Group #14546308 — subgroups

Subgroups — #14546308

Order 2187.

221 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 221 conjugacy classes of subgroups — too many to draw. Only its 41 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	#10624872	C₃	1	1	yes	maximal, characteristic
2	C₃×#6869	C₃	1	1	yes	maximal, characteristic
3	C₃×#6869	C₃	1	1	yes	maximal, characteristic
4	#10624683	C₃	1	1	yes	maximal, characteristic
5	C₃³×C₉	C₃²	1	1	yes	characteristic
6	C₃×C₂₇⋊C₃	—	3	1	no	—
7	C₃×C₂₇⋊C₃	—	3	1	no	—
8	C₃×C₂₇⋊C₃	—	3	1	no	—
9	#6895	—	3	1	no	—
10	#6895	C₉	1	3	yes	—
11	#6894	—	3	3	no	—
12	#6894	—	3	3	no	—
13	#6891	—	3	1	no	—
14	C₉×3₋¹⁺²	—	3	3	no	—
15	#6887	—	3	1	no	—
16	#6869	—	3	3	no	—
17	#6869	—	3	3	no	—
18	C₃²⋊C₂₇	—	3	3	no	—
19	C₃⁴	3₋¹⁺²	1	1	yes	characteristic
20	C₃×3₋¹⁺²	—	3	3	no	—
21	C₃²×C₉	—	3	1	no	—
22	C₃²×C₉	—	3	1	no	—
23	C₃²×C₉	—	3	1	no	—
24	C₃²×C₉	—	3	1	no	—
25	C₃²×C₉	—	3	1	no	—
26	C₃²×C₉	—	3	1	no	—
27	C₃²×C₉	—	3	1	no	—
28	C₃²×C₉	—	3	1	no	—
29	C₃²×C₉	—	3	1	no	—
30	C₃²×C₉	—	3	1	no	—
31	C₃²×C₉	—	3	1	no	—
32	C₃²×C₉	—	3	1	no	—
33	C₃²×C₉	C₃×C₉	1	1	yes	characteristic
34	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
35	C₃²×C₉	3₋¹⁺²	1	1	yes	characteristic
36	C₂₇⋊C₃	—	3	3	no	—
37	C₂₇⋊C₃	—	3	3	no	—
38	C₂₇⋊C₃	—	9	1	no	—
39	C₂₇⋊C₃	—	9	1	no	—
40	C₂₇⋊C₃	—	9	1	no	—
41	C₂₇⋊C₃	—	9	1	no	—
42	C₂₇⋊C₃	—	9	1	no	—
43	C₂₇⋊C₃	—	9	1	no	—
44	C₂₇⋊C₃	—	9	1	no	—
45	C₃×C₂₇	—	3	3	no	—
46	C₃×C₂₇	—	9	1	no	—
47	C₃×C₂₇	—	9	1	no	—
48	C₉⋊C₉	—	3	9	no	—
49	C₉⋊C₉	—	3	9	no	—
50	C₉⋊C₉	—	3	3	no	—
51	C₉⋊C₉	—	3	3	no	—
52	C₃²⋊C₉	—	3	6	no	—
53	C₃²⋊C₉	—	3	3	no	—
54	C₉²	—	3	3	no	—
55	C₃³	—	3	1	no	—
56	C₃³	—	3	1	no	—
57	C₃³	—	3	1	no	—
58	C₃³	—	3	1	no	—
59	C₃³	—	9	3	no	—
60	C₃³	C₃²⋊C₉	1	1	yes	characteristic
61	3₋¹⁺²	—	9	9	no	—
62	C₃×C₉	—	3	9	no	—
63	C₃×C₉	—	3	1	no	—
64	C₃×C₉	—	3	1	no	—
65	C₃×C₉	—	3	1	no	—
66	C₃×C₉	—	3	1	no	—
67	C₃×C₉	—	3	1	no	—
68	C₃×C₉	—	3	1	no	—
69	C₃×C₉	—	3	1	no	—
70	C₃×C₉	—	3	1	no	—
71	C₃×C₉	—	9	1	no	—
72	C₃×C₉	—	9	1	no	—
73	C₃×C₉	—	9	1	no	—
74	C₃×C₉	—	9	1	no	—
75	C₃×C₉	—	9	1	no	—
76	C₃×C₉	—	9	1	no	—
77	C₃×C₉	—	9	1	no	—
78	C₃×C₉	—	9	1	no	—
79	C₃×C₉	—	9	1	no	—
80	C₃×C₉	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
81	C₃×C₉	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
82	C₃×C₉	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
83	C₃×C₉	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
84	C₃×C₉	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
85	C₃×C₉	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
86	C₃×C₉	#731	1	1	yes	characteristic
87	C₃×C₉	#731	1	1	yes	characteristic
88	C₃×C₉	#731	1	1	yes	characteristic
89	C₃×C₉	#733	1	1	yes	characteristic
90	C₃×C₉	#733	1	1	yes	characteristic
91	C₃×C₉	#733	1	1	yes	characteristic
92	C₂₇	—	9	3	no	—
93	C₂₇	—	9	3	no	—
94	C₂₇	—	9	3	no	—
95	C₃²	—	3	3	no	—
96	C₃²	—	3	3	no	—
97	C₃²	—	9	1	no	—
98	C₃²	—	9	3	no	—
99	C₃²	—	9	3	no	—
100	C₃²	—	9	2	no	—
101	C₃²	—	9	2	no	—
102	C₃²	—	9	1	no	—
103	C₃²	#6866	1	1	yes	characteristic
104	C₃²	3₋¹⁺²⋊C₉	1	1	yes	characteristic
105	C₃²	3₋¹⁺²⋊C₉	1	1	yes	characteristic
106	C₃²	3₋¹⁺²⋊C₉	1	1	yes	characteristic
107	C₉	—	9	9	no	—
108	C₉	—	9	1	no	—
109	C₉	—	9	1	no	—
110	C₉	#6857	1	1	yes	characteristic
111	C₉	#6857	1	1	yes	characteristic
112	C₉	#6857	1	1	yes	characteristic
113	C₉	#6858	1	1	yes	characteristic
114	C₉	#6858	1	1	yes	characteristic
115	C₉	#6858	1	1	yes	characteristic
116	C₉	#6860	1	1	yes	characteristic
117	C₉	#6860	1	1	yes	characteristic
118	C₉	#6860	1	1	yes	characteristic
119	C₃	—	3	1	no	—
120	C₃	—	3	1	no	—
121	C₃	—	3	1	no	—
122	C₃	—	3	1	no	—
123	C₃	—	9	3	no	—
124	C₃	#10624452	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10624872

C₃

yes

maximal, characteristic

C₃×#6869

C₃

yes

maximal, characteristic

C₃×#6869

C₃

yes

maximal, characteristic

#10624683

C₃

yes

maximal, characteristic

C₃³×C₉

C₃²

yes

characteristic

C₃×C₂₇⋊C₃

—

C₃×C₂₇⋊C₃

—

C₃×C₂₇⋊C₃

—

#6895

—

#6895

C₉

yes

—

#6894

—

#6894

—

#6891

—

C₉×3₋¹⁺²

—

#6887

—

#6869

—

#6869

—

C₃²⋊C₂₇

—

C₃⁴

3₋¹⁺²

yes

characteristic

C₃×3₋¹⁺²

—

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

C₃×C₉

yes

characteristic

C₃²×C₉

H(3,𝔽₃)

yes

characteristic

C₃²×C₉

3₋¹⁺²

yes

characteristic

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₉²

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃²⋊C₉

yes

characteristic

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

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

AΣL(1,𝔽₂₇)

yes

characteristic

C₃×C₉

AΣL(1,𝔽₂₇)

yes

characteristic

C₃×C₉

AΣL(1,𝔽₂₇)

yes

characteristic

C₃×C₉

AΣL(1,𝔽₂₇)

yes

characteristic

C₃×C₉

AΣL(1,𝔽₂₇)

yes

characteristic

C₃×C₉

AΣL(1,𝔽₂₇)

yes

C₃×C₉

yes

C₃×C₉

yes

C₃×C₉

yes

C₃×C₉

yes

C₃×C₉

yes

C₃×C₉

yes

C₂₇

—

C₂₇

—

C₂₇

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

100

C₃²

—

101

C₃²

—

102

C₃²

—

103

C₃²

#6866

yes

characteristic

104

C₃²

3₋¹⁺²⋊C₉

yes

characteristic

105

C₃²

3₋¹⁺²⋊C₉

yes

characteristic

106

C₃²

3₋¹⁺²⋊C₉

yes

characteristic

107

C₉

—

108

C₉

—

109

C₉

—

110

C₉

#6857

yes

characteristic

111

C₉

#6857

yes

characteristic

112

C₉

#6857

yes

characteristic

113

C₉

#6858

yes

characteristic

114

C₉

#6858

yes

characteristic

115

C₉

#6858

yes

characteristic

116

C₉

#6860

yes

characteristic

117

C₉

#6860

yes

characteristic

118

C₉

#6860

yes

characteristic

119

C₃

—

120

C₃

—

121

C₃

—

122

C₃

—

123

C₃

—

124

C₃

#10624452

yes

characteristic

	subgroup	G/H
center	C₉	#6857
commutator	C₃²×C₉	C₃×C₉
Frattini, cosocle	C₃³×C₉	C₃²
socle	C₃	#10624452
prime radical	C₃⁴	3₋¹⁺²
LCS	G ▷ C₃²×C₉ ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₉ ◁ C₃²×C₉ ◁ C₃³×C₉ ◁ G
LECS	G ▷ C₃³×C₉ ▷ C₃³ ▷ C₃ ▷ C₁

subgroup

G/H

center

C₉

#6857

commutator

C₃²×C₉

C₃×C₉

Frattini, cosocle

C₃³×C₉

C₃²

socle

C₃

#10624452

prime radical

C₃⁴

3₋¹⁺²

LCS

G ▷ C₃²×C₉ ▷ C₃³ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₉ ◁ C₃²×C₉ ◁ C₃³×C₉ ◁ G

LECS

G ▷ C₃³×C₉ ▷ C₃³ ▷ C₃ ▷ C₁

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

subgroup	index
#10624872	3
C₃×#6869	3
#10624683	3

subgroup

index

#10624872

C₃×#6869

#10624683

Subgroups — #14546308

Conjugacy classes of subgroups

Maximal subgroups