Group #14517331 — subgroups

Subgroups — #14517331

Order 1944.

213 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 213 conjugacy classes of subgroups — too many to draw. Only its 72 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	C₄×#6900	C₂	1	1	yes	maximal, characteristic
2	#11757569	C₂	1	2	yes	maximal
3	Dic₆×3₋¹⁺²	—	3	1	no	maximal
4	C₃×3₋¹⁺²⋊Q₈	C₃	1	1	yes	maximal, characteristic
5	Dic₁₈⋊C₉	C₃	1	2	yes	maximal, complemented
6	C₉×Dic₁₈	C₃	1	1	yes	maximal, characteristic, complemented
7	C₂×#6900	C₂²	1	1	yes	characteristic
8	C₃×C₄×3₋¹⁺²	—	2	1	no	—
9	C₃×C₄×3₋¹⁺²	D₃	1	1	yes	characteristic
10	C₃×C₄×3₋¹⁺²	C₂×C₃	1	1	yes	characteristic
11	Dic₃×3₋¹⁺²	—	3	2	no	—
12	C₃×3₋¹⁺²⋊C₄	C₂×C₃	1	2	yes	—
13	C₄×C₉⋊C₉	—	2	2	no	—
14	C₄×C₉⋊C₉	—	2	1	no	—
15	C₄×C₉⋊C₉	C₂×C₃	1	2	yes	—
16	C₄×C₉²	C₂×C₃	1	1	yes	characteristic
17	Dic₉⋊C₉	C₂×C₃	1	4	yes	—
18	C₉×Dic₉	C₂×C₃	1	2	yes	—
19	#6900	Q₈	1	1	yes	characteristic, complemented, Sylow
20	C₃²×Dic₆	—	3	1	no	—
21	Q₈×3₋¹⁺²	—	9	1	no	—
22	3₋¹⁺²⋊Q₈	—	3	1	no	—
23	C₉×Dic₆	—	3	1	no	—
24	C₉×Dic₆	—	3	2	no	—
25	C₃×Dic₁₈	C₃²	1	1	yes	characteristic
26	C₂×C₃×3₋¹⁺²	—	2	1	no	—
27	C₂×C₃×3₋¹⁺²	C₂×D₃	1	1	yes	characteristic
28	C₂×C₃×3₋¹⁺²	C₂²×C₃	1	1	yes	characteristic
29	C₂×C₉⋊C₉	—	2	2	no	—
30	C₂×C₉⋊C₉	—	2	1	no	—
31	C₂×C₉⋊C₉	C₂²×C₃	1	2	yes	—
32	C₂×C₉²	C₂²×C₃	1	1	yes	characteristic
33	C₃³×C₄	C₃×D₃	1	1	yes	characteristic
34	C₃²×Dic₃	—	3	2	no	—
35	C₄×3₋¹⁺²	—	3	1	no	—
36	C₄×3₋¹⁺²	—	3	1	no	—
37	C₄×3₋¹⁺²	—	6	1	no	—
38	C₄×3₋¹⁺²	—	6	3	no	—
39	C₄×3₋¹⁺²	—	6	1	no	—
40	C₄×3₋¹⁺²	—	9	2	no	—
41	C₃×C₄×C₉	—	2	1	no	—
42	C₃×C₄×C₉	—	2	1	no	—
43	C₃×C₄×C₉	—	2	2	no	—
44	C₃×C₄×C₉	C₃×D₃	1	1	yes	characteristic
45	C₃×C₄×C₉	C₃×D₃	1	2	yes	—
46	C₃×C₄×C₉	C₂×C₃²	1	1	yes	characteristic
47	3₋¹⁺²⋊C₄	—	3	2	no	—
48	C₉×Dic₃	—	3	2	no	—
49	C₉×Dic₃	—	3	4	no	—
50	C₃×Dic₉	C₂×C₃²	1	2	yes	—
51	C₃×3₋¹⁺²	—	2	1	no	—
52	C₃×3₋¹⁺²	Dic₆	1	1	yes	characteristic
53	C₃×3₋¹⁺²	C₃×Q₈	1	1	yes	characteristic
54	C₉⋊C₉	—	2	2	no	—
55	C₉⋊C₉	—	2	1	no	—
56	C₉⋊C₉	C₃×Q₈	1	2	yes	complemented
57	C₉²	C₃×Q₈	1	1	yes	characteristic, complemented
58	C₃²×Q₈	—	9	1	no	—
59	C₃×Dic₆	—	3	1	no	—
60	C₃×Dic₆	—	9	1	no	—
61	Q₈×C₉	—	9	2	no	—
62	Q₈×C₉	—	9	1	no	—
63	Dic₁₈	3₋¹⁺²	1	1	yes	characteristic, complemented
64	C₂×C₃³	C₂×C₃×D₃	1	1	yes	characteristic
65	C₂×3₋¹⁺²	—	3	1	no	—
66	C₂×3₋¹⁺²	—	3	1	no	—
67	C₂×3₋¹⁺²	—	6	1	no	—
68	C₂×3₋¹⁺²	—	6	3	no	—
69	C₂×3₋¹⁺²	—	6	1	no	—
70	C₂×C₃×C₉	—	2	1	no	—
71	C₂×C₃×C₉	—	2	1	no	—
72	C₂×C₃×C₉	—	2	2	no	—
73	C₂×C₃×C₉	C₂×C₃×D₃	1	1	yes	characteristic
74	C₂×C₃×C₉	C₂×C₃×D₃	1	2	yes	—
75	C₂×C₃×C₉	C₂²×C₃²	1	1	yes	characteristic
76	C₃²×C₄	—	3	1	no	—
77	C₃²×C₄	—	3	1	no	—
78	C₃²×C₄	—	6	1	no	—
79	C₃²×C₄	—	9	2	no	—
80	C₃²×C₄	C₃²×D₃	1	1	yes	characteristic
81	C₃×Dic₃	—	3	2	no	—
82	C₃×Dic₃	—	9	2	no	—
83	C₄×C₉	—	2	3	no	—
84	C₄×C₉	—	2	1	no	—
85	C₄×C₉	—	2	1	no	—
86	C₄×C₉	—	3	2	no	—
87	C₄×C₉	—	6	2	no	—
88	C₄×C₉	—	6	1	no	—
89	C₄×C₉	—	9	4	no	—
90	C₄×C₉	—	9	2	no	—
91	C₄×C₉	AGL(1,ℤ/9)	1	1	yes	characteristic
92	C₄×C₉	C₂×3₋¹⁺²	1	1	yes	characteristic
93	Dic₉	C₂×3₋¹⁺²	1	2	yes	—
94	C₃³	C₃×Dic₆	1	1	yes	characteristic
95	3₋¹⁺²	—	3	1	no	—
96	3₋¹⁺²	—	3	1	no	—
97	3₋¹⁺²	—	6	1	no	—
98	3₋¹⁺²	—	6	3	no	—
99	3₋¹⁺²	—	6	1	no	—
100	C₃×C₉	—	2	1	no	—
101	C₃×C₉	—	2	1	no	—
102	C₃×C₉	—	2	2	no	—
103	C₃×C₉	C₃×Dic₆	1	1	yes	characteristic
104	C₃×C₉	C₃×Dic₆	1	2	yes	—
105	C₃×C₉	C₃²×Q₈	1	1	yes	characteristic
106	C₃×Q₈	—	9	1	no	—
107	C₃×Q₈	—	27	1	no	—
108	Dic₆	—	3	1	no	—
109	C₂×C₃²	—	3	1	no	—
110	C₂×C₃²	—	3	1	no	—
111	C₂×C₃²	—	6	1	no	—
112	C₂×C₃²	C₂×C₃²×D₃	1	1	yes	characteristic
113	C₂×C₉	—	2	3	no	—
114	C₂×C₉	—	2	1	no	—
115	C₂×C₉	—	2	1	no	—
116	C₂×C₉	—	3	2	no	—
117	C₂×C₉	—	6	2	no	—
118	C₂×C₉	—	6	1	no	—
119	C₂×C₉	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
120	C₂×C₉	C₂²×3₋¹⁺²	1	1	yes	characteristic
121	C₃×C₄	—	2	1	no	—
122	C₃×C₄	—	9	2	no	—
123	C₃×C₄	—	9	1	no	—
124	C₃×C₄	—	27	2	no	—
125	C₃×C₄	C₃×AGL(1,ℤ/9)	1	1	yes	characteristic
126	C₃×C₄	D₃×3₋¹⁺²	1	1	yes	characteristic
127	Dic₃	—	3	2	no	—
128	C₃²	—	3	1	no	—
129	C₃²	—	3	1	no	—
130	C₃²	—	6	1	no	—
131	C₃²	C₃²×Dic₆	1	1	yes	characteristic
132	C₉	—	2	3	no	—
133	C₉	—	2	1	no	—
134	C₉	—	2	1	no	—
135	C₉	—	3	2	no	—
136	C₉	—	6	2	no	—
137	C₉	—	6	1	no	—
138	C₉	3₋¹⁺²⋊Q₈	1	1	yes	characteristic, complemented
139	C₉	Q₈×3₋¹⁺²	1	1	yes	characteristic, complemented
140	Q₈	—	9	1	no	Sylow
141	C₂×C₃	—	2	1	no	—
142	C₂×C₃	—	9	1	no	—
143	C₂×C₃	C₂×C₃×AGL(1,ℤ/9)	1	1	yes	characteristic
144	C₂×C₃	C₂×D₃×3₋¹⁺²	1	1	yes	characteristic
145	C₄	—	9	2	no	—
146	C₄	#92265	1	1	yes	characteristic
147	C₃	—	2	1	no	—
148	C₃	—	9	1	no	—
149	C₃	C₃×3₋¹⁺²⋊Q₈	1	1	yes	characteristic
150	C₃	Dic₆×3₋¹⁺²	1	1	yes	characteristic
151	C₂	C₂×#92265	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#6900

C₂

yes

maximal, characteristic

#11757569

C₂

yes

maximal

Dic₆×3₋¹⁺²

—

maximal

C₃×3₋¹⁺²⋊Q₈

C₃

yes

maximal, characteristic

Dic₁₈⋊C₉

C₃

yes

maximal, complemented

C₉×Dic₁₈

C₃

yes

maximal, characteristic, complemented

C₂×#6900

C₂²

yes

characteristic

C₃×C₄×3₋¹⁺²

—

C₃×C₄×3₋¹⁺²

D₃

yes

characteristic

C₃×C₄×3₋¹⁺²

C₂×C₃

yes

characteristic

Dic₃×3₋¹⁺²

—

C₃×3₋¹⁺²⋊C₄

C₂×C₃

yes

—

C₄×C₉⋊C₉

—

C₄×C₉⋊C₉

—

C₄×C₉⋊C₉

C₂×C₃

yes

—

C₄×C₉²

C₂×C₃

yes

characteristic

Dic₉⋊C₉

C₂×C₃

yes

—

C₉×Dic₉

C₂×C₃

yes

—

#6900

Q₈

yes

characteristic, complemented, Sylow

C₃²×Dic₆

—

Q₈×3₋¹⁺²

—

3₋¹⁺²⋊Q₈

—

C₉×Dic₆

—

C₉×Dic₆

—

C₃×Dic₁₈

C₃²

yes

characteristic

C₂×C₃×3₋¹⁺²

—

C₂×C₃×3₋¹⁺²

C₂×D₃

yes

characteristic

C₂×C₃×3₋¹⁺²

C₂²×C₃

yes

characteristic

C₂×C₉⋊C₉

—

C₂×C₉⋊C₉

—

C₂×C₉⋊C₉

C₂²×C₃

yes

—

C₂×C₉²

C₂²×C₃

yes

characteristic

C₃³×C₄

C₃×D₃

yes

characteristic

C₃²×Dic₃

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

—

C₃×C₄×C₉

—

C₃×C₄×C₉

—

C₃×C₄×C₉

—

C₃×C₄×C₉

C₃×D₃

yes

characteristic

C₃×C₄×C₉

C₃×D₃

yes

—

C₃×C₄×C₉

C₂×C₃²

yes

characteristic

3₋¹⁺²⋊C₄

—

C₉×Dic₃

—

C₉×Dic₃

—

C₃×Dic₉

C₂×C₃²

yes

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

Dic₆

yes

characteristic

C₃×3₋¹⁺²

C₃×Q₈

yes

characteristic

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

C₃×Q₈

yes

complemented

C₉²

C₃×Q₈

yes

characteristic, complemented

C₃²×Q₈

—

C₃×Dic₆

—

C₃×Dic₆

—

Q₈×C₉

—

Q₈×C₉

—

Dic₁₈

3₋¹⁺²

yes

characteristic, complemented

C₂×C₃³

C₂×C₃×D₃

yes

characteristic

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

C₂×C₃×D₃

yes

characteristic

C₂×C₃×C₉

C₂×C₃×D₃

yes

—

C₂×C₃×C₉

C₂²×C₃²

yes

characteristic

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

C₃²×D₃

yes

characteristic

C₃×Dic₃

—

C₃×Dic₃

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

AGL(1,ℤ/9)

yes

characteristic

C₄×C₉

C₂×3₋¹⁺²

yes

characteristic

Dic₉

C₂×3₋¹⁺²

yes

—

C₃³

C₃×Dic₆

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

100

C₃×C₉

—

101

C₃×C₉

—

102

C₃×C₉

—

103

C₃×C₉

C₃×Dic₆

yes

characteristic

104

C₃×C₉

C₃×Dic₆

yes

—

105

C₃×C₉

C₃²×Q₈

yes

characteristic

106

C₃×Q₈

—

107

C₃×Q₈

—

108

Dic₆

—

109

C₂×C₃²

—

110

C₂×C₃²

—

111

C₂×C₃²

—

112

C₂×C₃²

C₂×C₃²×D₃

yes

characteristic

113

C₂×C₉

—

114

C₂×C₉

—

115

C₂×C₉

—

116

C₂×C₉

—

117

C₂×C₉

—

118

C₂×C₉

—

119

C₂×C₉

C₂×AGL(1,ℤ/9)

yes

characteristic

120

C₂×C₉

C₂²×3₋¹⁺²

yes

characteristic

121

C₃×C₄

—

122

C₃×C₄

—

123

C₃×C₄

—

124

C₃×C₄

—

125

C₃×C₄

C₃×AGL(1,ℤ/9)

yes

characteristic

126

C₃×C₄

D₃×3₋¹⁺²

yes

characteristic

127

Dic₃

—

128

C₃²

—

129

C₃²

—

130

C₃²

—

131

C₃²

C₃²×Dic₆

yes

characteristic

132

C₉

—

133

C₉

—

134

C₉

—

135

C₉

—

136

C₉

—

137

C₉

—

138

C₉

3₋¹⁺²⋊Q₈

yes

characteristic, complemented

139

C₉

Q₈×3₋¹⁺²

yes

characteristic, complemented

140

Q₈

—

Sylow

141

C₂×C₃

—

142

C₂×C₃

—

143

C₂×C₃

C₂×C₃×AGL(1,ℤ/9)

yes

characteristic

144

C₂×C₃

C₂×D₃×3₋¹⁺²

yes

characteristic

145

C₄

—

146

C₄

#92265

yes

characteristic

147

C₃

—

148

C₃

—

149

C₃

C₃×3₋¹⁺²⋊Q₈

yes

characteristic

150

C₃

Dic₆×3₋¹⁺²

yes

characteristic

151

C₂

C₂×#92265

yes

characteristic

	subgroup	G/H
center	C₂×C₃	C₂×C₃×AGL(1,ℤ/9)
commutator, cosocle	C₂×C₃×C₉	C₂²×C₃²
Frattini, socle	C₂×C₃²	C₂×C₃²×D₃
Fitting	C₄×#6900	C₂
prime radical	C₂×C₃³	C₂×C₃×D₃
hypercenter	C₄×C₉	AGL(1,ℤ/9)
hypocenter	C₉	Q₈×3₋¹⁺²
LCS	G ▷ C₂×C₃×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₃ ◁ C₄×C₉

subgroup

G/H

center

C₂×C₃

C₂×C₃×AGL(1,ℤ/9)

commutator, cosocle

C₂×C₃×C₉

C₂²×C₃²

Frattini, socle

C₂×C₃²

C₂×C₃²×D₃

Fitting

C₄×#6900

C₂

prime radical

C₂×C₃³

C₂×C₃×D₃

hypercenter

C₄×C₉

AGL(1,ℤ/9)

hypocenter

C₉

Q₈×3₋¹⁺²

LCS

G ▷ C₂×C₃×C₉ ▷ C₉

UCS

C₁ ◁ C₂×C₃ ◁ C₄×C₉

N	H ≅ G/N	type
5. Dic₁₈⋊C₉	C₃	semidirect
6. C₉×Dic₁₈	C₃	semidirect
19. #6900	Q₈	semidirect
56. C₉⋊C₉	C₃×Q₈	semidirect
57. C₉²	C₃×Q₈	semidirect
63. Dic₁₈	3₋¹⁺²	semidirect
138. C₉	3₋¹⁺²⋊Q₈	semidirect
139. C₉	Q₈×3₋¹⁺²	semidirect

H ≅ G/N

type

5. Dic₁₈⋊C₉

C₃

semidirect

6. C₉×Dic₁₈

C₃

semidirect

19. #6900

Q₈

semidirect

56. C₉⋊C₉

C₃×Q₈

semidirect

57. C₉²

C₃×Q₈

semidirect

63. Dic₁₈

3₋¹⁺²

semidirect

138. C₉

3₋¹⁺²⋊Q₈

semidirect

139. C₉

Q₈×3₋¹⁺²

semidirect

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

subgroup	index
C₄×#6900	2
#11757569	2
Dic₆×3₋¹⁺²	3
C₃×3₋¹⁺²⋊Q₈	3
Dic₁₈⋊C₉	3
C₉×Dic₁₈	3

subgroup

index

C₄×#6900

#11757569

Dic₆×3₋¹⁺²

C₃×3₋¹⁺²⋊Q₈

Dic₁₈⋊C₉

C₉×Dic₁₈

Subgroups — #14517331

Conjugacy classes of subgroups

Complements

Maximal subgroups