Group #14547205 — subgroups

Subgroups — #14547205

Order 2187.

554 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 59 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×#732	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×#6894	C₃	1	1	yes	maximal, characteristic, complemented
3	#10624768	C₃	1	1	yes	maximal, characteristic, complemented
4	#10624743	C₃	1	3	yes	maximal, complemented
5	#10624732	C₃	1	1	yes	maximal, characteristic, complemented
6	#10624596	C₃	1	6	yes	maximal, complemented
7	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
8	C₃²×H(3,𝔽₃)	C₃²	1	3	yes	—
9	C₃³×C₉	C₃²	1	1	yes	characteristic
10	C₃×#732	—	3	2	no	—
11	C₃×#732	—	3	1	no	—
12	C₃×#732	—	3	9	no	—
13	C₃×#732	—	3	2	no	—
14	C₃×#732	—	3	1	no	—
15	C₃×AΣL(1,𝔽₂₇)	—	3	9	no	—
16	#6894	—	3	3	no	—
17	#6894	—	3	6	no	—
18	#6891	C₃²	1	3	yes	—
19	C₃²⋊H(3,𝔽₃)	C₃²	1	3	yes	—
20	C₃×C₉⋊C₉	—	3	2	no	—
21	C₃×C₉⋊C₉	—	3	1	no	—
22	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
23	#6868	—	3	2	no	—
24	#6868	—	3	1	no	—
25	#6856	—	3	18	no	—
26	C₃⁴	—	3	3	no	—
27	C₃⁴	C₃³	1	1	yes	characteristic
28	C₃×3₋¹⁺²	—	3	2	no	—
29	C₃×3₋¹⁺²	—	3	3	no	—
30	C₃×3₋¹⁺²	—	9	3	no	—
31	C₃×3₋¹⁺²	H(3,𝔽₃)	1	3	yes	—
32	C₃×H(3,𝔽₃)	—	3	3	no	—
33	C₃×H(3,𝔽₃)	—	3	9	no	—
34	C₃×H(3,𝔽₃)	—	3	6	no	—
35	C₃×H(3,𝔽₃)	—	3	3	no	—
36	C₃×H(3,𝔽₃)	—	3	3	no	—
37	C₃×H(3,𝔽₃)	—	3	6	no	—
38	C₃×H(3,𝔽₃)	—	9	6	no	—
39	C₃²×C₉	—	3	2	no	—
40	C₃²×C₉	—	3	1	no	—
41	C₃²×C₉	—	3	1	no	—
42	C₃²×C₉	—	3	2	no	—
43	C₃²×C₉	—	3	3	no	—
44	C₃²×C₉	—	9	2	no	—
45	C₃²×C₉	H(3,𝔽₃)	1	2	yes	—
46	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
47	#732	—	3	18	no	—
48	#732	—	3	9	no	—
49	#732	—	9	6	no	—
50	#732	—	9	3	no	—
51	AΣL(1,𝔽₂₇)	—	9	27	no	—
52	C₉⋊C₉	—	3	9	no	—
53	C₉⋊C₉	—	3	18	no	—
54	C₃²⋊C₉	—	3	2	no	—
55	C₃²⋊C₉	—	3	2	no	—
56	C₃²⋊C₉	—	3	6	no	—
57	C₃²⋊C₉	—	3	3	no	—
58	C₃²⋊C₉	H(3,𝔽₃)	1	6	yes	—
59	C₃³	—	3	1	no	—
60	C₃³	—	3	2	no	—
61	C₃³	—	3	3	no	—
62	C₃³	—	9	2	no	—
63	C₃³	—	9	9	no	—
64	C₃³	—	9	3	no	—
65	C₃³	—	9	3	no	—
66	C₃³	—	9	9	no	—
67	C₃³	—	27	6	no	—
68	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
69	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
70	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
71	3₋¹⁺²	—	3	9	no	—
72	3₋¹⁺²	—	9	6	no	—
73	3₋¹⁺²	—	9	9	no	—
74	H(3,𝔽₃)	—	3	9	no	—
75	H(3,𝔽₃)	—	9	18	no	—
76	H(3,𝔽₃)	—	9	6	no	—
77	H(3,𝔽₃)	—	9	3	no	—
78	H(3,𝔽₃)	—	9	6	no	—
79	H(3,𝔽₃)	—	9	9	no	—
80	H(3,𝔽₃)	—	27	6	no	—
81	C₃×C₉	—	3	1	no	—
82	C₃×C₉	—	3	2	no	—
83	C₃×C₉	—	3	6	no	—
84	C₃×C₉	—	3	2	no	—
85	C₃×C₉	—	3	3	no	—
86	C₃×C₉	—	3	2	no	—
87	C₃×C₉	—	3	2	no	—
88	C₃×C₉	—	3	2	no	—
89	C₃×C₉	—	9	3	no	—
90	C₃×C₉	—	9	6	no	—
91	C₃×C₉	—	9	6	no	—
92	C₃×C₉	—	9	3	no	—
93	C₃×C₉	—	9	3	no	—
94	C₃×C₉	AΣL(1,𝔽₂₇)	1	3	yes	complemented
95	C₃²	—	3	1	no	—
96	C₃²	—	3	1	no	—
97	C₃²	—	3	2	no	—
98	C₃²	—	3	2	no	—
99	C₃²	—	3	2	no	—
100	C₃²	—	3	2	no	—
101	C₃²	—	3	2	no	—
102	C₃²	—	3	2	no	—
103	C₃²	—	9	3	no	—
104	C₃²	—	9	6	no	—
105	C₃²	—	9	9	no	—
106	C₃²	—	27	27	no	—
107	C₃²	—	27	3	no	—
108	C₃²	—	27	6	no	—
109	C₃²	—	27	3	no	—
110	C₃²	—	27	3	no	—
111	C₃²	—	27	6	no	—
112	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
113	C₃²	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
114	C₃²	#6909	1	2	yes	—
115	C₃²	#6911	1	3	yes	—
116	C₉	—	3	3	no	—
117	C₉	—	9	3	no	—
118	C₉	—	9	2	no	—
119	C₉	—	9	6	no	—
120	C₉	—	27	3	no	—
121	C₃	—	3	3	no	—
122	C₃	—	3	1	no	—
123	C₃	—	3	2	no	—
124	C₃	—	9	2	no	—
125	C₃	—	27	9	no	—
126	C₃	—	81	3	no	—
127	C₃	#10624737	1	1	yes	characteristic
128	C₃	#10624743	1	1	yes	characteristic
129	C₃	#10624757	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#732

C₃

yes

maximal, characteristic, complemented

C₃×#6894

C₃

yes

maximal, characteristic, complemented

#10624768

C₃

yes

maximal, characteristic, complemented

#10624743

C₃

yes

maximal, complemented

#10624732

C₃

yes

maximal, characteristic, complemented

#10624596

C₃

yes

maximal, complemented

C₃²×3₋¹⁺²

C₃²

yes

characteristic

C₃²×H(3,𝔽₃)

C₃²

yes

—

C₃³×C₉

C₃²

yes

—

—

—

—

—

C₃×AΣL(1,𝔽₂₇)

—

#6894

—

#6894

—

#6891

C₃²

yes

—

C₃²⋊H(3,𝔽₃)

C₃²

yes

—

C₃×C₉⋊C₉

—

C₃×C₉⋊C₉

—

C₃×C₃²⋊C₉

C₃²

yes

—

#6868

—

#6868

—

#6856

—

C₃⁴

—

C₃⁴

C₃³

yes

characteristic

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)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

H(3,𝔽₃)

yes

—

C₃²×C₉

yes

—

—

—

—

AΣL(1,𝔽₂₇)

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

H(3,𝔽₃)

yes

—

C₃³

—

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

—

3₋¹⁺²

—

—

—

—

—

—

—

—

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

complemented

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

100

C₃²

—

101

C₃²

—

102

C₃²

—

103

C₃²

—

104

C₃²

—

105

C₃²

—

106

C₃²

—

107

C₃²

—

108

C₃²

—

109

C₃²

—

110

C₃²

—

111

C₃²

—

112

C₃²

C₃²⋊H(3,𝔽₃)

yes

characteristic

113

C₃²

C₃×AΣL(1,𝔽₂₇)

yes

characteristic

114

C₃²

#6909

yes

—

115

C₃²

#6911

yes

—

116

C₉

—

117

C₉

—

118

C₉

—

119

C₉

—

120

C₉

—

121

C₃

—

122

C₃

—

123

C₃

—

124

C₃

—

125

C₃

—

126

C₃

—

127

C₃

#10624737

yes

characteristic

128

C₃

#10624743

yes

characteristic

129

C₃

#10624757

yes

—

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

LCS

G ▷ C₃⁴ ▷ C₃² ▷ C₁

UCS

C₁ ◁ C₃² ◁ C₃⁴ ◁ G

LECS

G ▷ C₃⁴ ▷ C₃² ▷ C₁

N	H ≅ G/N	type
1. C₃²×#732	C₃	semidirect
2. C₃×#6894	C₃	semidirect
3. #10624768	C₃	semidirect
4. #10624743	C₃	semidirect
5. #10624732	C₃	semidirect
6. #10624596	C₃	semidirect
94. C₃×C₉	AΣL(1,𝔽₂₇)	semidirect

H ≅ G/N

type

1. C₃²×#732

C₃

semidirect

2. C₃×#6894

C₃

semidirect

3. #10624768

C₃

semidirect

4. #10624743

C₃

semidirect

5. #10624732

C₃

semidirect

6. #10624596

C₃

semidirect

94. C₃×C₉

AΣL(1,𝔽₂₇)

semidirect

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

subgroup	index
C₃²×#732	3
C₃×#6894	3
#10624768	3
#10624743	3
#10624732	3
#10624596	3

subgroup

index

C₃²×#732

Subgroups — #14547205

Conjugacy classes of subgroups

Complements

Maximal subgroups