Group #14550232 — subgroups

Subgroups — #14550232

Order 2187.

584 conjugacy classes of subgroups; only the 314 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 314 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×C₂₇⋊C₃	C₃	1	2	yes	maximal
2	C₃²×C₂₇⋊C₃	C₃	1	1	yes	maximal, characteristic
3	C₃×#6887	C₃	1	1	yes	maximal, characteristic
4	#10624472	C₃	1	9	yes	maximal, complemented
5	C₃³×C₉	C₃²	1	1	yes	characteristic
6	C₃×C₂₇⋊C₃	—	3	6	no	—
7	C₃×C₂₇⋊C₃	—	3	3	no	—
8	C₃×C₂₇⋊C₃	—	3	6	no	—
9	C₃×C₂₇⋊C₃	—	3	6	no	—
10	C₃×C₂₇⋊C₃	—	3	2	no	—
11	C₃×C₂₇⋊C₃	—	3	2	no	—
12	C₃×C₂₇⋊C₃	C₉	1	18	yes	complemented
13	C₃×C₂₇⋊C₃	C₉	1	6	yes	complemented
14	C₃×C₂₇⋊C₃	C₃²	1	6	yes	—
15	C₃×C₂₇⋊C₃	C₃²	1	3	yes	—
16	C₃²×C₂₇	—	3	2	no	—
17	C₃²×C₂₇	C₉	1	3	yes	complemented
18	#6887	—	3	3	no	—
19	#6887	—	3	3	no	—
20	#6887	C₉	1	9	yes	—
21	C₃×C₉⋊C₉	—	3	1	no	—
22	C₃×C₉⋊C₉	C₃²	1	3	yes	—
23	C₃×C₃²⋊C₉	—	3	1	no	—
24	C₃×C₉²	—	3	1	no	—
25	C₃⁴	3₋¹⁺²	1	1	yes	characteristic
26	C₃²×C₉	—	3	3	no	—
27	C₃²×C₉	C₃×C₉	1	6	yes	—
28	C₃²×C₉	C₃×C₉	1	2	yes	—
29	C₃²×C₉	C₃×C₉	1	1	yes	characteristic
30	C₃²×C₉	C₃×C₉	1	3	yes	—
31	C₃²×C₉	H(3,𝔽₃)	1	3	yes	—
32	C₃²×C₉	3₋¹⁺²	1	1	yes	characteristic
33	C₃²×C₉	3₋¹⁺²	1	1	yes	characteristic
34	C₃²×C₉	3₋¹⁺²	1	6	yes	—
35	C₃²×C₉	3₋¹⁺²	1	6	yes	—
36	C₃²×C₉	3₋¹⁺²	1	2	yes	—
37	C₃²×C₉	3₋¹⁺²	1	2	yes	—
38	C₃²×C₉	3₋¹⁺²	1	2	yes	—
39	C₃²×C₉	3₋¹⁺²	1	3	yes	—
40	C₃²×C₉	C₃³	1	1	yes	characteristic
41	C₂₇⋊C₃	—	3	18	no	—
42	C₂₇⋊C₃	—	3	18	no	—
43	C₂₇⋊C₃	—	3	18	no	—
44	C₂₇⋊C₃	—	3	6	no	—
45	C₂₇⋊C₃	—	3	3	no	—
46	C₂₇⋊C₃	—	3	6	no	—
47	C₂₇⋊C₃	—	3	6	no	—
48	C₂₇⋊C₃	C₃×C₉	1	18	yes	—
49	C₃×C₂₇	—	3	18	no	—
50	C₃×C₂₇	—	3	9	no	—
51	C₃×C₂₇	—	3	6	no	—
52	C₃×C₂₇	C₃×C₉	1	9	yes	—
53	C₉⋊C₉	—	3	9	no	—
54	C₉⋊C₉	—	3	3	no	—
55	C₉⋊C₉	—	3	3	no	—
56	C₉⋊C₉	C₃×C₉	1	9	yes	—
57	C₃²⋊C₉	—	3	9	no	—
58	C₉²	—	3	9	no	—
59	C₃³	—	9	3	no	—
60	C₃³	C₃²⋊C₉	1	3	yes	—
61	C₃³	C₉⋊C₉	1	2	yes	—
62	C₃³	C₉⋊C₉	1	1	yes	characteristic
63	C₃³	C₉⋊C₉	1	6	yes	—
64	C₃³	C₃×3₋¹⁺²	1	1	yes	characteristic
65	C₃×C₉	—	3	9	no	—
66	C₃×C₉	—	9	9	no	—
67	C₃×C₉	C₉²	1	6	yes	—
68	C₃×C₉	C₉²	1	3	yes	—
69	C₃×C₉	C₃²⋊C₉	1	3	yes	—
70	C₃×C₉	C₃²⋊C₉	1	3	yes	—
71	C₃×C₉	C₃²⋊C₉	1	6	yes	—
72	C₃×C₉	C₃²⋊C₉	1	3	yes	—
73	C₃×C₉	C₃²⋊C₉	1	6	yes	—
74	C₃×C₉	C₃²⋊C₉	1	6	yes	—
75	C₃×C₉	C₃²⋊C₉	1	6	yes	—
76	C₃×C₉	C₉⋊C₉	1	6	yes	—
77	C₃×C₉	C₉⋊C₉	1	2	yes	—
78	C₃×C₉	C₉⋊C₉	1	2	yes	—
79	C₃×C₉	C₉⋊C₉	1	1	yes	characteristic
80	C₃×C₉	C₉⋊C₉	1	6	yes	—
81	C₃×C₉	C₉⋊C₉	1	6	yes	—
82	C₃×C₉	C₉⋊C₉	1	6	yes	—
83	C₃×C₉	C₉⋊C₉	1	6	yes	—
84	C₃×C₉	C₉⋊C₉	1	6	yes	—
85	C₃×C₉	C₉⋊C₉	1	6	yes	—
86	C₃×C₉	C₉⋊C₉	1	6	yes	—
87	C₃×C₉	C₉⋊C₉	1	3	yes	—
88	C₃×C₉	C₉⋊C₉	1	6	yes	—
89	C₃×C₉	C₉⋊C₉	1	1	yes	characteristic
90	C₃×C₉	C₃²×C₉	1	1	yes	characteristic
91	C₃×C₉	C₃²×C₉	1	2	yes	—
92	C₃×C₉	C₃²×C₉	1	1	yes	characteristic
93	C₃×C₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
94	C₃×C₉	C₃×3₋¹⁺²	1	2	yes	—
95	C₃×C₉	C₃×3₋¹⁺²	1	2	yes	—
96	C₃×C₉	C₃×3₋¹⁺²	1	2	yes	—
97	C₃×C₉	C₃×3₋¹⁺²	1	1	yes	characteristic
98	C₂₇	—	3	18	no	—
99	C₂₇	—	3	9	no	—
100	C₃²	—	3	3	no	—
101	C₃²	—	3	6	no	—
102	C₃²	—	3	3	no	—
103	C₃²	—	9	3	no	—
104	C₃²	—	9	6	no	—
105	C₃²	#6855	1	6	yes	—
106	C₃²	#6855	1	3	yes	—
107	C₃²	C₃×C₃²⋊C₉	1	1	yes	characteristic
108	C₃²	C₃×C₉⋊C₉	1	2	yes	—
109	C₃²	C₃×C₉⋊C₉	1	1	yes	characteristic
110	C₉	—	9	9	no	—
111	C₉	#6855	1	6	yes	—
112	C₉	#6855	1	6	yes	—
113	C₉	#6855	1	3	yes	—
114	C₉	#6855	1	3	yes	—
115	C₉	C₃×C₉²	1	1	yes	characteristic
116	C₉	C₃×C₃²⋊C₉	1	2	yes	—
117	C₉	C₃×C₃²⋊C₉	1	1	yes	characteristic
118	C₉	C₃×C₉⋊C₉	1	1	yes	characteristic
119	C₉	C₃×C₉⋊C₉	1	2	yes	—
120	C₉	C₃×C₉⋊C₉	1	2	yes	—
121	C₃	—	3	1	no	—
122	C₃	—	3	6	no	—
123	C₃	—	3	3	no	—
124	C₃	—	3	2	no	—
125	C₃	C₃×#6855	1	1	yes	characteristic
126	C₃	#10624709	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×C₂₇⋊C₃

C₃

yes

maximal

C₃²×C₂₇⋊C₃

C₃

yes

maximal, characteristic

C₃×#6887

C₃

yes

maximal, characteristic

#10624472

C₃

yes

maximal, complemented

C₃³×C₉

C₃²

yes

characteristic

C₃×C₂₇⋊C₃

—

C₃×C₂₇⋊C₃

—

C₃×C₂₇⋊C₃

—

C₃×C₂₇⋊C₃

—

C₃×C₂₇⋊C₃

—

C₃×C₂₇⋊C₃

—

C₃×C₂₇⋊C₃

C₉

yes

complemented

C₃×C₂₇⋊C₃

C₉

yes

complemented

C₃×C₂₇⋊C₃

C₃²

yes

—

C₃×C₂₇⋊C₃

C₃²

yes

—

C₃²×C₂₇

—

C₃²×C₂₇

C₉

yes

—

—

C₉

yes

—

C₃×C₉⋊C₉

—

C₃×C₉⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

—

C₃×C₉²

—

C₃⁴

3₋¹⁺²

yes

characteristic

C₃²×C₉

—

C₃²×C₉

C₃×C₉

yes

—

C₃²×C₉

C₃×C₉

yes

—

C₃²×C₉

C₃×C₉

yes

characteristic

C₃²×C₉

C₃×C₉

yes

—

C₃²×C₉

H(3,𝔽₃)

yes

—

C₃²×C₉

3₋¹⁺²

yes

characteristic

C₃²×C₉

3₋¹⁺²

yes

characteristic

C₃²×C₉

3₋¹⁺²

yes

—

C₃²×C₉

3₋¹⁺²

yes

—

C₃²×C₉

3₋¹⁺²

yes

—

C₃²×C₉

3₋¹⁺²

yes

—

C₃²×C₉

3₋¹⁺²

yes

—

C₃²×C₉

3₋¹⁺²

yes

—

C₃²×C₉

C₃³

yes

characteristic

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

C₃×C₉

yes

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

C₃×C₉

yes

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

C₃×C₉

yes

—

C₃²⋊C₉

—

C₉²

—

C₃³

—

C₃³

C₃²⋊C₉

yes

—

C₃³

C₉⋊C₉

yes

—

C₃³

C₉⋊C₉

yes

characteristic

C₃³

C₉⋊C₉

yes

—

C₃³

C₃×3₋¹⁺²

yes

characteristic

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₉²

yes

—

C₃×C₉

C₉²

yes

—

C₃×C₉

C₃²⋊C₉

yes

—

C₃×C₉

C₃²⋊C₉

yes

—

C₃×C₉

C₃²⋊C₉

yes

—

C₃×C₉

C₃²⋊C₉

yes

—

C₃×C₉

C₃²⋊C₉

yes

—

C₃×C₉

C₃²⋊C₉

yes

—

C₃×C₉

C₃²⋊C₉

yes

—

C₃×C₉

C₉⋊C₉

yes

—

C₃×C₉

C₉⋊C₉

yes

—

C₃×C₉

C₉⋊C₉

yes

—

C₃×C₉

C₉⋊C₉

yes

characteristic

C₃×C₉

C₉⋊C₉

yes

—

C₃×C₉

C₉⋊C₉

yes

—

C₃×C₉

C₉⋊C₉

yes

—

C₃×C₉

C₉⋊C₉

yes

—

C₃×C₉

C₉⋊C₉

yes

—

C₃×C₉

C₉⋊C₉

yes

—

C₃×C₉

C₉⋊C₉

yes

—

C₃×C₉

C₉⋊C₉

yes

—

C₃×C₉

C₉⋊C₉

yes

—

C₃×C₉

C₉⋊C₉

yes

characteristic

C₃×C₉

C₃²×C₉

yes

characteristic

C₃×C₉

C₃²×C₉

yes

—

C₃×C₉

C₃²×C₉

yes

characteristic

C₃×C₉

C₃×H(3,𝔽₃)

yes

characteristic

C₃×C₉

C₃×3₋¹⁺²

yes

—

C₃×C₉

C₃×3₋¹⁺²

yes

—

C₃×C₉

C₃×3₋¹⁺²

yes

—

C₃×C₉

C₃×3₋¹⁺²

yes

characteristic

C₂₇

—

C₂₇

—

100

C₃²

—

101

C₃²

—

102

C₃²

—

103

C₃²

—

104

C₃²

—

105

C₃²

#6855

yes

—

106

C₃²

#6855

yes

—

107

C₃²

C₃×C₃²⋊C₉

yes

characteristic

108

C₃²

C₃×C₉⋊C₉

yes

—

109

C₃²

C₃×C₉⋊C₉

yes

characteristic

110

C₉

—

111

C₉

#6855

yes

—

112

C₉

#6855

yes

—

113

C₉

#6855

yes

—

114

C₉

#6855

yes

—

115

C₉

C₃×C₉²

yes

characteristic

116

C₉

C₃×C₃²⋊C₉

yes

—

117

C₉

C₃×C₃²⋊C₉

yes

characteristic

118

C₉

C₃×C₉⋊C₉

yes

characteristic

119

C₉

C₃×C₉⋊C₉

yes

—

120

C₉

C₃×C₉⋊C₉

yes

—

121

C₃

—

122

C₃

—

123

C₃

—

124

C₃

—

125

C₃

C₃×#6855

yes

characteristic

126

C₃

#10624709

yes

—

	subgroup	G/H
center	C₃×C₉	C₉⋊C₉
commutator	C₉	C₃×C₉²
Frattini, cosocle	C₃²×C₉	C₃³
socle	C₃²	C₃×C₉⋊C₉
prime radical	C₃⁴	3₋¹⁺²
LCS	G ▷ C₉ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₉ ◁ C₃³×C₉ ◁ G
LECS	G ▷ C₃²×C₉ ▷ C₃ ▷ C₁

subgroup

G/H

center

C₃×C₉

C₉⋊C₉

commutator

C₉

C₃×C₉²

Frattini, cosocle

C₃²×C₉

C₃³

socle

C₃²

C₃×C₉⋊C₉

prime radical

C₃⁴

3₋¹⁺²

LCS

G ▷ C₉ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃×C₉ ◁ C₃³×C₉ ◁ G

LECS

G ▷ C₃²×C₉ ▷ C₃ ▷ C₁

N	H ≅ G/N	type
4. #10624472	C₃	semidirect
12. C₃×C₂₇⋊C₃	C₉	semidirect
13. C₃×C₂₇⋊C₃	C₉	semidirect
17. C₃²×C₂₇	C₉	semidirect

H ≅ G/N

type

4. #10624472

C₃

semidirect

12. C₃×C₂₇⋊C₃

C₉

semidirect

13. C₃×C₂₇⋊C₃

C₉

semidirect

17. C₃²×C₂₇

C₉

semidirect

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

subgroup	index
C₃²×C₂₇⋊C₃	3
C₃×#6887	3
#10624472	3

subgroup

index

C₃²×C₂₇⋊C₃

C₃×#6887

#10624472

Subgroups — #14550232

Conjugacy classes of subgroups

Complements

Maximal subgroups