Group #14550578 — subgroups

Subgroups — #14550578

Order 2187.

298 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 110 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	#10624879	C₃	1	1	yes	maximal, characteristic
2	#10624706	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×C₉⋊C₂₇	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×#6873	C₃	1	3	yes	maximal
5	C₉×C₂₇⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
6	#10624468	C₃	1	6	yes	maximal, complemented
7	C₃×#737	C₃²	1	1	yes	characteristic
8	C₃×C₂₇⋊C₃	C₃²	1	3	yes	—
9	C₃×C₂₇⋊C₃	C₃²	1	3	yes	—
10	C₃²×C₂₇	C₃²	1	3	yes	—
11	#6900	—	3	1	no	—
12	#6899	—	3	2	no	—
13	#6896	—	3	2	no	—
14	#6896	—	3	1	no	—
15	#6894	C₉	1	2	yes	—
16	#6894	C₉	1	4	yes	—
17	C₉⋊H(3,𝔽₃)	C₉	1	1	yes	characteristic
18	C₉⋊H(3,𝔽₃)	C₉	1	2	yes	—
19	C₉×3₋¹⁺²	—	3	1	no	—
20	C₉×3₋¹⁺²	—	3	2	no	—
21	#6887	C₃²	1	2	yes	—
22	C₃×C₉²	C₃²	1	1	yes	characteristic
23	C₉⋊C₂₇	—	3	1	no	—
24	C₉⋊C₂₇	—	3	2	no	—
25	C₉⋊C₂₇	—	3	1	no	—
26	C₉⋊C₂₇	—	3	2	no	—
27	#6873	—	3	9	no	—
28	#6864	—	3	2	no	—
29	C₉×C₂₇	—	3	1	no	—
30	#737	—	3	3	no	—
31	#737	—	3	6	no	—
32	C₃×3₋¹⁺²	—	3	2	no	—
33	C₃×3₋¹⁺²	—	3	1	no	—
34	C₃×3₋¹⁺²	—	3	1	no	—
35	C₃×3₋¹⁺²	—	3	1	no	—
36	C₃×3₋¹⁺²	C₃×C₉	1	2	yes	—
37	C₃×H(3,𝔽₃)	C₃×C₉	1	1	yes	characteristic
38	C₃²×C₉	—	3	1	no	—
39	C₃²×C₉	C₃×C₉	1	1	yes	characteristic
40	C₃²×C₉	C₃×C₉	1	2	yes	—
41	C₃²×C₉	C₃³	1	1	yes	characteristic
42	C₂₇⋊C₃	—	3	6	no	—
43	C₂₇⋊C₃	—	3	3	no	—
44	C₂₇⋊C₃	—	3	6	no	—
45	C₂₇⋊C₃	3₋¹⁺²	1	9	yes	complemented
46	C₃×C₂₇	—	3	3	no	—
47	C₃×C₂₇	—	3	3	no	—
48	C₃×C₂₇	—	3	3	no	—
49	C₃×C₂₇	—	3	3	no	—
50	C₃×C₂₇	—	3	6	no	—
51	C₉⋊C₉	—	3	4	no	—
52	C₉⋊C₉	—	3	4	no	—
53	C₉⋊C₉	—	3	4	no	—
54	C₉⋊C₉	—	3	4	no	—
55	C₉⋊C₉	—	3	2	no	—
56	C₉⋊C₉	—	3	2	no	—
57	C₉⋊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	C₃×C₉	1	2	yes	—
62	C₃²⋊C₉	C₃×C₉	1	4	yes	—
63	C₉²	—	3	2	no	—
64	C₉²	H(3,𝔽₃)	1	1	yes	characteristic
65	C₉²	H(3,𝔽₃)	1	2	yes	—
66	C₉²	3₋¹⁺²	1	2	yes	—
67	C₉²	3₋¹⁺²	1	1	yes	characteristic
68	C₉²	3₋¹⁺²	1	2	yes	—
69	C₉²	3₋¹⁺²	1	1	yes	characteristic
70	C₃³	—	3	1	no	—
71	C₃³	C₃²×C₉	1	1	yes	characteristic
72	3₋¹⁺²	—	3	6	no	—
73	3₋¹⁺²	—	9	6	no	—
74	3₋¹⁺²	—	9	3	no	—
75	3₋¹⁺²	—	9	3	no	—
76	3₋¹⁺²	—	9	2	no	—
77	3₋¹⁺²	—	9	1	no	—
78	H(3,𝔽₃)	—	3	3	no	—
79	C₃×C₉	—	3	3	no	—
80	C₃×C₉	—	3	2	no	—
81	C₃×C₉	—	3	4	no	—
82	C₃×C₉	—	3	2	no	—
83	C₃×C₉	—	9	2	no	—
84	C₃×C₉	—	9	3	no	—
85	C₃×C₉	C₃²⋊C₉	1	2	yes	—
86	C₃×C₉	C₃²⋊C₉	1	2	yes	—
87	C₃×C₉	C₃²⋊C₉	1	2	yes	—
88	C₃×C₉	C₃²⋊C₉	1	2	yes	—
89	C₃×C₉	C₃²⋊C₉	1	1	yes	characteristic
90	C₃×C₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
91	C₃×C₉	C₃×3₋¹⁺²	1	3	yes	—
92	C₃×C₉	C₃×3₋¹⁺²	1	1	yes	characteristic
93	C₃×C₉	C₃×3₋¹⁺²	1	1	yes	characteristic
94	C₃×C₉	#737	1	6	yes	—
95	C₂₇	—	3	9	no	—
96	C₂₇	—	9	3	no	—
97	C₂₇	—	9	3	no	—
98	C₃²	—	3	1	no	—
99	C₃²	—	3	2	no	—
100	C₃²	—	9	1	no	—
101	C₃²	—	9	1	no	—
102	C₃²	—	9	2	no	—
103	C₃²	C₃×C₃²⋊C₉	1	1	yes	characteristic
104	C₃²	C₉×3₋¹⁺²	1	3	yes	—
105	C₉	—	3	2	no	—
106	C₉	—	3	2	no	—
107	C₉	—	3	2	no	—
108	C₉	—	3	2	no	—
109	C₉	—	9	2	no	—
110	C₉	—	9	2	no	—
111	C₉	—	9	4	no	—
112	C₉	#6873	1	3	yes	complemented
113	C₉	C₉⋊H(3,𝔽₃)	1	3	yes	—
114	C₉	#6896	1	3	yes	—
115	C₉	#6899	1	2	yes	—
116	C₉	#6900	1	1	yes	characteristic
117	C₃	—	3	3	no	—
118	C₃	—	27	1	no	—
119	C₃	#10624560	1	1	yes	characteristic
120	C₃	C₃×#6873	1	1	yes	characteristic
121	C₃	#10624704	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10624879

C₃

yes

maximal, characteristic

#10624706

C₃

yes

maximal, characteristic, complemented

C₃×C₉⋊C₂₇

C₃

yes

maximal, characteristic, complemented

C₃×#6873

C₃

yes

maximal

C₉×C₂₇⋊C₃

C₃

yes

maximal, characteristic, complemented

#10624468

C₃

yes

maximal, complemented

C₃×#737

C₃²

yes

characteristic

C₃×C₂₇⋊C₃

C₃²

yes

—

C₃×C₂₇⋊C₃

C₃²

yes

—

C₃²×C₂₇

C₃²

yes

—

—

—

—

—

C₉

yes

—

#6894

C₉

yes

—

C₉⋊H(3,𝔽₃)

C₉

yes

characteristic

C₉⋊H(3,𝔽₃)

C₉

yes

—

C₉×3₋¹⁺²

—

C₉×3₋¹⁺²

—

#6887

C₃²

yes

—

C₃×C₉²

C₃²

yes

characteristic

C₉⋊C₂₇

—

C₉⋊C₂₇

—

C₉⋊C₂₇

—

C₉⋊C₂₇

—

#6873

—

#6864

—

C₉×C₂₇

—

#737

—

#737

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃×C₉

yes

—

C₃×H(3,𝔽₃)

C₃×C₉

yes

characteristic

C₃²×C₉

—

C₃²×C₉

C₃×C₉

yes

characteristic

C₃²×C₉

C₃×C₉

yes

—

C₃²×C₉

C₃³

yes

characteristic

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

3₋¹⁺²

yes

complemented

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×C₉

yes

—

C₃²⋊C₉

C₃×C₉

yes

—

C₉²

—

C₉²

H(3,𝔽₃)

yes

characteristic

C₉²

H(3,𝔽₃)

yes

—

C₉²

3₋¹⁺²

yes

—

C₉²

3₋¹⁺²

yes

characteristic

C₉²

3₋¹⁺²

yes

—

C₉²

3₋¹⁺²

yes

characteristic

C₃³

—

C₃³

C₃²×C₉

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

H(3,𝔽₃)

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

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

characteristic

C₃×C₉

C₃×H(3,𝔽₃)

yes

characteristic

C₃×C₉

C₃×3₋¹⁺²

yes

—

C₃×C₉

C₃×3₋¹⁺²

yes

characteristic

C₃×C₉

C₃×3₋¹⁺²

yes

characteristic

C₃×C₉

#737

yes

—

C₂₇

—

C₂₇

—

C₂₇

—

C₃²

—

C₃²

—

100

C₃²

—

101

C₃²

—

102

C₃²

—

103

C₃²

C₃×C₃²⋊C₉

yes

characteristic

104

C₃²

C₉×3₋¹⁺²

yes

—

105

C₉

—

106

C₉

—

107

C₉

—

108

C₉

—

109

C₉

—

110

C₉

—

111

C₉

—

112

C₉

#6873

yes

complemented

113

C₉

C₉⋊H(3,𝔽₃)

yes

—

114

C₉

#6896

yes

—

115

C₉

#6899

yes

—

116

C₉

#6900

yes

characteristic

117

C₃

—

118

C₃

—

119

C₃

#10624560

yes

characteristic

120

C₃

C₃×#6873

yes

characteristic

121

C₃

#10624704

yes

—

	subgroup	G/H
center	C₃×C₉	C₃×H(3,𝔽₃)
commutator	C₃³	C₃²×C₉
Frattini, cosocle	C₃²×C₉	C₃³
socle	C₃²	C₃×C₃²⋊C₉
prime radical	C₃×H(3,𝔽₃)	C₃×C₉
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₉ ◁ C₃×C₉² ◁ G
LECS	G ▷ C₃²×C₉ ▷ C₃ ▷ C₁

subgroup

G/H

center

C₃×C₉

C₃×H(3,𝔽₃)

commutator

C₃³

C₃²×C₉

Frattini, cosocle

C₃²×C₉

C₃³

socle

C₃²

C₃×C₃²⋊C₉

prime radical

C₃×H(3,𝔽₃)

C₃×C₉

LCS

G ▷ C₃³ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃×C₉ ◁ C₃×C₉² ◁ G

LECS

G ▷ C₃²×C₉ ▷ C₃ ▷ C₁

N	H ≅ G/N	type
2. #10624706	C₃	semidirect
3. C₃×C₉⋊C₂₇	C₃	semidirect
5. C₉×C₂₇⋊C₃	C₃	semidirect
6. #10624468	C₃	semidirect
45. C₂₇⋊C₃	3₋¹⁺²	semidirect
112. C₉	#6873	semidirect

H ≅ G/N

type

2. #10624706

C₃

semidirect

3. C₃×C₉⋊C₂₇

C₃

semidirect

5. C₉×C₂₇⋊C₃

C₃

semidirect

6. #10624468

C₃

semidirect

45. C₂₇⋊C₃

3₋¹⁺²

semidirect

112. C₉

#6873

semidirect

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

subgroup	index
#10624879	3
#10624706	3
C₃×C₉⋊C₂₇	3
C₃×#6873	3
C₉×C₂₇⋊C₃	3
#10624468	3

subgroup

index

C₃×C₉⋊C₂₇

C₉×C₂₇⋊C₃

Subgroups — #14550578

Conjugacy classes of subgroups

Complements

Maximal subgroups