Group #13091470 — subgroups

Subgroups — #13091470

Order 1458.

434 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 52 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	#10624881	C₂	1	1	yes	maximal, characteristic, complemented, Sylow
2	C₃×C₃⋊#272	C₃	1	1	yes	maximal, characteristic
3	C₃⋊D₃×3₋¹⁺²	—	3	1	no	maximal
4	#92305	C₃	1	3	yes	maximal, complemented
5	#92259	—	3	3	no	maximal
6	C₃²×3₋¹⁺²	—	2	1	no	—
7	C₃²×3₋¹⁺²	D₃	1	1	yes	characteristic, complemented
8	C₃²×H(3,𝔽₃)	C₂×C₃	1	1	yes	characteristic
9	#6893	—	2	3	no	—
10	#6893	—	2	9	no	—
11	#6893	C₂×C₃	1	3	yes	complemented
12	#6891	—	2	3	no	—
13	#6891	D₃	1	3	yes	complemented
14	C₃²×C₃⋊D₃	—	3	1	no	—
15	C₃²×C₃⋊D₃	—	3	3	no	—
16	C₃⋊#272	C₃²	1	1	yes	characteristic
17	C₃×#272	—	9	3	no	—
18	C₉×C₃⋊D₃	—	3	3	no	—
19	D₃×3₋¹⁺²	—	9	3	no	—
20	D₃×3₋¹⁺²	—	9	1	no	—
21	(C₃⋊D₃)⋊C₉	—	3	9	no	—
22	C₃⁴	C₃×D₃	1	1	yes	characteristic
23	C₃⁴	C₃×D₃	1	3	yes	—
24	C₃×3₋¹⁺²	—	2	3	no	—
25	C₃×3₋¹⁺²	—	2	3	no	—
26	C₃×3₋¹⁺²	—	2	3	no	—
27	C₃×3₋¹⁺²	—	2	1	no	—
28	C₃×3₋¹⁺²	—	2	3	no	—
29	C₃×3₋¹⁺²	—	3	3	no	—
30	C₃×3₋¹⁺²	—	6	3	no	—
31	C₃×3₋¹⁺²	—	6	6	no	—
32	C₃×3₋¹⁺²	—	6	3	no	—
33	C₃×3₋¹⁺²	C₃⋊D₃	1	1	yes	characteristic
34	C₃×H(3,𝔽₃)	—	2	3	no	—
35	C₃×H(3,𝔽₃)	—	2	1	no	—
36	C₃×H(3,𝔽₃)	—	3	3	no	—
37	C₃×H(3,𝔽₃)	—	6	3	no	—
38	C₃×H(3,𝔽₃)	C₂×C₃²	1	1	yes	characteristic
39	C₃²×C₉	—	2	3	no	—
40	C₃²×C₉	C₃×D₃	1	3	yes	complemented
41	C₃²⋊C₉	—	2	18	no	—
42	C₃²⋊C₉	—	2	9	no	—
43	C₃²⋊C₉	—	2	9	no	—
44	C₃²⋊C₉	C₃×D₃	1	9	yes	complemented
45	C₃×C₃⋊D₃	—	3	1	no	—
46	C₃×C₃⋊D₃	—	3	3	no	—
47	C₃×C₃⋊D₃	—	9	1	no	—
48	C₃×C₃⋊D₃	—	9	3	no	—
49	C₃²×D₃	—	9	3	no	—
50	C₃²×D₃	—	9	1	no	—
51	C₃²×D₃	—	27	3	no	—
52	C₂×3₋¹⁺²	—	27	1	no	—
53	#272	—	9	3	no	—
54	D₃×C₉	—	9	9	no	—
55	D₃×C₉	—	9	3	no	—
56	C₃³	—	2	1	no	—
57	C₃³	—	2	3	no	—
58	C₃³	—	2	3	no	—
59	C₃³	—	3	3	no	—
60	C₃³	—	6	3	no	—
61	C₃³	—	9	1	no	—
62	C₃³	—	9	3	no	—
63	C₃³	—	18	1	no	—
64	C₃³	—	18	3	no	—
65	C₃³	(C₃⋊D₃)⋊C₃	1	3	yes	—
66	C₃³	C₃²×D₃	1	3	yes	—
67	C₃³	C₃²×D₃	1	1	yes	characteristic
68	C₃³	C₃×C₃⋊D₃	1	1	yes	characteristic
69	3₋¹⁺²	—	3	1	no	—
70	3₋¹⁺²	—	6	1	no	—
71	3₋¹⁺²	—	6	9	no	—
72	3₋¹⁺²	—	6	2	no	—
73	3₋¹⁺²	—	6	9	no	—
74	3₋¹⁺²	—	6	6	no	—
75	3₋¹⁺²	—	6	9	no	—
76	3₋¹⁺²	—	6	1	no	—
77	3₋¹⁺²	—	6	3	no	—
78	H(3,𝔽₃)	—	3	3	no	—
79	H(3,𝔽₃)	—	6	9	no	—
80	H(3,𝔽₃)	—	6	3	no	—
81	C₃×C₉	—	2	3	no	—
82	C₃×C₉	—	2	9	no	—
83	C₃×C₉	—	3	9	no	—
84	C₃×C₉	—	6	9	no	—
85	C₃×C₉	C₃×C₃⋊D₃	1	3	yes	—
86	C₂×C₃²	—	27	1	no	—
87	C₃⋊D₃	—	3	1	no	—
88	C₃⋊D₃	—	3	3	no	—
89	C₃×D₃	—	9	3	no	—
90	C₃×D₃	—	9	1	no	—
91	C₃×D₃	—	27	1	no	—
92	C₃×D₃	—	27	3	no	—
93	C₃×D₃	—	27	3	no	—
94	C₃×D₃	—	27	9	no	—
95	C₂×C₉	—	27	3	no	—
96	C₃²	—	2	3	no	—
97	C₃²	—	2	1	no	—
98	C₃²	—	3	1	no	—
99	C₃²	—	3	1	no	—
100	C₃²	—	3	3	no	—
101	C₃²	—	3	3	no	—
102	C₃²	—	6	1	no	—
103	C₃²	—	6	1	no	—
104	C₃²	—	6	2	no	—
105	C₃²	—	6	3	no	—
106	C₃²	—	6	6	no	—
107	C₃²	—	6	1	no	—
108	C₃²	—	6	3	no	—
109	C₃²	—	6	3	no	—
110	C₃²	—	9	3	no	—
111	C₃²	—	9	9	no	—
112	C₃²	—	18	3	no	—
113	C₃²	—	18	9	no	—
114	C₃²	C₃×(C₃⋊D₃)⋊C₃	1	3	yes	—
115	C₃²	C₃⋊((C₃⋊D₃)⋊C₃)	1	1	yes	characteristic
116	C₃²	C₃²×C₃⋊D₃	1	1	yes	characteristic
117	C₉	—	3	3	no	—
118	C₉	—	6	3	no	—
119	C₉	—	6	9	no	—
120	C₂×C₃	—	27	1	no	—
121	C₂×C₃	—	81	1	no	—
122	D₃	—	9	3	no	—
123	D₃	—	9	1	no	—
124	D₃	—	27	3	no	—
125	C₃	—	2	1	no	—
126	C₃	—	3	3	no	—
127	C₃	—	3	1	no	—
128	C₃	—	6	1	no	—
129	C₃	—	6	3	no	—
130	C₃	—	9	3	no	—
131	C₃	—	18	3	no	—
132	C₃	C₃×C₃⋊((C₃⋊D₃)⋊C₃)	1	1	yes	characteristic
133	C₃	#92394	1	1	yes	characteristic
134	C₂	—	27	1	no	Sylow

subgroup

G/H

class size

multiplicity

normal?

properties

#10624881

C₂

yes

maximal, characteristic, complemented, Sylow

C₃×C₃⋊#272

C₃

yes

maximal, characteristic

C₃⋊D₃×3₋¹⁺²

—

maximal

#92305

C₃

yes

maximal, complemented

#92259

—

maximal

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

D₃

yes

characteristic, complemented

C₃²×H(3,𝔽₃)

C₂×C₃

yes

—

—

C₂×C₃

yes

complemented

#6891

—

#6891

D₃

yes

complemented

C₃²×C₃⋊D₃

—

C₃²×C₃⋊D₃

—

C₃⋊#272

C₃²

yes

characteristic

C₃×#272

—

C₉×C₃⋊D₃

—

D₃×3₋¹⁺²

—

D₃×3₋¹⁺²

—

(C₃⋊D₃)⋊C₉

—

C₃⁴

C₃×D₃

yes

characteristic

C₃⁴

C₃×D₃

yes

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃⋊D₃

yes

characteristic

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₂×C₃²

yes

characteristic

C₃²×C₉

—

C₃²×C₉

C₃×D₃

yes

complemented

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×D₃

yes

complemented

C₃×C₃⋊D₃

—

C₃×C₃⋊D₃

—

C₃×C₃⋊D₃

—

C₃×C₃⋊D₃

—

C₃²×D₃

—

C₃²×D₃

—

C₃²×D₃

—

C₂×3₋¹⁺²

—

#272

—

D₃×C₉

—

D₃×C₉

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

(C₃⋊D₃)⋊C₃

yes

—

C₃³

C₃²×D₃

yes

—

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

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₃×C₃⋊D₃

yes

—

C₂×C₃²

—

C₃⋊D₃

—

C₃⋊D₃

—

C₃×D₃

—

C₃×D₃

—

C₃×D₃

—

C₃×D₃

—

C₃×D₃

—

C₃×D₃

—

C₂×C₉

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

100

C₃²

—

101

C₃²

—

102

C₃²

—

103

C₃²

—

104

C₃²

—

105

C₃²

—

106

C₃²

—

107

C₃²

—

108

C₃²

—

109

C₃²

—

110

C₃²

—

111

C₃²

—

112

C₃²

—

113

C₃²

—

114

C₃²

C₃×(C₃⋊D₃)⋊C₃

yes

—

115

C₃²

C₃⋊((C₃⋊D₃)⋊C₃)

yes

characteristic

116

C₃²

C₃²×C₃⋊D₃

yes

characteristic

117

C₉

—

118

C₉

—

119

C₉

—

120

C₂×C₃

—

121

C₂×C₃

—

122

D₃

—

123

D₃

—

124

D₃

—

125

C₃

—

126

C₃

—

127

C₃

—

128

C₃

—

129

C₃

—

130

C₃

—

131

C₃

—

132

C₃

C₃×C₃⋊((C₃⋊D₃)⋊C₃)

yes

characteristic

133

C₃

#92394

yes

characteristic

134

C₂

—

Sylow

	subgroup	G/H
center	C₃	C₃×C₃⋊((C₃⋊D₃)⋊C₃)
commutator, cosocle, hypocenter	C₃×H(3,𝔽₃)	C₂×C₃²
Frattini, socle	C₃²	C₃²×C₃⋊D₃
Fitting	#10624881	C₂
prime radical	C₃×C₃⋊#272	C₃
hypercenter	C₃²	C₃⋊((C₃⋊D₃)⋊C₃)
LCS	G ▷ C₃×H(3,𝔽₃)
UCS	C₁ ◁ C₃ ◁ C₃²

subgroup

G/H

center

C₃

C₃×C₃⋊((C₃⋊D₃)⋊C₃)

commutator, cosocle, hypocenter

C₃×H(3,𝔽₃)

C₂×C₃²

Frattini, socle

C₃²

C₃²×C₃⋊D₃

Fitting

#10624881

C₂

prime radical

C₃×C₃⋊#272

C₃

hypercenter

C₃²

C₃⋊((C₃⋊D₃)⋊C₃)

LCS

G ▷ C₃×H(3,𝔽₃)

UCS

C₁ ◁ C₃ ◁ C₃²

N	H ≅ G/N	type
1. #10624881	C₂	semidirect
4. #92305	C₃	semidirect
7. C₃²×3₋¹⁺²	D₃	semidirect
11. #6893	C₂×C₃	semidirect
13. #6891	D₃	semidirect
40. C₃²×C₉	C₃×D₃	semidirect
44. C₃²⋊C₉	C₃×D₃	semidirect

H ≅ G/N

type

1. #10624881

C₂

semidirect

4. #92305

C₃

semidirect

7. C₃²×3₋¹⁺²

D₃

semidirect

11. #6893

C₂×C₃

semidirect

13. #6891

D₃

semidirect

40. C₃²×C₉

C₃×D₃

semidirect

44. C₃²⋊C₉

C₃×D₃

semidirect

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

subgroup	index
#10624881	2
C₃×C₃⋊#272	3
C₃⋊D₃×3₋¹⁺²	3
#92305	3
#92259	3

subgroup

index

#10624881

C₃×C₃⋊#272

C₃⋊D₃×3₋¹⁺²

#92305

#92259

Subgroups — #13091470

Conjugacy classes of subgroups

Complements

Maximal subgroups