Group #13091493 — subgroups

Subgroups — #13091493

Order 1458.

245 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 245 conjugacy classes of subgroups — too many to draw. Only its 46 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	#10624890	C₂	1	1	yes	maximal, characteristic, complemented, Sylow
2	#92311	C₃	1	3	yes	maximal, complemented
3	C₃×#4181	C₃	1	1	yes	maximal, characteristic
4	#6894⋊C₂	—	3	2	no	maximal
5	#92259	—	3	1	no	maximal
6	D₉×3₋¹⁺²	—	3	1	no	maximal
7	#6899	—	2	6	no	—
8	#6899	—	2	3	no	—
9	#6898	—	2	3	no	—
10	#6898	C₂×C₃	1	3	yes	complemented
11	#6894	—	2	2	no	—
12	#6894	D₃	1	2	yes	complemented
13	#6891	—	2	1	no	—
14	#6891	D₃	1	1	yes	characteristic
15	C₉×3₋¹⁺²	—	2	1	no	—
16	C₉×3₋¹⁺²	D₃	1	1	yes	characteristic, complemented
17	C₃×C₃²⋊C₉	C₂×C₃	1	1	yes	characteristic
18	C₃²×C₃⋊D₃	—	3	1	no	—
19	D₃×3₋¹⁺²	—	9	1	no	—
20	C₃²×D₉	—	3	1	no	—
21	C₃²×D₉	—	3	2	no	—
22	#4181	C₃²	1	1	yes	characteristic
23	D₉⋊C₉	—	3	6	no	—
24	(C₃⋊D₃)⋊C₉	—	3	3	no	—
25	C₉×D₉	—	3	3	no	—
26	C₃⁴	C₃×D₃	1	1	yes	characteristic
27	C₃×3₋¹⁺²	—	2	1	no	—
28	C₃×3₋¹⁺²	—	2	3	no	—
29	C₃×3₋¹⁺²	C₃⋊D₃	1	1	yes	characteristic
30	C₃²×C₉	C₃×D₃	1	1	yes	characteristic
31	C₃²×C₉	C₃×D₃	1	2	yes	—
32	C₉⋊C₉	—	2	3	no	—
33	C₉⋊C₉	—	2	12	no	—
34	C₉⋊C₉	—	2	3	no	—
35	C₉⋊C₉	—	2	6	no	—
36	C₉⋊C₉	—	2	3	no	—
37	C₉⋊C₉	—	2	6	no	—
38	C₉⋊C₉	C₃×D₃	1	6	yes	complemented
39	C₃²⋊C₉	—	2	3	no	—
40	C₃²⋊C₉	—	2	1	no	—
41	C₃²⋊C₉	—	2	3	no	—
42	C₃²⋊C₉	—	2	2	no	—
43	C₃²⋊C₉	—	2	1	no	—
44	C₃²⋊C₉	—	2	2	no	—
45	C₃²⋊C₉	—	2	1	no	—
46	C₃²⋊C₉	—	2	3	no	—
47	C₃²⋊C₉	C₃×D₃	1	3	yes	—
48	C₃²⋊C₉	C₂×C₃²	1	1	yes	characteristic
49	C₉²	—	2	3	no	—
50	C₉²	C₃×D₃	1	3	yes	complemented
51	C₃×C₃⋊D₃	—	3	1	no	—
52	C₃×C₃⋊D₃	—	9	1	no	—
53	C₃²×D₃	—	9	1	no	—
54	C₃²×D₃	—	27	1	no	—
55	C₂×3₋¹⁺²	—	27	1	no	—
56	D₃×C₉	—	9	3	no	—
57	C₃×D₉	—	3	1	no	—
58	C₃×D₉	—	3	2	no	—
59	C₃×D₉	—	9	1	no	—
60	C₃×D₉	—	9	2	no	—
61	C₃³	—	2	1	no	—
62	C₃³	—	3	1	no	—
63	C₃³	—	6	1	no	—
64	C₃³	—	9	1	no	—
65	C₃³	—	18	1	no	—
66	C₃³	C₃²×D₃	1	1	yes	characteristic
67	C₃³	C₃×C₃⋊D₃	1	1	yes	characteristic
68	3₋¹⁺²	—	3	1	no	—
69	3₋¹⁺²	—	6	1	no	—
70	3₋¹⁺²	—	6	2	no	—
71	3₋¹⁺²	—	6	1	no	—
72	3₋¹⁺²	—	18	3	no	—
73	C₃×C₉	—	2	1	no	—
74	C₃×C₉	—	2	6	no	—
75	C₃×C₉	—	2	3	no	—
76	C₃×C₉	—	2	3	no	—
77	C₃×C₉	—	2	2	no	—
78	C₃×C₉	—	9	1	no	—
79	C₃×C₉	—	9	2	no	—
80	C₃×C₉	C₃²×D₃	1	1	yes	characteristic
81	C₃×C₉	C₃²×D₃	1	2	yes	—
82	C₃×C₉	C₃×C₃⋊D₃	1	3	yes	—
83	C₂×C₃²	—	27	1	no	—
84	C₃⋊D₃	—	3	1	no	—
85	C₃×D₃	—	9	1	no	—
86	C₃×D₃	—	27	1	no	—
87	C₃×D₃	—	27	1	no	—
88	C₃×D₃	—	27	3	no	—
89	C₂×C₉	—	27	3	no	—
90	D₉	—	3	1	no	—
91	D₉	—	3	2	no	—
92	C₃²	—	2	1	no	—
93	C₃²	—	3	1	no	—
94	C₃²	—	3	1	no	—
95	C₃²	—	3	1	no	—
96	C₃²	—	6	1	no	—
97	C₃²	—	6	1	no	—
98	C₃²	—	6	1	no	—
99	C₃²	—	6	1	no	—
100	C₃²	—	6	1	no	—
101	C₃²	—	6	1	no	—
102	C₃²	—	9	3	no	—
103	C₃²	—	18	3	no	—
104	C₃²	#4207	1	1	yes	characteristic
105	C₃²	C₃²×C₃⋊D₃	1	1	yes	characteristic
106	C₉	—	3	3	no	—
107	C₉	—	3	1	no	—
108	C₉	—	3	2	no	—
109	C₉	—	6	6	no	—
110	C₉	—	6	3	no	—
111	C₉	—	6	3	no	—
112	C₉	—	6	1	no	—
113	C₉	—	6	2	no	—
114	C₂×C₃	—	27	1	no	—
115	C₂×C₃	—	81	1	no	—
116	D₃	—	9	1	no	—
117	D₃	—	27	1	no	—
118	C₃	—	2	1	no	—
119	C₃	—	3	1	no	—
120	C₃	—	6	1	no	—
121	C₃	—	9	1	no	—
122	C₃	—	18	1	no	—
123	C₃	C₃×#4207	1	1	yes	characteristic
124	C₃	#92394	1	1	yes	characteristic
125	C₂	—	27	1	no	Sylow

subgroup

G/H

class size

multiplicity

normal?

properties

#10624890

C₂

yes

maximal, characteristic, complemented, Sylow

#92311

C₃

yes

maximal, complemented

C₃×#4181

C₃

yes

maximal, characteristic

—

—

D₉×3₋¹⁺²

—

—

—

—

C₂×C₃

yes

complemented

#6894

—

#6894

D₃

yes

complemented

#6891

—

#6891

D₃

yes

characteristic

C₉×3₋¹⁺²

—

C₉×3₋¹⁺²

D₃

yes

characteristic, complemented

C₃×C₃²⋊C₉

C₂×C₃

yes

characteristic

C₃²×C₃⋊D₃

—

D₃×3₋¹⁺²

—

C₃²×D₉

—

C₃²×D₉

—

#4181

C₃²

yes

characteristic

D₉⋊C₉

—

(C₃⋊D₃)⋊C₉

—

C₉×D₉

—

C₃⁴

C₃×D₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃⋊D₃

yes

characteristic

C₃²×C₉

C₃×D₃

yes

characteristic

C₃²×C₉

C₃×D₃

yes

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

C₃×D₃

yes

complemented

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×D₃

yes

—

C₃²⋊C₉

C₂×C₃²

yes

characteristic

C₉²

—

C₉²

C₃×D₃

yes

complemented

C₃×C₃⋊D₃

—

C₃×C₃⋊D₃

—

C₃²×D₃

—

C₃²×D₃

—

C₂×3₋¹⁺²

—

D₃×C₉

—

C₃×D₉

—

C₃×D₉

—

C₃×D₉

—

C₃×D₉

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃²×D₃

yes

characteristic

C₃³

C₃×C₃⋊D₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₃²×D₃

yes

characteristic

C₃×C₉

C₃²×D₃

yes

—

C₃×C₉

C₃×C₃⋊D₃

yes

—

C₂×C₃²

—

C₃⋊D₃

—

C₃×D₃

—

C₃×D₃

—

C₃×D₃

—

C₃×D₃

—

C₂×C₉

—

D₉

—

D₉

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

100

C₃²

—

101

C₃²

—

102

C₃²

—

103

C₃²

—

104

C₃²

#4207

yes

characteristic

105

C₃²

C₃²×C₃⋊D₃

yes

characteristic

106

C₉

—

107

C₉

—

108

C₉

—

109

C₉

—

110

C₉

—

111

C₉

—

112

C₉

—

113

C₉

—

114

C₂×C₃

—

115

C₂×C₃

—

116

D₃

—

117

D₃

—

118

C₃

—

119

C₃

—

120

C₃

—

121

C₃

—

122

C₃

—

123

C₃

C₃×#4207

yes

characteristic

124

C₃

#92394

yes

characteristic

125

C₂

—

Sylow

	subgroup	G/H
center	C₃	C₃×#4207
commutator, cosocle, hypocenter	C₃²⋊C₉	C₂×C₃²
Frattini, socle	C₃²	C₃²×C₃⋊D₃
Fitting	#10624890	C₂
prime radical	C₃×#4181	C₃
hypercenter	C₃²	#4207
LCS	G ▷ C₃²⋊C₉
UCS	C₁ ◁ C₃ ◁ C₃²

subgroup

G/H

center

C₃

C₃×#4207

commutator, cosocle, hypocenter

C₃²⋊C₉

C₂×C₃²

Frattini, socle

C₃²

C₃²×C₃⋊D₃

C₂

C₃

C₃²

G ▷ C₃²⋊C₉

C₁ ◁ C₃ ◁ C₃²

N	H ≅ G/N	type
1. #10624890	C₂	semidirect
2. #92311	C₃	semidirect
10. #6898	C₂×C₃	semidirect
12. #6894	D₃	semidirect
16. C₉×3₋¹⁺²	D₃	semidirect
38. C₉⋊C₉	C₃×D₃	semidirect
50. C₉²	C₃×D₃	semidirect

H ≅ G/N

type

1. #10624890

C₂

semidirect

2. #92311

C₃

semidirect

10. #6898

C₂×C₃

semidirect

12. #6894

D₃

semidirect

16. C₉×3₋¹⁺²

D₃

semidirect

38. C₉⋊C₉

C₃×D₃

semidirect

50. C₉²

C₃×D₃

semidirect

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

subgroup	index
#10624890	2
#92311	3
C₃×#4181	3
#6894⋊C₂	3
#92259	3
D₉×3₋¹⁺²	3

subgroup

index

D₉×3₋¹⁺²

Subgroups — #13091493

Conjugacy classes of subgroups

Complements

Maximal subgroups