Group #14548370 — subgroups

Subgroups — #14548370

Order 2187.

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

normal subgroups only

This group has 355 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	#10624883	C₃	1	1	yes	maximal, characteristic, complemented
2	#10624820	C₃	1	4	yes	maximal, complemented
3	#10624808	C₃	1	2	yes	maximal, complemented
4	#10624777	C₃	1	2	yes	maximal, complemented
5	#10624740	C₃	1	1	yes	maximal, characteristic
6	#10624639	C₃	1	2	yes	maximal, complemented
7	C₃×#6862	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
9	C₃×#732	—	3	3	no	—
10	C₉⋊H(3,𝔽₃)	—	3	3	no	—
11	C₉⋊H(3,𝔽₃)	—	3	6	no	—
12	#6891	—	3	3	no	—
13	#6891	C₃²	1	4	yes	—
14	C₃²⋊H(3,𝔽₃)	C₃²	1	2	yes	—
15	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
16	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
17	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
18	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
19	H(3,𝔽₃)⋊C₉	—	3	6	no	—
20	H(3,𝔽₃)⋊C₉	—	3	12	no	—
21	#6867	—	3	6	no	—
22	#6862	—	3	3	no	—
23	#6857	—	3	6	no	—
24	C₃⁴	—	3	1	no	—
25	C₃⁴	C₃³	1	1	yes	characteristic
26	C₃×3₋¹⁺²	—	3	9	no	—
27	C₃×3₋¹⁺²	—	3	4	no	—
28	C₃×H(3,𝔽₃)	—	3	2	no	—
29	C₃×H(3,𝔽₃)	—	3	4	no	—
30	C₃×H(3,𝔽₃)	—	3	2	no	—
31	C₃×H(3,𝔽₃)	—	3	2	no	—
32	C₃×H(3,𝔽₃)	—	9	3	no	—
33	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	3	yes	—
34	C₃²×C₉	—	3	1	no	—
35	C₃²×C₉	—	3	1	no	—
36	C₃²×C₉	—	3	2	no	—
37	C₃²×C₉	H(3,𝔽₃)	1	6	yes	—
38	#732	—	9	9	no	—
39	C₃²⋊C₉	—	3	18	no	—
40	C₃²⋊C₉	—	3	2	no	—
41	C₃²⋊C₉	—	3	2	no	—
42	C₃²⋊C₉	—	3	2	no	—
43	C₃²⋊C₉	—	3	2	no	—
44	C₃²⋊C₉	—	3	2	no	—
45	C₃²⋊C₉	—	3	1	no	—
46	C₃²⋊C₉	—	3	2	no	—
47	C₃²⋊C₉	—	3	4	no	—
48	C₃²⋊C₉	—	3	4	no	—
49	C₃²⋊C₉	—	3	4	no	—
50	C₃²⋊C₉	—	3	4	no	—
51	C₃²⋊C₉	H(3,𝔽₃)	1	3	yes	—
52	C₃³	—	3	3	no	—
53	C₃³	—	3	3	no	—
54	C₃³	—	9	1	no	—
55	C₃³	—	9	2	no	—
56	C₃³	—	9	3	no	—
57	C₃³	—	9	2	no	—
58	C₃³	—	27	1	no	—
59	C₃³	—	27	2	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	—	27	9	no	—
64	3₋¹⁺²	—	27	4	no	—
65	H(3,𝔽₃)	—	3	3	no	—
66	H(3,𝔽₃)	—	3	6	no	—
67	H(3,𝔽₃)	—	9	6	no	—
68	H(3,𝔽₃)	—	9	12	no	—
69	H(3,𝔽₃)	—	9	6	no	—
70	H(3,𝔽₃)	—	27	2	no	—
71	C₃×C₉	—	3	6	no	—
72	C₃×C₉	—	3	3	no	—
73	C₃×C₉	—	9	6	no	—
74	C₃×C₉	—	9	1	no	—
75	C₃×C₉	—	9	2	no	—
76	C₃×C₉	—	9	4	no	—
77	C₃×C₉	—	9	3	no	—
78	C₃×C₉	—	27	1	no	—
79	C₃×C₉	—	27	2	no	—
80	C₃²	—	3	1	no	—
81	C₃²	—	3	2	no	—
82	C₃²	—	3	2	no	—
83	C₃²	—	3	1	no	—
84	C₃²	—	3	2	no	—
85	C₃²	—	3	1	no	—
86	C₃²	—	3	4	no	—
87	C₃²	—	3	2	no	—
88	C₃²	—	9	3	no	—
89	C₃²	—	9	3	no	—
90	C₃²	—	9	6	no	—
91	C₃²	—	27	3	no	—
92	C₃²	—	27	9	no	—
93	C₃²	—	27	2	no	—
94	C₃²	—	27	2	no	—
95	C₃²	—	27	4	no	—
96	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
97	C₃²	#6910	1	3	yes	—
98	C₉	—	9	6	no	—
99	C₉	—	9	3	no	—
100	C₉	—	27	1	no	—
101	C₉	—	27	2	no	—
102	C₉	—	27	4	no	—
103	C₃	—	3	3	no	—
104	C₃	—	9	2	no	—
105	C₃	—	9	1	no	—
106	C₃	—	27	3	no	—
107	C₃	—	81	2	no	—
108	C₃	#10624736	1	1	yes	characteristic
109	C₃	#10624748	1	1	yes	characteristic
110	C₃	#10624753	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10624883

C₃

yes

maximal, characteristic, complemented

#10624820

C₃

yes

maximal, complemented

#10624808

C₃

yes

maximal, complemented

#10624777

C₃

yes

maximal, complemented

#10624740

C₃

yes

maximal, characteristic

#10624639

C₃

yes

maximal, complemented

C₃×#6862

C₃

yes

maximal, characteristic, complemented

C₃²×H(3,𝔽₃)

C₃²

yes

characteristic

C₃×#732

—

C₉⋊H(3,𝔽₃)

—

C₉⋊H(3,𝔽₃)

—

#6891

—

#6891

C₃²

yes

—

C₃²⋊H(3,𝔽₃)

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

characteristic

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

characteristic

H(3,𝔽₃)⋊C₉

—

H(3,𝔽₃)⋊C₉

—

#6867

—

#6862

—

#6857

—

C₃⁴

—

C₃⁴

C₃³

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

H(3,𝔽₃)

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

H(3,𝔽₃)

yes

—

#732

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

H(3,𝔽₃)

yes

—

C₃³

—

C₃³

—

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

—

—

—

—

—

—

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

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

—

C₃²

C₃²⋊H(3,𝔽₃)

yes

characteristic

C₃²

#6910

yes

—

C₉

—

C₉

—

100

C₉

—

101

C₉

—

102

C₉

—

103

C₃

—

104

C₃

—

105

C₃

—

106

C₃

—

107

C₃

—

108

C₃

#10624736

yes

characteristic

109

C₃

#10624748

yes

characteristic

110

C₃

#10624753

yes

—

	subgroup	G/H
center, socle	C₃²	C₃²⋊H(3,𝔽₃)
commutator, Frattini, cosocle	C₃⁴	C₃³
prime radical	#10624740	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. #10624883	C₃	semidirect
2. #10624820	C₃	semidirect
3. #10624808	C₃	semidirect
4. #10624777	C₃	semidirect
6. #10624639	C₃	semidirect
7. C₃×#6862	C₃	semidirect

H ≅ G/N

type

1. #10624883

C₃

semidirect

2. #10624820

C₃

semidirect

3. #10624808

C₃

semidirect

4. #10624777

C₃

semidirect

6. #10624639

C₃

semidirect

7. C₃×#6862

C₃

semidirect

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

subgroup	index
#10624883	3
#10624820	3
#10624808	3
#10624777	3
#10624740	3
#10624639	3
C₃×#6862	3

subgroup

index

Subgroups — #14548370

Conjugacy classes of subgroups

Complements

Maximal subgroups