Group #14548491 — subgroups

Subgroups — #14548491

Order 2187.

294 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 50 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	#10624902	C₃	1	1	yes	maximal, characteristic
2	#10624776	C₃	1	1	yes	maximal, characteristic
3	#10624744	C₃	1	1	yes	maximal, characteristic
4	#10624673	C₃	1	2	yes	maximal
5	#10624668	C₃	1	1	yes	maximal, characteristic
6	#10624647	C₃	1	2	yes	maximal
7	#10624637	C₃	1	1	yes	maximal, characteristic
8	#10624615	C₃	1	2	yes	maximal
9	#10624611	C₃	1	2	yes	maximal
10	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
11	C₃×#733	—	3	3	no	—
12	#6898	—	3	3	no	—
13	#6896	—	3	3	no	—
14	#6895	—	3	4	no	—
15	#6891	C₃²	1	2	yes	—
16	#6891	C₃²	1	2	yes	—
17	#6891	C₃²	1	2	yes	—
18	#6891	C₃²	1	1	yes	characteristic
19	#6891	C₃²	1	1	yes	characteristic
20	#6891	C₃²	1	1	yes	characteristic
21	#6887	—	3	2	no	—
22	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
23	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
24	#6868	—	3	3	no	—
25	#6861	—	3	3	no	—
26	#6861	—	3	3	no	—
27	#6860	—	3	6	no	—
28	#6858	—	3	2	no	—
29	#6858	—	3	4	no	—
30	#6858	—	3	2	no	—
31	#6858	—	3	4	no	—
32	#6858	—	3	2	no	—
33	#6858	—	3	4	no	—
34	C₃⁴	C₃³	1	1	yes	characteristic
35	C₃×3₋¹⁺²	—	3	1	no	—
36	C₃×3₋¹⁺²	—	3	1	no	—
37	C₃×3₋¹⁺²	—	3	2	no	—
38	C₃×3₋¹⁺²	—	3	1	no	—
39	C₃×3₋¹⁺²	—	3	2	no	—
40	C₃×3₋¹⁺²	—	3	2	no	—
41	C₃×3₋¹⁺²	—	3	2	no	—
42	C₃×3₋¹⁺²	—	9	3	no	—
43	C₃×3₋¹⁺²	H(3,𝔽₃)	1	3	yes	—
44	C₃²×C₉	—	3	1	no	—
45	C₃²×C₉	—	3	2	no	—
46	C₃²×C₉	—	3	1	no	—
47	#733	—	9	9	no	—
48	C₉⋊C₉	—	3	12	no	—
49	C₉⋊C₉	—	3	12	no	—
50	C₃²⋊C₉	—	3	2	no	—
51	C₃²⋊C₉	—	3	2	no	—
52	C₃²⋊C₉	—	3	2	no	—
53	C₃²⋊C₉	—	3	2	no	—
54	C₃²⋊C₉	—	3	1	no	—
55	C₃²⋊C₉	—	3	2	no	—
56	C₃²⋊C₉	—	3	2	no	—
57	C₃²⋊C₉	—	3	2	no	—
58	C₃²⋊C₉	—	3	2	no	—
59	C₃²⋊C₉	—	3	2	no	—
60	C₃²⋊C₉	—	3	2	no	—
61	C₃²⋊C₉	—	3	2	no	—
62	C₃²⋊C₉	—	3	2	no	—
63	C₃²⋊C₉	—	3	1	no	—
64	C₃²⋊C₉	—	3	2	no	—
65	C₃²⋊C₉	—	3	2	no	—
66	C₃²⋊C₉	—	3	1	no	—
67	C₃²⋊C₉	—	3	2	no	—
68	C₃²⋊C₉	H(3,𝔽₃)	1	2	yes	—
69	C₃²⋊C₉	H(3,𝔽₃)	1	4	yes	—
70	C₃²⋊C₉	H(3,𝔽₃)	1	3	yes	—
71	C₉²	—	3	3	no	—
72	C₃³	—	9	1	no	—
73	C₃³	—	9	2	no	—
74	C₃³	—	9	1	no	—
75	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
76	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
77	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
78	3₋¹⁺²	—	3	3	no	—
79	3₋¹⁺²	—	9	2	no	—
80	3₋¹⁺²	—	9	6	no	—
81	3₋¹⁺²	—	9	3	no	—
82	3₋¹⁺²	—	9	3	no	—
83	3₋¹⁺²	—	27	1	no	—
84	3₋¹⁺²	—	27	2	no	—
85	3₋¹⁺²	—	27	2	no	—
86	3₋¹⁺²	—	27	2	no	—
87	C₃×C₉	—	3	3	no	—
88	C₃×C₉	—	3	2	no	—
89	C₃×C₉	—	3	3	no	—
90	C₃×C₉	—	3	3	no	—
91	C₃×C₉	—	3	4	no	—
92	C₃×C₉	—	9	6	no	—
93	C₃×C₉	—	9	2	no	—
94	C₃×C₉	—	9	1	no	—
95	C₃×C₉	—	9	3	no	—
96	C₃×C₉	—	9	1	no	—
97	C₃×C₉	—	9	1	no	—
98	C₃×C₉	—	9	2	no	—
99	C₃×C₉	—	9	2	no	—
100	C₃×C₉	—	9	2	no	—
101	C₃²	—	3	1	no	—
102	C₃²	—	3	1	no	—
103	C₃²	—	3	2	no	—
104	C₃²	—	3	2	no	—
105	C₃²	—	3	2	no	—
106	C₃²	—	3	2	no	—
107	C₃²	—	3	2	no	—
108	C₃²	—	3	1	no	—
109	C₃²	—	3	2	no	—
110	C₃²	—	3	1	no	—
111	C₃²	—	27	3	no	—
112	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
113	C₉	—	9	3	no	—
114	C₉	—	9	3	no	—
115	C₉	—	9	2	no	—
116	C₉	—	9	4	no	—
117	C₉	—	9	3	no	—
118	C₉	—	27	1	no	—
119	C₉	—	27	1	no	—
120	C₉	—	27	2	no	—
121	C₉	—	27	2	no	—
122	C₉	—	27	2	no	—
123	C₃	—	9	1	no	—
124	C₃	—	9	1	no	—
125	C₃	—	9	2	no	—
126	C₃	#10624749	1	1	yes	characteristic
127	C₃	#10624758	1	2	yes	—
128	C₃	#10624763	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10624902

C₃

yes

maximal, characteristic

#10624776

C₃

yes

maximal, characteristic

#10624744

C₃

yes

maximal, characteristic

#10624673

C₃

yes

maximal

#10624668

C₃

yes

maximal, characteristic

#10624647

C₃

yes

maximal

#10624637

C₃

yes

maximal, characteristic

#10624615

C₃

yes

maximal

#10624611

C₃

yes

maximal

C₃²×3₋¹⁺²

C₃²

yes

—

—

—

—

C₃²

yes

—

#6891

C₃²

yes

—

#6891

C₃²

yes

—

#6891

C₃²

yes

characteristic

#6891

C₃²

yes

characteristic

#6891

C₃²

yes

characteristic

#6887

—

C₃×C₃²⋊C₉

C₃²

yes

characteristic

C₃×C₃²⋊C₉

C₃²

yes

—

—

—

—

—

—

—

—

—

—

—

C₃⁴

C₃³

yes

characteristic

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

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

H(3,𝔽₃)

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

#733

—

C₉⋊C₉

—

C₉⋊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

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

H(3,𝔽₃)

yes

—

C₃²⋊C₉

H(3,𝔽₃)

yes

—

C₃²⋊C₉

H(3,𝔽₃)

yes

—

C₉²

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

C₃³

C₃×H(3,𝔽₃)

yes

—

C₃³

C₃×H(3,𝔽₃)

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

—

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

—

100

C₃×C₉

—

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

C₉

—

114

C₉

—

115

C₉

—

116

C₉

—

117

C₉

—

118

C₉

—

119

C₉

—

120

C₉

—

121

C₉

—

122

C₉

—

123

C₃

—

124

C₃

—

125

C₃

—

126

C₃

#10624749

yes

characteristic

127

C₃

#10624758

yes

—

128

C₃

#10624763

yes

characteristic

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

C₃⁴

C₃³

LCS

G ▷ C₃⁴ ▷ C₃² ▷ C₁

UCS

C₁ ◁ C₃² ◁ C₃⁴ ◁ G

LECS

G ▷ C₃⁴ ▷ C₃² ▷ C₁

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

subgroup	index
#10624902	3
#10624776	3
#10624744	3
#10624673	3
#10624668	3
#10624647	3
#10624637	3
#10624615	3
#10624611	3

subgroup

index

Subgroups — #14548491

Conjugacy classes of subgroups

Maximal subgroups