Group #72243192 — subgroups

Subgroups — #72243192

Order 4374.

1059 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1059 conjugacy classes of subgroups — too many to draw. Only its 158 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×#92265	C₃	1	4	yes	maximal
3	D₃×#6900	—	3	1	no	maximal
4	#13090635	C₃	1	8	yes	maximal, complemented
5	C₉²×D₉	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃×#6900	—	2	4	no	—
7	C₃×#6900	D₃	1	1	yes	characteristic
8	C₃×#6900	C₂×C₃	1	4	yes	—
9	#10624554	—	2	8	no	—
10	#10624554	—	2	1	no	—
11	#10624554	C₂×C₃	1	8	yes	complemented
12	C₉³	C₂×C₃	1	1	yes	characteristic, complemented
13	C₃×D₃×3₋¹⁺²	—	3	4	no	—
14	C₃²×AGL(1,ℤ/9)	C₃²	1	1	yes	characteristic
15	C₂×#6900	—	9	1	no	—
16	#92265	—	3	12	no	—
17	D₃×C₉⋊C₉	—	3	8	no	—
18	C₃×D₉⋊C₉	C₃²	1	8	yes	—
19	D₃×C₉²	—	3	1	no	—
20	C₃×C₉×D₉	C₃²	1	4	yes	—
21	C₃²×3₋¹⁺²	—	2	4	no	—
22	C₃²×3₋¹⁺²	C₃×D₃	1	4	yes	—
23	C₃²×3₋¹⁺²	C₂×C₃²	1	1	yes	characteristic
24	#6900	—	3	1	no	—
25	#6900	—	3	12	no	—
26	#6900	—	6	4	no	—
27	#6900	—	6	36	no	—
28	#6900	—	6	12	no	—
29	C₃×C₉⋊C₉	—	2	8	no	—
30	C₃×C₉⋊C₉	—	2	4	no	—
31	C₃×C₉⋊C₉	—	2	24	no	—
32	C₃×C₉⋊C₉	—	2	8	no	—
33	C₃×C₉⋊C₉	C₃×D₃	1	8	yes	—
34	C₃×C₉⋊C₉	C₂×C₃²	1	8	yes	—
35	C₃×C₉²	—	2	4	no	—
36	C₃×C₉²	C₃×D₃	1	1	yes	characteristic
37	C₃×C₉²	C₂×C₃²	1	4	yes	—
38	C₃³×D₃	—	3	1	no	—
39	C₂×C₃×3₋¹⁺²	—	9	4	no	—
40	D₃×3₋¹⁺²	—	9	12	no	—
41	C₃×AGL(1,ℤ/9)	—	3	4	no	—
42	C₃×D₃×C₉	—	3	4	no	—
43	C₃×D₃×C₉	—	3	8	no	—
44	C₃²×D₉	C₃³	1	1	yes	characteristic
45	C₂×C₉⋊C₉	—	9	8	no	—
46	C₂×C₉²	—	9	1	no	—
47	D₉⋊C₉	—	3	8	no	—
48	C₉×D₉	3₋¹⁺²	1	12	yes	complemented
49	C₃⁴	C₃²×D₃	1	1	yes	characteristic
50	C₃×3₋¹⁺²	—	3	4	no	—
51	C₃×3₋¹⁺²	—	3	12	no	—
52	C₃×3₋¹⁺²	—	3	4	no	—
53	C₃×3₋¹⁺²	—	6	4	no	—
54	C₃×3₋¹⁺²	—	6	12	no	—
55	C₃×3₋¹⁺²	—	6	12	no	—
56	C₃×3₋¹⁺²	—	6	36	no	—
57	C₃×3₋¹⁺²	—	6	4	no	—
58	C₃²×C₉	—	2	4	no	—
59	C₃²×C₉	—	2	8	no	—
60	C₃²×C₉	—	2	1	no	—
61	C₃²×C₉	C₃²×D₃	1	4	yes	—
62	C₃²×C₉	C₃²×D₃	1	8	yes	—
63	C₃²×C₉	C₂×C₃³	1	1	yes	characteristic
64	C₉⋊C₉	—	3	8	no	—
65	C₉⋊C₉	—	3	8	no	—
66	C₉⋊C₉	—	6	8	no	—
67	C₉⋊C₉	—	6	8	no	—
68	C₉⋊C₉	—	6	24	no	—
69	C₉⋊C₉	—	6	72	no	—
70	C₉⋊C₉	—	6	24	no	—
71	C₉⋊C₉	—	6	12	no	—
72	C₉²	—	2	36	no	—
73	C₉²	—	2	4	no	—
74	C₉²	—	2	12	no	—
75	C₉²	AGL(1,ℤ/9)	1	1	yes	characteristic, complemented
76	C₉²	C₂×3₋¹⁺²	1	12	yes	complemented
77	C₂×C₃³	—	9	1	no	—
78	C₃²×D₃	—	3	1	no	—
79	C₃²×D₃	—	9	4	no	—
80	C₂×3₋¹⁺²	—	27	12	no	—
81	C₂×C₃×C₉	—	9	4	no	—
82	C₂×C₃×C₉	—	9	8	no	—
83	AGL(1,ℤ/9)	—	9	1	no	—
84	D₃×C₉	—	3	12	no	—
85	D₃×C₉	—	9	8	no	—
86	C₃×D₉	C₃×3₋¹⁺²	1	4	yes	—
87	C₃³	—	3	1	no	—
88	C₃³	—	3	4	no	—
89	C₃³	—	6	4	no	—
90	C₃³	C₃³×D₃	1	1	yes	characteristic
91	3₋¹⁺²	—	9	12	no	—
92	3₋¹⁺²	—	9	1	no	—
93	3₋¹⁺²	—	18	12	no	—
94	3₋¹⁺²	—	18	36	no	—
95	3₋¹⁺²	—	18	4	no	—
96	C₃×C₉	—	2	36	no	—
97	C₃×C₉	—	2	4	no	—
98	C₃×C₉	—	2	4	no	—
99	C₃×C₉	—	2	12	no	—
100	C₃×C₉	—	2	12	no	—
101	C₃×C₉	—	3	8	no	—
102	C₃×C₉	—	3	8	no	—
103	C₃×C₉	—	6	8	no	—
104	C₃×C₉	—	6	24	no	—
105	C₃×C₉	—	6	8	no	—
106	C₃×C₉	—	6	4	no	—
107	C₃×C₉	C₃×AGL(1,ℤ/9)	1	4	yes	—
108	C₃×C₉	D₃×3₋¹⁺²	1	12	yes	—
109	C₃×C₉	C₂×C₃×3₋¹⁺²	1	4	yes	—
110	C₂×C₃²	—	9	1	no	—
111	C₂×C₃²	—	27	4	no	—
112	C₃×D₃	—	3	4	no	—
113	C₃×D₃	—	27	1	no	—
114	C₂×C₉	—	9	12	no	—
115	C₂×C₉	—	27	8	no	—
116	D₉	#6900	1	1	yes	characteristic, complemented
117	C₃²	—	2	4	no	—
118	C₃²	—	9	1	no	—
119	C₃²	—	9	4	no	—
120	C₃²	—	18	4	no	—
121	C₃²	C₃²×AGL(1,ℤ/9)	1	1	yes	characteristic
122	C₃²	C₃×D₃×3₋¹⁺²	1	4	yes	—
123	C₉	—	2	36	no	—
124	C₉	—	2	4	no	—
125	C₉	—	2	12	no	—
126	C₉	—	9	8	no	—
127	C₉	—	18	8	no	—
128	C₉	—	18	1	no	—
129	C₉	#92265	1	12	yes	complemented
130	C₉	C₂×#6900	1	1	yes	characteristic, complemented
131	C₂×C₃	—	9	4	no	—
132	C₂×C₃	—	81	1	no	—
133	D₃	—	3	1	no	—
134	C₃	—	2	4	no	—
135	C₃	—	27	1	no	—
136	C₃	D₃×#6900	1	1	yes	characteristic
137	C₃	C₃×#92265	1	4	yes	—
138	C₂	—	9	1	no	Sylow

subgroup

G/H

class size

multiplicity

normal?

properties

#14552587

C₂

yes

maximal, characteristic, complemented, Sylow

C₃×#92265

C₃

yes

—

C₃

yes

maximal, complemented

C₉²×D₉

C₃

yes

maximal, characteristic, complemented

C₃×#6900

—

C₃×#6900

D₃

yes

characteristic

C₃×#6900

C₂×C₃

yes

—

#10624554

—

#10624554

—

#10624554

C₂×C₃

yes

complemented

C₉³

C₂×C₃

yes

characteristic, complemented

C₃×D₃×3₋¹⁺²

—

C₃²×AGL(1,ℤ/9)

C₃²

yes

characteristic

C₂×#6900

—

#92265

—

D₃×C₉⋊C₉

—

C₃×D₉⋊C₉

C₃²

yes

—

D₃×C₉²

—

C₃×C₉×D₉

C₃²

yes

—

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

C₃×D₃

yes

—

C₃²×3₋¹⁺²

C₂×C₃²

yes

—

—

—

—

—

C₃×C₉⋊C₉

—

C₃×C₉⋊C₉

—

C₃×C₉⋊C₉

—

C₃×C₉⋊C₉

—

C₃×C₉⋊C₉

C₃×D₃

yes

—

C₃×C₉⋊C₉

C₂×C₃²

yes

—

C₃×C₉²

—

C₃×C₉²

C₃×D₃

yes

characteristic

C₃×C₉²

C₂×C₃²

yes

—

C₃³×D₃

—

C₂×C₃×3₋¹⁺²

—

D₃×3₋¹⁺²

—

C₃×AGL(1,ℤ/9)

—

C₃×D₃×C₉

—

C₃×D₃×C₉

—

C₃²×D₉

C₃³

yes

characteristic

C₂×C₉⋊C₉

—

C₂×C₉²

—

D₉⋊C₉

—

C₉×D₉

3₋¹⁺²

yes

complemented

C₃⁴

C₃²×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

C₃²×D₃

yes

—

C₃²×C₉

C₂×C₃³

yes

characteristic

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

characteristic, complemented

C₉²

C₂×3₋¹⁺²

yes

complemented

C₂×C₃³

—

C₃²×D₃

—

C₃²×D₃

—

C₂×3₋¹⁺²

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

AGL(1,ℤ/9)

—

D₃×C₉

—

D₃×C₉

—

C₃×D₉

C₃×3₋¹⁺²

yes

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃³×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

—

100

C₃×C₉

—

101

C₃×C₉

—

102

C₃×C₉

—

103

C₃×C₉

—

104

C₃×C₉

—

105

C₃×C₉

—

106

C₃×C₉

—

107

C₃×C₉

C₃×AGL(1,ℤ/9)

yes

—

108

C₃×C₉

D₃×3₋¹⁺²

yes

—

109

C₃×C₉

C₂×C₃×3₋¹⁺²

yes

—

110

C₂×C₃²

—

111

C₂×C₃²

—

112

C₃×D₃

—

113

C₃×D₃

—

114

C₂×C₉

—

115

C₂×C₉

—

116

D₉

#6900

yes

characteristic, complemented

117

C₃²

—

118

C₃²

—

119

C₃²

—

120

C₃²

—

121

C₃²

C₃²×AGL(1,ℤ/9)

yes

characteristic

122

C₃²

C₃×D₃×3₋¹⁺²

yes

—

123

C₉

—

124

C₉

—

125

C₉

—

126

C₉

—

127

C₉

—

128

C₉

—

129

C₉

#92265

yes

complemented

130

C₉

C₂×#6900

yes

characteristic, complemented

131

C₂×C₃

—

132

C₂×C₃

—

133

D₃

—

134

C₃

—

135

C₃

—

136

C₃

D₃×#6900

yes

characteristic

137

C₃

C₃×#92265

yes

—

138

C₂

—

Sylow

	subgroup	G/H
center	C₃²	C₃²×AGL(1,ℤ/9)
commutator, cosocle	C₃²×C₉	C₂×C₃³
Frattini, socle	C₃³	C₃³×D₃
Fitting	#14552587	C₂
prime radical	C₃²×AGL(1,ℤ/9)	C₃²
hypercenter	C₉²	AGL(1,ℤ/9)
hypocenter	C₉	C₂×#6900
LCS	G ▷ C₃²×C₉ ▷ C₉
UCS	C₁ ◁ C₃² ◁ C₉²

subgroup

G/H

center

C₃²

C₃²×AGL(1,ℤ/9)

commutator, cosocle

C₃²×C₉

C₂×C₃³

Frattini, socle

C₃³

C₃³×D₃

C₂

C₃²

C₉²

C₉

G ▷ C₃²×C₉ ▷ C₉

C₁ ◁ C₃² ◁ C₉²

N	H ≅ G/N	type
1. #14552587	C₂	semidirect
4. #13090635	C₃	semidirect
5. C₉²×D₉	C₃	semidirect
11. #10624554	C₂×C₃	semidirect
12. C₉³	C₂×C₃	semidirect
48. C₉×D₉	3₋¹⁺²	semidirect
75. C₉²	AGL(1,ℤ/9)	semidirect
76. C₉²	C₂×3₋¹⁺²	semidirect
116. D₉	#6900	semidirect
129. C₉	#92265	semidirect
130. C₉	C₂×#6900	semidirect

H ≅ G/N

type

1. #14552587

C₂

semidirect

4. #13090635

C₃

semidirect

5. C₉²×D₉

C₃

semidirect

11. #10624554

C₂×C₃

semidirect

12. C₉³

C₂×C₃

semidirect

48. C₉×D₉

3₋¹⁺²

semidirect

75. C₉²

AGL(1,ℤ/9)

semidirect

76. C₉²

C₂×3₋¹⁺²

semidirect

116. D₉

#6900

semidirect

129. C₉

#92265

semidirect

130. C₉

C₂×#6900

semidirect

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

subgroup	index
#14552587	2
C₃×#92265	3
D₃×#6900	3
#13090635	3
C₉²×D₉	3

subgroup

index

C₉²×D₉

Subgroups — #72243192

Conjugacy classes of subgroups

Complements

Maximal subgroups