Group #14548117 — subgroups

Subgroups — #14548117

Order 2187.

615 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 615 conjugacy classes of subgroups — too many to draw. Only its 53 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×#6891	C₃	1	1	yes	maximal, characteristic, complemented
2	#10624765	C₃	1	1	yes	maximal, characteristic, complemented
3	#10624765	C₃	1	1	yes	maximal, characteristic, complemented
4	#10624730	C₃	1	1	yes	maximal, characteristic
5	#10624633	C₃	1	3	yes	maximal, complemented
6	#10624631	C₃	1	3	yes	maximal, complemented
7	#10624629	C₃	1	3	yes	maximal, complemented
8	C₃⁵	C₃²	1	1	yes	characteristic
9	C₃²×3₋¹⁺²	C₃²	1	3	yes	—
10	C₃×AΣL(1,𝔽₂₇)	—	3	3	no	—
11	#6891	—	3	9	no	—
12	#6891	C₃²	1	3	yes	—
13	#6891	C₃²	1	3	yes	—
14	#6891	C₃²	1	1	yes	characteristic
15	#6891	C₃²	1	1	yes	characteristic
16	C₃²⋊H(3,𝔽₃)	C₃²	1	1	yes	characteristic
17	C₃×C₃²⋊C₉	—	3	3	no	—
18	#6866	—	3	3	no	—
19	#6866	—	3	3	no	—
20	#6860	—	3	9	no	—
21	#6858	—	3	9	no	—
22	#6857	—	3	9	no	—
23	C₃⁴	—	3	3	no	—
24	C₃⁴	—	9	3	no	—
25	C₃⁴	—	9	9	no	—
26	C₃⁴	H(3,𝔽₃)	1	3	yes	—
27	C₃⁴	C₃³	1	1	yes	characteristic
28	C₃×3₋¹⁺²	—	3	6	no	—
29	C₃×3₋¹⁺²	—	3	1	no	—
30	C₃×3₋¹⁺²	—	3	1	no	—
31	C₃×3₋¹⁺²	—	3	3	no	—
32	C₃×3₋¹⁺²	—	3	3	no	—
33	C₃×3₋¹⁺²	—	9	9	no	—
34	C₃×3₋¹⁺²	H(3,𝔽₃)	1	9	yes	—
35	C₃×H(3,𝔽₃)	—	3	1	no	—
36	C₃×H(3,𝔽₃)	—	3	3	no	—
37	C₃²×C₉	—	3	3	no	—
38	AΣL(1,𝔽₂₇)	—	9	9	no	—
39	C₃²⋊C₉	—	3	27	no	—
40	C₃²⋊C₉	—	3	3	no	—
41	C₃²⋊C₉	—	3	3	no	—
42	C₃²⋊C₉	—	3	3	no	—
43	C₃²⋊C₉	—	3	3	no	—
44	C₃²⋊C₉	—	3	3	no	—
45	C₃²⋊C₉	—	3	3	no	—
46	C₃²⋊C₉	—	3	3	no	—
47	C₃²⋊C₉	—	3	3	no	—
48	C₃³	—	3	9	no	—
49	C₃³	—	3	3	no	—
50	C₃³	—	3	3	no	—
51	C₃³	—	3	3	no	—
52	C₃³	—	3	6	no	—
53	C₃³	—	9	3	no	—
54	C₃³	—	9	9	no	—
55	C₃³	—	9	9	no	—
56	C₃³	—	9	9	no	—
57	C₃³	—	9	6	no	—
58	C₃³	—	9	9	no	—
59	C₃³	—	9	81	no	—
60	C₃³	—	9	1	no	—
61	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
62	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
63	3₋¹⁺²	—	9	18	no	—
64	3₋¹⁺²	—	9	9	no	—
65	3₋¹⁺²	—	9	3	no	—
66	3₋¹⁺²	—	9	3	no	—
67	3₋¹⁺²	—	27	3	no	—
68	3₋¹⁺²	—	27	3	no	—
69	H(3,𝔽₃)	—	9	3	no	—
70	H(3,𝔽₃)	—	27	3	no	—
71	C₃×C₉	—	3	9	no	—
72	C₃×C₉	—	9	9	no	—
73	C₃×C₉	—	9	1	no	—
74	C₃×C₉	—	9	1	no	—
75	C₃×C₉	—	9	3	no	—
76	C₃×C₉	—	9	3	no	—
77	C₃²	—	3	3	no	—
78	C₃²	—	3	3	no	—
79	C₃²	—	3	3	no	—
80	C₃²	—	3	3	no	—
81	C₃²	—	3	3	no	—
82	C₃²	—	3	9	no	—
83	C₃²	—	9	9	no	—
84	C₃²	—	9	18	no	—
85	C₃²	—	9	9	no	—
86	C₃²	—	9	81	no	—
87	C₃²	—	9	3	no	—
88	C₃²	—	9	6	no	—
89	C₃²	—	27	1	no	—
90	C₃²	—	27	3	no	—
91	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
92	C₃²	#6909	1	3	yes	—
93	C₉	—	9	9	no	—
94	C₉	—	27	1	no	—
95	C₉	—	27	1	no	—
96	C₉	—	27	3	no	—
97	C₉	—	27	3	no	—
98	C₃	—	3	3	no	—
99	C₃	—	9	3	no	—
100	C₃	—	9	9	no	—
101	C₃	—	81	1	no	—
102	C₃	#10624735	1	1	yes	characteristic
103	C₃	#10624755	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#6891

C₃

yes

maximal, characteristic, complemented

#10624765

C₃

yes

maximal, characteristic, complemented

#10624765

C₃

yes

maximal, characteristic, complemented

#10624730

C₃

yes

maximal, characteristic

#10624633

C₃

yes

maximal, complemented

#10624631

C₃

yes

maximal, complemented

#10624629

C₃

yes

maximal, complemented

C₃⁵

C₃²

yes

characteristic

C₃²×3₋¹⁺²

C₃²

yes

—

C₃×AΣL(1,𝔽₂₇)

—

#6891

—

#6891

C₃²

yes

—

#6891

C₃²

yes

—

#6891

C₃²

yes

characteristic

#6891

C₃²

yes

characteristic

C₃²⋊H(3,𝔽₃)

C₃²

yes

characteristic

C₃×C₃²⋊C₉

—

—

—

—

—

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

H(3,𝔽₃)

yes

—

C₃⁴

C₃³

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

H(3,𝔽₃)

yes

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃²×C₉

—

AΣL(1,𝔽₂₇)

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃×H(3,𝔽₃)

yes

—

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

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

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₃²⋊H(3,𝔽₃)

yes

characteristic

C₃²

#6909

yes

—

C₉

—

C₉

—

C₉

—

C₉

—

C₉

—

C₃

—

C₃

—

100

C₃

—

101

C₃

—

102

C₃

#10624735

yes

characteristic

103

C₃

#10624755

yes

—

	subgroup	G/H
center, socle	C₃²	C₃²⋊H(3,𝔽₃)
commutator, Frattini, cosocle	C₃⁴	C₃³
prime radical	#10624730	C₃
LCS	G ▷ C₃⁴ ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ G
LECS	G ▷ C₃⁴ ▷ C₃² ▷ C₁

subgroup

G/H

center, socle

C₃²

C₃²⋊H(3,𝔽₃)

commutator, Frattini, cosocle

C₃⁴

C₃³

C₃

G ▷ C₃⁴ ▷ C₃² ▷ C₁

C₁ ◁ C₃² ◁ C₃⁴ ◁ G

G ▷ C₃⁴ ▷ C₃² ▷ C₁

N	H ≅ G/N	type
1. C₃×#6891	C₃	semidirect
2. #10624765	C₃	semidirect
3. #10624765	C₃	semidirect
5. #10624633	C₃	semidirect
6. #10624631	C₃	semidirect
7. #10624629	C₃	semidirect

H ≅ G/N

type

1. C₃×#6891

C₃

semidirect

2. #10624765

C₃

semidirect

3. #10624765

C₃

semidirect

5. #10624633

C₃

semidirect

6. #10624631

C₃

semidirect

7. #10624629

C₃

semidirect

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

subgroup	index
C₃×#6891	3
#10624765	3
#10624730	3
#10624633	3
#10624631	3
#10624629	3

subgroup

index

Subgroups — #14548117

Conjugacy classes of subgroups

Complements

Maximal subgroups