Group #14550570 — subgroups

Subgroups — #14550570

Order 2187.

357 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 357 conjugacy classes of subgroups — too many to draw. Only its 110 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×C₉×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic
2	#10624706	C₃	1	2	yes	maximal, complemented
3	C₃×#6869	C₃	1	3	yes	maximal
4	C₉×C₂₇⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
5	#10624467	C₃	1	6	yes	maximal, complemented
6	C₃²×3₋¹⁺²	C₉	1	1	yes	characteristic
7	C₃³×C₉	C₃²	1	1	yes	characteristic
8	C₃×C₂₇⋊C₃	C₃²	1	6	yes	—
9	C₃×C₂₇⋊C₃	C₃²	1	3	yes	—
10	C₉×3₋¹⁺²	—	3	1	no	—
11	C₉×3₋¹⁺²	—	3	2	no	—
12	C₉×3₋¹⁺²	—	3	2	no	—
13	C₉×3₋¹⁺²	—	3	1	no	—
14	C₉×3₋¹⁺²	—	3	2	no	—
15	C₉×3₋¹⁺²	—	3	1	no	—
16	C₃×C₉⋊C₉	C₉	1	4	yes	—
17	C₃×C₉⋊C₉	C₉	1	2	yes	—
18	C₃×C₃²⋊C₉	C₉	1	2	yes	—
19	C₃×C₉²	C₃²	1	2	yes	—
20	C₃×C₉²	C₃²	1	1	yes	characteristic
21	C₉⋊C₂₇	—	3	2	no	—
22	C₉⋊C₂₇	—	3	4	no	—
23	#6869	—	3	9	no	—
24	#6864	—	3	2	no	—
25	C₉×C₂₇	—	3	1	no	—
26	C₃⁴	C₃×C₉	1	1	yes	characteristic
27	C₃×3₋¹⁺²	—	3	9	no	—
28	C₃×3₋¹⁺²	—	3	2	no	—
29	C₃×3₋¹⁺²	—	3	1	no	—
30	C₃²×C₉	—	3	1	no	—
31	C₃²×C₉	—	3	3	no	—
32	C₃²×C₉	—	3	1	no	—
33	C₃²×C₉	—	3	1	no	—
34	C₃²×C₉	—	3	6	no	—
35	C₃²×C₉	C₃×C₉	1	2	yes	—
36	C₃²×C₉	C₃×C₉	1	2	yes	—
37	C₃²×C₉	C₃×C₉	1	1	yes	characteristic
38	C₃²×C₉	C₃×C₉	1	4	yes	—
39	C₃²×C₉	C₃×C₉	1	2	yes	—
40	C₃²×C₉	C₃³	1	1	yes	characteristic
41	C₂₇⋊C₃	—	3	6	no	—
42	C₂₇⋊C₃	—	3	6	no	—
43	C₂₇⋊C₃	—	3	12	no	—
44	C₂₇⋊C₃	3₋¹⁺²	1	9	yes	complemented
45	C₃×C₂₇	—	3	3	no	—
46	C₃×C₂₇	—	3	6	no	—
47	C₉⋊C₉	—	3	4	no	—
48	C₉⋊C₉	—	3	4	no	—
49	C₉⋊C₉	—	3	4	no	—
50	C₉⋊C₉	—	3	4	no	—
51	C₉⋊C₉	—	3	2	no	—
52	C₃²⋊C₉	—	3	2	no	—
53	C₃²⋊C₉	—	3	2	no	—
54	C₃²⋊C₉	—	3	2	no	—
55	C₉²	—	3	2	no	—
56	C₉²	—	3	2	no	—
57	C₉²	—	3	2	no	—
58	C₉²	H(3,𝔽₃)	1	1	yes	characteristic
59	C₉²	H(3,𝔽₃)	1	2	yes	—
60	C₉²	3₋¹⁺²	1	2	yes	—
61	C₉²	3₋¹⁺²	1	2	yes	—
62	C₉²	3₋¹⁺²	1	1	yes	characteristic
63	C₉²	3₋¹⁺²	1	1	yes	characteristic
64	C₃³	—	3	3	no	—
65	C₃³	—	3	1	no	—
66	C₃³	—	3	3	no	—
67	C₃³	—	9	2	no	—
68	C₃³	C₃²×C₉	1	1	yes	characteristic
69	3₋¹⁺²	—	3	18	no	—
70	3₋¹⁺²	—	3	9	no	—
71	C₃×C₉	—	3	2	no	—
72	C₃×C₉	—	3	3	no	—
73	C₃×C₉	—	3	6	no	—
74	C₃×C₉	—	3	4	no	—
75	C₃×C₉	—	3	2	no	—
76	C₃×C₉	—	3	6	no	—
77	C₃×C₉	—	9	2	no	—
78	C₃×C₉	—	9	3	no	—
79	C₃×C₉	—	9	3	no	—
80	C₃×C₉	—	9	2	no	—
81	C₃×C₉	—	9	1	no	—
82	C₃×C₉	—	9	4	no	—
83	C₃×C₉	C₃²⋊C₉	1	2	yes	—
84	C₃×C₉	C₃²⋊C₉	1	2	yes	—
85	C₃×C₉	C₃²⋊C₉	1	2	yes	—
86	C₃×C₉	C₃²⋊C₉	1	2	yes	—
87	C₃×C₉	C₃²⋊C₉	1	1	yes	characteristic
88	C₃×C₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
89	C₃×C₉	C₃×3₋¹⁺²	1	3	yes	—
90	C₃×C₉	C₃×3₋¹⁺²	1	1	yes	characteristic
91	C₃×C₉	C₃×3₋¹⁺²	1	1	yes	characteristic
92	C₃×C₉	#737	1	6	yes	—
93	C₂₇	—	3	9	no	—
94	C₂₇	—	9	6	no	—
95	C₃²	—	3	1	no	—
96	C₃²	—	3	2	no	—
97	C₃²	—	3	3	no	—
98	C₃²	—	9	9	no	—
99	C₃²	—	9	1	no	—
100	C₃²	—	9	2	no	—
101	C₃²	C₃×C₃²⋊C₉	1	1	yes	characteristic
102	C₃²	C₉×3₋¹⁺²	1	3	yes	—
103	C₉	—	3	2	no	—
104	C₉	—	3	6	no	—
105	C₉	—	3	2	no	—
106	C₉	—	3	2	no	—
107	C₉	—	3	2	no	—
108	C₉	—	9	2	no	—
109	C₉	—	9	4	no	—
110	C₉	#6869	1	3	yes	complemented
111	C₉	C₉⋊H(3,𝔽₃)	1	3	yes	—
112	C₉	#6896	1	3	yes	—
113	C₉	#6899	1	2	yes	—
114	C₉	#6900	1	1	yes	characteristic
115	C₃	—	3	3	no	—
116	C₃	—	9	3	no	—
117	C₃	#10624560	1	1	yes	characteristic
118	C₃	C₃×#6869	1	1	yes	characteristic
119	C₃	#10624694	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₉×3₋¹⁺²

C₃

yes

maximal, characteristic

#10624706

C₃

yes

maximal, complemented

C₃×#6869

C₃

yes

maximal

C₉×C₂₇⋊C₃

C₃

yes

maximal, characteristic, complemented

#10624467

C₃

yes

maximal, complemented

C₃²×3₋¹⁺²

C₉

yes

characteristic

C₃³×C₉

C₃²

yes

characteristic

C₃×C₂₇⋊C₃

C₃²

yes

—

C₃×C₂₇⋊C₃

C₃²

yes

—

C₉×3₋¹⁺²

—

C₉×3₋¹⁺²

—

C₉×3₋¹⁺²

—

C₉×3₋¹⁺²

—

C₉×3₋¹⁺²

—

C₉×3₋¹⁺²

—

C₃×C₉⋊C₉

C₉

yes

—

C₃×C₉⋊C₉

C₉

yes

—

C₃×C₃²⋊C₉

C₉

yes

—

C₃×C₉²

C₃²

yes

—

C₃×C₉²

C₃²

yes

characteristic

C₉⋊C₂₇

—

C₉⋊C₂₇

—

#6869

—

#6864

—

C₉×C₂₇

—

C₃⁴

C₃×C₉

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃×C₉

yes

—

C₃²×C₉

C₃×C₉

yes

—

C₃²×C₉

C₃×C₉

yes

characteristic

C₃²×C₉

C₃×C₉

yes

—

C₃²×C₉

C₃×C₉

yes

—

C₃²×C₉

C₃³

yes

characteristic

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

3₋¹⁺²

yes

complemented

C₃×C₂₇

—

C₃×C₂₇

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₉²

—

C₉²

—

C₉²

—

C₉²

H(3,𝔽₃)

yes

characteristic

C₉²

H(3,𝔽₃)

yes

—

C₉²

3₋¹⁺²

yes

—

C₉²

3₋¹⁺²

yes

—

C₉²

3₋¹⁺²

yes

characteristic

C₉²

3₋¹⁺²

yes

characteristic

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃²×C₉

yes

characteristic

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

C₃²⋊C₉

yes

—

C₃×C₉

C₃²⋊C₉

yes

—

C₃×C₉

C₃²⋊C₉

yes

—

C₃×C₉

C₃²⋊C₉

yes

—

C₃×C₉

C₃²⋊C₉

yes

characteristic

C₃×C₉

C₃×H(3,𝔽₃)

yes

characteristic

C₃×C₉

C₃×3₋¹⁺²

yes

—

C₃×C₉

C₃×3₋¹⁺²

yes

characteristic

C₃×C₉

C₃×3₋¹⁺²

yes

characteristic

C₃×C₉

#737

yes

—

C₂₇

—

C₂₇

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

100

C₃²

—

101

C₃²

C₃×C₃²⋊C₉

yes

characteristic

102

C₃²

C₉×3₋¹⁺²

yes

—

103

C₉

—

104

C₉

—

105

C₉

—

106

C₉

—

107

C₉

—

108

C₉

—

109

C₉

—

110

C₉

#6869

yes

complemented

111

C₉

C₉⋊H(3,𝔽₃)

yes

—

112

C₉

#6896

yes

—

113

C₉

#6899

yes

—

114

C₉

#6900

yes

characteristic

115

C₃

—

116

C₃

—

117

C₃

#10624560

yes

characteristic

118

C₃

C₃×#6869

yes

characteristic

119

C₃

#10624694

yes

—

	subgroup	G/H
center	C₃×C₉	C₃×H(3,𝔽₃)
commutator	C₃³	C₃²×C₉
Frattini, cosocle	C₃²×C₉	C₃³
socle	C₃²	C₃×C₃²⋊C₉
prime radical	C₃⁴	C₃×C₉
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₉ ◁ C₃×C₉² ◁ G
LECS	G ▷ C₃²×C₉ ▷ C₃ ▷ C₁

subgroup

G/H

center

C₃×C₉

C₃×H(3,𝔽₃)

commutator

C₃³

C₃²×C₉

Frattini, cosocle

C₃²×C₉

C₃³

socle

C₃²

C₃×C₃²⋊C₉

prime radical

C₃⁴

C₃×C₉

LCS

G ▷ C₃³ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃×C₉ ◁ C₃×C₉² ◁ G

LECS

G ▷ C₃²×C₉ ▷ C₃ ▷ C₁

N	H ≅ G/N	type
2. #10624706	C₃	semidirect
4. C₉×C₂₇⋊C₃	C₃	semidirect
5. #10624467	C₃	semidirect
44. C₂₇⋊C₃	3₋¹⁺²	semidirect
110. C₉	#6869	semidirect

H ≅ G/N

type

2. #10624706

C₃

semidirect

4. C₉×C₂₇⋊C₃

C₃

semidirect

5. #10624467

C₃

semidirect

44. C₂₇⋊C₃

3₋¹⁺²

semidirect

110. C₉

#6869

semidirect

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

subgroup	index
C₃×C₉×3₋¹⁺²	3
#10624706	3
C₃×#6869	3
C₉×C₂₇⋊C₃	3
#10624467	3

subgroup

index

C₃×C₉×3₋¹⁺²

#10624706

C₃×#6869

C₉×C₂₇⋊C₃

#10624467

Subgroups — #14550570

Conjugacy classes of subgroups

Complements

Maximal subgroups