Group #14518459 — subgroups

Subgroups — C₂²×D₃×C₉⋊C₉ #14518459

Order 1944.

684 conjugacy classes of subgroups; only the 294 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 294 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	C₂²×C₃×C₉⋊C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₃×C₉⋊C₉	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₃×D₃×C₉	C₃	1	3	yes	maximal
4	C₂²×C₃×D₃×C₉	C₃	1	1	yes	maximal, characteristic
5	C₂³×C₉⋊C₉	—	3	1	no	maximal
6	C₂×C₃×C₉⋊C₉	C₂²	1	3	yes	complemented
7	D₃×C₉⋊C₉	C₂²	1	4	yes	direct product factor
8	C₂²×C₃²×C₉	C₂×C₃	1	1	yes	characteristic
9	C₂²×C₃²×C₉	C₂×C₃	1	3	yes	—
10	C₂×C₃×D₃×C₉	C₂×C₃	1	18	yes	—
11	C₂×C₃×D₃×C₉	C₂×C₃	1	6	yes	—
12	C₂²×C₉⋊C₉	—	2	3	no	—
13	C₂²×C₉⋊C₉	—	2	1	no	—
14	C₂²×C₉⋊C₉	—	3	6	no	—
15	C₂²×C₉⋊C₉	D₃	1	1	yes	characteristic, direct product factor
16	C₃×C₉⋊C₉	C₂³	1	1	yes	characteristic, complemented, Sylow
17	C₂²×C₃²×D₃	C₃²	1	1	yes	characteristic
18	C₂³×C₃×C₉	—	3	1	no	—
19	C₂³×C₃×C₉	—	3	3	no	—
20	C₂²×D₃×C₉	—	3	3	no	—
21	C₂²×D₃×C₉	C₉	1	3	yes	complemented
22	C₂×C₃²×C₉	C₂²×C₃	1	3	yes	—
23	C₂×C₃²×C₉	C₂²×C₃	1	9	yes	—
24	C₃×D₃×C₉	C₂²×C₃	1	12	yes	—
25	C₃×D₃×C₉	C₂²×C₃	1	4	yes	—
26	C₂×C₉⋊C₉	—	2	9	no	—
27	C₂×C₉⋊C₉	—	2	3	no	—
28	C₂×C₉⋊C₉	—	3	4	no	—
29	C₂×C₉⋊C₉	C₂×D₃	1	3	yes	direct product factor
30	C₂²×C₃³	C₂×C₃²	1	1	yes	characteristic
31	C₂×C₃²×D₃	C₂×C₃²	1	6	yes	—
32	C₂²×C₃×C₉	—	2	1	no	—
33	C₂²×C₃×C₉	—	2	3	no	—
34	C₂²×C₃×C₉	—	2	3	no	—
35	C₂²×C₃×C₉	—	3	3	no	—
36	C₂²×C₃×C₉	—	3	6	no	—
37	C₂²×C₃×C₉	—	3	18	no	—
38	C₂²×C₃×C₉	—	6	3	no	—
39	C₂²×C₃×C₉	C₂×C₉	1	3	yes	complemented
40	C₂²×C₃×C₉	C₃×D₃	1	3	yes	—
41	C₂²×C₃×C₉	C₃×D₃	1	1	yes	characteristic
42	C₂×D₃×C₉	—	3	18	no	—
43	C₂×D₃×C₉	C₂×C₉	1	18	yes	complemented
44	C₃²×C₉	C₂³×C₃	1	1	yes	characteristic
45	C₃²×C₉	C₂³×C₃	1	3	yes	—
46	C₉⋊C₉	—	2	3	no	—
47	C₉⋊C₉	—	2	1	no	—
48	C₉⋊C₉	C₂²×D₃	1	1	yes	characteristic, direct product factor
49	C₂³×C₃²	—	3	1	no	—
50	C₂²×C₃×D₃	C₃×C₉	1	1	yes	characteristic
51	C₂²×C₃×D₃	3₋¹⁺²	1	3	yes	—
52	C₂³×C₉	—	3	3	no	—
53	C₂³×C₉	—	9	3	no	—
54	C₂×C₃³	C₂²×C₃²	1	3	yes	—
55	C₃²×D₃	C₂²×C₃²	1	4	yes	—
56	C₂×C₃×C₉	—	2	3	no	—
57	C₂×C₃×C₉	—	2	9	no	—
58	C₂×C₃×C₉	—	2	9	no	—
59	C₂×C₃×C₉	—	3	4	no	—
60	C₂×C₃×C₉	—	3	9	no	—
61	C₂×C₃×C₉	—	3	12	no	—
62	C₂×C₃×C₉	—	6	9	no	—
63	C₂×C₃×C₉	C₂²×C₉	1	9	yes	complemented
64	C₂×C₃×C₉	C₂×C₃×D₃	1	9	yes	—
65	C₂×C₃×C₉	C₂×C₃×D₃	1	3	yes	—
66	D₃×C₉	—	3	12	no	—
67	D₃×C₉	C₂²×C₉	1	12	yes	complemented
68	C₂²×C₃²	—	2	1	no	—
69	C₂²×C₃²	—	2	3	no	—
70	C₂²×C₃²	—	3	6	no	—
71	C₂²×C₃²	C₂×C₃×C₉	1	1	yes	characteristic
72	C₂²×C₃²	C₂×3₋¹⁺²	1	3	yes	—
73	C₂²×C₃²	C₃²×D₃	1	1	yes	characteristic
74	C₂×C₃×D₃	C₂×C₃×C₉	1	6	yes	—
75	C₂×C₃×D₃	C₂×3₋¹⁺²	1	18	yes	—
76	C₂²×C₉	—	2	3	no	—
77	C₂²×C₉	—	3	18	no	—
78	C₂²×C₉	—	3	3	no	—
79	C₂²×C₉	—	6	3	no	—
80	C₂²×C₉	—	9	18	no	—
81	C₂²×C₉	D₃×C₉	1	3	yes	complemented
82	C₃³	C₂³×C₃²	1	1	yes	characteristic
83	C₃×C₉	—	2	1	no	—
84	C₃×C₉	—	2	3	no	—
85	C₃×C₉	—	2	3	no	—
86	C₃×C₉	—	3	3	no	—
87	C₃×C₉	—	6	3	no	—
88	C₃×C₉	C₂³×C₉	1	3	yes	complemented
89	C₃×C₉	C₂²×C₃×D₃	1	3	yes	—
90	C₃×C₉	C₂²×C₃×D₃	1	1	yes	characteristic
91	C₂³×C₃	—	3	1	no	—
92	C₂³×C₃	—	3	3	no	—
93	C₂²×D₃	C₉⋊C₉	1	1	yes	characteristic, direct product factor
94	C₂×C₃²	—	2	3	no	—
95	C₂×C₃²	—	2	9	no	—
96	C₂×C₃²	—	3	4	no	—
97	C₂×C₃²	C₂²×C₃×C₉	1	3	yes	—
98	C₂×C₃²	C₂²×3₋¹⁺²	1	9	yes	—
99	C₂×C₃²	C₂×C₃²×D₃	1	3	yes	—
100	C₃×D₃	C₂²×C₃×C₉	1	4	yes	—
101	C₃×D₃	C₂²×3₋¹⁺²	1	12	yes	—
102	C₂×C₉	—	2	9	no	—
103	C₂×C₉	—	3	12	no	—
104	C₂×C₉	—	3	9	no	—
105	C₂×C₉	—	6	9	no	—
106	C₂×C₉	—	9	12	no	—
107	C₂×C₉	C₂×D₃×C₉	1	9	yes	complemented
108	C₂²×C₃	—	2	1	no	—
109	C₂²×C₃	—	2	3	no	—
110	C₂²×C₃	—	3	6	no	—
111	C₂²×C₃	—	3	18	no	—
112	C₂²×C₃	C₂×C₉⋊C₉	1	1	yes	characteristic, complemented
113	C₂²×C₃	C₃×D₃×C₉	1	1	yes	characteristic
114	C₂²×C₃	D₃×3₋¹⁺²	1	3	yes	—
115	C₂×D₃	C₂×C₉⋊C₉	1	6	yes	direct product factor
116	C₃²	—	2	1	no	—
117	C₃²	—	2	3	no	—
118	C₃²	C₂³×C₃×C₉	1	1	yes	characteristic
119	C₃²	C₂³×3₋¹⁺²	1	3	yes	—
120	C₃²	C₂²×C₃²×D₃	1	1	yes	characteristic
121	C₉	—	2	3	no	—
122	C₉	—	3	3	no	—
123	C₉	—	6	3	no	—
124	C₉	C₂²×D₃×C₉	1	3	yes	complemented
125	C₂³	—	3	1	no	Sylow
126	C₂×C₃	—	2	3	no	—
127	C₂×C₃	—	2	9	no	—
128	C₂×C₃	—	3	4	no	—
129	C₂×C₃	—	3	12	no	—
130	C₂×C₃	C₂²×C₉⋊C₉	1	3	yes	complemented
131	C₂×C₃	C₂×C₃×D₃×C₉	1	3	yes	—
132	C₂×C₃	C₂×D₃×3₋¹⁺²	1	9	yes	—
133	D₃	C₂²×C₉⋊C₉	1	4	yes	direct product factor
134	C₂²	—	3	6	no	—
135	C₂²	D₃×C₉⋊C₉	1	1	yes	characteristic, direct product factor
136	C₃	—	2	1	no	—
137	C₃	—	2	3	no	—
138	C₃	C₂³×C₉⋊C₉	1	1	yes	characteristic, complemented
139	C₃	C₂²×C₃×D₃×C₉	1	1	yes	characteristic
140	C₃	C₂²×D₃×3₋¹⁺²	1	3	yes	—
141	C₂	—	3	4	no	—
142	C₂	C₂×D₃×C₉⋊C₉	1	3	yes	direct product factor

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₃×C₉⋊C₉

C₂

yes

maximal, characteristic, complemented

C₂×D₃×C₉⋊C₉

C₂

yes

maximal, direct product factor

C₂²×C₃×D₃×C₉

C₃

yes

maximal

C₂²×C₃×D₃×C₉

C₃

yes

maximal, characteristic

C₂³×C₉⋊C₉

—

maximal

C₂×C₃×C₉⋊C₉

C₂²

yes

complemented

D₃×C₉⋊C₉

C₂²

yes

direct product factor

C₂²×C₃²×C₉

C₂×C₃

yes

characteristic

C₂²×C₃²×C₉

C₂×C₃

yes

—

C₂×C₃×D₃×C₉

C₂×C₃

yes

—

C₂×C₃×D₃×C₉

C₂×C₃

yes

—

C₂²×C₉⋊C₉

—

C₂²×C₉⋊C₉

—

C₂²×C₉⋊C₉

—

C₂²×C₉⋊C₉

D₃

yes

characteristic, direct product factor

C₃×C₉⋊C₉

C₂³

yes

characteristic, complemented, Sylow

C₂²×C₃²×D₃

C₃²

yes

characteristic

C₂³×C₃×C₉

—

C₂³×C₃×C₉

—

C₂²×D₃×C₉

—

C₂²×D₃×C₉

C₉

yes

complemented

C₂×C₃²×C₉

C₂²×C₃

yes

—

C₂×C₃²×C₉

C₂²×C₃

yes

—

C₃×D₃×C₉

C₂²×C₃

yes

—

C₃×D₃×C₉

C₂²×C₃

yes

—

C₂×C₉⋊C₉

—

C₂×C₉⋊C₉

—

C₂×C₉⋊C₉

—

C₂×C₉⋊C₉

C₂×D₃

yes

direct product factor

C₂²×C₃³

C₂×C₃²

yes

characteristic

C₂×C₃²×D₃

C₂×C₃²

yes

—

C₂²×C₃×C₉

—

C₂²×C₃×C₉

—

C₂²×C₃×C₉

—

C₂²×C₃×C₉

—

C₂²×C₃×C₉

—

C₂²×C₃×C₉

—

C₂²×C₃×C₉

—

C₂²×C₃×C₉

C₂×C₉

yes

complemented

C₂²×C₃×C₉

C₃×D₃

yes

—

C₂²×C₃×C₉

C₃×D₃

yes

characteristic

C₂×D₃×C₉

—

C₂×D₃×C₉

C₂×C₉

yes

complemented

C₃²×C₉

C₂³×C₃

yes

characteristic

C₃²×C₉

C₂³×C₃

yes

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

C₂²×D₃

yes

characteristic, direct product factor

C₂³×C₃²

—

C₂²×C₃×D₃

C₃×C₉

yes

characteristic

C₂²×C₃×D₃

3₋¹⁺²

yes

—

C₂³×C₉

—

C₂³×C₉

—

C₂×C₃³

C₂²×C₃²

yes

—

C₃²×D₃

C₂²×C₃²

yes

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

C₂²×C₉

yes

complemented

C₂×C₃×C₉

C₂×C₃×D₃

yes

—

C₂×C₃×C₉

C₂×C₃×D₃

yes

—

D₃×C₉

—

D₃×C₉

C₂²×C₉

yes

complemented

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

C₂×C₃×C₉

yes

characteristic

C₂²×C₃²

C₂×3₋¹⁺²

yes

—

C₂²×C₃²

C₃²×D₃

yes

characteristic

C₂×C₃×D₃

C₂×C₃×C₉

yes

—

C₂×C₃×D₃

C₂×3₋¹⁺²

yes

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

D₃×C₉

yes

complemented

C₃³

C₂³×C₃²

yes

characteristic

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₂³×C₉

yes

complemented

C₃×C₉

C₂²×C₃×D₃

yes

—

C₃×C₉

C₂²×C₃×D₃

yes

characteristic

C₂³×C₃

—

C₂³×C₃

—

C₂²×D₃

C₉⋊C₉

yes

characteristic, direct product factor

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

C₂²×C₃×C₉

yes

—

C₂×C₃²

C₂²×3₋¹⁺²

yes

—

C₂×C₃²

C₂×C₃²×D₃

yes

—

100

C₃×D₃

C₂²×C₃×C₉

yes

—

101

C₃×D₃

C₂²×3₋¹⁺²

yes

—

102

C₂×C₉

—

103

C₂×C₉

—

104

C₂×C₉

—

105

C₂×C₉

—

106

C₂×C₉

—

107

C₂×C₉

C₂×D₃×C₉

yes

complemented

108

C₂²×C₃

—

109

C₂²×C₃

—

110

C₂²×C₃

—

111

C₂²×C₃

—

112

C₂²×C₃

C₂×C₉⋊C₉

yes

characteristic, complemented

113

C₂²×C₃

C₃×D₃×C₉

yes

characteristic

114

C₂²×C₃

D₃×3₋¹⁺²

yes

—

115

C₂×D₃

C₂×C₉⋊C₉

yes

direct product factor

116

C₃²

—

117

C₃²

—

118

C₃²

C₂³×C₃×C₉

yes

characteristic

119

C₃²

C₂³×3₋¹⁺²

yes

—

120

C₃²

C₂²×C₃²×D₃

yes

characteristic

121

C₉

—

122

C₉

—

123

C₉

—

124

C₉

C₂²×D₃×C₉

yes

complemented

125

C₂³

—

Sylow

126

C₂×C₃

—

127

C₂×C₃

—

128

C₂×C₃

—

129

C₂×C₃

—

130

C₂×C₃

C₂²×C₉⋊C₉

yes

complemented

131

C₂×C₃

C₂×C₃×D₃×C₉

yes

—

132

C₂×C₃

C₂×D₃×3₋¹⁺²

yes

—

133

D₃

C₂²×C₉⋊C₉

yes

direct product factor

134

C₂²

—

135

C₂²

D₃×C₉⋊C₉

yes

characteristic, direct product factor

136

C₃

—

137

C₃

—

138

C₃

C₂³×C₉⋊C₉

yes

characteristic, complemented

139

C₃

C₂²×C₃×D₃×C₉

yes

characteristic

140

C₃

C₂²×D₃×3₋¹⁺²

yes

—

141

C₂

—

142

C₂

C₂×D₃×C₉⋊C₉

yes

direct product factor

	subgroup	G/H
center	C₂²×C₃²	C₃²×D₃
commutator	C₃²	C₂³×C₃×C₉
Frattini	C₃²	C₂²×C₃²×D₃
Fitting	C₂²×C₃×C₉⋊C₉	C₂
socle	C₂²×C₃³	C₂×C₃²
prime radical	C₂²×C₃²×D₃	C₃²
cosocle	C₃³	C₂³×C₃²
hypercenter	C₂²×C₉⋊C₉	D₃
hypocenter	C₃	C₂³×C₉⋊C₉
LCS	G ▷ C₃² ▷ C₃
UCS	C₁ ◁ C₂²×C₃² ◁ C₂²×C₉⋊C₉

subgroup

G/H

center

C₂²×C₃²

C₃²×D₃

commutator

C₃²

C₂³×C₃×C₉

Frattini

C₃²

C₂²×C₃²×D₃

Fitting

C₂²×C₃×C₉⋊C₉

C₂

socle

C₂²×C₃³

C₂×C₃²

prime radical

C₂²×C₃²×D₃

C₃²

cosocle

C₃³

C₂³×C₃²

hypercenter

C₂²×C₉⋊C₉

D₃

hypocenter

C₃

C₂³×C₉⋊C₉

LCS

G ▷ C₃² ▷ C₃

UCS

C₁ ◁ C₂²×C₃² ◁ C₂²×C₉⋊C₉

N	H ≅ G/N	type
1. C₂²×C₃×C₉⋊C₉	C₂	semidirect
2. C₂×D₃×C₉⋊C₉	C₂	direct (N × H)
6. C₂×C₃×C₉⋊C₉	C₂²	semidirect
7. D₃×C₉⋊C₉	C₂²	direct (N × H)
15. C₂²×C₉⋊C₉	D₃	direct (N × H)
16. C₃×C₉⋊C₉	C₂³	semidirect
21. C₂²×D₃×C₉	C₉	semidirect
29. C₂×C₉⋊C₉	C₂×D₃	direct (N × H)
39. C₂²×C₃×C₉	C₂×C₉	semidirect
43. C₂×D₃×C₉	C₂×C₉	semidirect
48. C₉⋊C₉	C₂²×D₃	direct (N × H)
63. C₂×C₃×C₉	C₂²×C₉	semidirect
67. D₃×C₉	C₂²×C₉	semidirect
81. C₂²×C₉	D₃×C₉	semidirect
88. C₃×C₉	C₂³×C₉	semidirect
93. C₂²×D₃	C₉⋊C₉	direct (N × H)
107. C₂×C₉	C₂×D₃×C₉	semidirect
112. C₂²×C₃	C₂×C₉⋊C₉	semidirect
115. C₂×D₃	C₂×C₉⋊C₉	direct (N × H)
124. C₉	C₂²×D₃×C₉	semidirect
130. C₂×C₃	C₂²×C₉⋊C₉	semidirect
133. D₃	C₂²×C₉⋊C₉	direct (N × H)
135. C₂²	D₃×C₉⋊C₉	direct (N × H)
138. C₃	C₂³×C₉⋊C₉	semidirect
142. C₂	C₂×D₃×C₉⋊C₉	direct (N × H)

H ≅ G/N

type

1. C₂²×C₃×C₉⋊C₉

C₂

semidirect

2. C₂×D₃×C₉⋊C₉

C₂

direct (N × H)

6. C₂×C₃×C₉⋊C₉

C₂²

semidirect

7. D₃×C₉⋊C₉

C₂²

direct (N × H)

15. C₂²×C₉⋊C₉

D₃

direct (N × H)

16. C₃×C₉⋊C₉

C₂³

semidirect

21. C₂²×D₃×C₉

C₉

semidirect

29. C₂×C₉⋊C₉

C₂×D₃

direct (N × H)

39. C₂²×C₃×C₉

C₂×C₉

semidirect

43. C₂×D₃×C₉

C₂×C₉

semidirect

48. C₉⋊C₉

C₂²×D₃

direct (N × H)

63. C₂×C₃×C₉

C₂²×C₉

semidirect

67. D₃×C₉

C₂²×C₉

semidirect

81. C₂²×C₉

D₃×C₉

semidirect

88. C₃×C₉

C₂³×C₉

semidirect

93. C₂²×D₃

C₉⋊C₉

direct (N × H)

107. C₂×C₉

C₂×D₃×C₉

semidirect

112. C₂²×C₃

C₂×C₉⋊C₉

semidirect

115. C₂×D₃

C₂×C₉⋊C₉

direct (N × H)

124. C₉

C₂²×D₃×C₉

semidirect

130. C₂×C₃

C₂²×C₉⋊C₉

semidirect

133. D₃

C₂²×C₉⋊C₉

direct (N × H)

135. C₂²

D₃×C₉⋊C₉

direct (N × H)

138. C₃

C₂³×C₉⋊C₉

semidirect

142. C₂

C₂×D₃×C₉⋊C₉

direct (N × H)

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

subgroup	index
C₂²×C₃×C₉⋊C₉	2
C₂×D₃×C₉⋊C₉	2
C₂²×C₃×D₃×C₉	3
C₂³×C₉⋊C₉	3

subgroup

index

C₂²×C₃×C₉⋊C₉

C₂×D₃×C₉⋊C₉

C₂²×C₃×D₃×C₉

C₂³×C₉⋊C₉

Subgroups — C₂²×D₃×C₉⋊C₉ #14518459

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C22×D3×C9⋊C9 #14518459

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×D₃×C₉⋊C₉ #14518459