Group #14550739 — subgroups

Subgroups — #14550739

Order 2187.

258 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 77 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	#10624887	C₃	1	1	yes	maximal, characteristic
2	#10624709	C₃	1	2	yes	maximal, complemented
3	#10624709	C₃	1	1	yes	maximal, characteristic, complemented
4	#10624702	C₃	1	2	yes	maximal
5	#10624702	C₃	1	1	yes	maximal, characteristic
6	#10624473	C₃	1	2	yes	maximal, complemented
7	#10624473	C₃	1	1	yes	maximal, characteristic, complemented
8	#10624472	C₃	1	2	yes	maximal, complemented
9	#10624472	C₃	1	1	yes	maximal, characteristic, complemented
10	C₃×#737	C₃²	1	1	yes	characteristic
11	C₃×C₂₇⋊C₃	C₃²	1	2	yes	—
12	C₃×C₂₇⋊C₃	C₃²	1	2	yes	—
13	C₃×C₂₇⋊C₃	C₃²	1	2	yes	—
14	C₃×C₂₇⋊C₃	C₃²	1	2	yes	—
15	C₃×C₂₇⋊C₃	C₃²	1	1	yes	characteristic
16	#6900	—	3	1	no	—
17	#6899	—	3	2	no	—
18	#6896	—	3	3	no	—
19	#6894	—	3	2	no	—
20	#6894	—	3	3	no	—
21	C₉⋊H(3,𝔽₃)	—	3	1	no	—
22	#6887	C₃²	1	1	yes	characteristic
23	#6887	C₃²	1	1	yes	characteristic
24	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
25	#6875	—	3	6	no	—
26	#6875	—	3	3	no	—
27	#6872	—	3	6	no	—
28	#6872	—	3	3	no	—
29	#737	—	3	2	no	—
30	#737	—	3	1	no	—
31	#737	—	3	2	no	—
32	#737	—	3	2	no	—
33	#737	—	3	2	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	3₋¹⁺²	1	2	yes	—
38	C₃×H(3,𝔽₃)	3₋¹⁺²	1	1	yes	characteristic
39	C₃²×C₉	—	3	1	no	—
40	C₃²×C₉	—	3	1	no	—
41	C₃²×C₉	C₃³	1	1	yes	characteristic
42	C₂₇⋊C₃	—	3	2	no	—
43	C₂₇⋊C₃	—	3	1	no	—
44	C₂₇⋊C₃	—	3	2	no	—
45	C₂₇⋊C₃	—	3	2	no	—
46	C₂₇⋊C₃	—	3	2	no	—
47	C₂₇⋊C₃	—	3	2	no	—
48	C₂₇⋊C₃	—	3	2	no	—
49	C₂₇⋊C₃	—	3	2	no	—
50	C₂₇⋊C₃	—	3	2	no	—
51	C₂₇⋊C₃	—	3	2	no	—
52	C₂₇⋊C₃	—	3	2	no	—
53	C₂₇⋊C₃	—	3	2	no	—
54	C₂₇⋊C₃	—	3	2	no	—
55	C₂₇⋊C₃	3₋¹⁺²	1	6	yes	complemented
56	C₃×C₂₇	—	3	2	no	—
57	C₃×C₂₇	—	3	2	no	—
58	C₃×C₂₇	—	3	2	no	—
59	C₃×C₂₇	—	3	2	no	—
60	C₃×C₂₇	3₋¹⁺²	1	3	yes	complemented
61	C₉⋊C₉	—	3	12	no	—
62	C₉⋊C₉	—	3	6	no	—
63	C₉⋊C₉	—	3	1	no	—
64	C₉⋊C₉	—	3	1	no	—
65	C₉⋊C₉	—	3	1	no	—
66	C₉⋊C₉	—	3	1	no	—
67	C₉⋊C₉	H(3,𝔽₃)	1	3	yes	—
68	C₉⋊C₉	3₋¹⁺²	1	2	yes	—
69	C₉⋊C₉	3₋¹⁺²	1	1	yes	characteristic
70	C₉⋊C₉	3₋¹⁺²	1	3	yes	—
71	C₃²⋊C₉	—	3	6	no	—
72	C₃²⋊C₉	—	3	1	no	—
73	C₃²⋊C₉	—	3	1	no	—
74	C₉²	—	3	1	no	—
75	C₉²	—	3	1	no	—
76	C₃³	—	3	1	no	—
77	C₃³	C₃×3₋¹⁺²	1	1	yes	characteristic
78	3₋¹⁺²	—	3	4	no	—
79	3₋¹⁺²	—	3	2	no	—
80	3₋¹⁺²	—	9	2	no	—
81	3₋¹⁺²	—	9	1	no	—
82	3₋¹⁺²	—	9	2	no	—
83	3₋¹⁺²	—	9	1	no	—
84	3₋¹⁺²	—	27	3	no	—
85	H(3,𝔽₃)	—	3	2	no	—
86	H(3,𝔽₃)	—	3	1	no	—
87	C₃×C₉	—	3	3	no	—
88	C₃×C₉	—	3	3	no	—
89	C₃×C₉	—	3	2	no	—
90	C₃×C₉	—	3	2	no	—
91	C₃×C₉	—	3	1	no	—
92	C₃×C₉	—	9	2	no	—
93	C₃×C₉	—	9	1	no	—
94	C₃×C₉	—	27	1	no	—
95	C₃×C₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
96	C₃×C₉	C₃×3₋¹⁺²	1	2	yes	—
97	C₃×C₉	C₃×3₋¹⁺²	1	1	yes	characteristic
98	C₃×C₉	C₃×3₋¹⁺²	1	1	yes	characteristic
99	C₃×C₉	C₃×3₋¹⁺²	1	1	yes	characteristic
100	C₃×C₉	#737	1	2	yes	—
101	C₃×C₉	#737	1	2	yes	—
102	C₃×C₉	#737	1	2	yes	—
103	C₂₇	—	3	6	no	—
104	C₂₇	—	3	3	no	—
105	C₂₇	—	9	2	no	—
106	C₂₇	—	9	2	no	—
107	C₂₇	—	9	2	no	—
108	C₃²	—	9	1	no	—
109	C₃²	—	9	3	no	—
110	C₃²	—	9	1	no	—
111	C₃²	#6891	1	1	yes	characteristic
112	C₃²	#6894	1	2	yes	—
113	C₃²	#6894	1	1	yes	characteristic
114	C₉	—	9	2	no	—
115	C₉	—	9	1	no	—
116	C₉	—	9	3	no	—
117	C₉	—	9	3	no	—
118	C₉	—	9	2	no	—
119	C₉	C₉⋊H(3,𝔽₃)	1	2	yes	—
120	C₉	C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
121	C₉	#6896	1	2	yes	—
122	C₉	#6896	1	1	yes	characteristic
123	C₉	#6899	1	2	yes	—
124	C₉	#6900	1	1	yes	characteristic
125	C₃	—	3	1	no	—
126	C₃	—	3	2	no	—
127	C₃	—	3	1	no	—
128	C₃	—	27	1	no	—
129	C₃	#10624565	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10624887

C₃

yes

maximal, characteristic

#10624709

C₃

yes

maximal, complemented

#10624709

C₃

yes

maximal, characteristic, complemented

#10624702

C₃

yes

maximal

#10624702

C₃

yes

maximal, characteristic

#10624473

C₃

yes

maximal, complemented

#10624473

C₃

yes

maximal, characteristic, complemented

#10624472

C₃

yes

maximal, complemented

#10624472

C₃

yes

maximal, characteristic, complemented

C₃×#737

C₃²

yes

characteristic

C₃×C₂₇⋊C₃

C₃²

yes

—

C₃×C₂₇⋊C₃

C₃²

yes

—

C₃×C₂₇⋊C₃

C₃²

yes

—

C₃×C₂₇⋊C₃

C₃²

yes

—

C₃×C₂₇⋊C₃

C₃²

yes

—

—

—

—

—

C₉⋊H(3,𝔽₃)

—

#6887

C₃²

yes

characteristic

#6887

C₃²

yes

characteristic

C₃×C₉⋊C₉

C₃²

yes

—

—

—

—

—

—

—

—

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

3₋¹⁺²

yes

—

C₃×H(3,𝔽₃)

3₋¹⁺²

yes

characteristic

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃³

yes

characteristic

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

3₋¹⁺²

yes

complemented

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

3₋¹⁺²

yes

complemented

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

H(3,𝔽₃)

yes

—

C₉⋊C₉

3₋¹⁺²

yes

—

C₉⋊C₉

3₋¹⁺²

yes

characteristic

C₉⋊C₉

3₋¹⁺²

yes

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₉²

—

C₉²

—

C₃³

—

C₃³

C₃×3₋¹⁺²

yes

characteristic

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)

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₃×H(3,𝔽₃)

yes

characteristic

C₃×C₉

C₃×3₋¹⁺²

yes

—

C₃×C₉

C₃×3₋¹⁺²

yes

characteristic

C₃×C₉

C₃×3₋¹⁺²

yes

characteristic

C₃×C₉

C₃×3₋¹⁺²

yes

characteristic

100

C₃×C₉

#737

yes

—

101

C₃×C₉

#737

yes

—

102

C₃×C₉

#737

yes

—

103

C₂₇

—

104

C₂₇

—

105

C₂₇

—

106

C₂₇

—

107

C₂₇

—

108

C₃²

—

109

C₃²

—

110

C₃²

—

111

C₃²

#6891

yes

characteristic

112

C₃²

#6894

yes

—

113

C₃²

#6894

yes

characteristic

114

C₉

—

115

C₉

—

116

C₉

—

117

C₉

—

118

C₉

—

119

C₉

C₉⋊H(3,𝔽₃)

yes

—

120

C₉

C₉⋊H(3,𝔽₃)

yes

characteristic

121

C₉

#6896

yes

—

122

C₉

#6896

yes

characteristic

123

C₉

#6899

yes

—

124

C₉

#6900

yes

characteristic

125

C₃

—

126

C₃

—

127

C₃

—

128

C₃

—

129

C₃

#10624565

yes

characteristic

	subgroup	G/H
center, socle	C₃	#10624565
commutator, Frattini, cosocle	C₃²×C₉	C₃³
prime radical	C₃×H(3,𝔽₃)	3₋¹⁺²
LCS	G ▷ C₃²×C₉ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃ ◁ C₃²×C₉ ◁ G
LECS	G ▷ C₃²×C₉ ▷ C₃ ▷ C₁

subgroup

G/H

center, socle

C₃

#10624565

commutator, Frattini, cosocle

C₃²×C₉

C₃³

prime radical

C₃×H(3,𝔽₃)

3₋¹⁺²

LCS

G ▷ C₃²×C₉ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃ ◁ C₃²×C₉ ◁ G

LECS

G ▷ C₃²×C₉ ▷ C₃ ▷ C₁

N	H ≅ G/N	type
2. #10624709	C₃	semidirect
3. #10624709	C₃	semidirect
6. #10624473	C₃	semidirect
7. #10624473	C₃	semidirect
8. #10624472	C₃	semidirect
9. #10624472	C₃	semidirect
55. C₂₇⋊C₃	3₋¹⁺²	semidirect
60. C₃×C₂₇	3₋¹⁺²	semidirect

H ≅ G/N

type

2. #10624709

C₃

semidirect

3. #10624709

C₃

semidirect

6. #10624473

C₃

semidirect

7. #10624473

C₃

semidirect

8. #10624472

C₃

semidirect

9. #10624472

C₃

semidirect

55. C₂₇⋊C₃

3₋¹⁺²

semidirect

60. C₃×C₂₇

3₋¹⁺²

semidirect

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

subgroup	index
#10624887	3
#10624709	3
#10624702	3
#10624473	3
#10624472	3

subgroup

index

Subgroups — #14550739

Conjugacy classes of subgroups

Complements

Maximal subgroups