Group #14550303 — subgroups

Subgroups — #14550303

Order 2187.

1106 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 1106 conjugacy classes of subgroups — too many to draw. Only its 149 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	#10624943	C₃	1	1	yes	maximal, characteristic
2	#10624704	C₃	1	4	yes	maximal, complemented
3	C₃×#6873	C₃	1	2	yes	maximal, complemented
4	#10624702	C₃	1	4	yes	maximal, complemented
5	C₃×#6872	C₃	1	2	yes	maximal, complemented
6	3₋¹⁺⁴	—	3	12	no	—
7	3₋¹⁺⁴	—	3	12	no	—
8	3₋¹⁺⁴	C₉	1	4	yes	—
9	3₋¹⁺⁴	C₉	1	4	yes	—
10	H(5,𝔽₃)	C₉	1	1	yes	characteristic
11	C₃×#737	—	3	6	no	—
12	C₃×#737	—	3	6	no	—
13	C₃×#737	C₃²	1	2	yes	—
14	C₃×#737	C₃²	1	2	yes	—
15	C₃×C₂₇⋊C₃	C₃²	1	4	yes	complemented
16	C₃×C₂₇⋊C₃	C₃²	1	4	yes	complemented
17	C₃²×C₂₇	C₃²	1	1	yes	characteristic, complemented
18	#6873	—	3	6	no	—
19	#6873	—	3	12	no	—
20	#6872	—	3	6	no	—
21	#6872	—	3	12	no	—
22	#737	—	3	18	no	—
23	#737	—	3	72	no	—
24	#737	—	3	72	no	—
25	#737	—	3	24	no	—
26	#737	—	3	24	no	—
27	#737	—	3	12	no	—
28	#737	—	3	12	no	—
29	#737	—	3	4	no	—
30	#737	—	3	4	no	—
31	#737	—	3	8	no	—
32	#737	—	3	8	no	—
33	#737	H(3,𝔽₃)	1	6	yes	—
34	#737	H(3,𝔽₃)	1	6	yes	—
35	#737	3₋¹⁺²	1	12	yes	—
36	#737	3₋¹⁺²	1	12	yes	—
37	C₃×3₋¹⁺²	—	3	36	no	—
38	C₃×3₋¹⁺²	—	3	36	no	—
39	C₃×3₋¹⁺²	—	3	12	no	—
40	C₃×3₋¹⁺²	—	3	12	no	—
41	C₃×3₋¹⁺²	—	3	4	no	—
42	C₃×3₋¹⁺²	—	3	4	no	—
43	C₃×3₋¹⁺²	C₃×C₉	1	4	yes	—
44	C₃×3₋¹⁺²	C₃×C₉	1	4	yes	—
45	C₃×H(3,𝔽₃)	—	3	6	no	—
46	C₃×H(3,𝔽₃)	—	3	6	no	—
47	C₃×H(3,𝔽₃)	C₃×C₉	1	2	yes	—
48	C₃×H(3,𝔽₃)	C₃×C₉	1	2	yes	—
49	C₃²×C₉	—	3	9	no	—
50	C₃²×C₉	—	3	2	no	—
51	C₃²×C₉	—	3	2	no	—
52	C₃²×C₉	C₃³	1	1	yes	characteristic
53	C₂₇⋊C₃	—	3	4	no	—
54	C₂₇⋊C₃	—	3	8	no	—
55	C₂₇⋊C₃	—	3	4	no	—
56	C₂₇⋊C₃	—	3	8	no	—
57	C₃×C₂₇	—	3	2	no	—
58	C₃×C₂₇	—	3	4	no	—
59	C₃×C₂₇	—	3	2	no	—
60	C₃×C₂₇	—	3	4	no	—
61	C₃³	—	3	9	no	—
62	C₃³	—	3	2	no	—
63	C₃³	—	3	2	no	—
64	C₃³	C₃²×C₉	1	1	yes	characteristic
65	3₋¹⁺²	—	3	72	no	—
66	3₋¹⁺²	—	3	72	no	—
67	3₋¹⁺²	—	3	8	no	—
68	3₋¹⁺²	—	3	8	no	—
69	3₋¹⁺²	—	3	12	no	—
70	3₋¹⁺²	—	3	12	no	—
71	3₋¹⁺²	—	3	24	no	—
72	3₋¹⁺²	—	3	24	no	—
73	3₋¹⁺²	C₃²⋊C₉	1	12	yes	—
74	3₋¹⁺²	C₃²⋊C₉	1	12	yes	—
75	H(3,𝔽₃)	—	3	18	no	—
76	H(3,𝔽₃)	—	3	4	no	—
77	H(3,𝔽₃)	—	3	4	no	—
78	H(3,𝔽₃)	C₃²⋊C₉	1	6	yes	—
79	H(3,𝔽₃)	C₃²⋊C₉	1	6	yes	—
80	C₃×C₉	—	3	36	no	—
81	C₃×C₉	—	3	36	no	—
82	C₃×C₉	—	3	4	no	—
83	C₃×C₉	—	3	4	no	—
84	C₃×C₉	—	3	6	no	—
85	C₃×C₉	—	3	6	no	—
86	C₃×C₉	—	3	12	no	—
87	C₃×C₉	—	3	12	no	—
88	C₃×C₉	C₃×H(3,𝔽₃)	1	2	yes	—
89	C₃×C₉	C₃×H(3,𝔽₃)	1	2	yes	—
90	C₃×C₉	C₃×3₋¹⁺²	1	4	yes	—
91	C₃×C₉	C₃×3₋¹⁺²	1	4	yes	—
92	C₂₇	—	9	1	no	—
93	C₂₇	—	9	4	no	—
94	C₂₇	—	9	4	no	—
95	C₃²	—	3	6	no	—
96	C₃²	—	3	6	no	—
97	C₃²	—	9	1	no	—
98	C₃²	—	27	2	no	—
99	C₃²	—	27	2	no	—
100	C₃²	—	27	9	no	—
101	C₃²	C₃×C₃²⋊C₉	1	2	yes	—
102	C₃²	C₃×C₃²⋊C₉	1	2	yes	—
103	C₉	—	3	12	no	—
104	C₉	—	3	12	no	—
105	C₉	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
106	C₉	#6891	1	4	yes	—
107	C₉	#6891	1	4	yes	—
108	C₃	—	3	2	no	—
109	C₃	—	3	2	no	—
110	C₃	—	9	6	no	—
111	C₃	—	9	6	no	—
112	C₃	#10624547	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10624943

C₃

yes

maximal, characteristic

#10624704

C₃

yes

maximal, complemented

C₃×#6873

C₃

yes

maximal, complemented

#10624702

C₃

yes

maximal, complemented

C₃×#6872

C₃

yes

maximal, complemented

3₋¹⁺⁴

—

3₋¹⁺⁴

—

3₋¹⁺⁴

C₉

yes

—

3₋¹⁺⁴

C₉

yes

—

H(5,𝔽₃)

C₉

yes

—

—

C₃²

yes

—

C₃×#737

C₃²

yes

—

C₃×C₂₇⋊C₃

C₃²

yes

complemented

C₃×C₂₇⋊C₃

C₃²

yes

complemented

C₃²×C₂₇

C₃²

yes

characteristic, complemented

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

yes

—

#737

H(3,𝔽₃)

yes

—

#737

3₋¹⁺²

yes

—

#737

3₋¹⁺²

yes

—

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×C₉

yes

—

C₃×3₋¹⁺²

C₃×C₉

yes

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₃×C₉

yes

—

C₃×H(3,𝔽₃)

C₃×C₉

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃³

yes

characteristic

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃²×C₉

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

C₃²⋊C₉

yes

—

3₋¹⁺²

C₃²⋊C₉

yes

—

—

—

—

C₃²⋊C₉

yes

—

H(3,𝔽₃)

C₃²⋊C₉

yes

—

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

—

C₃×C₉

C₃×3₋¹⁺²

yes

—

C₃×C₉

C₃×3₋¹⁺²

yes

—

C₂₇

—

C₂₇

—

C₂₇

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

100

C₃²

—

101

C₃²

C₃×C₃²⋊C₉

yes

—

102

C₃²

C₃×C₃²⋊C₉

yes

—

103

C₉

—

104

C₉

—

105

C₉

C₃²⋊H(3,𝔽₃)

yes

characteristic

106

C₉

#6891

yes

—

107

C₉

#6891

yes

—

108

C₃

—

109

C₃

—

110

C₃

—

111

C₃

—

112

C₃

#10624547

yes

characteristic

	subgroup	G/H
center	C₉	C₃²⋊H(3,𝔽₃)
commutator	C₃³	C₃²×C₉
Frattini, cosocle	C₃²×C₉	C₃³
socle	C₃	#10624547
prime radical	H(5,𝔽₃)	C₉
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₉ ◁ C₃²×C₉ ◁ G
LECS	G ▷ C₃²×C₉ ▷ C₃ ▷ C₁

subgroup

G/H

center

C₉

C₃²⋊H(3,𝔽₃)

commutator

C₃³

C₃²×C₉

Frattini, cosocle

C₃²×C₉

C₃³

C₃

C₉

G ▷ C₃³ ▷ C₃ ▷ C₁

C₁ ◁ C₉ ◁ C₃²×C₉ ◁ G

G ▷ C₃²×C₉ ▷ C₃ ▷ C₁

N	H ≅ G/N	type
2. #10624704	C₃	semidirect
3. C₃×#6873	C₃	semidirect
4. #10624702	C₃	semidirect
5. C₃×#6872	C₃	semidirect
15. C₃×C₂₇⋊C₃	C₃²	semidirect
16. C₃×C₂₇⋊C₃	C₃²	semidirect
17. C₃²×C₂₇	C₃²	semidirect

H ≅ G/N

type

2. #10624704

C₃

semidirect

3. C₃×#6873

C₃

semidirect

4. #10624702

C₃

semidirect

5. C₃×#6872

C₃

semidirect

15. C₃×C₂₇⋊C₃

C₃²

semidirect

16. C₃×C₂₇⋊C₃

C₃²

semidirect

17. C₃²×C₂₇

C₃²

semidirect

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

subgroup	index
#10624943	3
#10624704	3
C₃×#6873	3
#10624702	3
C₃×#6872	3

subgroup

index

Subgroups — #14550303

Conjugacy classes of subgroups

Complements

Maximal subgroups