Group #14550544 — subgroups

Subgroups — #14550544

Order 2187.

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

normal subgroups only

This group has 362 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	C₃²×C₂₇⋊C₃	C₃	1	1	yes	maximal, characteristic
2	#10624872	C₃	1	1	yes	maximal, characteristic
3	#10624683	C₃	1	1	yes	maximal, characteristic
4	C₃×C₃²⋊C₂₇	C₃	1	1	yes	maximal, characteristic
5	#10624469	C₃	1	9	yes	maximal, complemented
6	C₃³×C₉	C₃²	1	1	yes	characteristic
7	C₃×C₂₇⋊C₃	—	3	2	no	—
8	C₃×C₂₇⋊C₃	—	3	9	no	—
9	C₃×C₂₇⋊C₃	C₃²	1	3	yes	—
10	C₃×C₂₇⋊C₃	C₃²	1	3	yes	—
11	C₃²×C₂₇	—	3	1	no	—
12	C₃²×C₂₇	C₃²	1	3	yes	—
13	#6895	C₉	1	6	yes	—
14	#6894	—	3	4	no	—
15	#6894	—	3	2	no	—
16	#6891	C₉	1	1	yes	characteristic
17	#6891	C₉	1	2	yes	—
18	C₉×3₋¹⁺²	—	3	1	no	—
19	C₉×3₋¹⁺²	—	3	2	no	—
20	#6887	C₃²	1	3	yes	—
21	C₃²⋊C₂₇	—	3	1	no	—
22	C₃²⋊C₂₇	—	3	2	no	—
23	C₃²⋊C₂₇	—	3	1	no	—
24	C₃²⋊C₂₇	—	3	2	no	—
25	C₃⁴	C₃×C₉	1	1	yes	characteristic
26	C₃×3₋¹⁺²	—	3	2	no	—
27	C₃×3₋¹⁺²	—	3	1	no	—
28	C₃²×C₉	—	3	3	no	—
29	C₃²×C₉	—	3	3	no	—
30	C₃²×C₉	—	3	3	no	—
31	C₃²×C₉	C₃×C₉	1	2	yes	—
32	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
33	C₃²×C₉	H(3,𝔽₃)	1	2	yes	—
34	C₃²×C₉	3₋¹⁺²	1	4	yes	—
35	C₃²×C₉	3₋¹⁺²	1	2	yes	—
36	C₃²×C₉	C₃³	1	1	yes	characteristic
37	C₂₇⋊C₃	—	3	18	no	—
38	C₂₇⋊C₃	—	3	6	no	—
39	C₂₇⋊C₃	—	3	9	no	—
40	C₂₇⋊C₃	3₋¹⁺²	1	9	yes	complemented
41	C₃×C₂₇	—	3	9	no	—
42	C₃×C₂₇	—	3	3	no	—
43	C₃×C₂₇	—	3	3	no	—
44	C₃×C₂₇	—	3	3	no	—
45	C₃×C₂₇	—	3	9	no	—
46	C₉⋊C₉	—	3	6	no	—
47	C₉⋊C₉	—	3	12	no	—
48	C₉⋊C₉	—	3	6	no	—
49	C₃²⋊C₉	—	3	2	no	—
50	C₃²⋊C₉	—	3	2	no	—
51	C₃²⋊C₉	—	3	2	no	—
52	C₃²⋊C₉	C₃×C₉	1	3	yes	—
53	C₃²⋊C₉	C₃×C₉	1	6	yes	—
54	C₉²	—	3	3	no	—
55	C₃³	—	3	3	no	—
56	C₃³	—	3	3	no	—
57	C₃³	—	9	2	no	—
58	C₃³	C₃²⋊C₉	1	2	yes	—
59	C₃³	C₃²⋊C₉	1	1	yes	characteristic
60	C₃³	C₃²×C₉	1	1	yes	characteristic
61	3₋¹⁺²	—	9	6	no	—
62	3₋¹⁺²	—	9	3	no	—
63	C₃×C₉	—	3	6	no	—
64	C₃×C₉	—	3	1	no	—
65	C₃×C₉	—	3	2	no	—
66	C₃×C₉	—	3	1	no	—
67	C₃×C₉	—	3	1	no	—
68	C₃×C₉	—	3	2	no	—
69	C₃×C₉	—	3	2	no	—
70	C₃×C₉	—	3	6	no	—
71	C₃×C₉	—	3	12	no	—
72	C₃×C₉	—	3	3	no	—
73	C₃×C₉	—	3	6	no	—
74	C₃×C₉	C₃²⋊C₉	1	2	yes	—
75	C₃×C₉	C₃²⋊C₉	1	4	yes	—
76	C₃×C₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
77	C₃×C₉	C₃×3₋¹⁺²	1	3	yes	—
78	C₃×C₉	C₃×3₋¹⁺²	1	2	yes	—
79	C₃×C₉	#737	1	3	yes	—
80	C₃×C₉	#737	1	3	yes	—
81	C₂₇	—	3	9	no	—
82	C₂₇	—	9	3	no	—
83	C₂₇	—	9	3	no	—
84	C₃²	—	3	1	no	—
85	C₃²	—	3	1	no	—
86	C₃²	—	3	2	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	—	9	9	no	—
93	C₃²	#6872	1	3	yes	—
94	C₃²	C₃×C₃²⋊C₉	1	1	yes	characteristic
95	C₃²	C₉×3₋¹⁺²	1	3	yes	—
96	C₉	—	3	2	no	—
97	C₉	—	3	4	no	—
98	C₉	—	9	3	no	—
99	C₉	—	9	6	no	—
100	C₉	#6891	1	1	yes	characteristic
101	C₉	#6893	1	1	yes	characteristic
102	C₉	#6893	1	1	yes	characteristic
103	C₉	#6894	1	6	yes	—
104	C₃	—	3	3	no	—
105	C₃	—	3	3	no	—
106	C₃	—	9	2	no	—
107	C₃	#10624558	1	1	yes	characteristic
108	C₃	C₃×#6872	1	1	yes	characteristic
109	C₃	#10624702	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×C₂₇⋊C₃

C₃

yes

maximal, characteristic

#10624872

C₃

yes

maximal, characteristic

#10624683

C₃

yes

maximal, characteristic

C₃×C₃²⋊C₂₇

C₃

yes

maximal, characteristic

#10624469

C₃

yes

maximal, complemented

C₃³×C₉

C₃²

yes

characteristic

C₃×C₂₇⋊C₃

—

C₃×C₂₇⋊C₃

—

C₃×C₂₇⋊C₃

C₃²

yes

—

C₃×C₂₇⋊C₃

C₃²

yes

—

C₃²×C₂₇

—

C₃²×C₂₇

C₃²

yes

—

#6895

C₉

yes

—

#6894

—

#6894

—

#6891

C₉

yes

characteristic

#6891

C₉

yes

—

C₉×3₋¹⁺²

—

C₉×3₋¹⁺²

—

#6887

C₃²

yes

—

C₃²⋊C₂₇

—

C₃²⋊C₂₇

—

C₃²⋊C₂₇

—

C₃²⋊C₂₇

—

C₃⁴

C₃×C₉

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃×C₉

yes

—

C₃²×C₉

H(3,𝔽₃)

yes

characteristic

C₃²×C₉

H(3,𝔽₃)

yes

—

C₃²×C₉

3₋¹⁺²

yes

—

C₃²×C₉

3₋¹⁺²

yes

—

C₃²×C₉

C₃³

yes

characteristic

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

3₋¹⁺²

yes

complemented

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×C₉

yes

—

C₃²⋊C₉

C₃×C₉

yes

—

C₉²

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃²⋊C₉

yes

—

C₃³

C₃²⋊C₉

yes

characteristic

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

C₃²⋊C₉

yes

—

C₃×C₉

C₃²⋊C₉

yes

—

C₃×C₉

C₃×H(3,𝔽₃)

yes

characteristic

C₃×C₉

C₃×3₋¹⁺²

yes

—

C₃×C₉

C₃×3₋¹⁺²

yes

—

C₃×C₉

#737

yes

—

C₃×C₉

#737

yes

—

C₂₇

—

C₂₇

—

C₂₇

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

#6872

yes

—

C₃²

C₃×C₃²⋊C₉

yes

characteristic

C₃²

C₉×3₋¹⁺²

yes

—

C₉

—

C₉

—

C₉

—

C₉

—

100

C₉

#6891

yes

characteristic

101

C₉

#6893

yes

characteristic

102

C₉

#6893

yes

characteristic

103

C₉

#6894

yes

—

104

C₃

—

105

C₃

—

106

C₃

—

107

C₃

#10624558

yes

characteristic

108

C₃

C₃×#6872

yes

characteristic

109

C₃

#10624702

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 ◁ C₃³×C₉ ◁ 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 ◁ C₃³×C₉ ◁ G

LECS

G ▷ C₃²×C₉ ▷ C₃ ▷ C₁

N	H ≅ G/N	type
5. #10624469	C₃	semidirect
40. C₂₇⋊C₃	3₋¹⁺²	semidirect

H ≅ G/N

type

5. #10624469

C₃

semidirect

40. C₂₇⋊C₃

3₋¹⁺²

semidirect

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

subgroup	index
C₃²×C₂₇⋊C₃	3
#10624872	3
#10624683	3
C₃×C₃²⋊C₂₇	3
#10624469	3

subgroup

index

C₃²×C₂₇⋊C₃

#10624872

#10624683

C₃×C₃²⋊C₂₇

#10624469

Subgroups — #14550544

Conjugacy classes of subgroups

Complements

Maximal subgroups