Group #72276233 — subgroups

Subgroups — #72276233

Order 2592.

546 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 153 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	C₂×C₄⋊(D₉⋊C₉)	C₂	1	6	yes	maximal, complemented
2	C₄²×C₉⋊C₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×#874	—	3	1	no	maximal
4	C₃×#63436	C₃	1	1	yes	maximal, characteristic
5	C₂²×D₉⋊C₉	C₂²	1	4	yes	—
6	C₂×C₄×C₉⋊C₉	C₂²	1	3	yes	—
7	C₄⋊(D₉⋊C₉)	—	2	12	no	—
8	C₂×C₉×D₁₂	—	3	6	no	—
9	C₂×C₃×D₃₆	C₂×C₃	1	6	yes	—
10	C₃×C₄²×C₉	—	2	1	no	—
11	C₃×C₄²×C₉	D₃	1	1	yes	characteristic
12	C₃×C₄²×C₉	C₂×C₃	1	1	yes	characteristic
13	C₂²×C₉⋊C₉	C₂³	1	1	yes	characteristic
14	C₂×D₉⋊C₉	—	2	12	no	—
15	C₄×C₉⋊C₉	D₄	1	6	yes	complemented
16	C₃×#874	—	3	1	no	—
17	C₉×#127	—	9	1	no	—
18	#63436	C₉	1	1	yes	characteristic, complemented
19	C₂²×D₃×C₉	—	3	4	no	—
20	C₂²×C₃×D₉	C₂²×C₃	1	4	yes	—
21	C₂×C₃×C₄×C₉	—	2	3	no	—
22	C₂×C₃×C₄×C₉	C₂×D₃	1	3	yes	—
23	C₂×C₃×C₄×C₉	C₂²×C₃	1	3	yes	—
24	C₉×D₁₂	—	6	12	no	—
25	C₃×D₃₆	—	2	12	no	—
26	C₂×C₉⋊C₉	C₂×D₄	1	3	yes	—
27	D₉⋊C₉	—	4	4	no	—
28	C₂×C₃×D₁₂	—	3	6	no	—
29	C₃²×C₄²	C₃×D₃	1	1	yes	characteristic
30	C₂×D₄×C₉	—	9	6	no	—
31	C₂×D₃₆	C₂×C₉	1	6	yes	complemented
32	C₄²×C₉	—	2	1	no	—
33	C₄²×C₉	—	3	1	no	—
34	C₄²×C₉	—	6	1	no	—
35	C₄²×C₉	C₂×C₉	1	1	yes	characteristic, complemented
36	C₂²×C₃×C₉	—	2	1	no	—
37	C₂²×C₃×C₉	C₂²×D₃	1	1	yes	characteristic
38	C₂²×C₃×C₉	C₂³×C₃	1	1	yes	characteristic
39	C₂×D₃×C₉	—	6	12	no	—
40	C₂×C₃×D₉	—	2	12	no	—
41	C₃×C₄×C₉	—	2	6	no	—
42	C₃×C₄×C₉	D₁₂	1	6	yes	—
43	C₃×C₄×C₉	C₃×D₄	1	6	yes	—
44	C₃×#127	—	9	1	no	—
45	#874	—	3	1	no	—
46	C₉⋊C₉	#127	1	1	yes	characteristic, complemented, Sylow
47	C₂²×C₃×D₃	—	3	4	no	—
48	C₂×C₃²×C₄	C₂×C₃×D₃	1	3	yes	—
49	C₃×D₁₂	—	6	12	no	—
50	C₂³×C₉	—	9	4	no	—
51	C₂²×D₉	C₂²×C₉	1	4	yes	—
52	D₄×C₉	—	18	12	no	—
53	C₂×C₄×C₉	—	2	3	no	—
54	C₂×C₄×C₉	—	3	3	no	—
55	C₂×C₄×C₉	—	6	3	no	—
56	C₂×C₄×C₉	C₂²×C₉	1	3	yes	—
57	D₃₆	—	2	12	no	—
58	C₂×C₃×C₉	—	2	3	no	—
59	C₂×C₃×C₉	C₂×D₁₂	1	3	yes	—
60	C₂×C₃×C₉	C₂×C₃×D₄	1	3	yes	—
61	D₃×C₉	—	12	4	no	—
62	C₃×D₉	—	4	4	no	—
63	C₂×C₃×D₄	—	9	6	no	—
64	C₂×D₁₂	—	3	6	no	—
65	C₃×C₄²	—	2	1	no	—
66	C₃×C₄²	D₃×C₉	1	1	yes	characteristic
67	C₃×C₄²	AGL(1,ℤ/9)	1	1	yes	characteristic
68	C₂²×C₃²	C₂²×C₃×D₃	1	1	yes	characteristic
69	C₂×C₃×D₃	—	6	12	no	—
70	C₃²×C₄	C₃×D₁₂	1	6	yes	—
71	C₂²×C₉	—	2	1	no	—
72	C₂²×C₉	—	3	1	no	—
73	C₂²×C₉	—	6	1	no	—
74	C₂²×C₉	—	18	12	no	—
75	C₂²×C₉	C₂³×C₉	1	1	yes	characteristic
76	C₂×D₉	—	2	12	no	—
77	C₄×C₉	—	2	6	no	—
78	C₄×C₉	—	3	6	no	—
79	C₄×C₉	—	6	6	no	—
80	C₄×C₉	D₄×C₉	1	6	yes	complemented
81	#127	—	9	1	no	Sylow
82	C₃×C₉	—	2	1	no	—
83	C₃×C₉	#874	1	1	yes	characteristic
84	C₃×C₉	C₃×#127	1	1	yes	characteristic
85	C₂³×C₃	—	9	4	no	—
86	C₂²×D₃	—	3	4	no	—
87	C₃×D₄	—	18	12	no	—
88	C₂×C₃×C₄	—	2	3	no	—
89	C₂×C₃×C₄	C₂×D₃×C₉	1	3	yes	—
90	C₂×C₃×C₄	C₂×AGL(1,ℤ/9)	1	3	yes	—
91	D₁₂	—	6	12	no	—
92	C₂×C₃²	C₂×C₃×D₁₂	1	3	yes	—
93	C₃×D₃	—	12	4	no	—
94	C₂×C₉	—	2	3	no	—
95	C₂×C₉	—	3	3	no	—
96	C₂×C₉	—	6	3	no	—
97	C₂×C₉	—	36	4	no	—
98	C₂×C₉	C₂×D₄×C₉	1	3	yes	—
99	D₉	—	4	4	no	—
100	C₂×D₄	—	9	6	no	—
101	C₄²	D₉⋊C₉	1	1	yes	characteristic, complemented
102	C₂²×C₃	—	2	1	no	—
103	C₂²×C₃	—	18	12	no	—
104	C₂²×C₃	C₂²×D₃×C₉	1	1	yes	characteristic
105	C₂²×C₃	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
106	C₂×D₃	—	6	12	no	—
107	C₃×C₄	—	2	6	no	—
108	C₃×C₄	C₉×D₁₂	1	6	yes	—
109	C₃×C₄	C₄⋊AGL(1,ℤ/9)	1	6	yes	—
110	C₃²	C₃×#874	1	1	yes	characteristic
111	C₉	—	2	1	no	—
112	C₉	—	3	1	no	—
113	C₉	—	6	1	no	—
114	C₉	C₉×#127	1	1	yes	characteristic, complemented
115	C₂³	—	9	4	no	—
116	D₄	—	18	12	no	—
117	C₂×C₄	C₂×D₉⋊C₉	1	3	yes	—
118	C₂×C₃	—	2	3	no	—
119	C₂×C₃	—	36	4	no	—
120	C₂×C₃	C₂×C₉×D₁₂	1	3	yes	—
121	C₂×C₃	C₂×C₄⋊AGL(1,ℤ/9)	1	3	yes	—
122	D₃	—	12	4	no	—
123	C₂²	—	18	12	no	—
124	C₂²	C₂²×D₉⋊C₉	1	1	yes	characteristic
125	C₄	C₄⋊(D₉⋊C₉)	1	6	yes	complemented
126	C₃	—	2	1	no	—
127	C₃	C₉×#874	1	1	yes	characteristic
128	C₃	#11721394	1	1	yes	characteristic
129	C₂	—	36	4	no	—
130	C₂	C₂×C₄⋊(D₉⋊C₉)	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(D₉⋊C₉)

C₂

yes

maximal, complemented

C₄²×C₉⋊C₉

C₂

yes

maximal, characteristic, complemented

C₉×#874

—

maximal

C₃×#63436

C₃

yes

maximal, characteristic

C₂²×D₉⋊C₉

C₂²

yes

—

C₂×C₄×C₉⋊C₉

C₂²

yes

—

C₄⋊(D₉⋊C₉)

—

C₂×C₉×D₁₂

—

C₂×C₃×D₃₆

C₂×C₃

yes

—

C₃×C₄²×C₉

—

C₃×C₄²×C₉

D₃

yes

characteristic

C₃×C₄²×C₉

C₂×C₃

yes

characteristic

C₂²×C₉⋊C₉

C₂³

yes

characteristic

C₂×D₉⋊C₉

—

C₄×C₉⋊C₉

D₄

yes

—

—

C₉

yes

characteristic, complemented

C₂²×D₃×C₉

—

C₂²×C₃×D₉

C₂²×C₃

yes

—

C₂×C₃×C₄×C₉

—

C₂×C₃×C₄×C₉

C₂×D₃

yes

—

C₂×C₃×C₄×C₉

C₂²×C₃

yes

—

C₉×D₁₂

—

C₃×D₃₆

—

C₂×C₉⋊C₉

C₂×D₄

yes

—

D₉⋊C₉

—

C₂×C₃×D₁₂

—

C₃²×C₄²

C₃×D₃

yes

characteristic

C₂×D₄×C₉

—

C₂×D₃₆

C₂×C₉

yes

complemented

C₄²×C₉

—

C₄²×C₉

—

C₄²×C₉

—

C₄²×C₉

C₂×C₉

yes

characteristic, complemented

C₂²×C₃×C₉

—

C₂²×C₃×C₉

C₂²×D₃

yes

characteristic

C₂²×C₃×C₉

C₂³×C₃

yes

characteristic

C₂×D₃×C₉

—

C₂×C₃×D₉

—

C₃×C₄×C₉

—

C₃×C₄×C₉

D₁₂

yes

—

C₃×C₄×C₉

C₃×D₄

yes

—

C₃×#127

—

#874

—

C₉⋊C₉

#127

yes

characteristic, complemented, Sylow

C₂²×C₃×D₃

—

C₂×C₃²×C₄

C₂×C₃×D₃

yes

—

C₃×D₁₂

—

C₂³×C₉

—

C₂²×D₉

C₂²×C₉

yes

—

D₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂²×C₉

yes

—

D₃₆

—

C₂×C₃×C₉

—

C₂×C₃×C₉

C₂×D₁₂

yes

—

C₂×C₃×C₉

C₂×C₃×D₄

yes

—

D₃×C₉

—

C₃×D₉

—

C₂×C₃×D₄

—

C₂×D₁₂

—

C₃×C₄²

—

C₃×C₄²

D₃×C₉

yes

characteristic

C₃×C₄²

AGL(1,ℤ/9)

yes

characteristic

C₂²×C₃²

C₂²×C₃×D₃

yes

characteristic

C₂×C₃×D₃

—

C₃²×C₄

C₃×D₁₂

yes

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

C₂³×C₉

yes

characteristic

C₂×D₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

D₄×C₉

yes

complemented

#127

—

Sylow

C₃×C₉

—

C₃×C₉

yes

C₃×C₉

yes

C₂³×C₃

—

C₂²×D₃

—

C₃×D₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×D₃×C₉

yes

—

C₂×C₃×C₄

C₂×AGL(1,ℤ/9)

yes

—

D₁₂

—

C₂×C₃²

C₂×C₃×D₁₂

yes

—

C₃×D₃

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

C₂×D₄×C₉

yes

—

D₉

—

100

C₂×D₄

—

101

C₄²

D₉⋊C₉

yes

characteristic, complemented

102

C₂²×C₃

—

103

C₂²×C₃

—

104

C₂²×C₃

C₂²×D₃×C₉

yes

characteristic

105

C₂²×C₃

C₂²×AGL(1,ℤ/9)

yes

characteristic

106

C₂×D₃

—

107

C₃×C₄

—

108

C₃×C₄

C₉×D₁₂

yes

—

109

C₃×C₄

C₄⋊AGL(1,ℤ/9)

yes

—

110

C₃²

C₃×#874

yes

characteristic

111

C₉

—

112

C₉

—

113

C₉

—

114

C₉

C₉×#127

yes

characteristic, complemented

115

C₂³

—

116

D₄

—

117

C₂×C₄

C₂×D₉⋊C₉

yes

—

118

C₂×C₃

—

119

C₂×C₃

—

120

C₂×C₃

C₂×C₉×D₁₂

yes

—

121

C₂×C₃

C₂×C₄⋊AGL(1,ℤ/9)

yes

—

122

D₃

—

123

C₂²

—

124

C₂²

C₂²×D₉⋊C₉

yes

characteristic

125

C₄

C₄⋊(D₉⋊C₉)

yes

complemented

126

C₃

—

127

C₃

C₉×#874

yes

characteristic

128

C₃

#11721394

yes

characteristic

129

C₂

—

130

C₂

C₂×C₄⋊(D₉⋊C₉)

yes

—

	subgroup	G/H
center	C₂²×C₃	C₂²×AGL(1,ℤ/9)
commutator	C₂²×C₉	C₂³×C₉
Frattini, socle	C₂²×C₃²	C₂²×C₃×D₃
Fitting	C₄²×C₉⋊C₉	C₂
prime radical	C₃×#63436	C₃
cosocle	C₂²×C₃×C₉	C₂³×C₃
hypercenter	C₃×C₄²	AGL(1,ℤ/9)
hypocenter	C₉	C₉×#127
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄²

subgroup

G/H

center

C₂²×C₃

C₂²×AGL(1,ℤ/9)

commutator

C₂²×C₉

C₂³×C₉

Frattini, socle

C₂²×C₃²

C₂²×C₃×D₃

Fitting

C₄²×C₉⋊C₉

C₂

prime radical

C₃×#63436

C₃

cosocle

C₂²×C₃×C₉

C₂³×C₃

C₃×C₄²

C₉

G ▷ C₂²×C₉ ▷ C₉

C₁ ◁ C₂²×C₃ ◁ C₃×C₄²

N	H ≅ G/N	type
1. C₂×C₄⋊(D₉⋊C₉)	C₂	semidirect
2. C₄²×C₉⋊C₉	C₂	semidirect
15. C₄×C₉⋊C₉	D₄	semidirect
18. #63436	C₉	semidirect
31. C₂×D₃₆	C₂×C₉	semidirect
35. C₄²×C₉	C₂×C₉	semidirect
46. C₉⋊C₉	#127	semidirect
80. C₄×C₉	D₄×C₉	semidirect
101. C₄²	D₉⋊C₉	semidirect
114. C₉	C₉×#127	semidirect
125. C₄	C₄⋊(D₉⋊C₉)	semidirect

H ≅ G/N

type

1. C₂×C₄⋊(D₉⋊C₉)

C₂

semidirect

2. C₄²×C₉⋊C₉

C₂

semidirect

15. C₄×C₉⋊C₉

D₄

semidirect

18. #63436

C₉

semidirect

31. C₂×D₃₆

C₂×C₉

semidirect

35. C₄²×C₉

C₂×C₉

semidirect

46. C₉⋊C₉

#127

semidirect

80. C₄×C₉

D₄×C₉

semidirect

101. C₄²

D₉⋊C₉

semidirect

114. C₉

C₉×#127

semidirect

125. C₄

C₄⋊(D₉⋊C₉)

semidirect

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

subgroup	index
C₂×C₄⋊(D₉⋊C₉)	2
C₄²×C₉⋊C₉	2
C₉×#874	3
C₃×#63436	3

subgroup

index

C₂×C₄⋊(D₉⋊C₉)

C₄²×C₉⋊C₉

C₉×#874

C₃×#63436

Subgroups — #72276233

Conjugacy classes of subgroups

Complements

Maximal subgroups