Group #14550401 — subgroups

Subgroups — #14550401

Order 2187.

581 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 581 conjugacy classes of subgroups — too many to draw. Only its 125 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	C₃²×#733	C₃	1	1	yes	maximal, characteristic
2	C₃²×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic
3	C₃×3₋¹⁺²⋊C₉	C₃	1	6	yes	maximal
4	C₃×#6868	C₃	1	2	yes	maximal
5	#10624558	C₃	1	3	yes	maximal
6	C₃²×3₋¹⁺²	C₃²	1	3	yes	—
7	C₃³×C₉	C₃²	1	1	yes	characteristic
8	C₃³×C₉	C₃²	1	2	yes	—
9	C₃³×C₉	C₃²	1	1	yes	characteristic
10	C₃×#733	—	3	3	no	—
11	C₃×#733	—	3	6	no	—
12	C₃×#733	C₉	1	9	yes	complemented
13	C₃×C₉⋊C₉	—	3	6	no	—
14	C₃×C₉⋊C₉	—	3	3	no	—
15	C₃×C₉⋊C₉	—	3	3	no	—
16	C₃×C₃²⋊C₉	C₃²	1	6	yes	—
17	3₋¹⁺²⋊C₉	—	3	18	no	—
18	#6868	—	3	6	no	—
19	#6855	—	3	9	no	—
20	C₃⁴	C₃³	1	1	yes	characteristic
21	C₃×3₋¹⁺²	—	3	6	no	—
22	C₃×3₋¹⁺²	—	3	6	no	—
23	C₃×3₋¹⁺²	—	3	3	no	—
24	C₃×3₋¹⁺²	—	3	6	no	—
25	C₃×3₋¹⁺²	—	3	12	no	—
26	C₃×3₋¹⁺²	C₃×C₉	1	9	yes	—
27	C₃²×C₉	—	3	2	no	—
28	C₃²×C₉	—	3	1	no	—
29	C₃²×C₉	—	3	4	no	—
30	C₃²×C₉	—	3	2	no	—
31	C₃²×C₉	—	3	2	no	—
32	C₃²×C₉	—	3	2	no	—
33	C₃²×C₉	—	3	1	no	—
34	C₃²×C₉	—	3	2	no	—
35	C₃²×C₉	—	3	2	no	—
36	C₃²×C₉	—	3	3	no	—
37	C₃²×C₉	—	3	6	no	—
38	C₃²×C₉	—	9	3	no	—
39	C₃²×C₉	—	9	6	no	—
40	C₃²×C₉	C₃×C₉	1	3	yes	—
41	C₃²×C₉	H(3,𝔽₃)	1	3	yes	—
42	C₃²×C₉	H(3,𝔽₃)	1	3	yes	—
43	C₃²×C₉	H(3,𝔽₃)	1	6	yes	—
44	C₃²×C₉	3₋¹⁺²	1	6	yes	—
45	#733	—	3	27	no	—
46	C₉⋊C₉	—	3	27	no	—
47	C₃²⋊C₉	—	3	6	no	—
48	C₃²⋊C₉	—	3	12	no	—
49	C₃³	—	3	3	no	—
50	C₃³	—	3	6	no	—
51	C₃³	C₃²×C₉	1	1	yes	characteristic
52	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
53	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
54	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
55	C₃³	C₃×3₋¹⁺²	1	2	yes	—
56	C₃³	#737	1	4	yes	—
57	C₃³	#737	1	2	yes	—
58	3₋¹⁺²	—	3	9	no	—
59	3₋¹⁺²	—	3	18	no	—
60	3₋¹⁺²	—	9	12	no	—
61	3₋¹⁺²	—	9	6	no	—
62	C₃×C₉	—	3	6	no	—
63	C₃×C₉	—	3	3	no	—
64	C₃×C₉	—	3	3	no	—
65	C₃×C₉	—	3	12	no	—
66	C₃×C₉	—	3	6	no	—
67	C₃×C₉	—	3	6	no	—
68	C₃×C₉	—	3	9	no	—
69	C₃×C₉	—	3	9	no	—
70	C₃×C₉	—	3	3	no	—
71	C₃×C₉	—	3	6	no	—
72	C₃×C₉	—	9	3	no	—
73	C₃×C₉	—	9	6	no	—
74	C₃×C₉	—	9	6	no	—
75	C₃×C₉	—	9	6	no	—
76	C₃×C₉	—	9	6	no	—
77	C₃×C₉	—	9	3	no	—
78	C₃×C₉	—	9	6	no	—
79	C₃×C₉	—	9	6	no	—
80	C₃×C₉	—	9	18	no	—
81	C₃×C₉	—	9	6	no	—
82	C₃×C₉	C₃²⋊C₉	1	9	yes	—
83	C₃²	—	3	6	no	—
84	C₃²	—	3	12	no	—
85	C₃²	—	3	3	no	—
86	C₃²	—	3	6	no	—
87	C₃²	—	3	1	no	—
88	C₃²	—	3	2	no	—
89	C₃²	—	3	2	no	—
90	C₃²	—	3	2	no	—
91	C₃²	—	3	2	no	—
92	C₃²	C₃×C₃²⋊C₉	1	1	yes	characteristic
93	C₃²	C₉×H(3,𝔽₃)	1	2	yes	—
94	C₃²	C₉×H(3,𝔽₃)	1	1	yes	characteristic
95	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
96	C₃²	C₉⋊H(3,𝔽₃)	1	4	yes	—
97	C₃²	C₉⋊H(3,𝔽₃)	1	2	yes	—
98	C₃²	#6893	1	2	yes	—
99	C₃²	#6912	1	3	yes	—
100	C₃²	#6913	1	6	yes	—
101	C₉	—	3	9	no	—
102	C₉	—	9	9	no	—
103	C₉	—	9	3	no	—
104	C₉	—	9	6	no	—
105	C₉	—	27	6	no	—
106	C₃	—	3	1	no	—
107	C₃	—	3	2	no	—
108	C₃	—	3	4	no	—
109	C₃	—	3	2	no	—
110	C₃	#10624555	1	1	yes	characteristic
111	C₃	#10624723	1	3	yes	—
112	C₃	#10624738	1	1	yes	characteristic
113	C₃	#10624739	1	2	yes	—
114	C₃	#10624791	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#733

C₃

yes

maximal, characteristic

C₃²×C₉⋊C₉

C₃

yes

maximal, characteristic

C₃×3₋¹⁺²⋊C₉

C₃

yes

maximal

C₃×#6868

C₃

yes

maximal

#10624558

C₃

yes

maximal

C₃²×3₋¹⁺²

C₃²

yes

—

C₃³×C₉

C₃²

yes

characteristic

C₃³×C₉

C₃²

yes

—

C₃³×C₉

C₃²

yes

—

—

C₉

yes

complemented

C₃×C₉⋊C₉

—

C₃×C₉⋊C₉

—

C₃×C₉⋊C₉

—

C₃×C₃²⋊C₉

C₃²

yes

—

3₋¹⁺²⋊C₉

—

#6868

—

#6855

—

C₃⁴

C₃³

yes

characteristic

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×C₉

yes

—

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

—

C₃²×C₉

C₃×C₉

yes

—

C₃²×C₉

H(3,𝔽₃)

yes

—

C₃²×C₉

H(3,𝔽₃)

yes

—

C₃²×C₉

H(3,𝔽₃)

yes

—

C₃²×C₉

3₋¹⁺²

yes

—

#733

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃³

—

C₃³

—

C₃³

C₃²×C₉

yes

characteristic

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

C₃³

C₃×H(3,𝔽₃)

yes

—

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

C₃³

C₃×3₋¹⁺²

yes

—

C₃³

#737

yes

—

C₃³

#737

yes

—

3₋¹⁺²

—

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

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₃²⋊C₉

yes

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₃×C₃²⋊C₉

yes

characteristic

C₃²

C₉×H(3,𝔽₃)

yes

—

C₃²

C₉×H(3,𝔽₃)

yes

characteristic

C₃²

C₃²⋊H(3,𝔽₃)

yes

characteristic

C₃²

C₉⋊H(3,𝔽₃)

yes

—

C₃²

C₉⋊H(3,𝔽₃)

yes

—

C₃²

#6893

yes

—

C₃²

#6912

yes

—

100

C₃²

#6913

yes

—

101

C₉

—

102

C₉

—

103

C₉

—

104

C₉

—

105

C₉

—

106

C₃

—

107

C₃

—

108

C₃

—

109

C₃

—

110

C₃

#10624555

yes

characteristic

111

C₃

#10624723

yes

—

112

C₃

#10624738

yes

characteristic

113

C₃

#10624739

yes

—

114

C₃

#10624791

yes

—

	subgroup	G/H
center, socle	C₃³	C₃×H(3,𝔽₃)
commutator	C₃³	C₃²×C₉
Frattini, prime radical, cosocle	C₃⁴	C₃³
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃³ ◁ C₃³×C₉ ◁ G
LECS	G ▷ C₃⁴ ▷ C₃ ▷ C₁

subgroup

G/H

center, socle

C₃³

C₃×H(3,𝔽₃)

commutator

C₃³

C₃²×C₉

Frattini, prime radical, cosocle

C₃⁴

C₃³

LCS

G ▷ C₃³ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃³ ◁ C₃³×C₉ ◁ G

LECS

G ▷ C₃⁴ ▷ C₃ ▷ C₁

N	H ≅ G/N	type
12. C₃×#733	C₉	semidirect

H ≅ G/N

type

12. C₃×#733

C₉

semidirect

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

subgroup	index
C₃²×#733	3
C₃²×C₉⋊C₉	3
C₃×3₋¹⁺²⋊C₉	3
C₃×#6868	3
#10624558	3

subgroup

index

C₃²×#733

C₃²×C₉⋊C₉

C₃×3₋¹⁺²⋊C₉

C₃×#6868

#10624558

Subgroups — #14550401

Conjugacy classes of subgroups

Complements

Maximal subgroups