Group #14554392 — subgroups

Subgroups — #14554392

Order 2187.

1466 conjugacy classes of subgroups; only the 314 normal subgroups are shown.

normal subgroups only

This group has 1466 conjugacy classes of subgroups — too many to draw. Only its 314 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	C₃²×#737	C₃	1	1	yes	maximal, characteristic
2	C₃×#6895	C₃	1	2	yes	maximal, complemented
3	C₃×#6895	C₃	1	1	yes	maximal, characteristic, complemented
4	#10624780	C₃	1	27	yes	maximal, complemented
5	#10624699	C₃	1	6	yes	maximal, complemented
6	#10624698	C₃	1	3	yes	maximal
7	C₃×#737	—	3	18	no	—
8	C₃×#737	—	3	9	no	—
9	C₃×#737	—	3	6	no	—
10	C₃×#737	C₃²	1	9	yes	—
11	C₃²×3₋¹⁺²	—	3	2	no	—
12	C₃²×3₋¹⁺²	C₃²	1	2	yes	—
13	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
14	C₃³×C₉	—	3	1	no	—
15	C₃³×C₉	C₃²	1	1	yes	characteristic
16	#6895	—	3	2	no	—
17	#6895	—	3	4	no	—
18	#6895	—	3	6	no	—
19	#6895	—	3	1	no	—
20	#6895	—	3	2	no	—
21	#6895	—	3	6	no	—
22	#6895	C₃²	1	18	yes	complemented
23	C₃×C₉⋊C₉	—	3	6	no	—
24	C₃×C₉⋊C₉	C₃²	1	9	yes	complemented
25	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
26	C₃×C₃²⋊C₉	C₃²	1	6	yes	—
27	3₋¹⁺²⋊C₉	C₃²	1	54	yes	complemented
28	H(3,𝔽₃)⋊C₉	C₃²	1	27	yes	—
29	C₃⁴	—	3	1	no	—
30	C₃⁴	C₃³	1	1	yes	characteristic
31	#737	—	3	162	no	—
32	#737	—	3	54	no	—
33	#737	—	3	27	no	—
34	C₃×3₋¹⁺²	—	3	54	no	—
35	C₃×3₋¹⁺²	—	3	4	no	—
36	C₃×3₋¹⁺²	—	3	6	no	—
37	C₃×3₋¹⁺²	—	3	6	no	—
38	C₃×3₋¹⁺²	—	3	18	no	—
39	C₃×3₋¹⁺²	3₋¹⁺²	1	18	yes	—
40	C₃×3₋¹⁺²	C₃³	1	6	yes	—
41	C₃×H(3,𝔽₃)	—	3	2	no	—
42	C₃×H(3,𝔽₃)	—	3	6	no	—
43	C₃×H(3,𝔽₃)	3₋¹⁺²	1	9	yes	—
44	C₃×H(3,𝔽₃)	C₃³	1	3	yes	—
45	C₃²×C₉	—	3	27	no	—
46	C₃²×C₉	—	3	2	no	—
47	C₃²×C₉	—	3	6	no	—
48	C₃²×C₉	—	3	3	no	—
49	C₃²×C₉	—	3	9	no	—
50	C₃²×C₉	—	3	6	no	—
51	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
52	C₃²×C₉	H(3,𝔽₃)	1	2	yes	—
53	C₃²×C₉	H(3,𝔽₃)	1	6	yes	—
54	C₃²×C₉	C₃³	1	3	yes	—
55	C₃²×C₉	C₃³	1	9	yes	—
56	C₉⋊C₉	—	3	18	no	—
57	C₉⋊C₉	—	3	36	no	—
58	C₉⋊C₉	—	3	9	no	—
59	C₉⋊C₉	—	3	18	no	—
60	C₃²⋊C₉	—	3	3	no	—
61	C₃²⋊C₉	—	3	6	no	—
62	C₃²⋊C₉	—	3	6	no	—
63	C₃²⋊C₉	—	3	6	no	—
64	C₃²⋊C₉	C₃³	1	18	yes	—
65	C₃³	—	3	3	no	—
66	C₃³	—	3	9	no	—
67	C₃³	—	9	3	no	—
68	C₃³	—	9	9	no	—
69	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
70	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
71	C₃³	C₃×3₋¹⁺²	1	6	yes	—
72	C₃³	C₃×3₋¹⁺²	1	3	yes	—
73	C₃³	C₃⁴	1	1	yes	characteristic
74	3₋¹⁺²	—	3	162	no	—
75	3₋¹⁺²	—	3	6	no	—
76	3₋¹⁺²	—	3	6	no	—
77	3₋¹⁺²	—	3	36	no	—
78	3₋¹⁺²	C₃×3₋¹⁺²	1	18	yes	—
79	H(3,𝔽₃)	—	3	6	no	—
80	H(3,𝔽₃)	—	3	18	no	—
81	H(3,𝔽₃)	C₃×3₋¹⁺²	1	9	yes	—
82	C₃×C₉	—	3	81	no	—
83	C₃×C₉	—	3	6	no	—
84	C₃×C₉	—	3	6	no	—
85	C₃×C₉	—	3	3	no	—
86	C₃×C₉	—	3	3	no	—
87	C₃×C₉	—	3	18	no	—
88	C₃×C₉	—	3	6	no	—
89	C₃×C₉	—	3	18	no	—
90	C₃×C₉	—	3	9	no	—
91	C₃×C₉	—	3	18	no	—
92	C₃×C₉	—	9	18	no	—
93	C₃×C₉	—	9	9	no	—
94	C₃×C₉	C₃×H(3,𝔽₃)	1	3	yes	—
95	C₃×C₉	C₃×H(3,𝔽₃)	1	6	yes	—
96	C₃²	—	3	6	no	—
97	C₃²	—	3	3	no	—
98	C₃²	—	3	3	no	—
99	C₃²	—	3	9	no	—
100	C₃²	—	9	6	no	—
101	C₃²	—	9	3	no	—
102	C₃²	—	9	27	no	—
103	C₃²	—	9	9	no	—
104	C₃²	#6891	1	3	yes	—
105	C₃²	#6891	1	6	yes	—
106	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
107	C₃²	C₃²×3₋¹⁺²	1	2	yes	—
108	C₃²	C₃²×3₋¹⁺²	1	1	yes	characteristic
109	C₉	—	3	3	no	—
110	C₉	—	3	18	no	—
111	C₉	—	3	6	no	—
112	C₉	—	9	9	no	—
113	C₉	—	9	18	no	—
114	C₃	—	3	3	no	—
115	C₃	—	3	2	no	—
116	C₃	—	3	1	no	—
117	C₃	—	3	6	no	—
118	C₃	—	9	9	no	—
119	C₃	C₃×#6891	1	1	yes	characteristic
120	C₃	#10624929	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#737

C₃

yes

maximal, characteristic

C₃×#6895

C₃

yes

maximal, complemented

C₃×#6895

C₃

yes

maximal, characteristic, complemented

#10624780

C₃

yes

maximal, complemented

#10624699

C₃

yes

maximal, complemented

#10624698

C₃

yes

—

—

—

C₃²

yes

—

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

C₃²

yes

—

C₃²×H(3,𝔽₃)

C₃²

yes

characteristic

C₃³×C₉

—

C₃³×C₉

C₃²

yes

—

—

—

—

—

—

C₃²

yes

complemented

C₃×C₉⋊C₉

—

C₃×C₉⋊C₉

C₃²

yes

complemented

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

—

3₋¹⁺²⋊C₉

C₃²

yes

complemented

H(3,𝔽₃)⋊C₉

C₃²

yes

—

C₃⁴

—

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

3₋¹⁺²

yes

—

C₃×3₋¹⁺²

C₃³

yes

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

3₋¹⁺²

yes

—

C₃×H(3,𝔽₃)

C₃³

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

H(3,𝔽₃)

yes

characteristic

C₃²×C₉

H(3,𝔽₃)

yes

—

C₃²×C₉

H(3,𝔽₃)

yes

—

C₃²×C₉

C₃³

yes

—

C₃²×C₉

C₃³

yes

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃³

yes

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

C₃³

C₃×H(3,𝔽₃)

yes

—

C₃³

C₃×3₋¹⁺²

yes

—

C₃³

C₃×3₋¹⁺²

yes

—

C₃³

C₃⁴

yes

characteristic

3₋¹⁺²

—

162

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

C₃×3₋¹⁺²

yes

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

H(3,𝔽₃)

C₃×3₋¹⁺²

yes

—

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×H(3,𝔽₃)

yes

—

C₃×C₉

C₃×H(3,𝔽₃)

yes

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

100

C₃²

—

101

C₃²

—

102

C₃²

—

103

C₃²

—

104

C₃²

#6891

yes

—

105

C₃²

#6891

yes

—

106

C₃²

C₃²×H(3,𝔽₃)

yes

characteristic

107

C₃²

C₃²×3₋¹⁺²

yes

—

108

C₃²

C₃²×3₋¹⁺²

yes

characteristic

109

C₉

—

110

C₉

—

111

C₉

—

112

C₉

—

113

C₉

—

114

C₃

—

115

C₃

—

116

C₃

—

117

C₃

—

118

C₃

—

119

C₃

C₃×#6891

yes

characteristic

120

C₃

#10624929

yes

—

	subgroup	G/H
center, socle	C₃²	C₃²×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃⁴
prime radical	C₃²×H(3,𝔽₃)	C₃²
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃³×C₉ ◁ G
LECS	G ▷ C₃³ ▷ C₃ ▷ C₁

subgroup

G/H

center, socle

C₃²

C₃²×H(3,𝔽₃)

commutator, Frattini, cosocle

C₃³

C₃⁴

prime radical

C₃²×H(3,𝔽₃)

C₃²

LCS

G ▷ C₃³ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃² ◁ C₃³×C₉ ◁ G

LECS

G ▷ C₃³ ▷ C₃ ▷ C₁

N	H ≅ G/N	type
2. C₃×#6895	C₃	semidirect
3. C₃×#6895	C₃	semidirect
4. #10624780	C₃	semidirect
5. #10624699	C₃	semidirect
22. #6895	C₃²	semidirect
24. C₃×C₉⋊C₉	C₃²	semidirect
27. 3₋¹⁺²⋊C₉	C₃²	semidirect

H ≅ G/N

type

2. C₃×#6895

C₃

semidirect

3. C₃×#6895

C₃

semidirect

4. #10624780

C₃

semidirect

5. #10624699

C₃

semidirect

22. #6895

C₃²

semidirect

24. C₃×C₉⋊C₉

C₃²

semidirect

27. 3₋¹⁺²⋊C₉

C₃²

semidirect

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

subgroup	index
C₃²×#737	3
C₃×#6895	3
#10624780	3
#10624699	3
#10624698	3

subgroup

index

C₃²×#737

Subgroups — #14554392

Conjugacy classes of subgroups

Complements

Maximal subgroups