Group #14518117 — subgroups

Subgroups — C₃⋊#10619756 #14518117

Order 1944.

345 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 345 conjugacy classes of subgroups — too many to draw. Only its 47 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	C₃×#66361	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₃⋊((C₂²⋊C₉)⋊C₂)	—	3	1	no	maximal
3	C₃⋊(((C₂²⋊C₉)⋊C₂)⋊C₃)	—	3	2	no	maximal
4	#10619977	—	3	1	no	maximal
5	#10619756	—	3	9	no	maximal
6	C₃⋊#4184	—	4	1	no	maximal
7	C₂²×C₃×H(3,𝔽₃)	D₃	1	1	yes	characteristic
8	C₂×C₃⋊((C₃⋊D₃)⋊C₃)	—	3	1	no	—
9	C₃×#1094	D₃	1	2	yes	complemented
10	C₃²×C₂²⋊C₉	D₃	1	1	yes	characteristic, complemented
11	C₃⋊((C₃⋊Dic₃)⋊C₃)	—	3	1	no	—
12	#66361	D₃	1	9	yes	complemented
13	C₃×#731	—	4	1	no	Sylow
14	#6312	—	3	1	no	—
15	C₃×#638	—	9	1	no	—
16	C₃⋊((C₂²⋊C₉)⋊C₂)	—	3	2	no	—
17	C₃⋊((C₂²⋊C₉)⋊C₂)	—	3	1	no	—
18	C₃×(C₂²⋊C₉)⋊C₂	—	9	3	no	—
19	((C₂²⋊C₉)⋊C₂)⋊C₃	—	9	6	no	—
20	#638⋊C₃	—	9	3	no	—
21	C₂×C₃×H(3,𝔽₃)	—	3	1	no	—
22	C₃⋊AGL(1,ℤ/9)	—	12	2	no	—
23	C₃⋊((C₃⋊D₃)⋊C₃)	—	6	1	no	—
24	C₃×C₃⋊D₉	—	12	1	no	—
25	#4184	—	12	9	no	—
26	C₂²×C₃³	—	3	1	no	—
27	C₂²×C₃³	C₃⋊D₃	1	1	yes	characteristic
28	C₂×C₃⋊(C₃⋊D₃)	—	3	1	no	—
29	C₂×C₃×C₃⋊D₃	—	9	1	no	—
30	C₃⋊(C₃⋊Dic₃)	—	3	1	no	—
31	C₃×C₃⋊Dic₃	—	9	1	no	—
32	C₂²×H(3,𝔽₃)	—	6	1	no	—
33	C₂²×H(3,𝔽₃)	C₃⋊D₃	1	3	yes	—
34	C₂×(C₃⋊D₃)⋊C₃	—	9	3	no	—
35	#1094	—	6	2	no	—
36	#1094	C₃⋊D₃	1	6	yes	complemented
37	C₃×C₂²⋊C₉	—	3	1	no	—
38	C₃×C₂²⋊C₉	—	3	2	no	—
39	C₃×C₂²⋊C₉	—	6	1	no	—
40	C₃×C₂²⋊C₉	C₃⋊D₃	1	3	yes	complemented
41	(C₃⋊Dic₃)⋊C₃	—	9	3	no	—
42	C₃×3₋¹⁺²	—	4	2	no	—
43	C₃×H(3,𝔽₃)	S₄	1	1	yes	characteristic
44	C₃²×C₉	—	4	1	no	—
45	#731	—	4	9	no	—
46	#638	—	9	1	no	—
47	#638	—	9	3	no	—
48	#638	—	27	3	no	—
49	C₃×#67	—	27	3	no	—
50	C₃×#67	—	27	1	no	—
51	(C₂²⋊C₉)⋊C₂	—	9	6	no	—
52	(C₂²⋊C₉)⋊C₂	—	9	3	no	—
53	C₂×C₃³	—	3	1	no	—
54	C₂×C₃³	—	3	1	no	—
55	C₂×C₃³	—	6	1	no	—
56	C₃⋊(C₃⋊D₃)	—	6	1	no	—
57	C₃×C₃⋊D₃	—	18	1	no	—
58	C₂×H(3,𝔽₃)	—	3	3	no	—
59	C₂×H(3,𝔽₃)	—	6	3	no	—
60	C₃⋊D₉	—	12	1	no	—
61	C₃⋊D₉	—	12	2	no	—
62	AGL(1,ℤ/9)	—	36	6	no	—
63	(C₃⋊D₃)⋊C₃	—	18	3	no	—
64	C₃×D₉	—	36	3	no	—
65	C₂²×C₃²	—	2	1	no	—
66	C₂²×C₃²	—	3	3	no	—
67	C₂²×C₃²	—	3	3	no	—
68	C₂²×C₃²	—	9	3	no	—
69	C₂²×C₃²	#272	1	1	yes	characteristic
70	C₂²×C₃²	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
71	C₂×C₃⋊D₃	—	9	1	no	—
72	C₂×C₃⋊D₃	—	9	3	no	—
73	C₂×C₃⋊D₃	—	27	3	no	—
74	C₂×C₃×D₃	—	27	1	no	—
75	C₂×C₃×D₃	—	27	3	no	—
76	C₃⋊Dic₃	—	9	1	no	—
77	C₃⋊Dic₃	—	9	3	no	—
78	C₃⋊Dic₃	—	27	3	no	—
79	C₃×Dic₃	—	27	1	no	—
80	C₃×Dic₃	—	27	3	no	—
81	C₂²⋊C₉	—	3	3	no	—
82	C₂²⋊C₉	—	3	6	no	—
83	C₃³	—	3	1	no	—
84	C₃³	C₃⋊S₄	1	1	yes	characteristic
85	3₋¹⁺²	—	4	6	no	—
86	3₋¹⁺²	—	24	2	no	—
87	H(3,𝔽₃)	—	6	1	no	—
88	H(3,𝔽₃)	C₃⋊S₄	1	3	yes	—
89	C₃×C₉	—	4	3	no	—
90	C₃×C₉	—	12	1	no	—
91	C₃×C₉	—	12	2	no	—
92	C₃×C₉	—	24	1	no	—
93	C₃×D₄	—	81	1	no	—
94	#67	—	27	3	no	—
95	#67	—	27	1	no	—
96	#67	—	81	3	no	—
97	C₂×C₃²	—	3	1	no	—
98	C₂×C₃²	—	3	3	no	—
99	C₂×C₃²	—	3	1	no	—
100	C₂×C₃²	—	6	1	no	—
101	C₂×C₃²	—	6	3	no	—
102	C₂×C₃²	—	9	3	no	—
103	C₂×C₃²	—	9	3	no	—
104	C₂×C₃²	—	18	3	no	—
105	C₃⋊D₃	—	18	1	no	—
106	C₃⋊D₃	—	18	3	no	—
107	C₃⋊D₃	—	54	3	no	—
108	C₃×D₃	—	54	3	no	—
109	C₃×D₃	—	54	1	no	—
110	D₉	—	36	3	no	—
111	D₉	—	36	6	no	—
112	C₂²×C₃	—	3	1	no	—
113	C₂²×C₃	—	6	1	no	—
114	C₂²×C₃	—	9	3	no	—
115	C₂²×C₃	—	81	1	no	—
116	C₂²×C₃	#4184	1	3	yes	complemented
117	C₂²×C₃	C₃⋊#272	1	1	yes	characteristic
118	C₂×D₃	—	27	1	no	—
119	C₂×D₃	—	27	3	no	—
120	C₂×D₃	—	81	3	no	—
121	C₃×C₄	—	81	1	no	—
122	Dic₃	—	27	1	no	—
123	Dic₃	—	27	3	no	—
124	Dic₃	—	81	3	no	—
125	C₃²	—	2	1	no	—
126	C₃²	—	3	3	no	—
127	C₃²	—	3	3	no	—
128	C₃²	—	9	3	no	—
129	C₃²	#6258	1	1	yes	characteristic
130	C₃²	C₃⋊(C₃⋊S₄)	1	1	yes	characteristic
131	C₉	—	12	3	no	—
132	C₉	—	12	6	no	—
133	D₄	—	81	1	no	Sylow
134	C₂×C₃	—	3	3	no	—
135	C₂×C₃	—	3	1	no	—
136	C₂×C₃	—	9	1	no	—
137	C₂×C₃	—	9	3	no	—
138	C₂×C₃	—	18	1	no	—
139	C₂×C₃	—	18	3	no	—
140	C₂×C₃	—	162	1	no	—
141	D₃	—	54	3	no	—
142	D₃	—	54	1	no	—
143	D₃	—	162	3	no	—
144	C₂²	—	81	1	no	—
145	C₂²	C₃⋊#4184	1	1	yes	characteristic, complemented
146	C₄	—	81	1	no	—
147	C₃	—	3	1	no	—
148	C₃	—	6	1	no	—
149	C₃	—	9	3	no	—
150	C₃	#10619756	1	3	yes	complemented
151	C₃	C₃⋊#6258	1	1	yes	characteristic
152	C₂	—	3	1	no	—
153	C₂	—	162	1	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#66361

C₂

yes

maximal, characteristic, complemented

C₃×C₃⋊((C₂²⋊C₉)⋊C₂)

—

maximal

C₃⋊(((C₂²⋊C₉)⋊C₂)⋊C₃)

—

—

—

—

C₂²×C₃×H(3,𝔽₃)

D₃

yes

characteristic

C₂×C₃⋊((C₃⋊D₃)⋊C₃)

—

C₃×#1094

D₃

yes

complemented

C₃²×C₂²⋊C₉

D₃

yes

characteristic, complemented

C₃⋊((C₃⋊Dic₃)⋊C₃)

—

#66361

D₃

yes

—

—

—

C₃⋊((C₂²⋊C₉)⋊C₂)

—

C₃⋊((C₂²⋊C₉)⋊C₂)

—

C₃×(C₂²⋊C₉)⋊C₂

—

((C₂²⋊C₉)⋊C₂)⋊C₃

—

#638⋊C₃

—

C₂×C₃×H(3,𝔽₃)

—

C₃⋊AGL(1,ℤ/9)

—

C₃⋊((C₃⋊D₃)⋊C₃)

—

C₃×C₃⋊D₉

—

#4184

—

C₂²×C₃³

—

C₂²×C₃³

C₃⋊D₃

yes

characteristic

C₂×C₃⋊(C₃⋊D₃)

—

C₂×C₃×C₃⋊D₃

—

C₃⋊(C₃⋊Dic₃)

—

C₃×C₃⋊Dic₃

—

C₂²×H(3,𝔽₃)

—

C₂²×H(3,𝔽₃)

C₃⋊D₃

yes

—

C₂×(C₃⋊D₃)⋊C₃

—

#1094

—

#1094

C₃⋊D₃

yes

complemented

C₃×C₂²⋊C₉

—

C₃×C₂²⋊C₉

—

C₃×C₂²⋊C₉

—

C₃×C₂²⋊C₉

C₃⋊D₃

yes

complemented

(C₃⋊Dic₃)⋊C₃

—

C₃×3₋¹⁺²

—

C₃×H(3,𝔽₃)

S₄

yes

characteristic

C₃²×C₉

—

—

—

—

—

C₃×#67

—

C₃×#67

—

(C₂²⋊C₉)⋊C₂

—

(C₂²⋊C₉)⋊C₂

—

C₂×C₃³

—

C₂×C₃³

—

C₂×C₃³

—

C₃⋊(C₃⋊D₃)

—

C₃×C₃⋊D₃

—

C₂×H(3,𝔽₃)

—

C₂×H(3,𝔽₃)

—

C₃⋊D₉

—

C₃⋊D₉

—

AGL(1,ℤ/9)

—

(C₃⋊D₃)⋊C₃

—

C₃×D₉

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

#272

yes

characteristic

C₂²×C₃²

C₃⋊(C₃⋊D₃)

yes

characteristic

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₂×C₃×D₃

—

C₂×C₃×D₃

—

C₃⋊Dic₃

—

C₃⋊Dic₃

—

C₃⋊Dic₃

—

C₃×Dic₃

—

C₃×Dic₃

—

C₂²⋊C₉

—

C₂²⋊C₉

—

C₃³

—

C₃³

C₃⋊S₄

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

H(3,𝔽₃)

—

H(3,𝔽₃)

C₃⋊S₄

yes

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×D₄

—

#67

—

#67

—

#67

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

100

C₂×C₃²

—

101

C₂×C₃²

—

102

C₂×C₃²

—

103

C₂×C₃²

—

104

C₂×C₃²

—

105

C₃⋊D₃

—

106

C₃⋊D₃

—

107

C₃⋊D₃

—

108

C₃×D₃

—

109

C₃×D₃

—

110

D₉

—

111

D₉

—

112

C₂²×C₃

—

113

C₂²×C₃

—

114

C₂²×C₃

—

115

C₂²×C₃

—

116

C₂²×C₃

#4184

yes

complemented

117

C₂²×C₃

C₃⋊#272

yes

characteristic

118

C₂×D₃

—

119

C₂×D₃

—

120

C₂×D₃

—

121

C₃×C₄

—

122

Dic₃

—

123

Dic₃

—

124

Dic₃

—

125

C₃²

—

126

C₃²

—

127

C₃²

—

128

C₃²

—

129

C₃²

#6258

yes

characteristic

130

C₃²

C₃⋊(C₃⋊S₄)

yes

characteristic

131

C₉

—

132

C₉

—

133

D₄

—

Sylow

134

C₂×C₃

—

135

C₂×C₃

—

136

C₂×C₃

—

137

C₂×C₃

—

138

C₂×C₃

—

139

C₂×C₃

—

140

C₂×C₃

—

162

—

141

D₃

—

142

D₃

—

143

D₃

—

162

—

144

C₂²

—

145

C₂²

C₃⋊#4184

yes

characteristic, complemented

146

C₄

—

147

C₃

—

148

C₃

—

149

C₃

—

150

C₃

#10619756

yes

complemented

151

C₃

C₃⋊#6258

yes

characteristic

152

C₂

—

153

C₂

—

162

—

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃×#66361	C₂
Frattini	C₃²	C₃⋊(C₃⋊S₄)
Fitting	C₂²×C₃×H(3,𝔽₃)	D₃
socle	C₂²×C₃²	#272
LCS	G ▷ C₃×#66361
UCS	C₁

subgroup

G/H

center, hypercenter

C₁

commutator, cosocle, hypocenter

C₃×#66361

C₂

Frattini

C₃²

C₃⋊(C₃⋊S₄)

Fitting

C₂²×C₃×H(3,𝔽₃)

D₃

C₂²×C₃²

G ▷ C₃×#66361

C₁

N	H ≅ G/N	type
1. C₃×#66361	C₂	semidirect
9. C₃×#1094	D₃	semidirect
10. C₃²×C₂²⋊C₉	D₃	semidirect
12. #66361	D₃	semidirect
36. #1094	C₃⋊D₃	semidirect
40. C₃×C₂²⋊C₉	C₃⋊D₃	semidirect
116. C₂²×C₃	#4184	semidirect
145. C₂²	C₃⋊#4184	semidirect
150. C₃	#10619756	semidirect

H ≅ G/N

type

1. C₃×#66361

C₂

semidirect

9. C₃×#1094

D₃

semidirect

10. C₃²×C₂²⋊C₉

D₃

semidirect

12. #66361

D₃

semidirect

36. #1094

C₃⋊D₃

semidirect

40. C₃×C₂²⋊C₉

C₃⋊D₃

semidirect

116. C₂²×C₃

#4184

semidirect

145. C₂²

C₃⋊#4184

semidirect

150. C₃

#10619756

semidirect

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

subgroup	index
C₃×#66361	2
C₃×C₃⋊((C₂²⋊C₉)⋊C₂)	3
C₃⋊(((C₂²⋊C₉)⋊C₂)⋊C₃)	3
#10619977	3
#10619756	3
C₃⋊#4184	4

subgroup

index

C₃×#66361

C₃×C₃⋊((C₂²⋊C₉)⋊C₂)

C₃⋊(((C₂²⋊C₉)⋊C₂)⋊C₃)

#10619977

#10619756

C₃⋊#4184

Subgroups — C₃⋊#10619756 #14518117

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3⋊#10619756 #14518117

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃⋊#10619756 #14518117