Group #14550994 — subgroups

Subgroups — #14550994

Order 2187.

313 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 313 conjugacy classes of subgroups — too many to draw. Only its 47 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	#10624734	C₃	1	2	yes	maximal, complemented
2	#10624688	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×#6855	C₃	1	1	yes	maximal, characteristic, complemented
4	#10624483	C₃	1	2	yes	maximal, complemented
5	#10624482	C₃	1	2	yes	maximal, complemented
6	#10624481	C₃	1	4	yes	maximal, complemented
7	#10624480	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃³×C₉	—	3	1	no	—
9	C₃³×C₉	C₃²	1	1	yes	characteristic
10	C₃×#733	C₃²	1	4	yes	—
11	C₃×#732	C₃²	1	2	yes	—
12	C₃×#731	—	3	2	no	—
13	C₃×AΣL(1,𝔽₂₇)	—	3	2	no	—
14	#6893	—	3	2	no	—
15	#6887	—	3	1	no	—
16	#6868	C₃²	1	1	yes	characteristic
17	#6868	C₃²	1	2	yes	—
18	#6867	—	3	1	no	—
19	#6866	—	3	1	no	—
20	#6855	—	3	1	no	—
21	#6855	—	3	1	no	—
22	#6855	C₃²	1	1	yes	characteristic
23	#6855	C₃²	1	2	yes	—
24	C₃⁴	H(3,𝔽₃)	1	1	yes	characteristic
25	C₃×3₋¹⁺²	—	3	4	no	—
26	C₃×H(3,𝔽₃)	—	3	2	no	—
27	C₃²×C₉	—	3	9	no	—
28	C₃²×C₉	—	3	3	no	—
29	C₃²×C₉	—	3	1	no	—
30	C₃²×C₉	—	3	2	no	—
31	C₃²×C₉	—	3	1	no	—
32	C₃²×C₉	—	3	2	no	—
33	C₃²×C₉	—	9	2	no	—
34	C₃²×C₉	—	9	6	no	—
35	C₃²×C₉	—	9	3	no	—
36	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
37	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
38	C₃²×C₉	C₃³	1	1	yes	characteristic
39	#733	—	9	4	no	—
40	#732	—	9	2	no	—
41	#731	—	9	6	no	—
42	AΣL(1,𝔽₂₇)	—	9	6	no	—
43	C₉⋊C₉	—	9	2	no	—
44	C₃²⋊C₉	—	3	1	no	—
45	C₃²⋊C₉	—	3	2	no	—
46	C₃²⋊C₉	—	9	2	no	—
47	C₃²⋊C₉	—	9	4	no	—
48	C₉²	—	9	1	no	—
49	C₃³	—	3	3	no	—
50	C₃³	—	3	6	no	—
51	C₃³	—	3	1	no	—
52	C₃³	—	3	2	no	—
53	C₃³	—	9	2	no	—
54	C₃³	#732	1	3	yes	—
55	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
56	3₋¹⁺²	—	9	4	no	—
57	3₋¹⁺²	—	9	4	no	—
58	3₋¹⁺²	—	9	4	no	—
59	H(3,𝔽₃)	—	9	2	no	—
60	H(3,𝔽₃)	—	9	2	no	—
61	H(3,𝔽₃)	—	9	2	no	—
62	C₃×C₉	—	3	1	no	—
63	C₃×C₉	—	3	2	no	—
64	C₃×C₉	—	3	3	no	—
65	C₃×C₉	—	3	6	no	—
66	C₃×C₉	—	3	1	no	—
67	C₃×C₉	—	3	1	no	—
68	C₃×C₉	—	3	1	no	—
69	C₃×C₉	—	3	1	no	—
70	C₃×C₉	—	3	2	no	—
71	C₃×C₉	—	3	2	no	—
72	C₃×C₉	—	9	3	no	—
73	C₃×C₉	—	9	6	no	—
74	C₃×C₉	—	9	27	no	—
75	C₃×C₉	—	9	2	no	—
76	C₃×C₉	—	9	4	no	—
77	C₃×C₉	—	9	1	no	—
78	C₃×C₉	—	9	2	no	—
79	C₃×C₉	—	9	1	no	—
80	C₃×C₉	—	9	2	no	—
81	C₃×C₉	—	9	4	no	—
82	C₃×C₉	#732	1	1	yes	characteristic
83	C₃×C₉	#732	1	2	yes	—
84	C₃²	—	3	9	no	—
85	C₃²	—	3	18	no	—
86	C₃²	—	3	1	no	—
87	C₃²	—	3	3	no	—
88	C₃²	—	3	2	no	—
89	C₃²	—	3	6	no	—
90	C₃²	—	3	3	no	—
91	C₃²	—	27	2	no	—
92	C₃²	—	27	4	no	—
93	C₃²	—	27	2	no	—
94	C₃²	C₃×#732	1	1	yes	characteristic
95	C₃²	C₃×#732	1	1	yes	characteristic
96	C₃²	#6909	1	2	yes	—
97	C₉	—	3	1	no	—
98	C₉	—	3	2	no	—
99	C₉	—	9	3	no	—
100	C₉	—	9	6	no	—
101	C₉	—	9	1	no	—
102	C₉	—	9	1	no	—
103	C₉	—	27	1	no	—
104	C₉	—	27	2	no	—
105	C₉	—	27	4	no	—
106	C₃	—	3	1	no	—
107	C₃	—	3	9	no	—
108	C₃	—	3	2	no	—
109	C₃	—	81	2	no	—
110	C₃	#10624599	1	1	yes	characteristic
111	C₃	#10624837	1	1	yes	characteristic
112	C₃	#10624850	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10624734

C₃

yes

maximal, complemented

#10624688

C₃

yes

maximal, characteristic, complemented

C₃×#6855

C₃

yes

maximal, characteristic, complemented

#10624483

C₃

yes

maximal, complemented

#10624482

C₃

yes

maximal, complemented

#10624481

C₃

yes

maximal, complemented

#10624480

C₃

yes

maximal, characteristic, complemented

C₃³×C₉

—

C₃³×C₉

C₃²

yes

characteristic

C₃×#733

C₃²

yes

—

C₃×#732

C₃²

yes

—

C₃×#731

—

C₃×AΣL(1,𝔽₂₇)

—

#6893

—

#6887

—

#6868

C₃²

yes

characteristic

#6868

C₃²

yes

—

—

—

—

—

C₃²

yes

characteristic

#6855

C₃²

yes

—

C₃⁴

H(3,𝔽₃)

yes

characteristic

C₃×3₋¹⁺²

—

C₃×H(3,𝔽₃)

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

yes

C₃²×C₉

yes

C₃²×C₉

C₃³

yes

—

—

—

AΣL(1,𝔽₂₇)

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₉²

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

#732

yes

—

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

H(3,𝔽₃)

—

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

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

#732

yes

characteristic

C₃×C₉

#732

yes

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

yes

C₃²

yes

C₃²

yes

—

C₉

—

C₉

—

C₉

—

100

C₉

—

101

C₉

—

102

C₉

—

103

C₉

—

104

C₉

—

105

C₉

—

106

C₃

—

107

C₃

—

108

C₃

—

109

C₃

—

110

C₃

#10624599

yes

characteristic

111

C₃

#10624837

yes

characteristic

112

C₃

#10624850

yes

—

	subgroup	G/H
center, socle	C₃²	C₃×#732
commutator, Frattini, cosocle	C₃²×C₉	C₃³
LCS	G ▷ C₃²×C₉ ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃²×C₉ ◁ C₃³×C₉ ◁ G
LECS	G ▷ C₃²×C₉ ▷ C₃³ ▷ C₃ ▷ C₁

subgroup

G/H

center, socle

C₃²

C₃×#732

commutator, Frattini, cosocle

C₃²×C₉

C₃³

LCS

G ▷ C₃²×C₉ ▷ C₃³ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃² ◁ C₃²×C₉ ◁ C₃³×C₉ ◁ G

LECS

G ▷ C₃²×C₉ ▷ C₃³ ▷ C₃ ▷ C₁

N	H ≅ G/N	type
1. #10624734	C₃	semidirect
2. #10624688	C₃	semidirect
3. C₃×#6855	C₃	semidirect
4. #10624483	C₃	semidirect
5. #10624482	C₃	semidirect
6. #10624481	C₃	semidirect
7. #10624480	C₃	semidirect

H ≅ G/N

type

1. #10624734

C₃

semidirect

2. #10624688

C₃

semidirect

3. C₃×#6855

C₃

semidirect

4. #10624483

C₃

semidirect

5. #10624482

C₃

semidirect

6. #10624481

C₃

semidirect

7. #10624480

C₃

semidirect

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

subgroup	index
#10624734	3
#10624688	3
C₃×#6855	3
#10624483	3
#10624482	3
#10624481	3
#10624480	3

subgroup

index

Subgroups — #14550994

Conjugacy classes of subgroups

Complements

Maximal subgroups