Group #14547296 — subgroups

Subgroups — #14547296

Order 2187.

417 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 417 conjugacy classes of subgroups — too many to draw. Only its 56 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×#6912	C₃	1	1	yes	maximal, characteristic, complemented
2	#10624885	C₃	1	1	yes	maximal, characteristic, complemented
3	#10624823	C₃	1	3	yes	maximal, complemented
4	#10624736	C₃	1	1	yes	maximal, characteristic
5	#10624612	C₃	1	3	yes	maximal, complemented
6	#10624609	C₃	1	3	yes	maximal
7	#10624570	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃²×3₋¹⁺²	C₃²	1	3	yes	—
9	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
10	#6912	—	3	9	no	—
11	C₃×#733	—	3	2	no	—
12	C₃×#733	—	3	1	no	—
13	C₃×#731	—	3	2	no	—
14	C₃×#731	—	3	1	no	—
15	#6896	—	3	9	no	—
16	#6895	—	3	2	no	—
17	#6891	C₃²	1	1	yes	characteristic
18	#6891	C₃²	1	1	yes	characteristic
19	#6891	C₃²	1	3	yes	—
20	C₃²⋊H(3,𝔽₃)	C₃²	1	1	yes	characteristic
21	#6887	—	3	1	no	—
22	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
23	3₋¹⁺²⋊C₉	—	3	9	no	—
24	#6860	—	3	9	no	—
25	#6857	—	3	9	no	—
26	#6855	—	3	2	no	—
27	#6855	—	3	1	no	—
28	C₃⁴	C₃³	1	1	yes	characteristic
29	C₃×3₋¹⁺²	—	3	3	no	—
30	C₃×3₋¹⁺²	—	3	3	no	—
31	C₃×3₋¹⁺²	—	3	3	no	—
32	C₃×3₋¹⁺²	—	3	9	no	—
33	C₃×3₋¹⁺²	—	3	3	no	—
34	C₃×3₋¹⁺²	—	3	3	no	—
35	C₃×3₋¹⁺²	—	3	1	no	—
36	C₃×3₋¹⁺²	—	3	1	no	—
37	C₃×3₋¹⁺²	—	9	2	no	—
38	C₃×3₋¹⁺²	—	9	6	no	—
39	C₃×3₋¹⁺²	H(3,𝔽₃)	1	9	yes	—
40	C₃×H(3,𝔽₃)	—	3	1	no	—
41	C₃×H(3,𝔽₃)	—	3	3	no	—
42	C₃²×C₉	—	3	3	no	—
43	C₃²×C₉	—	3	3	no	—
44	C₃²×C₉	H(3,𝔽₃)	1	2	yes	—
45	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
46	#733	—	3	18	no	—
47	#733	—	3	9	no	—
48	#731	—	9	6	no	—
49	#731	—	9	3	no	—
50	C₉⋊C₉	—	3	6	no	—
51	C₉⋊C₉	—	3	18	no	—
52	C₃²⋊C₉	—	3	3	no	—
53	C₃²⋊C₉	—	3	6	no	—
54	C₃²⋊C₉	—	3	3	no	—
55	C₃²⋊C₉	—	3	3	no	—
56	C₃²⋊C₉	—	3	3	no	—
57	C₃²⋊C₉	—	3	3	no	—
58	C₃²⋊C₉	—	3	3	no	—
59	C₉²	—	3	3	no	—
60	C₃³	—	3	1	no	—
61	C₃³	—	3	2	no	—
62	C₃³	—	9	3	no	—
63	C₃³	—	9	1	no	—
64	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
65	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
66	3₋¹⁺²	—	3	9	no	—
67	3₋¹⁺²	—	9	9	no	—
68	3₋¹⁺²	—	9	6	no	—
69	3₋¹⁺²	—	9	9	no	—
70	3₋¹⁺²	—	9	9	no	—
71	3₋¹⁺²	—	9	9	no	—
72	3₋¹⁺²	—	9	1	no	—
73	3₋¹⁺²	—	9	1	no	—
74	3₋¹⁺²	—	9	2	no	—
75	3₋¹⁺²	—	9	2	no	—
76	H(3,𝔽₃)	—	9	1	no	—
77	H(3,𝔽₃)	—	9	2	no	—
78	H(3,𝔽₃)	—	27	3	no	—
79	C₃×C₉	—	3	1	no	—
80	C₃×C₉	—	3	2	no	—
81	C₃×C₉	—	3	9	no	—
82	C₃×C₉	—	3	2	no	—
83	C₃×C₉	—	3	2	no	—
84	C₃×C₉	—	3	2	no	—
85	C₃×C₉	—	3	2	no	—
86	C₃×C₉	—	9	3	no	—
87	C₃×C₉	—	9	9	no	—
88	C₃×C₉	—	9	9	no	—
89	C₃×C₉	—	9	3	no	—
90	C₃×C₉	—	9	1	no	—
91	C₃×C₉	—	9	1	no	—
92	C₃×C₉	AΣL(1,𝔽₂₇)	1	3	yes	—
93	C₃²	—	3	3	no	—
94	C₃²	—	3	3	no	—
95	C₃²	—	3	3	no	—
96	C₃²	—	3	3	no	—
97	C₃²	—	3	3	no	—
98	C₃²	—	27	1	no	—
99	C₃²	—	27	2	no	—
100	C₃²	—	27	1	no	—
101	C₃²	—	27	3	no	—
102	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
103	C₃²	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
104	C₃²	#6909	1	2	yes	—
105	C₉	—	3	3	no	—
106	C₉	—	9	2	no	—
107	C₉	—	9	9	no	—
108	C₉	—	9	9	no	—
109	C₉	—	27	1	no	—
110	C₉	—	27	1	no	—
111	C₉	—	27	3	no	—
112	C₃	—	3	1	no	—
113	C₃	—	3	2	no	—
114	C₃	—	9	3	no	—
115	C₃	—	81	1	no	—
116	C₃	#10624743	1	1	yes	characteristic
117	C₃	#10624750	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#6912

C₃

yes

maximal, characteristic, complemented

#10624885

C₃

yes

maximal, characteristic, complemented

#10624823

C₃

yes

maximal, complemented

#10624736

C₃

yes

maximal, characteristic

#10624612

C₃

yes

maximal, complemented

#10624609

C₃

yes

maximal

#10624570

C₃

yes

maximal, characteristic, complemented

C₃²×3₋¹⁺²

C₃²

yes

—

C₃²×3₋¹⁺²

C₃²

yes

—

—

—

—

—

—

—

C₃²

yes

characteristic

#6891

C₃²

yes

characteristic

#6891

C₃²

yes

—

C₃²⋊H(3,𝔽₃)

C₃²

yes

characteristic

#6887

—

C₃×C₃²⋊C₉

C₃²

yes

—

3₋¹⁺²⋊C₉

—

—

—

—

—

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

—

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

—

C₃²×C₉

—

C₃²×C₉

H(3,𝔽₃)

yes

—

C₃²×C₉

yes

—

—

—

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₉²

—

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

—

3₋¹⁺²

—

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

AΣL(1,𝔽₂₇)

yes

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

100

C₃²

—

101

C₃²

—

102

C₃²

C₃²⋊H(3,𝔽₃)

yes

characteristic

103

C₃²

C₃×AΣL(1,𝔽₂₇)

yes

characteristic

104

C₃²

#6909

yes

—

105

C₉

—

106

C₉

—

107

C₉

—

108

C₉

—

109

C₉

—

110

C₉

—

111

C₉

—

112

C₃

—

113

C₃

—

114

C₃

—

115

C₃

—

116

C₃

#10624743

yes

characteristic

117

C₃

#10624750

yes

—

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

prime radical

C₃²⋊H(3,𝔽₃)

C₃²

LCS

G ▷ C₃⁴ ▷ C₃² ▷ C₁

UCS

C₁ ◁ C₃² ◁ C₃⁴ ◁ G

LECS

G ▷ C₃⁴ ▷ C₃² ▷ C₁

N	H ≅ G/N	type
1. C₃×#6912	C₃	semidirect
2. #10624885	C₃	semidirect
3. #10624823	C₃	semidirect
5. #10624612	C₃	semidirect
7. #10624570	C₃	semidirect

H ≅ G/N

type

1. C₃×#6912

C₃

semidirect

2. #10624885

C₃

semidirect

3. #10624823

C₃

semidirect

5. #10624612

C₃

semidirect

7. #10624570

C₃

semidirect

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

subgroup	index
C₃×#6912	3
#10624885	3
#10624823	3
#10624736	3
#10624612	3
#10624609	3
#10624570	3

subgroup

index

Subgroups — #14547296

Conjugacy classes of subgroups

Complements

Maximal subgroups