Group #13047296 — subgroups

Subgroups — C₂×C₄×D₉⋊C₉ #13047296

Order 1296.

275 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 275 conjugacy classes of subgroups — too many to draw. Only its 121 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×D₉⋊C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₉⋊C₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×Dic₉⋊C₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×D₉⋊C₉	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₄×D₃×C₉	—	3	1	no	maximal
6	C₂×C₃×C₄×D₉	C₃	1	1	yes	maximal, characteristic
7	C₂²×C₉⋊C₉	C₂²	1	1	yes	characteristic
8	C₂×D₉⋊C₉	C₄	1	4	yes	direct product factor
9	C₂×D₉⋊C₉	C₂²	1	2	yes	—
10	C₄×C₉⋊C₉	C₂²	1	2	yes	complemented
11	Dic₉⋊C₉	C₂²	1	2	yes	complemented
12	C₂²×D₃×C₉	—	3	1	no	—
13	C₂²×C₃×D₉	C₂×C₃	1	1	yes	characteristic
14	C₂×C₃×C₄×C₉	—	2	1	no	—
15	C₂×C₃×C₄×C₉	D₃	1	1	yes	characteristic
16	C₂×C₃×C₄×C₉	C₂×C₃	1	1	yes	characteristic
17	C₂×C₉×Dic₃	—	3	1	no	—
18	C₂×C₃×Dic₉	C₂×C₃	1	1	yes	characteristic
19	C₄×D₃×C₉	—	3	4	no	—
20	C₃×C₄×D₉	C₂×C₃	1	4	yes	—
21	C₂×C₉⋊C₉	C₂×C₄	1	2	yes	complemented
22	C₂×C₉⋊C₉	C₂³	1	1	yes	characteristic
23	D₉⋊C₉	C₂×C₄	1	4	yes	direct product factor
24	C₂×C₃×C₄×D₃	—	3	1	no	—
25	C₂²×C₄×C₉	—	9	1	no	—
26	C₂×C₄×D₉	C₉	1	1	yes	characteristic, complemented
27	C₂²×C₃×C₉	—	2	1	no	—
28	C₂²×C₃×C₉	C₂×D₃	1	1	yes	characteristic
29	C₂²×C₃×C₉	C₂²×C₃	1	1	yes	characteristic
30	C₂×D₃×C₉	—	3	4	no	—
31	C₂×D₃×C₉	—	3	2	no	—
32	C₂×C₃×D₉	C₃×C₄	1	4	yes	—
33	C₂×C₃×D₉	C₂²×C₃	1	2	yes	—
34	C₃×C₄×C₉	—	2	2	no	—
35	C₃×C₄×C₉	C₂×D₃	1	2	yes	—
36	C₃×C₄×C₉	C₂²×C₃	1	2	yes	—
37	C₉×Dic₃	—	3	2	no	—
38	C₃×Dic₉	C₂²×C₃	1	2	yes	—
39	C₉⋊C₉	C₂²×C₄	1	1	yes	characteristic, complemented, Sylow
40	C₂²×C₃×D₃	—	3	1	no	—
41	C₂×C₃²×C₄	C₃×D₃	1	1	yes	characteristic
42	C₂×C₃×Dic₃	—	3	1	no	—
43	C₃×C₄×D₃	—	3	4	no	—
44	C₂³×C₉	—	9	1	no	—
45	C₂²×D₉	C₂×C₉	1	1	yes	characteristic
46	C₂×C₄×C₉	—	2	1	no	—
47	C₂×C₄×C₉	—	3	1	no	—
48	C₂×C₄×C₉	—	6	1	no	—
49	C₂×C₄×C₉	—	9	4	no	—
50	C₂×C₄×C₉	—	9	1	no	—
51	C₂×C₄×C₉	C₂×C₉	1	1	yes	characteristic, complemented
52	C₂×Dic₉	C₂×C₉	1	1	yes	characteristic, complemented
53	C₄×D₉	C₂×C₉	1	4	yes	complemented
54	C₂×C₃×C₉	—	2	2	no	—
55	C₂×C₃×C₉	—	2	1	no	—
56	C₂×C₃×C₉	C₄×D₃	1	2	yes	—
57	C₂×C₃×C₉	C₂×C₃×C₄	1	2	yes	—
58	C₂×C₃×C₉	C₂²×D₃	1	1	yes	characteristic
59	C₂×C₃×C₉	C₂³×C₃	1	1	yes	characteristic
60	D₃×C₉	—	3	4	no	—
61	C₃×D₉	C₂×C₃×C₄	1	4	yes	—
62	C₂²×C₃×C₄	—	9	1	no	—
63	C₂×C₄×D₃	—	3	1	no	—
64	C₂²×C₃²	C₂×C₃×D₃	1	1	yes	characteristic
65	C₂×C₃×D₃	—	3	4	no	—
66	C₂×C₃×D₃	—	3	2	no	—
67	C₃²×C₄	C₂×C₃×D₃	1	2	yes	—
68	C₃×Dic₃	—	3	2	no	—
69	C₂²×C₉	—	2	1	no	—
70	C₂²×C₉	—	3	1	no	—
71	C₂²×C₉	—	6	1	no	—
72	C₂²×C₉	—	9	2	no	—
73	C₂²×C₉	—	9	4	no	—
74	C₂²×C₉	C₂²×C₉	1	1	yes	characteristic
75	C₂×D₉	C₄×C₉	1	4	yes	complemented
76	C₂×D₉	C₂²×C₉	1	2	yes	—
77	C₄×C₉	—	2	2	no	—
78	C₄×C₉	—	3	2	no	—
79	C₄×C₉	—	6	2	no	—
80	C₄×C₉	—	9	2	no	—
81	C₄×C₉	C₂²×C₉	1	2	yes	complemented
82	Dic₉	C₂²×C₉	1	2	yes	complemented
83	C₃×C₉	—	2	1	no	—
84	C₃×C₉	C₂×C₄×D₃	1	1	yes	characteristic
85	C₃×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
86	C₂³×C₃	—	9	1	no	—
87	C₂²×D₃	—	3	1	no	—
88	C₂×C₃×C₄	—	2	1	no	—
89	C₂×C₃×C₄	—	9	1	no	—
90	C₂×C₃×C₄	—	9	4	no	—
91	C₂×C₃×C₄	D₃×C₉	1	1	yes	characteristic
92	C₂×C₃×C₄	AGL(1,ℤ/9)	1	1	yes	characteristic
93	C₂×Dic₃	—	3	1	no	—
94	C₄×D₃	—	3	4	no	—
95	C₂×C₃²	C₃×C₄×D₃	1	2	yes	—
96	C₂×C₃²	C₂²×C₃×D₃	1	1	yes	characteristic
97	C₃×D₃	—	3	4	no	—
98	C₂×C₉	—	2	2	no	—
99	C₂×C₉	—	2	1	no	—
100	C₂×C₉	—	3	2	no	—
101	C₂×C₉	—	3	1	no	—
102	C₂×C₉	—	6	1	no	—
103	C₂×C₉	—	6	2	no	—
104	C₂×C₉	—	9	4	no	—
105	C₂×C₉	C₂×C₄×C₉	1	2	yes	complemented
106	C₂×C₉	C₂³×C₉	1	1	yes	characteristic
107	D₉	C₂×C₄×C₉	1	4	yes	complemented
108	C₂²×C₄	—	9	1	no	Sylow
109	C₂²×C₃	—	2	1	no	—
110	C₂²×C₃	—	9	4	no	—
111	C₂²×C₃	—	9	2	no	—
112	C₂²×C₃	C₂×D₃×C₉	1	1	yes	characteristic
113	C₂²×C₃	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
114	C₂×D₃	—	3	4	no	—
115	C₂×D₃	—	3	2	no	—
116	C₃×C₄	—	2	2	no	—
117	C₃×C₄	—	9	2	no	—
118	C₃×C₄	C₂×D₃×C₉	1	2	yes	—
119	C₃×C₄	C₂×AGL(1,ℤ/9)	1	2	yes	—
120	Dic₃	—	3	2	no	—
121	C₃²	C₂×C₃×C₄×D₃	1	1	yes	characteristic
122	C₉	—	2	1	no	—
123	C₉	—	3	1	no	—
124	C₉	—	6	1	no	—
125	C₉	C₂²×C₄×C₉	1	1	yes	characteristic, complemented
126	C₂³	—	9	1	no	—
127	C₂×C₄	—	9	1	no	—
128	C₂×C₄	—	9	4	no	—
129	C₂×C₄	D₉⋊C₉	1	1	yes	characteristic, direct product factor
130	C₂×C₃	—	2	1	no	—
131	C₂×C₃	—	2	2	no	—
132	C₂×C₃	—	9	4	no	—
133	C₂×C₃	C₄×D₃×C₉	1	2	yes	—
134	C₂×C₃	C₄×AGL(1,ℤ/9)	1	2	yes	—
135	C₂×C₃	C₂²×D₃×C₉	1	1	yes	characteristic
136	C₂×C₃	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
137	D₃	—	3	4	no	—
138	C₂²	—	9	2	no	—
139	C₂²	—	9	4	no	—
140	C₂²	C₂×D₉⋊C₉	1	1	yes	characteristic
141	C₄	—	9	2	no	—
142	C₄	C₂×D₉⋊C₉	1	2	yes	direct product factor
143	C₃	—	2	1	no	—
144	C₃	C₂×C₄×D₃×C₉	1	1	yes	characteristic
145	C₃	C₂×C₄×AGL(1,ℤ/9)	1	1	yes	characteristic
146	C₂	—	9	4	no	—
147	C₂	C₄×D₉⋊C₉	1	2	yes	direct product factor
148	C₂	C₂²×D₉⋊C₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₉⋊C₉

C₂

yes

maximal, characteristic

C₂×C₄×C₉⋊C₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₉⋊C₉

C₂

yes

maximal, characteristic, complemented

C₄×D₉⋊C₉

C₂

yes

maximal, direct product factor

C₂×C₄×D₃×C₉

—

maximal

C₂×C₃×C₄×D₉

C₃

yes

maximal, characteristic

C₂²×C₉⋊C₉

C₂²

yes

characteristic

C₂×D₉⋊C₉

C₄

yes

direct product factor

C₂×D₉⋊C₉

C₂²

yes

—

C₄×C₉⋊C₉

C₂²

yes

complemented

Dic₉⋊C₉

C₂²

yes

complemented

C₂²×D₃×C₉

—

C₂²×C₃×D₉

C₂×C₃

yes

characteristic

C₂×C₃×C₄×C₉

—

C₂×C₃×C₄×C₉

D₃

yes

characteristic

C₂×C₃×C₄×C₉

C₂×C₃

yes

characteristic

C₂×C₉×Dic₃

—

C₂×C₃×Dic₉

C₂×C₃

yes

characteristic

C₄×D₃×C₉

—

C₃×C₄×D₉

C₂×C₃

yes

—

C₂×C₉⋊C₉

C₂×C₄

yes

complemented

C₂×C₉⋊C₉

C₂³

yes

characteristic

D₉⋊C₉

C₂×C₄

yes

direct product factor

C₂×C₃×C₄×D₃

—

C₂²×C₄×C₉

—

C₂×C₄×D₉

C₉

yes

characteristic, complemented

C₂²×C₃×C₉

—

C₂²×C₃×C₉

C₂×D₃

yes

characteristic

C₂²×C₃×C₉

C₂²×C₃

yes

characteristic

C₂×D₃×C₉

—

C₂×D₃×C₉

—

C₂×C₃×D₉

C₃×C₄

yes

—

C₂×C₃×D₉

C₂²×C₃

yes

—

C₃×C₄×C₉

—

C₃×C₄×C₉

C₂×D₃

yes

—

C₃×C₄×C₉

C₂²×C₃

yes

—

C₉×Dic₃

—

C₃×Dic₉

C₂²×C₃

yes

—

C₉⋊C₉

C₂²×C₄

yes

characteristic, complemented, Sylow

C₂²×C₃×D₃

—

C₂×C₃²×C₄

C₃×D₃

yes

characteristic

C₂×C₃×Dic₃

—

C₃×C₄×D₃

—

C₂³×C₉

—

C₂²×D₉

C₂×C₉

yes

characteristic

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂×C₉

yes

characteristic, complemented

C₂×Dic₉

C₂×C₉

yes

characteristic, complemented

C₄×D₉

C₂×C₉

yes

complemented

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

C₄×D₃

yes

—

C₂×C₃×C₉

C₂×C₃×C₄

yes

—

C₂×C₃×C₉

C₂²×D₃

yes

characteristic

C₂×C₃×C₉

C₂³×C₃

yes

characteristic

D₃×C₉

—

C₃×D₉

C₂×C₃×C₄

yes

—

C₂²×C₃×C₄

—

C₂×C₄×D₃

—

C₂²×C₃²

C₂×C₃×D₃

yes

characteristic

C₂×C₃×D₃

—

C₂×C₃×D₃

—

C₃²×C₄

C₂×C₃×D₃

yes

—

C₃×Dic₃

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

yes

characteristic

C₂×D₉

C₄×C₉

yes

complemented

C₂×D₉

C₂²×C₉

yes

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

C₂²×C₉

yes

complemented

Dic₉

C₂²×C₉

yes

complemented

C₃×C₉

—

C₃×C₉

C₂×C₄×D₃

yes

characteristic

C₃×C₉

C₂²×C₃×C₄

yes

characteristic

C₂³×C₃

—

C₂²×D₃

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

D₃×C₉

yes

characteristic

C₂×C₃×C₄

AGL(1,ℤ/9)

yes

characteristic

C₂×Dic₃

—

C₄×D₃

—

C₂×C₃²

C₃×C₄×D₃

yes

—

C₂×C₃²

C₂²×C₃×D₃

yes

characteristic

C₃×D₃

—

C₂×C₉

—

C₂×C₉

—

100

C₂×C₉

—

101

C₂×C₉

—

102

C₂×C₉

—

103

C₂×C₉

—

104

C₂×C₉

—

105

C₂×C₉

C₂×C₄×C₉

yes

complemented

106

C₂×C₉

C₂³×C₉

yes

characteristic

107

D₉

C₂×C₄×C₉

yes

complemented

108

C₂²×C₄

—

Sylow

109

C₂²×C₃

—

110

C₂²×C₃

—

111

C₂²×C₃

—

112

C₂²×C₃

C₂×D₃×C₉

yes

characteristic

113

C₂²×C₃

C₂×AGL(1,ℤ/9)

yes

characteristic

114

C₂×D₃

—

115

C₂×D₃

—

116

C₃×C₄

—

117

C₃×C₄

—

118

C₃×C₄

C₂×D₃×C₉

yes

—

119

C₃×C₄

C₂×AGL(1,ℤ/9)

yes

—

120

Dic₃

—

121

C₃²

C₂×C₃×C₄×D₃

yes

characteristic

122

C₉

—

123

C₉

—

124

C₉

—

125

C₉

C₂²×C₄×C₉

yes

characteristic, complemented

126

C₂³

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

D₉⋊C₉

yes

characteristic, direct product factor

130

C₂×C₃

—

131

C₂×C₃

—

132

C₂×C₃

—

133

C₂×C₃

C₄×D₃×C₉

yes

—

134

C₂×C₃

C₄×AGL(1,ℤ/9)

yes

—

135

C₂×C₃

C₂²×D₃×C₉

yes

characteristic

136

C₂×C₃

C₂²×AGL(1,ℤ/9)

yes

characteristic

137

D₃

—

138

C₂²

—

139

C₂²

—

140

C₂²

C₂×D₉⋊C₉

yes

characteristic

141

C₄

—

142

C₄

C₂×D₉⋊C₉

yes

direct product factor

143

C₃

—

144

C₃

C₂×C₄×D₃×C₉

yes

characteristic

145

C₃

C₂×C₄×AGL(1,ℤ/9)

yes

characteristic

146

C₂

—

147

C₂

C₄×D₉⋊C₉

yes

direct product factor

148

C₂

C₂²×D₉⋊C₉

yes

characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃×C₄	AGL(1,ℤ/9)
commutator, hypocenter	C₉	C₂²×C₄×C₉
Frattini	C₂×C₃²	C₂²×C₃×D₃
Fitting	C₂×C₄×C₉⋊C₉	C₂
socle	C₂²×C₃²	C₂×C₃×D₃
prime radical	C₂²×C₃×D₉	C₂×C₃
cosocle	C₂×C₃×C₉	C₂³×C₃
LCS	G ▷ C₉
UCS	C₁ ◁ C₂×C₃×C₄

subgroup

G/H

center, hypercenter

C₂×C₃×C₄

AGL(1,ℤ/9)

commutator, hypocenter

C₉

C₂²×C₄×C₉

Frattini

C₂×C₃²

C₂²×C₃×D₃

Fitting

C₂×C₄×C₉⋊C₉

C₂

socle

C₂²×C₃²

C₂×C₃×D₃

prime radical

C₂²×C₃×D₉

C₂×C₃

cosocle

C₂×C₃×C₉

C₂³×C₃

LCS

G ▷ C₉

UCS

C₁ ◁ C₂×C₃×C₄

N	H ≅ G/N	type
2. C₂×C₄×C₉⋊C₉	C₂	semidirect
3. C₂×Dic₉⋊C₉	C₂	semidirect
4. C₄×D₉⋊C₉	C₂	direct (N × H)
8. C₂×D₉⋊C₉	C₄	direct (N × H)
10. C₄×C₉⋊C₉	C₂²	semidirect
11. Dic₉⋊C₉	C₂²	semidirect
21. C₂×C₉⋊C₉	C₂×C₄	semidirect
23. D₉⋊C₉	C₂×C₄	direct (N × H)
26. C₂×C₄×D₉	C₉	semidirect
39. C₉⋊C₉	C₂²×C₄	semidirect
51. C₂×C₄×C₉	C₂×C₉	semidirect
52. C₂×Dic₉	C₂×C₉	semidirect
53. C₄×D₉	C₂×C₉	semidirect
75. C₂×D₉	C₄×C₉	semidirect
81. C₄×C₉	C₂²×C₉	semidirect
82. Dic₉	C₂²×C₉	semidirect
105. C₂×C₉	C₂×C₄×C₉	semidirect
107. D₉	C₂×C₄×C₉	semidirect
125. C₉	C₂²×C₄×C₉	semidirect
129. C₂×C₄	D₉⋊C₉	direct (N × H)
142. C₄	C₂×D₉⋊C₉	direct (N × H)
147. C₂	C₄×D₉⋊C₉	direct (N × H)

H ≅ G/N

type

2. C₂×C₄×C₉⋊C₉

C₂

semidirect

3. C₂×Dic₉⋊C₉

C₂

semidirect

4. C₄×D₉⋊C₉

C₂

direct (N × H)

8. C₂×D₉⋊C₉

C₄

direct (N × H)

10. C₄×C₉⋊C₉

C₂²

semidirect

11. Dic₉⋊C₉

C₂²

semidirect

21. C₂×C₉⋊C₉

C₂×C₄

semidirect

23. D₉⋊C₉

C₂×C₄

direct (N × H)

26. C₂×C₄×D₉

C₉

semidirect

39. C₉⋊C₉

C₂²×C₄

semidirect

51. C₂×C₄×C₉

C₂×C₉

semidirect

52. C₂×Dic₉

C₂×C₉

semidirect

53. C₄×D₉

C₂×C₉

semidirect

75. C₂×D₉

C₄×C₉

semidirect

81. C₄×C₉

C₂²×C₉

semidirect

82. Dic₉

C₂²×C₉

semidirect

105. C₂×C₉

C₂×C₄×C₉

semidirect

107. D₉

C₂×C₄×C₉

semidirect

125. C₉

C₂²×C₄×C₉

semidirect

129. C₂×C₄

D₉⋊C₉

direct (N × H)

142. C₄

C₂×D₉⋊C₉

direct (N × H)

147. C₂

C₄×D₉⋊C₉

direct (N × H)

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

subgroup	index
C₂²×D₉⋊C₉	2
C₂×C₄×C₉⋊C₉	2
C₂×Dic₉⋊C₉	2
C₄×D₉⋊C₉	2
C₂×C₄×D₃×C₉	3
C₂×C₃×C₄×D₉	3

subgroup

index

C₂²×D₉⋊C₉

C₂×C₄×C₉⋊C₉

C₂×Dic₉⋊C₉

C₄×D₉⋊C₉

C₂×C₄×D₃×C₉

C₂×C₃×C₄×D₉

Subgroups — C₂×C₄×D₉⋊C₉ #13047296

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×C4×D9⋊C9 #13047296

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×D₉⋊C₉ #13047296