Group #14549772 — subgroups

Subgroups — #14549772

Order 2187.

377 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 377 conjugacy classes of subgroups — too many to draw. Only its 71 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×#6893	C₃	1	1	yes	maximal, characteristic
2	#10624556	C₃	1	3	yes	maximal, complemented
3	#10624452	C₃	1	6	yes	maximal, complemented
4	#10624449	C₃	1	1	yes	maximal, characteristic
5	#10624449	C₃	1	1	yes	maximal, characteristic
6	#10624449	C₃	1	1	yes	maximal, characteristic
7	C₃²×3₋¹⁺²	C₃²	1	2	yes	—
8	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
9	C₃³×C₉	C₃²	1	1	yes	characteristic
10	#6893	—	3	6	no	—
11	#6893	—	3	3	no	—
12	C₃×C₉⋊C₉	—	3	3	no	—
13	C₃×C₃²⋊C₉	—	3	3	no	—
14	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
15	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
16	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
17	#6855	—	3	6	no	—
18	C₃⁴	—	3	1	no	—
19	C₃⁴	C₃³	1	1	yes	characteristic
20	C₃×3₋¹⁺²	—	3	2	no	—
21	C₃×3₋¹⁺²	—	3	2	no	—
22	C₃×3₋¹⁺²	—	9	6	no	—
23	C₃×3₋¹⁺²	3₋¹⁺²	1	6	yes	—
24	C₃×H(3,𝔽₃)	—	3	2	no	—
25	C₃×H(3,𝔽₃)	—	9	3	no	—
26	C₃×H(3,𝔽₃)	3₋¹⁺²	1	3	yes	—
27	C₃²×C₉	—	3	2	no	—
28	C₃²×C₉	—	3	3	no	—
29	C₃²×C₉	—	3	2	no	—
30	C₃²×C₉	—	3	1	no	—
31	C₃²×C₉	—	3	3	no	—
32	C₃²×C₉	—	3	3	no	—
33	C₃²×C₉	—	9	3	no	—
34	C₃²×C₉	H(3,𝔽₃)	1	3	yes	—
35	C₉⋊C₉	—	9	9	no	—
36	C₃²⋊C₉	—	3	27	no	—
37	C₃²⋊C₉	—	3	9	no	—
38	C₃²⋊C₉	—	9	3	no	—
39	C₃²⋊C₉	—	9	3	no	—
40	C₃³	—	3	3	no	—
41	C₃³	—	3	3	no	—
42	C₃³	—	3	3	no	—
43	C₃³	—	3	3	no	—
44	C₃³	—	3	3	no	—
45	C₃³	—	9	3	no	—
46	C₃³	—	9	3	no	—
47	C₃³	—	9	3	no	—
48	C₃³	—	9	3	no	—
49	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
50	C₃³	C₃×3₋¹⁺²	1	2	yes	—
51	C₃³	C₃×3₋¹⁺²	1	1	yes	characteristic
52	3₋¹⁺²	—	9	6	no	—
53	3₋¹⁺²	—	9	6	no	—
54	3₋¹⁺²	—	9	6	no	—
55	H(3,𝔽₃)	—	9	3	no	—
56	H(3,𝔽₃)	—	9	6	no	—
57	C₃×C₉	—	3	3	no	—
58	C₃×C₉	—	3	6	no	—
59	C₃×C₉	—	3	9	no	—
60	C₃×C₉	—	3	3	no	—
61	C₃×C₉	—	3	6	no	—
62	C₃×C₉	—	9	3	no	—
63	C₃×C₉	—	9	6	no	—
64	C₃×C₉	—	9	6	no	—
65	C₃×C₉	—	9	3	no	—
66	C₃×C₉	—	9	6	no	—
67	C₃×C₉	—	9	3	no	—
68	C₃×C₉	—	9	3	no	—
69	C₃×C₉	—	9	6	no	—
70	C₃×C₉	—	9	6	no	—
71	C₃×C₉	—	9	3	no	—
72	C₃×C₉	—	9	3	no	—
73	C₃²	—	3	1	no	—
74	C₃²	—	3	2	no	—
75	C₃²	—	3	6	no	—
76	C₃²	—	3	3	no	—
77	C₃²	—	9	3	no	—
78	C₃²	—	9	3	no	—
79	C₃²	—	9	3	no	—
80	C₃²	—	9	3	no	—
81	C₃²	—	9	9	no	—
82	C₃²	—	27	3	no	—
83	C₃²	—	27	3	no	—
84	C₃²	—	27	3	no	—
85	C₃²	#6891	1	1	yes	characteristic
86	C₃²	#6908	1	3	yes	—
87	C₃²	#6909	1	3	yes	—
88	C₃²	#6913	1	3	yes	—
89	C₃²	#6913	1	3	yes	—
90	C₉	—	9	3	no	—
91	C₉	—	9	6	no	—
92	C₉	—	9	9	no	—
93	C₉	—	27	3	no	—
94	C₉	—	27	3	no	—
95	C₃	—	9	1	no	—
96	C₃	—	9	2	no	—
97	C₃	—	27	3	no	—
98	C₃	#10624627	1	3	yes	—
99	C₃	#10624650	1	3	yes	—
100	C₃	#10624677	1	3	yes	—
101	C₃	#10624769	1	1	yes	characteristic
102	C₃	#10624780	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#6893

C₃

yes

maximal, characteristic

#10624556

C₃

yes

maximal, complemented

#10624452

C₃

yes

maximal, complemented

#10624449

C₃

yes

maximal, characteristic

#10624449

C₃

yes

maximal, characteristic

#10624449

C₃

yes

maximal, characteristic

C₃²×3₋¹⁺²

C₃²

yes

—

C₃²×H(3,𝔽₃)

C₃²

yes

characteristic

C₃³×C₉

C₃²

yes

characteristic

#6893

—

#6893

—

C₃×C₉⋊C₉

—

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

—

#6855

—

C₃⁴

—

C₃⁴

C₃³

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

3₋¹⁺²

yes

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

3₋¹⁺²

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

H(3,𝔽₃)

yes

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

C₃³

C₃×3₋¹⁺²

yes

—

C₃³

C₃×3₋¹⁺²

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

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

—

C₃×C₉

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

#6891

yes

characteristic

C₃²

#6908

yes

—

C₃²

#6909

yes

—

C₃²

#6913

yes

—

C₃²

#6913

yes

—

C₉

—

C₉

—

C₉

—

C₉

—

C₉

—

C₃

—

C₃

—

C₃

—

C₃

#10624627

yes

—

C₃

#10624650

yes

—

100

C₃

#10624677

yes

—

101

C₃

#10624769

yes

characteristic

102

C₃

#10624780

yes

—

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

subgroup

G/H

center, socle

C₃³

C₃×H(3,𝔽₃)

commutator, Frattini, cosocle

C₃⁴

C₃³

prime radical

C₃²×H(3,𝔽₃)

C₃²

LCS

G ▷ C₃⁴ ▷ C₃² ▷ C₁

UCS

C₁ ◁ C₃³ ◁ C₃³×C₉ ◁ G

LECS

G ▷ C₃⁴ ▷ C₃² ▷ C₁

N	H ≅ G/N	type
2. #10624556	C₃	semidirect
3. #10624452	C₃	semidirect

H ≅ G/N

type

2. #10624556

C₃

semidirect

3. #10624452

C₃

semidirect

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

subgroup	index
C₃×#6893	3
#10624556	3
#10624452	3
#10624449	3

subgroup

index

Subgroups — #14549772

Conjugacy classes of subgroups

Complements

Maximal subgroups