Group #13091235 — subgroups

Subgroups — #10624797⋊C₂ #13091235

Order 1458.

165 conjugacy classes of subgroups; only the 28 normal subgroups are shown.

normal subgroups only

This group has 165 conjugacy classes of subgroups — too many to draw. Only its 28 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	#10624797	C₂	1	1	yes	maximal, characteristic, complemented, Sylow
2	C₉×#272	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×#4174	C₃	1	1	yes	maximal, characteristic
4	D₃×C₃²⋊C₉	—	3	1	no	maximal
5	#272⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
6	#272⋊C₉	C₃	1	1	yes	maximal, 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	C₃×C₃²⋊C₉	D₃	1	1	yes	characteristic, complemented
11	3₋¹⁺²⋊C₉	—	2	1	no	—
12	3₋¹⁺²⋊C₉	—	2	1	no	—
13	H(3,𝔽₃)⋊C₉	C₂×C₃	1	1	yes	characteristic, complemented
14	H(3,𝔽₃)⋊C₉	C₂×C₃	1	1	yes	characteristic, complemented
15	#6866	—	2	1	no	—
16	C₃³×D₃	—	3	1	no	—
17	C₃×#272	C₃²	1	1	yes	characteristic
18	C₃×D₃×C₉	—	3	1	no	—
19	C₃×D₃×C₉	—	3	1	no	—
20	C₃×D₃×C₉	—	3	1	no	—
21	C₃×D₃×C₉	—	9	1	no	—
22	C₂×C₃²⋊C₉	—	9	1	no	—
23	#4174	—	3	1	no	—
24	C₃⁴	C₃×D₃	1	1	yes	characteristic
25	C₃×3₋¹⁺²	—	2	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	1	yes	characteristic
30	C₃²×C₉	C₃×D₃	1	1	yes	characteristic
31	AΣL(1,𝔽₂₇)	—	3	1	no	—
32	AΣL(1,𝔽₂₇)	—	6	1	no	—
33	C₉⋊C₉	—	6	2	no	—
34	C₉⋊C₉	—	6	1	no	—
35	C₃²⋊C₉	—	2	1	no	—
36	C₃²⋊C₉	—	2	1	no	—
37	C₃²⋊C₉	—	2	1	no	—
38	C₃²⋊C₉	—	3	1	no	—
39	C₃²⋊C₉	—	6	1	no	—
40	C₃²⋊C₉	—	6	1	no	—
41	C₂×C₃³	—	9	1	no	—
42	C₃²×D₃	—	3	1	no	—
43	C₃²×D₃	—	3	3	no	—
44	C₃²×D₃	—	9	1	no	—
45	C₃²×D₃	—	9	1	no	—
46	C₃²×D₃	—	9	1	no	—
47	C₃²×D₃	—	9	1	no	—
48	C₂×C₃×C₉	—	9	1	no	—
49	C₂×C₃×C₉	—	9	1	no	—
50	C₂×C₃×C₉	—	9	1	no	—
51	#272	3₋¹⁺²	1	1	yes	characteristic
52	D₃×C₉	—	9	1	no	—
53	D₃×C₉	—	9	1	no	—
54	D₃×C₉	—	9	1	no	—
55	D₃×C₉	—	9	3	no	—
56	C₃³	—	3	1	no	—
57	C₃³	—	3	3	no	—
58	C₃³	—	3	1	no	—
59	C₃³	—	3	1	no	—
60	C₃³	—	6	1	no	—
61	C₃³	—	9	1	no	—
62	C₃³	—	9	1	no	—
63	C₃³	C₃²×D₃	1	1	yes	characteristic
64	3₋¹⁺²	—	6	2	no	—
65	3₋¹⁺²	—	6	1	no	—
66	H(3,𝔽₃)	—	2	1	no	—
67	H(3,𝔽₃)	—	6	1	no	—
68	H(3,𝔽₃)	C₂×3₋¹⁺²	1	1	yes	characteristic
69	C₃×C₉	—	2	1	no	—
70	C₃×C₉	—	3	1	no	—
71	C₃×C₉	—	3	1	no	—
72	C₃×C₉	—	6	1	no	—
73	C₃×C₉	—	6	1	no	—
74	C₃×C₉	—	6	1	no	—
75	C₃×C₉	—	6	1	no	—
76	C₃×C₉	—	9	1	no	—
77	C₃×C₉	—	9	3	no	—
78	C₃×C₉	—	9	1	no	—
79	C₃×C₉	—	9	1	no	—
80	C₃×C₉	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
81	C₂×C₃²	—	9	1	no	—
82	C₂×C₃²	—	9	3	no	—
83	C₂×C₃²	—	27	1	no	—
84	C₂×C₃²	—	27	1	no	—
85	C₂×C₃²	—	27	1	no	—
86	C₃×D₃	—	3	1	no	—
87	C₃×D₃	—	3	1	no	—
88	C₃×D₃	—	3	1	no	—
89	C₃×D₃	—	3	1	no	—
90	C₃×D₃	—	9	3	no	—
91	C₃×D₃	—	9	1	no	—
92	C₃×D₃	—	9	1	no	—
93	C₃×D₃	—	9	1	no	—
94	C₃×D₃	—	9	1	no	—
95	C₂×C₉	—	27	1	no	—
96	C₂×C₉	—	27	1	no	—
97	C₂×C₉	—	27	1	no	—
98	C₃²	—	2	1	no	—
99	C₃²	—	3	1	no	—
100	C₃²	—	3	1	no	—
101	C₃²	—	3	1	no	—
102	C₃²	—	3	1	no	—
103	C₃²	—	3	3	no	—
104	C₃²	—	6	1	no	—
105	C₃²	—	9	3	no	—
106	C₃²	—	9	1	no	—
107	C₃²	—	9	1	no	—
108	C₃²	—	9	1	no	—
109	C₃²	—	9	1	no	—
110	C₃²	—	9	1	no	—
111	C₃²	—	9	1	no	—
112	C₃²	—	18	3	no	—
113	C₃²	C₃×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
114	C₃²	D₃×3₋¹⁺²	1	1	yes	characteristic
115	C₉	—	3	1	no	—
116	C₉	—	6	1	no	—
117	C₉	—	9	1	no	—
118	C₉	—	9	1	no	—
119	C₉	—	18	1	no	—
120	C₉	—	18	1	no	—
121	C₉	—	18	1	no	—
122	C₉	—	18	1	no	—
123	C₂×C₃	—	9	1	no	—
124	C₂×C₃	—	9	1	no	—
125	C₂×C₃	—	9	1	no	—
126	C₂×C₃	—	9	1	no	—
127	C₂×C₃	—	27	3	no	—
128	D₃	—	3	1	no	—
129	D₃	—	9	1	no	—
130	C₃	—	3	1	no	—
131	C₃	—	6	1	no	—
132	C₃	—	9	1	no	—
133	C₃	—	9	3	no	—
134	C₃	—	18	1	no	—
135	C₃	#92262	1	1	yes	characteristic
136	C₃	#92280	1	1	yes	characteristic
137	C₃	#92281	1	1	yes	characteristic
138	C₃	#6913⋊C₂	1	1	yes	characteristic
139	C₂	—	9	1	no	Sylow

subgroup

G/H

class size

multiplicity

normal?

properties

#10624797

C₂

yes

maximal, characteristic, complemented, Sylow

C₉×#272

C₃

yes

maximal, characteristic, complemented

C₃×#4174

C₃

yes

maximal, characteristic

D₃×C₃²⋊C₉

—

maximal

#272⋊C₉

C₃

yes

maximal, characteristic, complemented

#272⋊C₉

C₃

yes

maximal, characteristic, complemented

C₃×AΣL(1,𝔽₂₇)

C₂×C₃

yes

characteristic

#6894

—

C₉×H(3,𝔽₃)

C₂×C₃

yes

characteristic, complemented

C₃×C₃²⋊C₉

D₃

yes

characteristic, complemented

3₋¹⁺²⋊C₉

—

3₋¹⁺²⋊C₉

—

H(3,𝔽₃)⋊C₉

C₂×C₃

yes

characteristic, complemented

H(3,𝔽₃)⋊C₉

C₂×C₃

yes

characteristic, complemented

#6866

—

C₃³×D₃

—

C₃×#272

C₃²

yes

characteristic

C₃×D₃×C₉

—

C₃×D₃×C₉

—

C₃×D₃×C₉

—

C₃×D₃×C₉

—

C₂×C₃²⋊C₉

—

#4174

—

C₃⁴

C₃×D₃

yes

characteristic

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

characteristic

C₃²×C₉

C₃×D₃

yes

characteristic

AΣL(1,𝔽₂₇)

—

AΣL(1,𝔽₂₇)

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₂×C₃³

—

C₃²×D₃

—

C₃²×D₃

—

C₃²×D₃

—

C₃²×D₃

—

C₃²×D₃

—

C₃²×D₃

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

#272

3₋¹⁺²

yes

characteristic

D₃×C₉

—

D₃×C₉

—

D₃×C₉

—

D₃×C₉

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃²×D₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

H(3,𝔽₃)

C₂×3₋¹⁺²

yes

characteristic

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

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

—

C₂×C₃²

—

C₃×D₃

—

C₃×D₃

—

C₃×D₃

—

C₃×D₃

—

C₃×D₃

—

C₃×D₃

—

C₃×D₃

—

C₃×D₃

—

C₃×D₃

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

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

C₃×(C₃⋊D₃)⋊C₃

yes

characteristic

114

C₃²

D₃×3₋¹⁺²

yes

characteristic

115

C₉

—

116

C₉

—

117

C₉

—

118

C₉

—

119

C₉

—

120

C₉

—

121

C₉

—

122

C₉

—

123

C₂×C₃

—

124

C₂×C₃

—

125

C₂×C₃

—

126

C₂×C₃

—

127

C₂×C₃

—

128

D₃

—

129

D₃

—

130

C₃

—

131

C₃

—

132

C₃

—

133

C₃

—

134

C₃

—

135

C₃

#92262

yes

characteristic

136

C₃

#92280

yes

characteristic

137

C₃

#92281

yes

characteristic

138

C₃

#6913⋊C₂

yes

characteristic

139

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	#10624797	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. #10624797	C₂	semidirect
2. C₉×#272	C₃	semidirect
5. #272⋊C₉	C₃	semidirect
6. #272⋊C₉	C₃	semidirect
9. C₉×H(3,𝔽₃)	C₂×C₃	semidirect
10. C₃×C₃²⋊C₉	D₃	semidirect
13. H(3,𝔽₃)⋊C₉	C₂×C₃	semidirect
14. H(3,𝔽₃)⋊C₉	C₂×C₃	semidirect

H ≅ G/N

type

1. #10624797

C₂

semidirect

2. C₉×#272

C₃

semidirect

5. #272⋊C₉

C₃

semidirect

6. #272⋊C₉

C₃

semidirect

9. C₉×H(3,𝔽₃)

C₂×C₃

semidirect

10. C₃×C₃²⋊C₉

D₃

semidirect

13. H(3,𝔽₃)⋊C₉

C₂×C₃

semidirect

14. H(3,𝔽₃)⋊C₉

C₂×C₃

semidirect

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

subgroup	index
#10624797	2
C₉×#272	3
C₃×#4174	3
D₃×C₃²⋊C₉	3
#272⋊C₉	3

subgroup

index

D₃×C₃²⋊C₉

Subgroups — #10624797⋊C₂ #13091235

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10624797⋊C2 #13091235

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10624797⋊C₂ #13091235