Group #14554479 — subgroups

Subgroups — #14554479

Order 2187.

740 conjugacy classes of subgroups; only the 260 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 260 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×#6895	C₃	1	2	yes	maximal, complemented
2	C₃×#6893	C₃	1	1	yes	maximal, characteristic
3	C₃×#6887	C₃	1	1	yes	maximal, characteristic, complemented
4	#10624810	C₃	1	27	yes	maximal, complemented
5	#10624691	C₃	1	3	yes	maximal, complemented
6	#10624690	C₃	1	3	yes	maximal, complemented
7	#10624689	C₃	1	3	yes	maximal
8	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
9	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
10	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
11	C₃³×C₉	C₃²	1	1	yes	characteristic
12	#6895	—	3	2	no	—
13	#6895	—	3	6	no	—
14	#6895	—	3	4	no	—
15	#6895	C₃²	1	18	yes	—
16	#6893	—	3	1	no	—
17	#6893	—	3	6	no	—
18	#6893	—	3	2	no	—
19	#6887	—	3	6	no	—
20	#6887	C₃²	1	9	yes	—
21	C₃×C₉⋊C₉	—	3	2	no	—
22	C₃×C₉⋊C₉	—	3	6	no	—
23	C₃×C₃²⋊C₉	C₃²	1	9	yes	—
24	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
25	C₃×C₃²⋊C₉	C₃²	1	6	yes	—
26	C₃×C₉²	—	3	1	no	—
27	#6868	C₃²	1	27	yes	—
28	#6867	C₃²	1	27	yes	—
29	#6866	C₃²	1	27	yes	—
30	C₃⁴	C₃³	1	3	yes	—
31	C₃⁴	C₃³	1	1	yes	characteristic
32	C₃×3₋¹⁺²	—	3	2	no	—
33	C₃×3₋¹⁺²	—	3	1	no	—
34	C₃×3₋¹⁺²	—	3	2	no	—
35	C₃×3₋¹⁺²	—	3	1	no	—
36	C₃×3₋¹⁺²	—	3	9	no	—
37	C₃×3₋¹⁺²	—	3	9	no	—
38	C₃×H(3,𝔽₃)	—	3	2	no	—
39	C₃×H(3,𝔽₃)	—	3	1	no	—
40	C₃×H(3,𝔽₃)	—	3	9	no	—
41	C₃²×C₉	—	3	9	no	—
42	C₃²×C₉	—	3	3	no	—
43	C₃²×C₉	—	3	6	no	—
44	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
45	C₃²×C₉	H(3,𝔽₃)	1	6	yes	—
46	C₃²×C₉	H(3,𝔽₃)	1	2	yes	—
47	C₃²×C₉	C₃³	1	3	yes	—
48	C₃²×C₉	C₃³	1	3	yes	—
49	C₃²×C₉	C₃³	1	3	yes	—
50	C₉⋊C₉	—	3	18	no	—
51	C₉⋊C₉	—	3	18	no	—
52	C₉⋊C₉	—	3	36	no	—
53	C₃²⋊C₉	—	3	6	no	—
54	C₃²⋊C₉	—	3	6	no	—
55	C₃²⋊C₉	—	3	6	no	—
56	C₃²⋊C₉	—	3	9	no	—
57	C₃²⋊C₉	—	3	18	no	—
58	C₃²⋊C₉	C₃³	1	9	yes	—
59	C₃²⋊C₉	C₃³	1	18	yes	—
60	C₉²	—	3	9	no	—
61	C₃³	—	3	3	no	—
62	C₃³	—	3	9	no	—
63	C₃³	—	9	3	no	—
64	C₃³	—	9	9	no	—
65	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
66	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
67	C₃³	#737	1	9	yes	—
68	C₃³	C₃⁴	1	1	yes	characteristic
69	3₋¹⁺²	—	9	3	no	—
70	3₋¹⁺²	—	9	6	no	—
71	3₋¹⁺²	—	9	3	no	—
72	3₋¹⁺²	—	9	6	no	—
73	H(3,𝔽₃)	—	9	3	no	—
74	H(3,𝔽₃)	—	9	6	no	—
75	C₃×C₉	—	3	9	no	—
76	C₃×C₉	—	3	3	no	—
77	C₃×C₉	—	3	18	no	—
78	C₃×C₉	—	3	6	no	—
79	C₃×C₉	—	3	9	no	—
80	C₃×C₉	—	3	3	no	—
81	C₃×C₉	—	3	3	no	—
82	C₃×C₉	—	3	18	no	—
83	C₃×C₉	—	3	9	no	—
84	C₃×C₉	—	3	9	no	—
85	C₃×C₉	—	9	9	no	—
86	C₃×C₉	—	9	18	no	—
87	C₃×C₉	C₃×H(3,𝔽₃)	1	3	yes	—
88	C₃×C₉	C₃×H(3,𝔽₃)	1	6	yes	—
89	C₃²	—	3	3	no	—
90	C₃²	—	3	6	no	—
91	C₃²	—	3	3	no	—
92	C₃²	—	9	9	no	—
93	C₃²	—	9	3	no	—
94	C₃²	—	9	9	no	—
95	C₃²	—	27	9	no	—
96	C₃²	#6893	1	3	yes	—
97	C₃²	#6893	1	6	yes	—
98	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
99	C₃²	C₃×#737	1	3	yes	—
100	C₉	—	3	3	no	—
101	C₉	—	3	6	no	—
102	C₉	—	9	3	no	—
103	C₉	—	9	3	no	—
104	C₉	—	9	9	no	—
105	C₉	—	9	18	no	—
106	C₃	—	3	3	no	—
107	C₃	—	3	3	no	—
108	C₃	—	3	6	no	—
109	C₃	—	27	3	no	—
110	C₃	C₃×#6893	1	1	yes	characteristic
111	C₃	#10624929	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#6895

C₃

yes

maximal, complemented

C₃×#6893

C₃

yes

maximal, characteristic

C₃×#6887

C₃

yes

maximal, characteristic, complemented

#10624810

C₃

yes

maximal, complemented

#10624691

C₃

yes

maximal, complemented

#10624690

C₃

yes

maximal, complemented

#10624689

C₃

yes

maximal

C₃²×3₋¹⁺²

C₃²

yes

characteristic

C₃²×3₋¹⁺²

C₃²

yes

characteristic

C₃²×H(3,𝔽₃)

C₃²

yes

characteristic

C₃³×C₉

C₃²

yes

—

—

—

C₃²

yes

—

—

—

—

—

C₃²

yes

—

C₃×C₉⋊C₉

—

C₃×C₉⋊C₉

—

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₉²

—

#6868

C₃²

yes

—

#6867

C₃²

yes

—

#6866

C₃²

yes

—

C₃⁴

C₃³

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×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

H(3,𝔽₃)

yes

characteristic

C₃²×C₉

H(3,𝔽₃)

yes

—

C₃²×C₉

H(3,𝔽₃)

yes

—

C₃²×C₉

C₃³

yes

—

C₃²×C₉

C₃³

yes

—

C₃²×C₉

C₃³

yes

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃³

yes

—

C₃²⋊C₉

C₃³

yes

—

C₉²

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃×H(3,𝔽₃)

yes

—

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

C₃³

#737

yes

—

C₃³

C₃⁴

yes

characteristic

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

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₃×H(3,𝔽₃)

yes

—

C₃×C₉

C₃×H(3,𝔽₃)

yes

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

#6893

yes

—

C₃²

#6893

yes

—

C₃²

C₃²×H(3,𝔽₃)

yes

characteristic

C₃²

C₃×#737

yes

—

100

C₉

—

101

C₉

—

102

C₉

—

103

C₉

—

104

C₉

—

105

C₉

—

106

C₃

—

107

C₃

—

108

C₃

—

109

C₃

—

110

C₃

C₃×#6893

yes

characteristic

111

C₃

#10624929

yes

—

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

subgroup

G/H

center, socle

C₃²

C₃²×H(3,𝔽₃)

commutator, Frattini, cosocle

C₃³

C₃⁴

prime radical

C₃²×H(3,𝔽₃)

C₃²

LCS

G ▷ C₃³ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃² ◁ C₃³×C₉ ◁ G

LECS

G ▷ C₃³ ▷ C₃ ▷ C₁

N	H ≅ G/N	type
1. C₃×#6895	C₃	semidirect
3. C₃×#6887	C₃	semidirect
4. #10624810	C₃	semidirect
5. #10624691	C₃	semidirect
6. #10624690	C₃	semidirect

H ≅ G/N

type

1. C₃×#6895

C₃

semidirect

3. C₃×#6887

C₃

semidirect

4. #10624810

C₃

semidirect

5. #10624691

C₃

semidirect

6. #10624690

C₃

semidirect

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

subgroup	index
C₃×#6895	3
C₃×#6893	3
C₃×#6887	3
#10624810	3
#10624691	3
#10624690	3
#10624689	3

subgroup

index

Subgroups — #14554479

Conjugacy classes of subgroups

Complements

Maximal subgroups