Group #14550661 — subgroups

Subgroups — #14550661

Order 2187.

1577 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 1577 conjugacy classes of subgroups — too many to draw. Only its 107 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	C₃³×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic
2	C₃×#6891	C₃	1	1	yes	maximal, characteristic
3	C₃×3₋¹⁺²⋊C₉	C₃	1	9	yes	maximal, complemented
4	#10624547	C₃	1	1	yes	maximal, characteristic
5	#10624547	C₃	1	1	yes	maximal, characteristic
6	C₃⁵	C₃²	1	1	yes	characteristic
7	C₃²×3₋¹⁺²	—	3	3	no	—
8	C₃²×3₋¹⁺²	—	3	27	no	—
9	C₃²×3₋¹⁺²	—	3	9	no	—
10	C₃²×3₋¹⁺²	C₃²	1	3	yes	—
11	C₃³×C₉	C₃²	1	3	yes	—
12	#6891	—	3	9	no	—
13	C₃×C₃²⋊C₉	—	3	3	no	—
14	C₃×C₃²⋊C₉	—	3	3	no	—
15	C₃×C₃²⋊C₉	—	3	3	no	—
16	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
17	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
18	3₋¹⁺²⋊C₉	—	3	27	no	—
19	C₃⁴	—	3	9	no	—
20	C₃⁴	—	3	3	no	—
21	C₃⁴	—	3	3	no	—
22	C₃⁴	—	3	3	no	—
23	C₃⁴	—	3	3	no	—
24	C₃⁴	—	9	3	no	—
25	C₃⁴	—	9	3	no	—
26	C₃⁴	H(3,𝔽₃)	1	3	yes	—
27	C₃⁴	C₃³	1	1	yes	characteristic
28	C₃×3₋¹⁺²	—	3	243	no	—
29	C₃×3₋¹⁺²	—	3	27	no	—
30	C₃×3₋¹⁺²	—	3	27	no	—
31	C₃×3₋¹⁺²	—	3	27	no	—
32	C₃×3₋¹⁺²	—	3	27	no	—
33	C₃×3₋¹⁺²	—	3	6	no	—
34	C₃×3₋¹⁺²	—	3	9	no	—
35	C₃×3₋¹⁺²	—	3	9	no	—
36	C₃×3₋¹⁺²	—	3	9	no	—
37	C₃×3₋¹⁺²	3₋¹⁺²	1	9	yes	complemented
38	C₃²×C₉	—	3	6	no	—
39	C₃²×C₉	—	3	3	no	—
40	C₃²×C₉	—	3	27	no	—
41	C₃²×C₉	—	3	3	no	—
42	C₃²×C₉	—	3	3	no	—
43	C₃²×C₉	—	3	3	no	—
44	C₃²×C₉	H(3,𝔽₃)	1	9	yes	—
45	C₃²⋊C₉	—	3	27	no	—
46	C₃²⋊C₉	—	3	9	no	—
47	C₃²⋊C₉	—	3	9	no	—
48	C₃²⋊C₉	—	9	9	no	—
49	C₃²⋊C₉	—	9	9	no	—
50	C₃³	—	3	3	no	—
51	C₃³	—	3	3	no	—
52	C₃³	—	3	3	no	—
53	C₃³	—	3	27	no	—
54	C₃³	—	3	3	no	—
55	C₃³	—	3	3	no	—
56	C₃³	—	3	3	no	—
57	C₃³	—	3	3	no	—
58	C₃³	—	3	3	no	—
59	C₃³	—	3	3	no	—
60	C₃³	—	3	3	no	—
61	C₃³	—	3	3	no	—
62	C₃³	—	3	3	no	—
63	C₃³	—	3	3	no	—
64	C₃³	—	3	3	no	—
65	C₃³	—	3	3	no	—
66	C₃³	—	9	27	no	—
67	C₃³	—	9	27	no	—
68	C₃³	—	9	27	no	—
69	C₃³	—	9	9	no	—
70	C₃³	—	9	9	no	—
71	C₃³	—	9	9	no	—
72	C₃³	AΣL(1,𝔽₂₇)	1	9	yes	—
73	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
74	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
75	C₃³	C₃×3₋¹⁺²	1	3	yes	—
76	C₃³	#737	1	3	yes	—
77	C₃³	#737	1	3	yes	—
78	3₋¹⁺²	—	3	243	no	—
79	3₋¹⁺²	—	3	27	no	—
80	3₋¹⁺²	—	9	18	no	—
81	C₃×C₉	—	3	54	no	—
82	C₃×C₉	—	3	9	no	—
83	C₃×C₉	—	3	27	no	—
84	C₃×C₉	—	3	27	no	—
85	C₃×C₉	—	3	9	no	—
86	C₃×C₉	—	9	3	no	—
87	C₃×C₉	—	9	3	no	—
88	C₃×C₉	—	9	3	no	—
89	C₃×C₉	—	9	3	no	—
90	C₃×C₉	—	9	3	no	—
91	C₃×C₉	—	9	9	no	—
92	C₃×C₉	—	9	9	no	—
93	C₃²	—	3	9	no	—
94	C₃²	—	3	9	no	—
95	C₃²	—	3	9	no	—
96	C₃²	—	3	3	no	—
97	C₃²	—	3	3	no	—
98	C₃²	—	3	3	no	—
99	C₃²	—	3	9	no	—
100	C₃²	—	3	9	no	—
101	C₃²	—	3	9	no	—
102	C₃²	—	3	9	no	—
103	C₃²	—	9	81	no	—
104	C₃²	—	9	3	no	—
105	C₃²	—	9	3	no	—
106	C₃²	—	9	3	no	—
107	C₃²	—	9	3	no	—
108	C₃²	—	9	3	no	—
109	C₃²	—	9	3	no	—
110	C₃²	—	9	3	no	—
111	C₃²	—	9	3	no	—
112	C₃²	—	9	3	no	—
113	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
114	C₃²	#6891	1	1	yes	characteristic
115	C₃²	C₉⋊H(3,𝔽₃)	1	9	yes	—
116	C₃²	#6893	1	1	yes	characteristic
117	C₃²	#6893	1	1	yes	characteristic
118	C₃²	C₃×AΣL(1,𝔽₂₇)	1	3	yes	—
119	C₃²	#6910	1	3	yes	—
120	C₃²	#6912	1	3	yes	—
121	C₉	—	3	27	no	—
122	C₉	—	9	9	no	—
123	C₉	—	27	3	no	—
124	C₉	—	27	3	no	—
125	C₃	—	3	9	no	—
126	C₃	—	9	9	no	—
127	C₃	#10624563	1	1	yes	characteristic
128	C₃	#10624730	1	1	yes	characteristic
129	C₃	#10624736	1	1	yes	characteristic
130	C₃	#10624738	1	1	yes	characteristic
131	C₃	#10624781	1	3	yes	—
132	C₃	#10624823	1	3	yes	—
133	C₃	#10624824	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃³×3₋¹⁺²

C₃

yes

maximal, characteristic

C₃×#6891

C₃

yes

maximal, characteristic

C₃×3₋¹⁺²⋊C₉

C₃

yes

maximal, complemented

#10624547

C₃

yes

maximal, characteristic

#10624547

C₃

yes

maximal, characteristic

C₃⁵

C₃²

yes

characteristic

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

C₃²

yes

—

C₃³×C₉

C₃²

yes

—

#6891

—

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

—

3₋¹⁺²⋊C₉

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

H(3,𝔽₃)

yes

—

C₃⁴

C₃³

yes

characteristic

C₃×3₋¹⁺²

—

243

—

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

3₋¹⁺²

yes

complemented

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

H(3,𝔽₃)

yes

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

AΣL(1,𝔽₂₇)

yes

—

C₃³

C₃×H(3,𝔽₃)

yes

—

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

C₃³

C₃×3₋¹⁺²

yes

—

C₃³

#737

yes

—

C₃³

#737

yes

—

3₋¹⁺²

—

243

—

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

—

C₃²

—

C₃²

—

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

—

113

C₃²

C₃²⋊H(3,𝔽₃)

yes

characteristic

114

C₃²

#6891

yes

characteristic

115

C₃²

C₉⋊H(3,𝔽₃)

yes

—

116

C₃²

#6893

yes

characteristic

117

C₃²

#6893

yes

characteristic

118

C₃²

C₃×AΣL(1,𝔽₂₇)

yes

—

119

C₃²

#6910

yes

—

120

C₃²

#6912

yes

—

121

C₉

—

122

C₉

—

123

C₉

—

124

C₉

—

125

C₃

—

126

C₃

—

127

C₃

#10624563

yes

characteristic

128

C₃

#10624730

yes

characteristic

129

C₃

#10624736

yes

characteristic

130

C₃

#10624738

yes

characteristic

131

C₃

#10624781

yes

—

132

C₃

#10624823

yes

—

133

C₃

#10624824

yes

—

	subgroup	G/H
center, socle	C₃³	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃⁴	C₃³
prime radical	C₃⁵	C₃²
LCS	G ▷ C₃⁴ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃³ ◁ C₃⁵ ◁ G
LECS	G ▷ C₃⁴ ▷ C₃ ▷ C₁

subgroup

G/H

center, socle

C₃³

C₃×H(3,𝔽₃)

commutator, Frattini, cosocle

C₃⁴

C₃³

prime radical

C₃⁵

C₃²

LCS

G ▷ C₃⁴ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃³ ◁ C₃⁵ ◁ G

LECS

G ▷ C₃⁴ ▷ C₃ ▷ C₁

N	H ≅ G/N	type
3. C₃×3₋¹⁺²⋊C₉	C₃	semidirect
37. C₃×3₋¹⁺²	3₋¹⁺²	semidirect

H ≅ G/N

type

3. C₃×3₋¹⁺²⋊C₉

C₃

semidirect

37. C₃×3₋¹⁺²

3₋¹⁺²

semidirect

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

subgroup	index
C₃³×3₋¹⁺²	3
C₃×#6891	3
C₃×3₋¹⁺²⋊C₉	3
#10624547	3

subgroup

index

C₃³×3₋¹⁺²

C₃×#6891

C₃×3₋¹⁺²⋊C₉

#10624547

Subgroups — #14550661

Conjugacy classes of subgroups

Complements

Maximal subgroups