Group #14547611 — subgroups

Subgroups — #14547611

Order 2187.

1232 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 1232 conjugacy classes of subgroups — too many to draw. Only its 62 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	C₃×3₋¹⁺⁴	C₃	1	1	yes	maximal, characteristic, complemented
2	#10624771	C₃	1	1	yes	maximal, characteristic, complemented
3	#10624743	C₃	1	3	yes	maximal, complemented
4	#10624737	C₃	1	1	yes	maximal, characteristic, complemented
5	#10624608	C₃	1	6	yes	maximal, complemented
6	#10624571	C₃	1	1	yes	maximal, characteristic, complemented
7	3₋¹⁺⁴	—	3	27	no	—
8	C₃×#737	—	3	6	no	—
9	C₃×#737	—	3	3	no	—
10	C₃²×3₋¹⁺²	—	3	3	no	—
11	C₃²×3₋¹⁺²	C₃²	1	2	yes	—
12	C₃²×3₋¹⁺²	C₃²	1	3	yes	—
13	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
14	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
15	C₃×#732	—	3	3	no	—
16	C₃×AΣL(1,𝔽₂₇)	—	3	9	no	—
17	#6891	C₃²	1	3	yes	—
18	C₃²⋊H(3,𝔽₃)	C₃²	1	3	yes	—
19	#6867	—	3	3	no	—
20	#6857	—	3	18	no	—
21	#6855	—	3	3	no	—
22	C₃⁴	H(3,𝔽₃)	1	3	yes	—
23	C₃⁴	C₃³	1	1	yes	characteristic
24	#737	—	3	162	no	—
25	#737	—	3	54	no	—
26	#737	—	3	27	no	—
27	C₃×3₋¹⁺²	—	3	18	no	—
28	C₃×3₋¹⁺²	—	3	81	no	—
29	C₃×3₋¹⁺²	—	3	2	no	—
30	C₃×3₋¹⁺²	—	3	1	no	—
31	C₃×3₋¹⁺²	—	3	2	no	—
32	C₃×3₋¹⁺²	—	3	2	no	—
33	C₃×3₋¹⁺²	—	3	18	no	—
34	C₃×3₋¹⁺²	—	3	9	no	—
35	C₃×3₋¹⁺²	—	3	3	no	—
36	C₃×3₋¹⁺²	—	3	6	no	—
37	C₃×3₋¹⁺²	—	3	3	no	—
38	C₃×3₋¹⁺²	—	9	9	no	—
39	C₃×3₋¹⁺²	H(3,𝔽₃)	1	6	yes	—
40	C₃×H(3,𝔽₃)	—	3	2	no	—
41	C₃×H(3,𝔽₃)	—	3	1	no	—
42	C₃×H(3,𝔽₃)	—	3	9	no	—
43	C₃×H(3,𝔽₃)	—	3	3	no	—
44	C₃×H(3,𝔽₃)	—	3	3	no	—
45	C₃×H(3,𝔽₃)	—	3	6	no	—
46	C₃²×C₉	—	3	9	no	—
47	C₃²×C₉	—	3	2	no	—
48	C₃²×C₉	—	3	3	no	—
49	C₃²×C₉	H(3,𝔽₃)	1	3	yes	—
50	#732	—	9	9	no	—
51	AΣL(1,𝔽₂₇)	—	9	27	no	—
52	C₃²⋊C₉	—	3	6	no	—
53	C₃²⋊C₉	—	3	3	no	—
54	C₃³	—	3	3	no	—
55	C₃³	—	3	9	no	—
56	C₃³	—	3	3	no	—
57	C₃³	—	9	3	no	—
58	C₃³	—	9	9	no	—
59	C₃³	—	9	3	no	—
60	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
61	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
62	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
63	3₋¹⁺²	—	3	18	no	—
64	3₋¹⁺²	—	3	9	no	—
65	3₋¹⁺²	—	3	162	no	—
66	3₋¹⁺²	—	3	18	no	—
67	3₋¹⁺²	—	3	18	no	—
68	3₋¹⁺²	—	3	18	no	—
69	3₋¹⁺²	—	3	9	no	—
70	3₋¹⁺²	—	9	6	no	—
71	3₋¹⁺²	—	9	18	no	—
72	3₋¹⁺²	—	9	9	no	—
73	H(3,𝔽₃)	—	3	18	no	—
74	H(3,𝔽₃)	—	3	9	no	—
75	H(3,𝔽₃)	—	9	9	no	—
76	H(3,𝔽₃)	—	9	9	no	—
77	H(3,𝔽₃)	—	27	6	no	—
78	C₃×C₉	—	3	18	no	—
79	C₃×C₉	—	3	81	no	—
80	C₃×C₉	—	3	6	no	—
81	C₃×C₉	—	3	3	no	—
82	C₃×C₉	—	3	6	no	—
83	C₃×C₉	—	3	9	no	—
84	C₃×C₉	—	9	6	no	—
85	C₃×C₉	—	9	3	no	—
86	C₃×C₉	—	9	3	no	—
87	C₃×C₉	AΣL(1,𝔽₂₇)	1	9	yes	complemented
88	C₃²	—	3	3	no	—
89	C₃²	—	3	9	no	—
90	C₃²	—	3	2	no	—
91	C₃²	—	3	1	no	—
92	C₃²	—	3	3	no	—
93	C₃²	—	3	6	no	—
94	C₃²	—	9	9	no	—
95	C₃²	—	27	3	no	—
96	C₃²	—	27	9	no	—
97	C₃²	—	27	3	no	—
98	C₃²	—	27	3	no	—
99	C₃²	—	27	6	no	—
100	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
101	C₃²	C₃×AΣL(1,𝔽₂₇)	1	3	yes	—
102	C₉	—	3	18	no	—
103	C₉	—	3	9	no	—
104	C₉	—	9	9	no	—
105	C₉	—	27	3	no	—
106	C₃	—	3	3	no	—
107	C₃	—	9	2	no	—
108	C₃	—	9	1	no	—
109	C₃	—	9	9	no	—
110	C₃	—	81	3	no	—
111	C₃	#10624730	1	1	yes	characteristic
112	C₃	#10624749	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×3₋¹⁺⁴

C₃

yes

maximal, characteristic, complemented

#10624771

C₃

yes

maximal, characteristic, complemented

#10624743

C₃

yes

maximal, complemented

#10624737

C₃

yes

maximal, characteristic, complemented

#10624608

C₃

yes

maximal, complemented

#10624571

C₃

yes

maximal, characteristic, complemented

3₋¹⁺⁴

—

C₃×#737

—

C₃×#737

—

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

C₃²

yes

—

C₃²×3₋¹⁺²

C₃²

yes

—

C₃²×3₋¹⁺²

C₃²

yes

characteristic

C₃²×H(3,𝔽₃)

C₃²

yes

characteristic

C₃×#732

—

C₃×AΣL(1,𝔽₂₇)

—

#6891

C₃²

yes

—

C₃²⋊H(3,𝔽₃)

C₃²

yes

—

#6867

—

#6857

—

#6855

—

C₃⁴

H(3,𝔽₃)

yes

—

C₃⁴

C₃³

yes

characteristic

#737

—

162

—

#737

—

#737

—

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

—

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)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

H(3,𝔽₃)

yes

—

#732

—

AΣL(1,𝔽₂₇)

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃×H(3,𝔽₃)

yes

—

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

162

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

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

AΣL(1,𝔽₂₇)

yes

complemented

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

100

C₃²

C₃²⋊H(3,𝔽₃)

yes

characteristic

101

C₃²

C₃×AΣL(1,𝔽₂₇)

yes

—

102

C₉

—

103

C₉

—

104

C₉

—

105

C₉

—

106

C₃

—

107

C₃

—

108

C₃

—

109

C₃

—

110

C₃

—

111

C₃

#10624730

yes

characteristic

112

C₃

#10624749

yes

—

	subgroup	G/H
center, socle	C₃²	C₃²⋊H(3,𝔽₃)
commutator, Frattini, cosocle	C₃⁴	C₃³
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

LCS

G ▷ C₃⁴ ▷ C₃² ▷ C₁

UCS

C₁ ◁ C₃² ◁ C₃⁴ ◁ G

LECS

G ▷ C₃⁴ ▷ C₃² ▷ C₁

N	H ≅ G/N	type
1. C₃×3₋¹⁺⁴	C₃	semidirect
2. #10624771	C₃	semidirect
3. #10624743	C₃	semidirect
4. #10624737	C₃	semidirect
5. #10624608	C₃	semidirect
6. #10624571	C₃	semidirect
87. C₃×C₉	AΣL(1,𝔽₂₇)	semidirect

H ≅ G/N

type

1. C₃×3₋¹⁺⁴

C₃

semidirect

2. #10624771

C₃

semidirect

3. #10624743

C₃

semidirect

4. #10624737

C₃

semidirect

5. #10624608

C₃

semidirect

6. #10624571

C₃

semidirect

87. C₃×C₉

AΣL(1,𝔽₂₇)

semidirect

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

subgroup	index
C₃×3₋¹⁺⁴	3
#10624771	3
#10624743	3
#10624737	3
#10624608	3
#10624571	3

subgroup

index

C₃×3₋¹⁺⁴

Subgroups — #14547611

Conjugacy classes of subgroups

Complements

Maximal subgroups