no	—
86	C₃²	—	9	3	no	—
87	C₃²	—	9	3	no	—
88	C₃²	—	9	6	no	—
89	C₃²	—	27	3	no	—
90	C₃²	—	27	6	no	—
91	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
92	C₃²	C₃×#731	1	1	yes	characteristic
93	C₃²	#6910	1	2	yes	—
94	C₃²	#6911	1	3	yes	—
95	C₉	—	3	3	no	—
96	C₉	—	9	2	no	—
97	C₉	—	9	6	no	—
98	C₉	—	27	3	no	—
99	C₉	—	27	6	no	—
100	C₃	—	3	1	no	—
101	C₃	—	3	2	no	—
102	C₃	—	3	3	no	—
103	C₃	—	9	2	no	—
104	C₃	—	27	3	no	—
105	C₃	#10624737	1	1	yes	characteristic
106	C₃	#10624741	1	1	yes	characteristic
107	C₃	#10624758	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₉⋊H(3,𝔽₃)

C₃

yes

maximal, characteristic

#10624768

C₃

yes

maximal, complemented

#10624741

C₃

yes

maximal

C₃×#6868

C₃

yes

maximal, characteristic, complemented

#10624597

C₃

yes

maximal, complemented

C₃²×3₋¹⁺²

—

C₃²×H(3,𝔽₃)

C₃²

yes

characteristic

C₃³×C₉

C₃²

yes

characteristic

C₃×#731

—

C₉⋊H(3,𝔽₃)

—

C₉⋊H(3,𝔽₃)

—

#6891

C₃²

yes

—

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

—

—

—

—

—

—

C₃⁴

—

C₃⁴

C₃³

yes

characteristic

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)

H(3,𝔽₃)

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

H(3,𝔽₃)

yes

—

C₃²×C₉

H(3,𝔽₃)

yes

characteristic

#731

—

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

C₃×H(3,𝔽₃)

yes

—

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

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

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

#731

yes

complemented

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₃²⋊H(3,𝔽₃)

yes

C₃²

yes

C₃²

yes

—

C₃²

#6911

yes

—

C₉

—

C₉

—

C₉

—

C₉

—

C₉

—

100

C₃

—

101

C₃

—

102

C₃

—

103

C₃

—

104

C₃

—

105

C₃

#10624737

yes

characteristic

106

C₃

#10624741

yes

characteristic

107

C₃

#10624758

yes

—

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

prime radical

C₃²×H(3,𝔽₃)

C₃²

LCS

G ▷ C₃⁴ ▷ C₃² ▷ C₁

UCS

C₁ ◁ C₃² ◁ C₃⁴ ◁ G

LECS

G ▷ C₃⁴ ▷ C₃² ▷ C₁

N	H ≅ G/N	type
2. #10624768	C₃	semidirect
4. C₃×#6868	C₃	semidirect
5. #10624597	C₃	semidirect
76. C₃×C₉	#731	semidirect

H ≅ G/N

type

2. #10624768

C₃

semidirect

4. C₃×#6868

C₃

semidirect

5. #10624597

C₃

semidirect

76. C₃×C₉

#731

semidirect

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

subgroup	index
C₃×C₉⋊H(3,𝔽₃)	3
#10624768	3
#10624741	3
C₃×#6868	3
#10624597	3

subgroup

index

C₃×C₉⋊H(3,𝔽₃)

Subgroups — #14547196

Conjugacy classes of subgroups

Complements

Maximal subgroups