Group #14550737 — subgroups

Subgroups — #14550737

Order 2187.

258 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 77 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	#10624887	C₃	1	1	yes	maximal, characteristic
2	#10624709	C₃	1	3	yes	maximal, complemented
3	#10624704	C₃	1	3	yes	maximal
4	#10624472	C₃	1	3	yes	maximal, complemented
5	#10624471	C₃	1	1	yes	maximal, characteristic, complemented
6	#10624471	C₃	1	1	yes	maximal, characteristic, complemented
7	#10624471	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃×#737	C₃²	1	1	yes	characteristic
9	C₃×C₂₇⋊C₃	C₃²	1	3	yes	—
10	C₃×C₂₇⋊C₃	C₃²	1	3	yes	—
11	C₃×C₂₇⋊C₃	C₃²	1	3	yes	—
12	#6900	—	3	1	no	—
13	#6899	—	3	2	no	—
14	#6896	—	3	3	no	—
15	#6894	—	3	2	no	—
16	#6894	—	3	3	no	—
17	C₉⋊H(3,𝔽₃)	—	3	1	no	—
18	#6887	C₃²	1	1	yes	characteristic
19	#6887	C₃²	1	1	yes	characteristic
20	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
21	#6875	—	3	9	no	—
22	#6873	—	3	9	no	—
23	#737	—	3	3	no	—
24	#737	—	3	3	no	—
25	#737	—	3	3	no	—
26	C₃×3₋¹⁺²	—	3	3	no	—
27	C₃×3₋¹⁺²	—	3	1	no	—
28	C₃×3₋¹⁺²	—	3	1	no	—
29	C₃×3₋¹⁺²	3₋¹⁺²	1	2	yes	—
30	C₃×H(3,𝔽₃)	3₋¹⁺²	1	1	yes	characteristic
31	C₃²×C₉	—	3	1	no	—
32	C₃²×C₉	—	3	1	no	—
33	C₃²×C₉	C₃³	1	1	yes	characteristic
34	C₂₇⋊C₃	—	3	3	no	—
35	C₂₇⋊C₃	—	3	3	no	—
36	C₂₇⋊C₃	—	3	3	no	—
37	C₂₇⋊C₃	—	3	3	no	—
38	C₂₇⋊C₃	—	3	3	no	—
39	C₂₇⋊C₃	—	3	3	no	—
40	C₂₇⋊C₃	—	3	3	no	—
41	C₂₇⋊C₃	—	3	3	no	—
42	C₂₇⋊C₃	3₋¹⁺²	1	9	yes	complemented
43	C₃×C₂₇	—	3	3	no	—
44	C₃×C₂₇	—	3	3	no	—
45	C₃×C₂₇	—	3	3	no	—
46	C₉⋊C₉	—	3	12	no	—
47	C₉⋊C₉	—	3	6	no	—
48	C₉⋊C₉	—	3	1	no	—
49	C₉⋊C₉	—	3	1	no	—
50	C₉⋊C₉	—	3	1	no	—
51	C₉⋊C₉	—	3	1	no	—
52	C₉⋊C₉	—	3	1	no	—
53	C₉⋊C₉	H(3,𝔽₃)	1	3	yes	—
54	C₉⋊C₉	3₋¹⁺²	1	2	yes	—
55	C₉⋊C₉	3₋¹⁺²	1	1	yes	characteristic
56	C₃²⋊C₉	—	3	6	no	—
57	C₃²⋊C₉	—	3	1	no	—
58	C₃²⋊C₉	—	3	1	no	—
59	C₉²	—	3	1	no	—
60	C₉²	3₋¹⁺²	1	3	yes	—
61	C₃³	—	3	1	no	—
62	C₃³	C₃×3₋¹⁺²	1	1	yes	characteristic
63	3₋¹⁺²	—	3	6	no	—
64	3₋¹⁺²	—	9	3	no	—
65	3₋¹⁺²	—	9	1	no	—
66	3₋¹⁺²	—	9	1	no	—
67	3₋¹⁺²	—	9	1	no	—
68	3₋¹⁺²	—	27	3	no	—
69	H(3,𝔽₃)	—	3	3	no	—
70	C₃×C₉	—	3	3	no	—
71	C₃×C₉	—	3	3	no	—
72	C₃×C₉	—	3	2	no	—
73	C₃×C₉	—	3	2	no	—
74	C₃×C₉	—	3	1	no	—
75	C₃×C₉	—	9	3	no	—
76	C₃×C₉	—	27	1	no	—
77	C₃×C₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
78	C₃×C₉	C₃×3₋¹⁺²	1	3	yes	—
79	C₃×C₉	C₃×3₋¹⁺²	1	1	yes	characteristic
80	C₃×C₉	C₃×3₋¹⁺²	1	1	yes	characteristic
81	C₃×C₉	#737	1	3	yes	—
82	C₃×C₉	#737	1	3	yes	—
83	C₂₇	—	3	9	no	—
84	C₂₇	—	9	3	no	—
85	C₂₇	—	9	3	no	—
86	C₃²	—	9	1	no	—
87	C₃²	—	9	3	no	—
88	C₃²	—	9	1	no	—
89	C₃²	#6891	1	1	yes	characteristic
90	C₃²	#6894	1	3	yes	—
91	C₉	—	9	2	no	—
92	C₉	—	9	1	no	—
93	C₉	—	9	3	no	—
94	C₉	—	9	3	no	—
95	C₉	—	9	2	no	—
96	C₉	C₉⋊H(3,𝔽₃)	1	3	yes	—
97	C₉	#6896	1	3	yes	—
98	C₉	#6899	1	1	yes	characteristic
99	C₉	#6899	1	1	yes	characteristic
100	C₉	#6900	1	1	yes	characteristic
101	C₃	—	3	1	no	—
102	C₃	—	3	3	no	—
103	C₃	—	27	1	no	—
104	C₃	#10624565	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10624887

C₃

yes

maximal, characteristic

#10624709

C₃

yes

maximal, complemented

#10624704

C₃

yes

maximal

#10624472

C₃

yes

maximal, complemented

#10624471

C₃

yes

maximal, characteristic, complemented

#10624471

C₃

yes

maximal, characteristic, complemented

#10624471

C₃

yes

maximal, characteristic, complemented

C₃×#737

C₃²

yes

characteristic

C₃×C₂₇⋊C₃

C₃²

yes

—

C₃×C₂₇⋊C₃

C₃²

yes

—

C₃×C₂₇⋊C₃

C₃²

yes

—

—

—

—

—

—

C₉⋊H(3,𝔽₃)

—

#6887

C₃²

yes

characteristic

#6887

C₃²

yes

characteristic

C₃×C₉⋊C₉

C₃²

yes

—

—

—

—

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

3₋¹⁺²

yes

—

C₃×H(3,𝔽₃)

3₋¹⁺²

yes

characteristic

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃³

yes

characteristic

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

3₋¹⁺²

yes

complemented

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

H(3,𝔽₃)

yes

—

C₉⋊C₉

3₋¹⁺²

yes

—

C₉⋊C₉

3₋¹⁺²

yes

characteristic

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₉²

—

C₉²

3₋¹⁺²

yes

—

C₃³

—

C₃³

C₃×3₋¹⁺²

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

H(3,𝔽₃)

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

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

—

C₃×C₉

#737

yes

—

C₂₇

—

C₂₇

—

C₂₇

—

C₃²

—

C₃²

—

C₃²

—

C₃²

#6891

yes

characteristic

C₃²

#6894

yes

—

C₉

—

C₉

—

C₉

—

C₉

—

C₉

—

C₉

C₉⋊H(3,𝔽₃)

yes

—

C₉

#6896

yes

—

C₉

yes

C₉

yes

100

C₉

#6900

yes

characteristic

101

C₃

—

102

C₃

—

103

C₃

—

104

C₃

#10624565

yes

characteristic

	subgroup	G/H
center, socle	C₃	#10624565
commutator, Frattini, cosocle	C₃²×C₉	C₃³
prime radical	C₃×H(3,𝔽₃)	3₋¹⁺²
LCS	G ▷ C₃²×C₉ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃ ◁ C₃²×C₉ ◁ G
LECS	G ▷ C₃²×C₉ ▷ C₃ ▷ C₁

subgroup

G/H

center, socle

C₃

#10624565

commutator, Frattini, cosocle

C₃²×C₉

C₃³

prime radical

C₃×H(3,𝔽₃)

3₋¹⁺²

LCS

G ▷ C₃²×C₉ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃ ◁ C₃²×C₉ ◁ G

LECS

G ▷ C₃²×C₉ ▷ C₃ ▷ C₁

N	H ≅ G/N	type
2. #10624709	C₃	semidirect
4. #10624472	C₃	semidirect
5. #10624471	C₃	semidirect
6. #10624471	C₃	semidirect
7. #10624471	C₃	semidirect
42. C₂₇⋊C₃	3₋¹⁺²	semidirect

H ≅ G/N

type

2. #10624709

C₃

semidirect

4. #10624472

C₃

semidirect

5. #10624471

C₃

semidirect

6. #10624471

C₃

semidirect

7. #10624471

C₃

semidirect

42. 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
#10624887	3
#10624709	3
#10624704	3
#10624472	3
#10624471	3

subgroup

index

Subgroups — #14550737

Conjugacy classes of subgroups

Complements

Maximal subgroups