Group #14550555 — subgroups

Subgroups — #14550555

Order 2187.

395 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 395 conjugacy classes of subgroups — too many to draw. Only its 134 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	C₃×C₉⋊C₂₇	C₃	1	2	yes	maximal, complemented
3	C₃×C₃²⋊C₂₇	C₃	1	3	yes	maximal
4	C₃×#6864	C₃	1	1	yes	maximal, characteristic, complemented
5	#10624463	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	C₃²×C₂₇	C₃²	1	6	yes	—
9	C₃²×C₂₇	C₃²	1	3	yes	—
10	C₉×3₋¹⁺²	—	3	3	no	—
11	C₉×3₋¹⁺²	—	3	3	no	—
12	C₉×3₋¹⁺²	—	3	3	no	—
13	C₃×C₉⋊C₉	C₉	1	2	yes	—
14	C₃×C₉⋊C₉	C₉	1	4	yes	—
15	C₃×C₃²⋊C₉	C₉	1	2	yes	—
16	C₃×C₉²	C₃²	1	2	yes	—
17	C₃×C₉²	C₃²	1	1	yes	characteristic
18	C₉⋊C₂₇	—	3	6	no	—
19	C₃²⋊C₂₇	—	3	9	no	—
20	#6864	—	3	3	no	—
21	C₃⁴	C₃×C₉	1	1	yes	characteristic
22	C₃×3₋¹⁺²	—	3	3	no	—
23	C₃×3₋¹⁺²	—	3	3	no	—
24	C₃×3₋¹⁺²	—	3	6	no	—
25	C₃²×C₉	—	3	1	no	—
26	C₃²×C₉	—	3	3	no	—
27	C₃²×C₉	—	3	1	no	—
28	C₃²×C₉	—	3	1	no	—
29	C₃²×C₉	—	3	6	no	—
30	C₃²×C₉	C₃×C₉	1	2	yes	—
31	C₃²×C₉	C₃×C₉	1	4	yes	—
32	C₃²×C₉	C₃×C₉	1	1	yes	characteristic
33	C₃²×C₉	C₃×C₉	1	2	yes	—
34	C₃²×C₉	C₃×C₉	1	2	yes	—
35	C₃²×C₉	C₃³	1	1	yes	characteristic
36	C₃×C₂₇	—	3	3	no	—
37	C₃×C₂₇	—	3	6	no	—
38	C₃×C₂₇	—	3	6	no	—
39	C₃×C₂₇	—	3	6	no	—
40	C₃×C₂₇	—	3	12	no	—
41	C₃×C₂₇	3₋¹⁺²	1	9	yes	complemented
42	C₉⋊C₉	—	3	12	no	—
43	C₉⋊C₉	—	3	6	no	—
44	C₃²⋊C₉	—	3	6	no	—
45	C₉²	—	3	2	no	—
46	C₉²	—	3	2	no	—
47	C₉²	—	3	2	no	—
48	C₉²	H(3,𝔽₃)	1	3	yes	—
49	C₉²	3₋¹⁺²	1	3	yes	—
50	C₉²	3₋¹⁺²	1	3	yes	—
51	C₃³	—	3	1	no	—
52	C₃³	—	3	3	no	—
53	C₃³	—	3	2	no	—
54	C₃³	—	3	6	no	—
55	C₃³	C₂₇⋊C₃	1	3	yes	—
56	C₃³	C₃²×C₉	1	1	yes	characteristic
57	3₋¹⁺²	—	3	27	no	—
58	C₃×C₉	—	3	2	no	—
59	C₃×C₉	—	3	1	no	—
60	C₃×C₉	—	3	2	no	—
61	C₃×C₉	—	3	2	no	—
62	C₃×C₉	—	3	2	no	—
63	C₃×C₉	—	3	6	no	—
64	C₃×C₉	—	3	4	no	—
65	C₃×C₉	—	3	2	no	—
66	C₃×C₉	—	3	6	no	—
67	C₃×C₉	—	3	12	no	—
68	C₃×C₉	—	9	3	no	—
69	C₃×C₉	—	9	3	no	—
70	C₃×C₉	—	9	2	no	—
71	C₃×C₉	—	9	1	no	—
72	C₃×C₉	C₃²⋊C₉	1	6	yes	—
73	C₃×C₉	C₃²⋊C₉	1	3	yes	—
74	C₃×C₉	C₂₇⋊C₃	1	6	yes	—
75	C₃×C₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
76	C₃×C₉	C₃×3₋¹⁺²	1	3	yes	—
77	C₃×C₉	C₃×3₋¹⁺²	1	1	yes	characteristic
78	C₃×C₉	C₃×3₋¹⁺²	1	1	yes	characteristic
79	C₃×C₉	#737	1	6	yes	—
80	C₂₇	—	3	9	no	—
81	C₂₇	—	9	6	no	—
82	C₃²	—	3	3	no	—
83	C₃²	—	3	9	no	—
84	C₃²	—	9	1	no	—
85	C₃²	—	9	2	no	—
86	C₃²	—	9	6	no	—
87	C₃²	C₃×C₃²⋊C₉	1	1	yes	characteristic
88	C₃²	C₉×3₋¹⁺²	1	3	yes	—
89	C₃²	C₃×C₂₇⋊C₃	1	1	yes	characteristic
90	C₃²	C₃×C₂₇⋊C₃	1	2	yes	—
91	C₃²	#6903	1	6	yes	—
92	C₉	—	3	3	no	—
93	C₉	—	3	6	no	—
94	C₉	—	3	6	no	—
95	C₉	—	9	2	no	—
96	C₉	—	9	4	no	—
97	C₉	C₉⋊H(3,𝔽₃)	1	3	yes	—
98	C₉	#6896	1	3	yes	—
99	C₉	#6899	1	2	yes	—
100	C₉	#6900	1	1	yes	characteristic
101	C₃	—	9	3	no	—
102	C₃	#10624560	1	1	yes	characteristic
103	C₃	#10624683	1	1	yes	characteristic
104	C₃	#10624684	1	2	yes	—
105	C₃	#10624727	1	3	yes	—
106	C₃	#10624729	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₉×3₋¹⁺²

C₃

yes

maximal, characteristic

C₃×C₉⋊C₂₇

C₃

yes

maximal, complemented

C₃×C₃²⋊C₂₇

C₃

yes

maximal

C₃×#6864

C₃

yes

maximal, characteristic, complemented

#10624463

C₃

yes

maximal, complemented

C₃²×3₋¹⁺²

C₉

yes

characteristic

C₃³×C₉

C₃²

yes

characteristic

C₃²×C₂₇

C₃²

yes

—

C₃²×C₂₇

C₃²

yes

—

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

—

C₃²⋊C₂₇

—

#6864

—

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

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

3₋¹⁺²

yes

complemented

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₉²

—

C₉²

—

C₉²

—

C₉²

H(3,𝔽₃)

yes

—

C₉²

3₋¹⁺²

yes

—

C₉²

3₋¹⁺²

yes

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₂₇⋊C₃

yes

—

C₃³

C₃²×C₉

yes

characteristic

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

C₃²⋊C₉

yes

—

C₃×C₉

C₃²⋊C₉

yes

—

C₃×C₉

C₂₇⋊C₃

yes

—

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

—

C₃²

C₃×C₃²⋊C₉

yes

characteristic

C₃²

C₉×3₋¹⁺²

yes

—

C₃²

C₃×C₂₇⋊C₃

yes

characteristic

C₃²

C₃×C₂₇⋊C₃

yes

—

C₃²

#6903

yes

—

C₉

—

C₉

—

C₉

—

C₉

—

C₉

—

C₉

C₉⋊H(3,𝔽₃)

yes

—

C₉

#6896

yes

—

C₉

#6899

yes

—

100

C₉

#6900

yes

characteristic

101

C₃

—

102

C₃

#10624560

yes

characteristic

103

C₃

#10624683

yes

characteristic

104

C₃

#10624684

yes

—

105

C₃

#10624727

yes

—

106

C₃

#10624729

yes

—

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

subgroup

G/H

center, Frattini, cosocle

C₃²×C₉

C₃³

commutator, socle

C₃³

C₃²×C₉

prime radical

C₃⁴

C₃×C₉

LCS

G ▷ C₃³ ▷ C₁

UCS

C₁ ◁ C₃²×C₉ ◁ G

LECS

G ▷ C₃²×C₉ ▷ C₃ ▷ C₁

N	H ≅ G/N	type
2. C₃×C₉⋊C₂₇	C₃	semidirect
4. C₃×#6864	C₃	semidirect
5. #10624463	C₃	semidirect
41. C₃×C₂₇	3₋¹⁺²	semidirect

H ≅ G/N

type

2. C₃×C₉⋊C₂₇

C₃

semidirect

4. C₃×#6864

C₃

semidirect

5. #10624463

C₃

semidirect

41. C₃×C₂₇

3₋¹⁺²

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
C₃×C₉⋊C₂₇	3
C₃×C₃²⋊C₂₇	3
C₃×#6864	3
#10624463	3

subgroup

index

C₃×C₉×3₋¹⁺²

C₃×C₉⋊C₂₇

C₃×C₃²⋊C₂₇

C₃×#6864

#10624463

Subgroups — #14550555

Conjugacy classes of subgroups

Complements

Maximal subgroups