Group #14550539 — subgroups

Subgroups — #14550539

Order 2187.

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

normal subgroups only

This group has 410 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	C₃×C₉×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic
3	#10624683	C₃	1	2	yes	maximal
4	#10624467	C₃	1	9	yes	maximal, complemented
5	C₃²×3₋¹⁺²	C₉	1	1	yes	characteristic
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	6	yes	—
10	C₃×C₂₇⋊C₃	C₃²	1	3	yes	—
11	C₃²×C₂₇	—	3	1	no	—
12	C₉×3₋¹⁺²	—	3	1	no	—
13	C₉×3₋¹⁺²	—	3	2	no	—
14	C₉×3₋¹⁺²	—	3	4	no	—
15	C₉×3₋¹⁺²	—	3	2	no	—
16	C₃×C₉⋊C₉	C₉	1	6	yes	—
17	C₃×C₃²⋊C₉	C₉	1	2	yes	—
18	C₃×C₉²	C₃²	1	3	yes	—
19	C₃²⋊C₂₇	—	3	2	no	—
20	C₃²⋊C₂₇	—	3	4	no	—
21	C₃⁴	C₃×C₉	1	1	yes	characteristic
22	C₃×3₋¹⁺²	—	3	2	no	—
23	C₃×3₋¹⁺²	—	3	1	no	—
24	C₃×3₋¹⁺²	—	3	9	no	—
25	C₃²×C₉	—	3	3	no	—
26	C₃²×C₉	—	3	6	no	—
27	C₃²×C₉	C₃×C₉	1	3	yes	—
28	C₃²×C₉	C₃×C₉	1	2	yes	—
29	C₃²×C₉	C₃×C₉	1	6	yes	—
30	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
31	C₃²×C₉	H(3,𝔽₃)	1	2	yes	—
32	C₃²×C₉	3₋¹⁺²	1	4	yes	—
33	C₃²×C₉	3₋¹⁺²	1	2	yes	—
34	C₃²×C₉	C₃³	1	1	yes	characteristic
35	C₂₇⋊C₃	—	3	18	no	—
36	C₂₇⋊C₃	—	3	6	no	—
37	C₂₇⋊C₃	—	3	18	no	—
38	C₂₇⋊C₃	3₋¹⁺²	1	9	yes	complemented
39	C₃×C₂₇	—	3	9	no	—
40	C₃×C₂₇	—	3	3	no	—
41	C₃×C₂₇	—	3	6	no	—
42	C₉⋊C₉	—	3	6	no	—
43	C₉⋊C₉	—	3	12	no	—
44	C₃²⋊C₉	—	3	4	no	—
45	C₃²⋊C₉	—	3	2	no	—
46	C₉²	—	3	3	no	—
47	C₉²	—	3	6	no	—
48	C₃³	—	3	3	no	—
49	C₃³	—	3	3	no	—
50	C₃³	—	9	2	no	—
51	C₃³	C₃²⋊C₉	1	2	yes	—
52	C₃³	C₃²⋊C₉	1	1	yes	characteristic
53	C₃³	C₃²×C₉	1	1	yes	characteristic
54	3₋¹⁺²	—	3	18	no	—
55	3₋¹⁺²	—	3	9	no	—
56	C₃×C₉	—	3	3	no	—
57	C₃×C₉	—	3	6	no	—
58	C₃×C₉	—	3	1	no	—
59	C₃×C₉	—	3	2	no	—
60	C₃×C₉	—	3	2	no	—
61	C₃×C₉	—	3	4	no	—
62	C₃×C₉	—	3	6	no	—
63	C₃×C₉	—	3	12	no	—
64	C₃×C₉	—	3	9	no	—
65	C₃×C₉	—	3	6	no	—
66	C₃×C₉	—	9	6	no	—
67	C₃×C₉	C₃²⋊C₉	1	2	yes	—
68	C₃×C₉	C₃²⋊C₉	1	4	yes	—
69	C₃×C₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
70	C₃×C₉	C₃×3₋¹⁺²	1	3	yes	—
71	C₃×C₉	C₃×3₋¹⁺²	1	2	yes	—
72	C₃×C₉	#737	1	6	yes	—
73	C₂₇	—	3	9	no	—
74	C₂₇	—	9	6	no	—
75	C₃²	—	3	1	no	—
76	C₃²	—	3	1	no	—
77	C₃²	—	3	2	no	—
78	C₃²	—	3	2	no	—
79	C₃²	—	3	2	no	—
80	C₃²	—	3	2	no	—
81	C₃²	—	3	4	no	—
82	C₃²	—	9	9	no	—
83	C₃²	#6869	1	3	yes	—
84	C₃²	C₃×C₃²⋊C₉	1	1	yes	characteristic
85	C₃²	C₉×3₋¹⁺²	1	3	yes	—
86	C₉	—	3	2	no	—
87	C₉	—	3	4	no	—
88	C₉	—	3	9	no	—
89	C₉	—	9	6	no	—
90	C₉	#6891	1	1	yes	characteristic
91	C₉	#6893	1	2	yes	—
92	C₉	#6894	1	6	yes	—
93	C₃	—	3	3	no	—
94	C₃	—	3	3	no	—
95	C₃	—	9	2	no	—
96	C₃	#10624558	1	1	yes	characteristic
97	C₃	C₃×#6869	1	1	yes	characteristic
98	C₃	#10624694	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×C₂₇⋊C₃

C₃

yes

maximal, characteristic

C₃×C₉×3₋¹⁺²

C₃

yes

maximal, characteristic

#10624683

C₃

yes

maximal

#10624467

C₃

yes

maximal, complemented

C₃²×3₋¹⁺²

C₉

yes

characteristic

C₃³×C₉

C₃²

yes

characteristic

C₃×C₂₇⋊C₃

—

C₃×C₂₇⋊C₃

—

C₃×C₂₇⋊C₃

C₃²

yes

—

C₃×C₂₇⋊C₃

C₃²

yes

—

C₃²×C₂₇

—

C₉×3₋¹⁺²

—

C₉×3₋¹⁺²

—

C₉×3₋¹⁺²

—

C₉×3₋¹⁺²

—

C₃×C₉⋊C₉

C₉

yes

—

C₃×C₃²⋊C₉

C₉

yes

—

C₃×C₉²

C₃²

yes

—

C₃²⋊C₂₇

—

C₃²⋊C₂₇

—

C₃⁴

C₃×C₉

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃×C₉

yes

—

C₃²×C₉

C₃×C₉

yes

—

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

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₉²

—

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

—

C₂₇

—

C₂₇

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

#6869

yes

—

C₃²

C₃×C₃²⋊C₉

yes

characteristic

C₃²

C₉×3₋¹⁺²

yes

—

C₉

—

C₉

—

C₉

—

C₉

—

C₉

#6891

yes

characteristic

C₉

#6893

yes

—

C₉

#6894

yes

—

C₃

—

C₃

—

C₃

—

C₃

yes

C₃

yes

C₃

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
4. #10624467	C₃	semidirect
38. C₂₇⋊C₃	3₋¹⁺²	semidirect

H ≅ G/N

type

4. #10624467

C₃

semidirect

38. 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
C₃×C₉×3₋¹⁺²	3
#10624683	3
#10624467	3

subgroup

index

C₃²×C₂₇⋊C₃

C₃×C₉×3₋¹⁺²

#10624683

#10624467

Subgroups — #14550539

Conjugacy classes of subgroups

Complements

Maximal subgroups