Group #14547223 — subgroups

Subgroups — #14547223

Order 2187.

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

normal subgroups only

This group has 404 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×#6894	C₃	1	1	yes	maximal, characteristic, complemented
2	#10624768	C₃	1	1	yes	maximal, characteristic, complemented
3	#10624745	C₃	1	3	yes	maximal, complemented
4	#10624732	C₃	1	1	yes	maximal, characteristic
5	C₃×#6868	C₃	1	1	yes	maximal, characteristic, complemented
6	#10624598	C₃	1	6	yes	maximal, complemented
7	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
8	C₃³×C₉	C₃²	1	1	yes	characteristic
9	C₃×#732	—	3	2	no	—
10	C₃×#732	—	3	1	no	—
11	C₃×#731	—	3	9	no	—
12	#6894	—	3	3	no	—
13	#6894	—	3	6	no	—
14	#6891	C₃²	1	3	yes	—
15	C₃²⋊H(3,𝔽₃)	C₃²	1	3	yes	—
16	C₃×C₉⋊C₉	—	3	2	no	—
17	C₃×C₉⋊C₉	—	3	1	no	—
18	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
19	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
20	#6868	—	3	2	no	—
21	#6868	—	3	1	no	—
22	#6868	—	3	2	no	—
23	#6868	—	3	1	no	—
24	#6859	—	3	18	no	—
25	C₃⁴	C₃³	1	1	yes	characteristic
26	C₃×3₋¹⁺²	—	3	2	no	—
27	C₃×3₋¹⁺²	—	3	3	no	—
28	C₃×3₋¹⁺²	—	9	3	no	—
29	C₃×3₋¹⁺²	H(3,𝔽₃)	1	3	yes	—
30	C₃×H(3,𝔽₃)	—	3	3	no	—
31	C₃×H(3,𝔽₃)	—	3	3	no	—
32	C₃×H(3,𝔽₃)	—	3	6	no	—
33	C₃²×C₉	—	3	2	no	—
34	C₃²×C₉	—	3	1	no	—
35	C₃²×C₉	—	3	1	no	—
36	C₃²×C₉	—	3	2	no	—
37	C₃²×C₉	—	3	3	no	—
38	C₃²×C₉	—	3	3	no	—
39	C₃²×C₉	—	9	2	no	—
40	C₃²×C₉	H(3,𝔽₃)	1	2	yes	—
41	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
42	#732	—	9	6	no	—
43	#732	—	9	3	no	—
44	#731	—	9	27	no	—
45	C₉⋊C₉	—	3	9	no	—
46	C₉⋊C₉	—	3	18	no	—
47	C₃²⋊C₉	—	3	2	no	—
48	C₃²⋊C₉	—	3	2	no	—
49	C₃²⋊C₉	—	3	6	no	—
50	C₃²⋊C₉	—	3	6	no	—
51	C₃²⋊C₉	—	3	3	no	—
52	C₃²⋊C₉	—	3	3	no	—
53	C₃²⋊C₉	H(3,𝔽₃)	1	6	yes	—
54	C₃³	—	3	1	no	—
55	C₃³	—	3	2	no	—
56	C₃³	—	3	3	no	—
57	C₃³	—	9	2	no	—
58	C₃³	—	9	3	no	—
59	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
60	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
61	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
62	3₋¹⁺²	—	3	9	no	—
63	3₋¹⁺²	—	9	6	no	—
64	3₋¹⁺²	—	9	9	no	—
65	H(3,𝔽₃)	—	9	3	no	—
66	H(3,𝔽₃)	—	9	6	no	—
67	H(3,𝔽₃)	—	9	9	no	—
68	H(3,𝔽₃)	—	27	6	no	—
69	C₃×C₉	—	3	1	no	—
70	C₃×C₉	—	3	2	no	—
71	C₃×C₉	—	3	6	no	—
72	C₃×C₉	—	3	2	no	—
73	C₃×C₉	—	3	3	no	—
74	C₃×C₉	—	3	2	no	—
75	C₃×C₉	—	3	2	no	—
76	C₃×C₉	—	3	2	no	—
77	C₃×C₉	—	9	3	no	—
78	C₃×C₉	—	9	6	no	—
79	C₃×C₉	—	9	6	no	—
80	C₃×C₉	—	9	3	no	—
81	C₃×C₉	—	9	3	no	—
82	C₃×C₉	—	9	3	no	—
83	C₃×C₉	—	9	9	no	—
84	C₃×C₉	#731	1	3	yes	complemented
85	C₃²	—	3	1	no	—
86	C₃²	—	3	1	no	—
87	C₃²	—	3	2	no	—
88	C₃²	—	3	2	no	—
89	C₃²	—	3	2	no	—
90	C₃²	—	3	2	no	—
91	C₃²	—	3	2	no	—
92	C₃²	—	3	2	no	—
93	C₃²	—	9	3	no	—
94	C₃²	—	9	6	no	—
95	C₃²	—	27	3	no	—
96	C₃²	—	27	3	no	—
97	C₃²	—	27	6	no	—
98	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
99	C₃²	C₃×#731	1	1	yes	characteristic
100	C₃²	#6910	1	2	yes	—
101	C₃²	#6911	1	3	yes	—
102	C₉	—	3	3	no	—
103	C₉	—	9	3	no	—
104	C₉	—	9	2	no	—
105	C₉	—	9	6	no	—
106	C₉	—	27	3	no	—
107	C₉	—	27	3	no	—
108	C₃	—	3	1	no	—
109	C₃	—	3	2	no	—
110	C₃	—	3	3	no	—
111	C₃	—	9	2	no	—
112	C₃	—	81	3	no	—
113	C₃	#10624737	1	1	yes	characteristic
114	C₃	#10624745	1	1	yes	characteristic
115	C₃	#10624759	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#6894

C₃

yes

maximal, characteristic, complemented

#10624768

C₃

yes

maximal, characteristic, complemented

#10624745

C₃

yes

maximal, complemented

#10624732

C₃

yes

maximal, characteristic

C₃×#6868

C₃

yes

maximal, characteristic, complemented

#10624598

C₃

yes

maximal, complemented

C₃²×3₋¹⁺²

C₃²

yes

characteristic

C₃³×C₉

C₃²

yes

—

—

—

—

—

C₃²

yes

—

C₃²⋊H(3,𝔽₃)

C₃²

yes

—

C₃×C₉⋊C₉

—

C₃×C₉⋊C₉

—

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

—

—

—

—

—

—

C₃⁴

C₃³

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

H(3,𝔽₃)

yes

—

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

yes

—

—

—

C₉⋊C₉

—

C₉⋊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

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

—

3₋¹⁺²

—

—

—

—

—

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

—

C₃×C₉

—

C₃×C₉

#731

yes

complemented

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₃²⋊H(3,𝔽₃)

yes

characteristic

C₃²

C₃×#731

yes

characteristic

100

C₃²

#6910

yes

—

101

C₃²

#6911

yes

—

102

C₉

—

103

C₉

—

104

C₉

—

105

C₉

—

106

C₉

—

107

C₉

—

108

C₃

—

109

C₃

—

110

C₃

—

111

C₃

—

112

C₃

—

113

C₃

#10624737

yes

characteristic

114

C₃

#10624745

yes

characteristic

115

C₃

#10624759

yes

—

	subgroup	G/H
center, socle	C₃²	C₃²⋊H(3,𝔽₃)
commutator, Frattini, cosocle	C₃⁴	C₃³
prime radical	#10624732	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×#6894	C₃	semidirect
2. #10624768	C₃	semidirect
3. #10624745	C₃	semidirect
5. C₃×#6868	C₃	semidirect
6. #10624598	C₃	semidirect
84. C₃×C₉	#731	semidirect

H ≅ G/N

type

1. C₃×#6894

C₃

semidirect

2. #10624768

C₃

semidirect

3. #10624745

C₃

semidirect

5. C₃×#6868

C₃

semidirect

6. #10624598

C₃

semidirect

84. C₃×C₉

#731

semidirect

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

subgroup	index
C₃×#6894	3
#10624768	3
#10624745	3
#10624732	3
C₃×#6868	3
#10624598	3

subgroup

index

Subgroups — #14547223

Conjugacy classes of subgroups

Complements

Maximal subgroups