Group #14517109 — subgroups

Subgroups — #14517109

Order 1944.

366 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 46 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×#92297	C₂	1	2	yes	maximal, complemented
2	C₉×#1084	C₂	1	1	yes	maximal, characteristic, complemented
3	#10619902	—	3	1	no	maximal
4	#10619874	—	3	2	no	maximal
5	C₂×C₉×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
6	#92297	—	2	2	no	—
7	C₂×C₃⋊(C₃⋊D₉)	D₃	1	2	yes	complemented
8	C₂×C₃⋊((C₃⋊D₃)⋊C₃)	—	3	2	no	—
9	C₂×C₃×C₃⋊D₉	—	3	2	no	—
10	C₃×#1084	D₃	1	1	yes	characteristic
11	C₃×C₉×Dic₃	—	3	2	no	—
12	C₃×C₉×Dic₃	—	6	1	no	—
13	C₂×#4169	—	6	2	no	—
14	C₉×H(3,𝔽₃)	D₄	1	1	yes	characteristic, complemented, Sylow
15	#6292	—	9	2	no	—
16	C₄⋊(C₃⋊D₉)	—	9	1	no	—
17	#6198	—	9	1	no	—
18	#6192	—	9	6	no	—
19	C₂×C₃×H(3,𝔽₃)	C₂×D₃	1	1	yes	characteristic
20	C₂×C₃²×C₉	—	2	1	no	—
21	C₂×C₃²×C₉	C₂×D₃	1	2	yes	—
22	C₃⋊(C₃⋊D₉)	—	2	2	no	—
23	C₃⋊((C₃⋊D₃)⋊C₃)	—	6	2	no	—
24	C₃×C₃⋊D₉	—	6	2	no	—
25	C₂×C₃²⋊C₉	—	2	2	no	—
26	C₂×C₃²⋊C₉	—	4	1	no	—
27	#4169	—	6	4	no	—
28	C₂×C₃⋊(C₃⋊D₃)	—	3	2	no	—
29	C₂×C₃×C₃⋊D₃	—	9	2	no	—
30	C₃²×Dic₃	—	3	2	no	—
31	C₃²×Dic₃	—	6	1	no	—
32	C₂×C₃⋊D₉	—	3	2	no	—
33	C₂×C₃⋊D₉	—	6	2	no	—
34	C₂×C₃⋊D₉	—	9	6	no	—
35	C₂×(C₃⋊D₃)⋊C₃	—	9	2	no	—
36	C₂×(C₃⋊D₃)⋊C₃	—	18	2	no	—
37	C₂×C₃×D₉	—	9	6	no	—
38	C₃×C₄×C₉	—	9	1	no	—
39	#1084	D₉	1	1	yes	characteristic, complemented
40	C₉×Dic₃	—	3	6	no	—
41	C₉×Dic₃	—	6	3	no	—
42	C₃×H(3,𝔽₃)	D₁₂	1	1	yes	characteristic
43	C₃²×C₉	—	2	1	no	—
44	C₃²×C₉	#67	1	2	yes	complemented
45	C₃²⋊C₉	—	2	2	no	—
46	C₃²⋊C₉	—	4	1	no	—
47	C₄⋊(C₃⋊D₃)	—	27	1	no	—
48	#626	—	27	2	no	—
49	#626	—	27	2	no	—
50	#626	—	27	4	no	—
51	D₃₆	—	27	3	no	—
52	C₂×C₃³	—	2	1	no	—
53	C₂×C₃³	D₃²	1	2	yes	—
54	C₃⋊(C₃⋊D₃)	—	6	2	no	—
55	C₃×C₃⋊D₃	—	18	2	no	—
56	C₂×H(3,𝔽₃)	—	2	2	no	—
57	C₂×H(3,𝔽₃)	—	4	1	no	—
58	C₂×H(3,𝔽₃)	C₂×D₉	1	1	yes	characteristic
59	C₂×C₃×C₉	—	2	2	no	—
60	C₂×C₃×C₉	—	3	6	no	—
61	C₂×C₃×C₉	—	4	1	no	—
62	C₂×C₃×C₉	—	6	3	no	—
63	C₂×C₃×C₉	D₃²	1	1	yes	characteristic
64	C₃⋊D₉	—	6	2	no	—
65	C₃⋊D₉	—	6	4	no	—
66	C₃⋊D₉	—	18	6	no	—
67	(C₃⋊D₃)⋊C₃	—	18	4	no	—
68	(C₃⋊D₃)⋊C₃	—	18	2	no	—
69	C₃×D₉	—	18	6	no	—
70	C₂×C₃⋊D₃	—	9	2	no	—
71	C₂×C₃⋊D₃	—	9	2	no	—
72	C₂×C₃⋊D₃	—	18	2	no	—
73	C₂×C₃⋊D₃	—	27	2	no	—
74	C₂×C₃⋊D₃	—	27	4	no	—
75	C₂×C₃×D₃	—	27	2	no	—
76	C₂×C₃×D₃	—	27	4	no	—
77	C₂×C₃×D₃	—	27	2	no	—
78	C₃²×C₄	—	9	1	no	—
79	C₃×Dic₃	—	3	2	no	—
80	C₃×Dic₃	—	3	4	no	—
81	C₃×Dic₃	—	3	2	no	—
82	C₃×Dic₃	—	6	1	no	—
83	C₃×Dic₃	—	6	1	no	—
84	C₃×Dic₃	—	6	2	no	—
85	C₂×D₉	—	9	6	no	—
86	C₂×D₉	—	54	2	no	—
87	C₄×C₉	—	9	3	no	—
88	C₃³	—	2	1	no	—
89	C₃³	#626	1	2	yes	—
90	H(3,𝔽₃)	—	2	2	no	—
91	H(3,𝔽₃)	—	4	1	no	—
92	H(3,𝔽₃)	D₃₆	1	1	yes	characteristic, complemented
93	C₃×C₉	—	2	2	no	—
94	C₃×C₉	—	3	6	no	—
95	C₃×C₉	—	4	1	no	—
96	C₃×C₉	—	6	3	no	—
97	C₃×C₉	#625	1	1	yes	characteristic
98	#67	—	81	2	no	—
99	D₁₂	—	81	1	no	—
100	D₁₂	—	81	1	no	—
101	D₁₂	—	81	2	no	—
102	C₂×C₃²	—	2	2	no	—
103	C₂×C₃²	—	2	1	no	—
104	C₂×C₃²	—	3	2	no	—
105	C₂×C₃²	—	3	4	no	—
106	C₂×C₃²	—	4	1	no	—
107	C₂×C₃²	—	6	1	no	—
108	C₂×C₃²	—	6	2	no	—
109	C₂×C₃²	D₃×D₉	1	2	yes	—
110	C₂×C₃²	#1113	1	1	yes	characteristic
111	C₃⋊D₃	—	18	2	no	—
112	C₃⋊D₃	—	18	4	no	—
113	C₃⋊D₃	—	18	2	no	—
114	C₃⋊D₃	—	54	2	no	—
115	C₃⋊D₃	—	54	4	no	—
116	C₃×D₃	—	54	2	no	—
117	C₃×D₃	—	54	4	no	—
118	C₃×D₃	—	54	2	no	—
119	C₂×C₉	—	6	2	no	—
120	C₂×C₉	—	12	1	no	—
121	C₂×C₉	((C₃⋊D₃)⋊C₃)⋊C₂	1	3	yes	—
122	D₉	—	18	6	no	—
123	D₉	—	54	4	no	—
124	C₂²×C₃	—	81	2	no	—
125	C₂×D₃	—	27	2	no	—
126	C₂×D₃	—	27	2	no	—
127	C₂×D₃	—	27	4	no	—
128	C₂×D₃	—	81	2	no	—
129	C₂×D₃	—	162	2	no	—
130	C₃×C₄	—	9	1	no	—
131	C₃×C₄	—	9	1	no	—
132	C₃×C₄	—	9	2	no	—
133	Dic₃	—	3	2	no	—
134	Dic₃	—	6	1	no	—
135	C₃²	—	2	2	no	—
136	C₃²	—	2	1	no	—
137	C₃²	—	3	4	no	—
138	C₃²	—	3	2	no	—
139	C₃²	—	4	1	no	—
140	C₃²	—	6	2	no	—
141	C₃²	—	6	1	no	—
142	C₃²	#6192	1	2	yes	complemented
143	C₃²	#6295	1	1	yes	characteristic
144	C₉	—	6	2	no	—
145	C₉	—	12	1	no	—
146	C₉	#6198	1	3	yes	complemented
147	D₄	—	243	1	no	Sylow
148	C₂×C₃	—	3	2	no	—
149	C₂×C₃	—	6	1	no	—
150	C₂×C₃	—	6	2	no	—
151	C₂×C₃	—	12	1	no	—
152	C₂×C₃	—	162	2	no	—
153	C₂×C₃	#66431	1	1	yes	characteristic
154	C₂×C₃	#66432	1	1	yes	characteristic
155	C₂×C₃	#4207⋊C₂	1	2	yes	—
156	D₃	—	54	4	no	—
157	D₃	—	54	2	no	—
158	D₃	—	54	2	no	—
159	D₃	—	162	2	no	—
160	D₃	—	162	4	no	—
161	C₂²	—	81	2	no	—
162	C₄	—	9	1	no	—
163	C₃	—	3	2	no	—
164	C₃	—	6	2	no	—
165	C₃	—	6	1	no	—
166	C₃	—	12	1	no	—
167	C₃	#10619897	1	1	yes	characteristic
168	C₃	#10619902	1	1	yes	characteristic
169	C₃	#10619907	1	2	yes	—
170	C₂	—	162	2	no	—
171	C₂	#11757748	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#92297

C₂

yes

maximal, complemented

C₉×#1084

C₂

yes

maximal, characteristic, complemented

—

—

C₂×C₉×H(3,𝔽₃)

C₂²

yes

characteristic

#92297

—

C₂×C₃⋊(C₃⋊D₉)

D₃

yes

complemented

C₂×C₃⋊((C₃⋊D₃)⋊C₃)

—

C₂×C₃×C₃⋊D₉

—

C₃×#1084

D₃

yes

characteristic

C₃×C₉×Dic₃

—

C₃×C₉×Dic₃

—

C₂×#4169

—

C₉×H(3,𝔽₃)

D₄

yes

characteristic, complemented, Sylow

#6292

—

C₄⋊(C₃⋊D₉)

—

#6198

—

#6192

—

C₂×C₃×H(3,𝔽₃)

C₂×D₃

yes

characteristic

C₂×C₃²×C₉

—

C₂×C₃²×C₉

C₂×D₃

yes

—

C₃⋊(C₃⋊D₉)

—

C₃⋊((C₃⋊D₃)⋊C₃)

—

C₃×C₃⋊D₉

—

C₂×C₃²⋊C₉

—

C₂×C₃²⋊C₉

—

#4169

—

C₂×C₃⋊(C₃⋊D₃)

—

C₂×C₃×C₃⋊D₃

—

C₃²×Dic₃

—

C₃²×Dic₃

—

C₂×C₃⋊D₉

—

C₂×C₃⋊D₉

—

C₂×C₃⋊D₉

—

C₂×(C₃⋊D₃)⋊C₃

—

C₂×(C₃⋊D₃)⋊C₃

—

C₂×C₃×D₉

—

C₃×C₄×C₉

—

#1084

D₉

yes

characteristic, complemented

C₉×Dic₃

—

C₉×Dic₃

—

C₃×H(3,𝔽₃)

D₁₂

yes

characteristic

C₃²×C₉

—

C₃²×C₉

#67

yes

complemented

C₃²⋊C₉

—

C₃²⋊C₉

—

C₄⋊(C₃⋊D₃)

—

#626

—

#626

—

#626

—

D₃₆

—

C₂×C₃³

—

C₂×C₃³

D₃²

yes

—

C₃⋊(C₃⋊D₃)

—

C₃×C₃⋊D₃

—

C₂×H(3,𝔽₃)

—

C₂×H(3,𝔽₃)

—

C₂×H(3,𝔽₃)

C₂×D₉

yes

characteristic

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

D₃²

yes

characteristic

C₃⋊D₉

—

C₃⋊D₉

—

C₃⋊D₉

—

(C₃⋊D₃)⋊C₃

—

(C₃⋊D₃)⋊C₃

—

C₃×D₉

—

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₂×C₃×D₃

—

C₂×C₃×D₃

—

C₂×C₃×D₃

—

C₃²×C₄

—

C₃×Dic₃

—

C₃×Dic₃

—

C₃×Dic₃

—

C₃×Dic₃

—

C₃×Dic₃

—

C₃×Dic₃

—

C₂×D₉

—

C₂×D₉

—

C₄×C₉

—

C₃³

—

C₃³

#626

yes

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

H(3,𝔽₃)

D₃₆

yes

characteristic, complemented

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

#625

yes

characteristic

#67

—

D₁₂

—

100

D₁₂

—

101

D₁₂

—

102

C₂×C₃²

—

103

C₂×C₃²

—

104

C₂×C₃²

—

105

C₂×C₃²

—

106

C₂×C₃²

—

107

C₂×C₃²

—

108

C₂×C₃²

—

109

C₂×C₃²

D₃×D₉

yes

—

110

C₂×C₃²

#1113

yes

characteristic

111

C₃⋊D₃

—

112

C₃⋊D₃

—

113

C₃⋊D₃

—

114

C₃⋊D₃

—

115

C₃⋊D₃

—

116

C₃×D₃

—

117

C₃×D₃

—

118

C₃×D₃

—

119

C₂×C₉

—

120

C₂×C₉

—

121

C₂×C₉

((C₃⋊D₃)⋊C₃)⋊C₂

yes

—

122

D₉

—

123

D₉

—

124

C₂²×C₃

—

125

C₂×D₃

—

126

C₂×D₃

—

127

C₂×D₃

—

128

C₂×D₃

—

129

C₂×D₃

—

162

—

130

C₃×C₄

—

131

C₃×C₄

—

132

C₃×C₄

—

133

Dic₃

—

134

Dic₃

—

135

C₃²

—

136

C₃²

—

137

C₃²

—

138

C₃²

—

139

C₃²

—

140

C₃²

—

141

C₃²

—

142

C₃²

#6192

yes

complemented

143

C₃²

#6295

yes

characteristic

144

C₉

—

145

C₉

—

146

C₉

#6198

yes

complemented

147

D₄

—

243

Sylow

148

C₂×C₃

—

149

C₂×C₃

—

150

C₂×C₃

—

151

C₂×C₃

—

152

C₂×C₃

—

162

—

153

C₂×C₃

#66431

yes

characteristic

154

C₂×C₃

#66432

yes

characteristic

155

C₂×C₃

#4207⋊C₂

yes

—

156

D₃

—

157

D₃

—

158

D₃

—

159

D₃

—

162

—

160

D₃

—

162

—

161

C₂²

—

162

C₄

—

163

C₃

—

164

C₃

—

165

C₃

—

166

C₃

—

167

C₃

#10619897

yes

characteristic

168

C₃

#10619902

yes

characteristic

169

C₃

#10619907

yes

—

170

C₂

—

162

—

171

C₂

#11757748

yes

characteristic

	subgroup	G/H
center, hypercenter	C₂	#11757748
commutator, Fitting, cosocle	C₂×C₉×H(3,𝔽₃)	C₂²
Frattini, socle	C₂×C₃²	#1113
hypocenter	C₉×H(3,𝔽₃)	D₄
LCS	G ▷ C₂×C₉×H(3,𝔽₃) ▷ C₉×H(3,𝔽₃)
UCS	C₁ ◁ C₂

subgroup

G/H

center, hypercenter

C₂

#11757748

commutator, Fitting, cosocle

C₂×C₉×H(3,𝔽₃)

C₂²

Frattini, socle

C₂×C₃²

#1113

hypocenter

C₉×H(3,𝔽₃)

D₄

LCS

G ▷ C₂×C₉×H(3,𝔽₃) ▷ C₉×H(3,𝔽₃)

UCS

C₁ ◁ C₂

N	H ≅ G/N	type
1. C₂×#92297	C₂	semidirect
2. C₉×#1084	C₂	semidirect
7. C₂×C₃⋊(C₃⋊D₉)	D₃	semidirect
14. C₉×H(3,𝔽₃)	D₄	semidirect
39. #1084	D₉	semidirect
44. C₃²×C₉	#67	semidirect
92. H(3,𝔽₃)	D₃₆	semidirect
142. C₃²	#6192	semidirect
146. C₉	#6198	semidirect

H ≅ G/N

type

1. C₂×#92297

C₂

semidirect

2. C₉×#1084

C₂

semidirect

7. C₂×C₃⋊(C₃⋊D₉)

D₃

semidirect

14. C₉×H(3,𝔽₃)

D₄

semidirect

39. #1084

D₉

semidirect

44. C₃²×C₉

#67

semidirect

92. H(3,𝔽₃)

D₃₆

semidirect

142. C₃²

#6192

semidirect

146. C₉

#6198

semidirect

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

subgroup	index
C₂×#92297	2
C₉×#1084	2
#10619902	3
#10619874	3

subgroup

index

Subgroups — #14517109

Conjugacy classes of subgroups

Complements

Maximal subgroups