Group #72243159 — subgroups

Subgroups — #72243159

Order 4374.

1020 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 1020 conjugacy classes of subgroups — too many to draw. Only its 100 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	#14552587	C₂	1	1	yes	maximal, characteristic, complemented, Sylow
2	C₃×#92314	C₃	1	1	yes	maximal, characteristic
3	C₃⋊#92265	—	3	4	no	maximal
4	#13090636	C₃	1	2	yes	maximal, complemented
5	C₉×C₉⋊D₉	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃×#6900	—	2	4	no	—
7	C₃×#6900	D₃	1	4	yes	—
8	C₃×#6900	C₂×C₃	1	1	yes	characteristic
9	#10624554	—	2	8	no	—
10	#10624554	—	2	4	no	—
11	#10624554	C₂×C₃	1	2	yes	complemented
12	C₉³	C₂×C₃	1	1	yes	characteristic, complemented
13	C₃⋊D₃×3₋¹⁺²	—	9	1	no	—
14	C₃×C₃⋊AGL(1,ℤ/9)	—	3	4	no	—
15	#92314	—	3	1	no	—
16	C₃⋊(D₉⋊C₉)	—	3	8	no	—
17	C₃×C₉⋊D₉	C₃²	1	1	yes	characteristic
18	C₉×C₃⋊D₉	—	3	4	no	—
19	#92265	—	9	12	no	—
20	C₃²×3₋¹⁺²	—	2	4	no	—
21	C₃²×3₋¹⁺²	C₃×D₃	1	4	yes	—
22	C₃²×3₋¹⁺²	C₃⋊D₃	1	1	yes	characteristic
23	#6900	—	3	12	no	—
24	#6900	—	3	1	no	—
25	#6900	—	6	12	no	—
26	#6900	—	6	36	no	—
27	#6900	—	6	4	no	—
28	C₃×C₉⋊C₉	—	2	24	no	—
29	C₃×C₉⋊C₉	—	2	4	no	—
30	C₃×C₉⋊C₉	—	2	8	no	—
31	C₃×C₉⋊C₉	—	2	8	no	—
32	C₃×C₉⋊C₉	—	2	4	no	—
33	C₃×C₉⋊C₉	C₃×D₃	1	8	yes	—
34	C₃×C₉²	—	2	4	no	—
35	C₃×C₉²	C₃×D₃	1	4	yes	—
36	C₃×C₉²	C₂×C₃²	1	1	yes	characteristic
37	C₃²×C₃⋊D₃	—	9	1	no	—
38	C₃⋊AGL(1,ℤ/9)	—	9	4	no	—
39	C₉×C₃⋊D₃	—	9	1	no	—
40	C₉×C₃⋊D₃	—	9	2	no	—
41	C₃×C₃⋊D₉	—	3	4	no	—
42	D₃×3₋¹⁺²	—	27	4	no	—
43	C₃×AGL(1,ℤ/9)	—	9	12	no	—
44	C₉⋊D₉	3₋¹⁺²	1	1	yes	characteristic, complemented
45	D₉⋊C₉	—	9	24	no	—
46	C₉×D₉	—	9	12	no	—
47	C₃⁴	C₃×C₃⋊D₃	1	1	yes	characteristic
48	C₃×3₋¹⁺²	—	3	4	no	—
49	C₃×3₋¹⁺²	—	3	12	no	—
50	C₃×3₋¹⁺²	—	3	4	no	—
51	C₃×3₋¹⁺²	—	6	4	no	—
52	C₃×3₋¹⁺²	—	6	12	no	—
53	C₃×3₋¹⁺²	—	6	12	no	—
54	C₃×3₋¹⁺²	—	6	36	no	—
55	C₃×3₋¹⁺²	—	6	4	no	—
56	C₃²×C₉	—	2	4	no	—
57	C₃²×C₉	—	2	8	no	—
58	C₃²×C₉	—	2	4	no	—
59	C₃²×C₉	C₃²×D₃	1	4	yes	—
60	C₃²×C₉	C₃×C₃⋊D₃	1	1	yes	characteristic
61	C₃²×C₉	C₃×C₃⋊D₃	1	2	yes	—
62	C₉⋊C₉	—	3	24	no	—
63	C₉⋊C₉	—	6	24	no	—
64	C₉⋊C₉	—	6	72	no	—
65	C₉⋊C₉	—	6	8	no	—
66	C₉⋊C₉	—	6	8	no	—
67	C₉⋊C₉	—	6	24	no	—
68	C₉⋊C₉	—	6	4	no	—
69	C₉⋊C₉	—	6	4	no	—
70	C₉²	—	2	36	no	—
71	C₉²	—	2	12	no	—
72	C₉²	—	2	4	no	—
73	C₉²	AGL(1,ℤ/9)	1	12	yes	complemented
74	C₉²	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
75	C₃×C₃⋊D₃	—	9	1	no	—
76	C₃×C₃⋊D₃	—	27	1	no	—
77	C₃²×D₃	—	27	4	no	—
78	C₂×3₋¹⁺²	—	81	1	no	—
79	C₃⋊D₉	—	3	4	no	—
80	AGL(1,ℤ/9)	—	27	12	no	—
81	D₃×C₉	—	27	8	no	—
82	D₃×C₉	—	27	4	no	—
83	C₃×D₉	—	9	12	no	—
84	C₃³	—	3	1	no	—
85	C₃³	—	3	4	no	—
86	C₃³	—	6	4	no	—
87	C₃³	C₃²×C₃⋊D₃	1	1	yes	characteristic
88	3₋¹⁺²	—	9	1	no	—
89	3₋¹⁺²	—	9	12	no	—
90	3₋¹⁺²	—	18	4	no	—
91	3₋¹⁺²	—	18	36	no	—
92	3₋¹⁺²	—	18	12	no	—
93	C₃×C₉	—	2	36	no	—
94	C₃×C₉	—	2	4	no	—
95	C₃×C₉	—	2	12	no	—
96	C₃×C₉	—	2	4	no	—
97	C₃×C₉	—	2	12	no	—
98	C₃×C₉	—	3	8	no	—
99	C₃×C₉	—	6	24	no	—
100	C₃×C₉	—	6	8	no	—
101	C₃×C₉	—	6	8	no	—
102	C₃×C₉	—	6	4	no	—
103	C₃×C₉	—	6	4	no	—
104	C₃×C₉	C₃×AGL(1,ℤ/9)	1	12	yes	—
105	C₃×C₉	D₃×3₋¹⁺²	1	4	yes	—
106	C₃×C₉	C₃⋊AGL(1,ℤ/9)	1	4	yes	—
107	C₂×C₃²	—	81	1	no	—
108	C₃⋊D₃	—	9	1	no	—
109	C₃×D₃	—	27	4	no	—
110	C₃×D₃	—	81	4	no	—
111	C₂×C₉	—	81	2	no	—
112	C₂×C₉	—	81	1	no	—
113	D₉	—	9	12	no	—
114	C₃²	—	2	4	no	—
115	C₃²	—	9	4	no	—
116	C₃²	—	9	1	no	—
117	C₃²	—	18	4	no	—
118	C₃²	C₃×C₃⋊AGL(1,ℤ/9)	1	4	yes	—
119	C₃²	C₃⋊D₃×3₋¹⁺²	1	1	yes	characteristic
120	C₉	—	2	36	no	—
121	C₉	—	2	12	no	—
122	C₉	—	2	4	no	—
123	C₉	—	9	2	no	—
124	C₉	—	18	8	no	—
125	C₉	—	18	4	no	—
126	C₉	#92265	1	12	yes	complemented
127	C₉	#92314	1	1	yes	characteristic, complemented
128	C₂×C₃	—	81	1	no	—
129	C₂×C₃	—	243	1	no	—
130	D₃	—	27	4	no	—
131	C₃	—	2	4	no	—
132	C₃	—	27	1	no	—
133	C₃	C₃⋊#92265	1	4	yes	—
134	C₃	C₃×#92314	1	1	yes	characteristic
135	C₂	—	81	1	no	Sylow

subgroup

G/H

class size

multiplicity

normal?

properties

#14552587

C₂

yes

maximal, characteristic, complemented, Sylow

C₃×#92314

C₃

yes

maximal, characteristic

C₃⋊#92265

—

maximal

#13090636

C₃

yes

maximal, complemented

C₉×C₉⋊D₉

C₃

yes

maximal, characteristic, complemented

C₃×#6900

—

C₃×#6900

D₃

yes

—

C₃×#6900

C₂×C₃

yes

—

—

C₂×C₃

yes

complemented

C₉³

C₂×C₃

yes

characteristic, complemented

C₃⋊D₃×3₋¹⁺²

—

C₃×C₃⋊AGL(1,ℤ/9)

—

#92314

—

C₃⋊(D₉⋊C₉)

—

C₃×C₉⋊D₉

C₃²

yes

characteristic

C₉×C₃⋊D₉

—

#92265

—

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

C₃×D₃

yes

—

C₃²×3₋¹⁺²

C₃⋊D₃

yes

—

—

—

—

—

C₃×C₉⋊C₉

—

C₃×C₉⋊C₉

—

C₃×C₉⋊C₉

—

C₃×C₉⋊C₉

—

C₃×C₉⋊C₉

—

C₃×C₉⋊C₉

C₃×D₃

yes

—

C₃×C₉²

—

C₃×C₉²

C₃×D₃

yes

—

C₃×C₉²

C₂×C₃²

yes

characteristic

C₃²×C₃⋊D₃

—

C₃⋊AGL(1,ℤ/9)

—

C₉×C₃⋊D₃

—

C₉×C₃⋊D₃

—

C₃×C₃⋊D₉

—

D₃×3₋¹⁺²

—

C₃×AGL(1,ℤ/9)

—

C₉⋊D₉

3₋¹⁺²

yes

characteristic, complemented

D₉⋊C₉

—

C₉×D₉

—

C₃⁴

C₃×C₃⋊D₃

yes

characteristic

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

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃²×D₃

yes

—

C₃²×C₉

C₃×C₃⋊D₃

yes

characteristic

C₃²×C₉

C₃×C₃⋊D₃

yes

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉²

—

C₉²

—

C₉²

—

C₉²

AGL(1,ℤ/9)

yes

complemented

C₉²

C₂×3₋¹⁺²

yes

characteristic, complemented

C₃×C₃⋊D₃

—

C₃×C₃⋊D₃

—

C₃²×D₃

—

C₂×3₋¹⁺²

—

C₃⋊D₉

—

AGL(1,ℤ/9)

—

D₃×C₉

—

D₃×C₉

—

C₃×D₉

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃²×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

—

100

C₃×C₉

—

101

C₃×C₉

—

102

C₃×C₉

—

103

C₃×C₉

—

104

C₃×C₉

C₃×AGL(1,ℤ/9)

yes

—

105

C₃×C₉

D₃×3₋¹⁺²

yes

—

106

C₃×C₉

C₃⋊AGL(1,ℤ/9)

yes

—

107

C₂×C₃²

—

108

C₃⋊D₃

—

109

C₃×D₃

—

110

C₃×D₃

—

111

C₂×C₉

—

112

C₂×C₉

—

113

D₉

—

114

C₃²

—

115

C₃²

—

116

C₃²

—

117

C₃²

—

118

C₃²

C₃×C₃⋊AGL(1,ℤ/9)

yes

—

119

C₃²

C₃⋊D₃×3₋¹⁺²

yes

characteristic

120

C₉

—

121

C₉

—

122

C₉

—

123

C₉

—

124

C₉

—

125

C₉

—

126

C₉

#92265

yes

complemented

127

C₉

#92314

yes

characteristic, complemented

128

C₂×C₃

—

129

C₂×C₃

—

243

—

130

D₃

—

131

C₃

—

132

C₃

—

133

C₃

C₃⋊#92265

yes

—

134

C₃

C₃×#92314

yes

characteristic

135

C₂

—

Sylow

	subgroup	G/H
center	C₃	C₃×#92314
commutator, cosocle	C₃×C₉²	C₂×C₃²
Frattini, socle	C₃³	C₃²×C₃⋊D₃
Fitting	#14552587	C₂
prime radical	C₃×#92314	C₃
hypercenter	C₉	#92314
hypocenter	C₉²	C₂×3₋¹⁺²
LCS	G ▷ C₃×C₉² ▷ C₉²
UCS	C₁ ◁ C₃ ◁ C₉

subgroup

G/H

center

C₃

C₃×#92314

commutator, cosocle

C₃×C₉²

C₂×C₃²

Frattini, socle

C₃³

C₃²×C₃⋊D₃

C₂

C₃

C₉

C₉²

C₂×3₋¹⁺²

LCS

G ▷ C₃×C₉² ▷ C₉²

UCS

C₁ ◁ C₃ ◁ C₉

N	H ≅ G/N	type
1. #14552587	C₂	semidirect
4. #13090636	C₃	semidirect
5. C₉×C₉⋊D₉	C₃	semidirect
11. #10624554	C₂×C₃	semidirect
12. C₉³	C₂×C₃	semidirect
44. C₉⋊D₉	3₋¹⁺²	semidirect
73. C₉²	AGL(1,ℤ/9)	semidirect
74. C₉²	C₂×3₋¹⁺²	semidirect
126. C₉	#92265	semidirect
127. C₉	#92314	semidirect

H ≅ G/N

type

1. #14552587

C₂

semidirect

4. #13090636

C₃

semidirect

5. C₉×C₉⋊D₉

C₃

semidirect

11. #10624554

C₂×C₃

semidirect

12. C₉³

C₂×C₃

semidirect

44. C₉⋊D₉

3₋¹⁺²

semidirect

73. C₉²

AGL(1,ℤ/9)

semidirect

74. C₉²

C₂×3₋¹⁺²

semidirect

126. C₉

#92265

semidirect

127. C₉

#92314

semidirect

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

subgroup	index
#14552587	2
C₃×#92314	3
C₃⋊#92265	3
#13090636	3
C₉×C₉⋊D₉	3

subgroup

index

C₉×C₉⋊D₉

Subgroups — #72243159

Conjugacy classes of subgroups

Complements

Maximal subgroups