Group #13091231 — subgroups

Subgroups — #13091231

Order 1458.

234 conjugacy classes of subgroups; only the 31 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 31 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	#10624794	C₂	1	1	yes	maximal, characteristic, complemented, Sylow
2	C₃×D₃×3₋¹⁺²	—	3	1	no	maximal
3	C₉×#272	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×#4174	C₃	1	1	yes	maximal, characteristic
5	#272⋊C₉	C₃	1	2	yes	maximal, complemented
6	C₃²×3₋¹⁺²	D₃	1	1	yes	characteristic, complemented
7	C₃×AΣL(1,𝔽₂₇)	C₂×C₃	1	1	yes	characteristic
8	#6894	—	2	1	no	—
9	C₉×H(3,𝔽₃)	C₂×C₃	1	1	yes	characteristic, complemented
10	3₋¹⁺²⋊C₉	—	2	2	no	—
11	H(3,𝔽₃)⋊C₉	C₂×C₃	1	2	yes	complemented
12	#6866	—	2	1	no	—
13	C₃³×D₃	—	3	1	no	—
14	C₂×C₃×3₋¹⁺²	—	9	1	no	—
15	C₃×#272	C₃²	1	1	yes	characteristic
16	D₃×3₋¹⁺²	—	3	9	no	—
17	C₃×D₃×C₉	—	3	2	no	—
18	C₃×D₃×C₉	—	3	1	no	—
19	C₃×D₃×C₉	—	9	1	no	—
20	#4174	—	3	1	no	—
21	C₃⁴	C₃×D₃	1	1	yes	characteristic
22	C₃×3₋¹⁺²	—	2	1	no	—
23	C₃×3₋¹⁺²	—	3	9	no	—
24	C₃×3₋¹⁺²	—	3	1	no	—
25	C₃×3₋¹⁺²	—	6	1	no	—
26	C₃×H(3,𝔽₃)	C₂×C₃²	1	1	yes	characteristic
27	C₃²×C₉	—	3	1	no	—
28	C₃²×C₉	C₃×D₃	1	1	yes	characteristic
29	C₃²×C₉	C₃×D₃	1	2	yes	—
30	AΣL(1,𝔽₂₇)	—	3	1	no	—
31	AΣL(1,𝔽₂₇)	—	6	1	no	—
32	C₉⋊C₉	—	6	2	no	—
33	C₉⋊C₉	—	6	1	no	—
34	C₃²⋊C₉	—	2	2	no	—
35	C₃²⋊C₉	—	2	1	no	—
36	C₃²⋊C₉	—	6	1	no	—
37	C₂×C₃³	—	9	1	no	—
38	C₃²×D₃	—	3	1	no	—
39	C₃²×D₃	—	3	3	no	—
40	C₃²×D₃	—	9	1	no	—
41	C₃²×D₃	—	9	2	no	—
42	C₃²×D₃	—	9	1	no	—
43	C₂×3₋¹⁺²	—	9	9	no	—
44	C₂×C₃×C₉	—	9	2	no	—
45	C₂×C₃×C₉	—	9	1	no	—
46	#272	3₋¹⁺²	1	1	yes	characteristic, complemented
47	D₃×C₉	—	3	6	no	—
48	D₃×C₉	—	3	3	no	—
49	D₃×C₉	—	9	3	no	—
50	C₃³	—	3	1	no	—
51	C₃³	—	3	3	no	—
52	C₃³	—	3	1	no	—
53	C₃³	—	3	1	no	—
54	C₃³	—	6	1	no	—
55	C₃³	—	9	2	no	—
56	C₃³	C₃²×D₃	1	1	yes	characteristic
57	3₋¹⁺²	—	3	9	no	—
58	3₋¹⁺²	—	6	9	no	—
59	3₋¹⁺²	—	6	2	no	—
60	3₋¹⁺²	—	6	1	no	—
61	H(3,𝔽₃)	—	2	1	no	—
62	H(3,𝔽₃)	—	6	1	no	—
63	H(3,𝔽₃)	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
64	C₃×C₉	—	2	1	no	—
65	C₃×C₉	—	3	2	no	—
66	C₃×C₉	—	3	6	no	—
67	C₃×C₉	—	3	3	no	—
68	C₃×C₉	—	6	1	no	—
69	C₃×C₉	—	6	2	no	—
70	C₃×C₉	—	6	1	no	—
71	C₃×C₉	—	9	3	no	—
72	C₃×C₉	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
73	C₂×C₃²	—	9	1	no	—
74	C₂×C₃²	—	9	3	no	—
75	C₂×C₃²	—	27	1	no	—
76	C₂×C₃²	—	27	2	no	—
77	C₃×D₃	—	3	1	no	—
78	C₃×D₃	—	3	2	no	—
79	C₃×D₃	—	3	1	no	—
80	C₃×D₃	—	9	3	no	—
81	C₃×D₃	—	9	1	no	—
82	C₃×D₃	—	9	1	no	—
83	C₃×D₃	—	9	2	no	—
84	C₂×C₉	—	9	6	no	—
85	C₂×C₉	—	9	3	no	—
86	C₃²	—	2	1	no	—
87	C₃²	—	3	1	no	—
88	C₃²	—	3	2	no	—
89	C₃²	—	3	1	no	—
90	C₃²	—	3	3	no	—
91	C₃²	—	6	1	no	—
92	C₃²	—	9	3	no	—
93	C₃²	—	9	1	no	—
94	C₃²	—	9	2	no	—
95	C₃²	—	9	1	no	—
96	C₃²	—	9	2	no	—
97	C₃²	—	18	3	no	—
98	C₃²	C₃×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
99	C₃²	D₃×3₋¹⁺²	1	1	yes	characteristic
100	C₉	—	3	6	no	—
101	C₉	—	6	1	no	—
102	C₉	—	18	1	no	—
103	C₉	—	18	1	no	—
104	C₉	—	18	2	no	—
105	C₉	#4174	1	3	yes	complemented
106	C₂×C₃	—	9	1	no	—
107	C₂×C₃	—	9	1	no	—
108	C₂×C₃	—	9	2	no	—
109	C₂×C₃	—	27	3	no	—
110	D₃	—	3	1	no	—
111	D₃	—	9	1	no	—
112	C₃	—	3	1	no	—
113	C₃	—	6	1	no	—
114	C₃	—	9	1	no	—
115	C₃	—	9	3	no	—
116	C₃	—	18	1	no	—
117	C₃	#92262	1	1	yes	characteristic
118	C₃	C₃×#4174	1	1	yes	characteristic
119	C₃	#6913⋊C₂	1	2	yes	—
120	C₂	—	9	1	no	Sylow

subgroup

G/H

class size

multiplicity

normal?

properties

#10624794

C₂

yes

maximal, characteristic, complemented, Sylow

C₃×D₃×3₋¹⁺²

—

maximal

C₉×#272

C₃

yes

maximal, characteristic, complemented

C₃×#4174

C₃

yes

maximal, characteristic

#272⋊C₉

C₃

yes

maximal, complemented

C₃²×3₋¹⁺²

D₃

yes

characteristic, complemented

C₃×AΣL(1,𝔽₂₇)

C₂×C₃

yes

characteristic

#6894

—

C₉×H(3,𝔽₃)

C₂×C₃

yes

characteristic, complemented

3₋¹⁺²⋊C₉

—

H(3,𝔽₃)⋊C₉

C₂×C₃

yes

complemented

#6866

—

C₃³×D₃

—

C₂×C₃×3₋¹⁺²

—

C₃×#272

C₃²

yes

characteristic

D₃×3₋¹⁺²

—

C₃×D₃×C₉

—

C₃×D₃×C₉

—

C₃×D₃×C₉

—

#4174

—

C₃⁴

C₃×D₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×H(3,𝔽₃)

C₂×C₃²

yes

characteristic

C₃²×C₉

—

C₃²×C₉

C₃×D₃

yes

characteristic

C₃²×C₉

C₃×D₃

yes

—

AΣL(1,𝔽₂₇)

—

AΣL(1,𝔽₂₇)

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₂×C₃³

—

C₃²×D₃

—

C₃²×D₃

—

C₃²×D₃

—

C₃²×D₃

—

C₃²×D₃

—

C₂×3₋¹⁺²

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

#272

3₋¹⁺²

yes

characteristic, complemented

D₃×C₉

—

D₃×C₉

—

D₃×C₉

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃²×D₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

H(3,𝔽₃)

C₂×3₋¹⁺²

yes

characteristic, complemented

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

(C₃⋊D₃)⋊C₃

yes

characteristic

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₃×D₃

—

C₃×D₃

—

C₃×D₃

—

C₃×D₃

—

C₃×D₃

—

C₃×D₃

—

C₃×D₃

—

C₂×C₉

—

C₂×C₉

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₃×(C₃⋊D₃)⋊C₃

yes

characteristic

C₃²

D₃×3₋¹⁺²

yes

characteristic

100

C₉

—

101

C₉

—

102

C₉

—

103

C₉

—

104

C₉

—

105

C₉

#4174

yes

complemented

106

C₂×C₃

—

107

C₂×C₃

—

108

C₂×C₃

—

109

C₂×C₃

—

110

D₃

—

111

D₃

—

112

C₃

—

113

C₃

—

114

C₃

—

115

C₃

—

116

C₃

—

117

C₃

#92262

yes

characteristic

118

C₃

C₃×#4174

yes

characteristic

119

C₃

#6913⋊C₂

yes

—

120

C₂

—

Sylow

	subgroup	G/H
center, socle	C₃²	C₃×(C₃⋊D₃)⋊C₃
commutator, cosocle	C₃×H(3,𝔽₃)	C₂×C₃²
Frattini	C₃³	C₃²×D₃
Fitting	#10624794	C₂
prime radical	C₃×#4174	C₃
hypercenter	C₃×C₉	(C₃⋊D₃)⋊C₃
hypocenter	H(3,𝔽₃)	C₂×3₋¹⁺²
LCS	G ▷ C₃×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₃² ◁ C₃×C₉

subgroup

G/H

center, socle

C₃²

C₃×(C₃⋊D₃)⋊C₃

commutator, cosocle

C₃×H(3,𝔽₃)

C₂×C₃²

Frattini

C₃³

C₃²×D₃

C₂

C₃

C₃×C₉

(C₃⋊D₃)⋊C₃

hypocenter

H(3,𝔽₃)

C₂×3₋¹⁺²

LCS

G ▷ C₃×H(3,𝔽₃) ▷ H(3,𝔽₃)

UCS

C₁ ◁ C₃² ◁ C₃×C₉

N	H ≅ G/N	type
1. #10624794	C₂	semidirect
3. C₉×#272	C₃	semidirect
5. #272⋊C₉	C₃	semidirect
6. C₃²×3₋¹⁺²	D₃	semidirect
9. C₉×H(3,𝔽₃)	C₂×C₃	semidirect
11. H(3,𝔽₃)⋊C₉	C₂×C₃	semidirect
46. #272	3₋¹⁺²	semidirect
63. H(3,𝔽₃)	C₂×3₋¹⁺²	semidirect
105. C₉	#4174	semidirect

H ≅ G/N

type

1. #10624794

C₂

semidirect

3. C₉×#272

C₃

semidirect

5. #272⋊C₉

C₃

semidirect

6. C₃²×3₋¹⁺²

D₃

semidirect

9. C₉×H(3,𝔽₃)

C₂×C₃

semidirect

11. H(3,𝔽₃)⋊C₉

C₂×C₃

semidirect

46. #272

3₋¹⁺²

semidirect

63. H(3,𝔽₃)

C₂×3₋¹⁺²

semidirect

105. C₉

#4174

semidirect

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

subgroup	index
#10624794	2
C₃×D₃×3₋¹⁺²	3
C₉×#272	3
C₃×#4174	3
#272⋊C₉	3

subgroup

index

C₃×D₃×3₋¹⁺²

Subgroups — #13091231

Conjugacy classes of subgroups

Complements

Maximal subgroups