Group #14548181 — subgroups

Subgroups — #14548181

Order 2187.

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

normal subgroups only

This group has 300 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	#10624913	C₃	1	1	yes	maximal, characteristic
2	#10624812	C₃	1	2	yes	maximal
3	#10624767	C₃	1	1	yes	maximal, characteristic
4	#10624733	C₃	1	1	yes	maximal, characteristic
5	#10624668	C₃	1	1	yes	maximal, characteristic
6	#10624668	C₃	1	1	yes	maximal, characteristic
7	#10624660	C₃	1	2	yes	maximal
8	#10624659	C₃	1	2	yes	maximal
9	#10624659	C₃	1	2	yes	maximal
10	C₃³×C₉	C₃²	1	1	yes	characteristic
11	C₃×#733	—	3	3	no	—
12	#6898	—	3	6	no	—
13	#6896	—	3	3	no	—
14	#6896	—	3	3	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	#6891	C₃²	1	2	yes	—
22	#6891	C₃²	1	1	yes	characteristic
23	3₋¹⁺²⋊C₉	—	3	6	no	—
24	#6867	—	3	3	no	—
25	#6861	—	3	3	no	—
26	#6861	—	3	3	no	—
27	#6858	—	3	6	no	—
28	#6858	—	3	6	no	—
29	#6858	—	3	6	no	—
30	C₃⁴	C₃³	1	1	yes	characteristic
31	C₃×3₋¹⁺²	—	3	2	no	—
32	C₃×3₋¹⁺²	—	3	1	no	—
33	C₃×3₋¹⁺²	—	3	2	no	—
34	C₃×3₋¹⁺²	—	3	1	no	—
35	C₃×3₋¹⁺²	—	3	1	no	—
36	C₃×3₋¹⁺²	—	3	2	no	—
37	C₃×3₋¹⁺²	—	3	2	no	—
38	C₃×3₋¹⁺²	H(3,𝔽₃)	1	3	yes	—
39	C₃²×C₉	—	3	1	no	—
40	C₃²×C₉	—	3	1	no	—
41	C₃²×C₉	—	3	2	no	—
42	C₃²×C₉	—	9	3	no	—
43	#733	—	9	9	no	—
44	C₉⋊C₉	—	3	24	no	—
45	C₃²⋊C₉	—	3	1	no	—
46	C₃²⋊C₉	—	3	2	no	—
47	C₃²⋊C₉	—	3	2	no	—
48	C₃²⋊C₉	—	3	2	no	—
49	C₃²⋊C₉	—	3	2	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	2	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	1	no	—
60	C₃²⋊C₉	—	3	1	no	—
61	C₃²⋊C₉	—	3	2	no	—
62	C₃²⋊C₉	—	3	2	no	—
63	C₃²⋊C₉	H(3,𝔽₃)	1	6	yes	—
64	C₃²⋊C₉	H(3,𝔽₃)	1	3	yes	—
65	C₉²	—	3	3	no	—
66	C₃³	—	9	1	no	—
67	C₃³	—	9	1	no	—
68	C₃³	—	9	2	no	—
69	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
70	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
71	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
72	3₋¹⁺²	—	9	3	no	—
73	3₋¹⁺²	—	9	3	no	—
74	3₋¹⁺²	—	9	3	no	—
75	3₋¹⁺²	—	9	6	no	—
76	3₋¹⁺²	—	27	1	no	—
77	3₋¹⁺²	—	27	2	no	—
78	3₋¹⁺²	—	27	2	no	—
79	3₋¹⁺²	—	27	2	no	—
80	C₃×C₉	—	3	3	no	—
81	C₃×C₉	—	3	3	no	—
82	C₃×C₉	—	3	3	no	—
83	C₃×C₉	—	3	3	no	—
84	C₃×C₉	—	3	6	no	—
85	C₃×C₉	—	9	2	no	—
86	C₃×C₉	—	9	2	no	—
87	C₃×C₉	—	9	1	no	—
88	C₃×C₉	—	9	6	no	—
89	C₃×C₉	—	9	1	no	—
90	C₃×C₉	—	9	1	no	—
91	C₃×C₉	—	9	2	no	—
92	C₃×C₉	—	9	2	no	—
93	C₃×C₉	—	9	2	no	—
94	C₃²	—	3	1	no	—
95	C₃²	—	3	1	no	—
96	C₃²	—	3	2	no	—
97	C₃²	—	3	2	no	—
98	C₃²	—	3	1	no	—
99	C₃²	—	3	1	no	—
100	C₃²	—	3	2	no	—
101	C₃²	—	3	2	no	—
102	C₃²	—	3	2	no	—
103	C₃²	—	3	2	no	—
104	C₃²	—	9	9	no	—
105	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
106	C₉	—	9	3	no	—
107	C₉	—	9	3	no	—
108	C₉	—	9	3	no	—
109	C₉	—	9	6	no	—
110	C₉	—	27	1	no	—
111	C₉	—	27	1	no	—
112	C₉	—	27	2	no	—
113	C₉	—	27	2	no	—
114	C₉	—	27	2	no	—
115	C₃	—	9	1	no	—
116	C₃	—	9	1	no	—
117	C₃	—	9	2	no	—
118	C₃	#10624749	1	1	yes	characteristic
119	C₃	#10624751	1	1	yes	characteristic
120	C₃	#10624764	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10624913

C₃

yes

maximal, characteristic

#10624812

C₃

yes

maximal

#10624767

C₃

yes

maximal, characteristic

#10624733

C₃

yes

maximal, characteristic

#10624668

C₃

yes

maximal, characteristic

#10624668

C₃

yes

maximal, characteristic

#10624660

C₃

yes

maximal

#10624659

C₃

yes

maximal

#10624659

C₃

yes

maximal

C₃³×C₉

C₃²

yes

—

—

—

—

C₃²

yes

—

#6891

C₃²

yes

—

#6891

C₃²

yes

—

#6891

C₃²

yes

characteristic

#6891

C₃²

yes

characteristic

#6891

C₃²

yes

characteristic

#6891

C₃²

yes

—

#6891

C₃²

yes

characteristic

3₋¹⁺²⋊C₉

—

—

—

—

—

—

—

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

H(3,𝔽₃)

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

#733

—

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

—

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

—

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

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

100

C₃²

—

101

C₃²

—

102

C₃²

—

103

C₃²

—

104

C₃²

—

105

C₃²

C₃²⋊H(3,𝔽₃)

yes

characteristic

106

C₉

—

107

C₉

—

108

C₉

—

109

C₉

—

110

C₉

—

111

C₉

—

112

C₉

—

113

C₉

—

114

C₉

—

115

C₃

—

116

C₃

—

117

C₃

—

118

C₃

#10624749

yes

characteristic

119

C₃

#10624751

yes

characteristic

120

C₃

#10624764

yes

—

	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
#10624913	3
#10624812	3
#10624767	3
#10624733	3
#10624668	3
#10624660	3
#10624659	3

subgroup

index

Subgroups — #14548181

Conjugacy classes of subgroups

Maximal subgroups