Group #14554455 — subgroups

Subgroups — #14554455

Order 2187.

752 conjugacy classes of subgroups; only the 278 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 278 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×#733	C₃	1	1	yes	maximal, characteristic
2	#10624886	C₃	1	2	yes	maximal
3	C₃×#6895	C₃	1	1	yes	maximal, characteristic
4	#10624863	C₃	1	1	yes	maximal, characteristic
5	#10624802	C₃	1	18	yes	maximal, complemented
6	#10624799	C₃	1	9	yes	maximal, complemented
7	#10624699	C₃	1	4	yes	maximal
8	#10624699	C₃	1	2	yes	maximal
9	C₃×#6868	C₃	1	2	yes	maximal
10	C₃²×3₋¹⁺²	C₃²	1	2	yes	—
11	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
12	C₃³×C₉	C₃²	1	1	yes	characteristic
13	C₃×#733	—	3	9	no	—
14	C₃×#733	C₃²	1	3	yes	—
15	C₃×#733	C₃²	1	6	yes	—
16	#6895	—	3	3	no	—
17	#6895	—	3	6	no	—
18	#6895	—	3	6	no	—
19	#6894	—	3	12	no	—
20	#6894	C₃²	1	18	yes	—
21	C₉×3₋¹⁺²	—	3	6	no	—
22	C₉×3₋¹⁺²	C₃²	1	9	yes	—
23	#6887	—	3	3	no	—
24	C₃×C₉⋊C₉	C₃²	1	3	yes	—
25	C₃×C₉⋊C₉	C₃²	1	6	yes	—
26	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
27	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
28	C₃×C₃²⋊C₉	C₃²	1	4	yes	—
29	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
30	3₋¹⁺²⋊C₉	C₃²	1	36	yes	—
31	3₋¹⁺²⋊C₉	C₃²	1	18	yes	—
32	#6868	C₃²	1	6	yes	—
33	#6868	C₃²	1	12	yes	—
34	C₃⁴	C₃³	1	1	yes	characteristic
35	C₃×3₋¹⁺²	—	3	2	no	—
36	C₃×3₋¹⁺²	—	3	6	no	—
37	C₃×3₋¹⁺²	—	3	4	no	—
38	C₃×3₋¹⁺²	—	3	6	no	—
39	C₃×3₋¹⁺²	—	3	12	no	—
40	C₃×3₋¹⁺²	3₋¹⁺²	1	9	yes	—
41	C₃×3₋¹⁺²	C₃³	1	6	yes	—
42	C₃×3₋¹⁺²	C₃³	1	3	yes	—
43	C₃²×C₉	—	3	1	no	—
44	C₃²×C₉	—	3	1	no	—
45	C₃²×C₉	—	3	2	no	—
46	C₃²×C₉	—	3	2	no	—
47	C₃²×C₉	—	3	3	no	—
48	C₃²×C₉	—	3	6	no	—
49	C₃²×C₉	—	3	2	no	—
50	C₃²×C₉	—	3	4	no	—
51	C₃²×C₉	C₃³	1	1	yes	characteristic
52	C₃²×C₉	C₃³	1	2	yes	—
53	C₃²×C₉	C₃³	1	6	yes	—
54	C₃²×C₉	C₃³	1	3	yes	—
55	#733	—	3	9	no	—
56	#733	—	3	18	no	—
57	C₉⋊C₉	—	3	9	no	—
58	C₉⋊C₉	—	3	18	no	—
59	C₉⋊C₉	—	3	18	no	—
60	C₉⋊C₉	—	3	54	no	—
61	C₃²⋊C₉	—	3	2	no	—
62	C₃²⋊C₉	—	3	2	no	—
63	C₃²⋊C₉	—	3	4	no	—
64	C₃²⋊C₉	—	3	2	no	—
65	C₃²⋊C₉	—	3	4	no	—
66	C₃²⋊C₉	—	3	4	no	—
67	C₃²⋊C₉	H(3,𝔽₃)	1	3	yes	—
68	C₃²⋊C₉	H(3,𝔽₃)	1	6	yes	—
69	C₃²⋊C₉	C₃³	1	6	yes	—
70	C₃²⋊C₉	C₃³	1	12	yes	—
71	C₉²	—	3	9	no	—
72	C₃³	—	9	1	no	—
73	C₃³	—	9	2	no	—
74	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
75	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
76	C₃³	C₃×3₋¹⁺²	1	3	yes	—
77	C₃³	#737	1	6	yes	—
78	C₃³	C₃⁴	1	1	yes	characteristic
79	3₋¹⁺²	—	3	6	no	—
80	3₋¹⁺²	—	3	18	no	—
81	3₋¹⁺²	—	3	6	no	—
82	3₋¹⁺²	—	3	12	no	—
83	3₋¹⁺²	—	9	12	no	—
84	3₋¹⁺²	C₃×3₋¹⁺²	1	9	yes	—
85	C₃×C₉	—	3	3	no	—
86	C₃×C₉	—	3	9	no	—
87	C₃×C₉	—	3	6	no	—
88	C₃×C₉	—	3	6	no	—
89	C₃×C₉	—	3	12	no	—
90	C₃×C₉	—	3	1	no	—
91	C₃×C₉	—	3	2	no	—
92	C₃×C₉	—	3	6	no	—
93	C₃×C₉	—	3	1	no	—
94	C₃×C₉	—	3	2	no	—
95	C₃×C₉	—	3	2	no	—
96	C₃×C₉	—	3	4	no	—
97	C₃×C₉	—	9	3	no	—
98	C₃×C₉	—	9	3	no	—
99	C₃×C₉	—	9	6	no	—
100	C₃×C₉	—	9	6	no	—
101	C₃×C₉	—	9	6	no	—
102	C₃×C₉	—	27	2	no	—
103	C₃×C₉	—	27	4	no	—
104	C₃×C₉	C₃×H(3,𝔽₃)	1	9	yes	—
105	C₃²	—	3	1	no	—
106	C₃²	—	3	1	no	—
107	C₃²	—	3	2	no	—
108	C₃²	—	3	2	no	—
109	C₃²	—	3	2	no	—
110	C₃²	—	3	2	no	—
111	C₃²	—	3	2	no	—
112	C₃²	—	9	3	no	—
113	C₃²	—	9	6	no	—
114	C₃²	C₉⋊H(3,𝔽₃)	1	3	yes	—
115	C₃²	C₉⋊H(3,𝔽₃)	1	6	yes	—
116	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
117	C₃²	C₃²×3₋¹⁺²	1	1	yes	characteristic
118	C₃²	C₃×#737	1	2	yes	—
119	C₉	—	3	9	no	—
120	C₉	—	3	9	no	—
121	C₉	—	9	1	no	—
122	C₉	—	9	2	no	—
123	C₉	—	9	6	no	—
124	C₉	—	9	6	no	—
125	C₉	—	9	6	no	—
126	C₉	—	9	12	no	—
127	C₃	—	3	1	no	—
128	C₃	—	3	2	no	—
129	C₃	—	3	6	no	—
130	C₃	—	3	3	no	—
131	C₃	C₃×C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
132	C₃	#10624933	1	1	yes	characteristic
133	C₃	#10624936	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#733

C₃

yes

maximal, characteristic

#10624886

C₃

yes

maximal

C₃×#6895

C₃

yes

maximal, characteristic

#10624863

C₃

yes

maximal, characteristic

#10624802

C₃

yes

maximal, complemented

#10624799

C₃

yes

maximal, complemented

#10624699

C₃

yes

maximal

#10624699

C₃

yes

maximal

C₃×#6868

C₃

yes

maximal

C₃²×3₋¹⁺²

C₃²

yes

—

C₃²×3₋¹⁺²

C₃²

yes

characteristic

C₃³×C₉

C₃²

yes

characteristic

C₃×#733

—

C₃×#733

C₃²

yes

—

C₃×#733

C₃²

yes

—

—

—

—

—

C₃²

yes

—

C₉×3₋¹⁺²

—

C₉×3₋¹⁺²

C₃²

yes

—

#6887

—

C₃×C₉⋊C₉

C₃²

yes

—

C₃×C₉⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

characteristic

3₋¹⁺²⋊C₉

C₃²

yes

—

3₋¹⁺²⋊C₉

C₃²

yes

—

#6868

C₃²

yes

—

#6868

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

3₋¹⁺²

yes

—

C₃×3₋¹⁺²

C₃³

yes

—

C₃×3₋¹⁺²

C₃³

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃³

yes

characteristic

C₃²×C₉

C₃³

yes

—

C₃²×C₉

C₃³

yes

—

C₃²×C₉

C₃³

yes

—

#733

—

#733

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊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

C₃³

yes

—

C₃²⋊C₉

C₃³

yes

—

C₉²

—

C₃³

—

C₃³

—

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

C₃³

C₃×H(3,𝔽₃)

yes

—

C₃³

C₃×3₋¹⁺²

yes

—

C₃³

#737

yes

—

C₃³

C₃⁴

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

C₃×3₋¹⁺²

yes

—

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

—

100

C₃×C₉

—

101

C₃×C₉

—

102

C₃×C₉

—

103

C₃×C₉

—

104

C₃×C₉

C₃×H(3,𝔽₃)

yes

—

105

C₃²

—

106

C₃²

—

107

C₃²

—

108

C₃²

—

109

C₃²

—

110

C₃²

—

111

C₃²

—

112

C₃²

—

113

C₃²

—

114

C₃²

C₉⋊H(3,𝔽₃)

yes

—

115

C₃²

C₉⋊H(3,𝔽₃)

yes

—

116

C₃²

C₃²×H(3,𝔽₃)

yes

characteristic

117

C₃²

C₃²×3₋¹⁺²

yes

characteristic

118

C₃²

C₃×#737

yes

—

119

C₉

—

120

C₉

—

121

C₉

—

122

C₉

—

123

C₉

—

124

C₉

—

125

C₉

—

126

C₉

—

127

C₃

—

128

C₃

—

129

C₃

—

130

C₃

—

131

C₃

C₃×C₉⋊H(3,𝔽₃)

yes

characteristic

132

C₃

#10624933

yes

characteristic

133

C₃

#10624936

yes

—

	subgroup	G/H
center, socle	C₃²	C₃²×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃⁴
prime radical	C₃⁴	C₃³
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃×C₃²⋊C₉ ◁ G
LECS	G ▷ C₃³ ▷ C₃ ▷ C₁

subgroup

G/H

center, socle

C₃²

C₃²×H(3,𝔽₃)

commutator, Frattini, cosocle

C₃³

C₃⁴

prime radical

C₃⁴

C₃³

LCS

G ▷ C₃³ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃² ◁ C₃×C₃²⋊C₉ ◁ G

LECS

G ▷ C₃³ ▷ C₃ ▷ C₁

N	H ≅ G/N	type
5. #10624802	C₃	semidirect
6. #10624799	C₃	semidirect

H ≅ G/N

type

5. #10624802

C₃

semidirect

6. #10624799

C₃

semidirect

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

subgroup	index
C₃²×#733	3
#10624886	3
C₃×#6895	3
#10624863	3
#10624802	3
#10624799	3
#10624699	3
C₃×#6868	3

subgroup

index

C₃²×#733

Subgroups — #14554455

Conjugacy classes of subgroups

Complements

Maximal subgroups