Group #28355064 — subgroups

Subgroups — #28355064

Order 13851.

402 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 61 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×#13095332	C₃	1	1	yes	maximal, characteristic
2	#17987519	C₃	1	3	yes	maximal, complemented
3	#17987518	C₃	1	3	yes	maximal, complemented
4	#17987511	C₃	1	3	yes	maximal
5	C₃×#13095322	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃×#13095322	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃×C₁₉×#733	C₃	1	1	yes	maximal, characteristic, complemented
8	#13095332	—	3	3	no	—
9	#13095332	—	3	3	no	—
10	#13095332	—	3	3	no	—
11	C₃×#10587025	C₃²	1	3	yes	—
12	C₃×#10587025	C₃²	1	3	yes	—
13	C₃×#10587025	C₃²	1	1	yes	characteristic
14	C₃×#10587025	C₃²	1	1	yes	characteristic
15	C₃×#10587022	—	3	1	no	—
16	C₃×#10587023	—	3	1	no	—
17	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	1	yes	characteristic
18	C₃×C₉×C₁₉⋊C₃	—	3	1	no	—
19	#13095322	—	3	1	no	—
20	#13095322	—	3	1	no	—
21	#13095322	—	3	1	no	—
22	#13095322	—	3	1	no	—
23	#13095322	—	3	1	no	—
24	#13095322	—	3	1	no	—
25	#13095322	—	3	9	no	—
26	#13095321	—	3	9	no	—
27	#13095318	—	3	9	no	—
28	C₃×C₁₉×3₋¹⁺²	C₃²	1	3	yes	—
29	C₃²×C₉×C₁₉	C₃²	1	1	yes	characteristic
30	C₁₉×#733	—	3	1	no	—
31	C₁₉×#733	—	3	1	no	—
32	C₁₉×#733	—	3	1	no	—
33	#10624758	—	19	1	no	maximal, Sylow
34	C₃³×C₁₉	C₃³	1	1	yes	characteristic
35	C₃²×C₁₉⋊C₃	—	3	1	no	—
36	C₁₉×3₋¹⁺²	—	3	3	no	—
37	C₁₉×3₋¹⁺²	—	3	3	no	—
38	C₁₉×3₋¹⁺²	—	3	3	no	—
39	C₃×C₉×C₁₉	—	3	1	no	—
40	C₃×C₉×C₁₉	—	3	1	no	—
41	C₃×C₉×C₁₉	—	3	1	no	—
42	C₃×C₉×C₁₉	—	3	3	no	—
43	C₃×C₉×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
44	C₃×C₉×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
45	C₃×C₉×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
46	C₃²⋊(C₁₉⋊C₃)	—	3	1	no	—
47	C₃²⋊(C₁₉⋊C₃)	—	3	1	no	—
48	C₃²⋊(C₁₉⋊C₃)	H(3,𝔽₃)	1	3	yes	—
49	#10587025	—	3	1	no	—
50	#10587025	—	3	1	no	—
51	#10587025	—	3	1	no	—
52	#10587025	—	3	1	no	—
53	#10587025	—	3	3	no	—
54	#10587025	—	3	3	no	—
55	#10587025	—	3	3	no	—
56	#10587025	—	3	3	no	—
57	#10587025	—	3	3	no	—
58	#10587025	—	3	3	no	—
59	#10587025	H(3,𝔽₃)	1	3	yes	—
60	#10587025	H(3,𝔽₃)	1	3	yes	—
61	C₃×#4294	—	3	1	no	—
62	C₃×#4294	—	3	1	no	—
63	C₃×#4294	—	3	3	no	—
64	C₃×#4294	—	3	3	no	—
65	#10587023	—	3	9	no	—
66	#10587022	—	3	9	no	—
67	C₉×C₁₉⋊C₃	—	3	9	no	—
68	C₃×#737	—	19	1	no	—
69	#6912	—	19	3	no	—
70	#6912	—	19	3	no	—
71	#6910	—	19	3	no	—
72	C₃×#733	—	19	1	no	—
73	C₃×#733	—	19	1	no	—
74	C₃×#733	C₁₉⋊C₃	1	1	yes	characteristic, complemented
75	C₃²×C₁₉	—	9	1	no	—
76	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
77	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
78	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
79	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
80	C₃×C₁₉⋊C₃	—	3	3	no	—
81	C₃×C₁₉⋊C₃	—	9	3	no	—
82	C₉×C₁₉	—	3	1	no	—
83	C₉×C₁₉	—	3	1	no	—
84	C₉×C₁₉	—	3	1	no	—
85	C₉×C₁₉	—	9	3	no	—
86	#4294	—	3	3	no	—
87	#4294	—	3	3	no	—
88	#4294	—	9	3	no	—
89	#4294	—	9	3	no	—
90	#737	—	57	3	no	—
91	#737	—	57	3	no	—
92	#737	—	57	3	no	—
93	C₃×3₋¹⁺²	—	19	1	no	—
94	C₃×3₋¹⁺²	—	19	1	no	—
95	C₃×3₋¹⁺²	—	19	3	no	—
96	C₃×3₋¹⁺²	—	19	3	no	—
97	C₃×3₋¹⁺²	—	57	1	no	—
98	C₃×3₋¹⁺²	—	57	1	no	—
99	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	3	yes	—
100	C₃×H(3,𝔽₃)	—	19	1	no	—
101	C₃²×C₉	—	57	1	no	—
102	C₃²×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
103	#733	—	3	1	no	—
104	#733	—	3	1	no	—
105	#733	—	3	1	no	—
106	#733	—	57	9	no	—
107	#733	—	57	9	no	—
108	#733	—	57	1	no	—
109	#733	—	57	1	no	—
110	#733	—	57	1	no	—
111	#733	—	57	1	no	—
112	#733	—	57	1	no	—
113	#733	—	57	1	no	—
114	#731	—	57	9	no	—
115	C₃×C₁₉	—	3	1	no	—
116	C₃×C₁₉	—	3	1	no	—
117	C₃×C₁₉	—	3	1	no	—
118	C₃×C₁₉	—	3	1	no	—
119	C₃×C₁₉	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
120	C₁₉⋊C₃	—	9	3	no	—
121	C₃³	—	57	1	no	—
122	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
123	3₋¹⁺²	—	3	3	no	—
124	3₋¹⁺²	—	3	3	no	—
125	3₋¹⁺²	—	3	3	no	—
126	3₋¹⁺²	—	19	3	no	—
127	3₋¹⁺²	—	19	3	no	—
128	3₋¹⁺²	—	57	9	no	—
129	3₋¹⁺²	—	57	9	no	—
130	3₋¹⁺²	—	57	1	no	—
131	3₋¹⁺²	—	57	1	no	—
132	3₋¹⁺²	—	57	1	no	—
133	3₋¹⁺²	—	57	1	no	—
134	3₋¹⁺²	—	57	3	no	—
135	3₋¹⁺²	—	57	3	no	—
136	3₋¹⁺²	—	57	3	no	—
137	3₋¹⁺²	—	57	3	no	—
138	3₋¹⁺²	—	57	3	no	—
139	3₋¹⁺²	—	57	3	no	—
140	H(3,𝔽₃)	—	19	3	no	—
141	H(3,𝔽₃)	—	57	1	no	—
142	H(3,𝔽₃)	—	57	1	no	—
143	C₃×C₉	—	3	1	no	—
144	C₃×C₉	—	3	1	no	—
145	C₃×C₉	—	3	1	no	—
146	C₃×C₉	—	3	3	no	—
147	C₃×C₉	—	57	9	no	—
148	C₃×C₉	—	57	1	no	—
149	C₃×C₉	—	57	1	no	—
150	C₃×C₉	—	57	3	no	—
151	C₃×C₉	—	57	3	no	—
152	C₃×C₉	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
153	C₃×C₉	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
154	C₃×C₉	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
155	C₁₉	#10624758	1	1	yes	characteristic, complemented, Sylow
156	C₃²	—	9	1	no	—
157	C₃²	—	57	3	no	—
158	C₃²	—	171	3	no	—
159	C₃²	H(3,𝔽₃)×C₁₉⋊C₃	1	1	yes	characteristic
160	C₃²	H(3,𝔽₃)×C₁₉⋊C₃	1	1	yes	characteristic
161	C₃²	H(3,𝔽₃)×C₁₉⋊C₃	1	1	yes	characteristic
162	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
163	C₉	—	3	1	no	—
164	C₉	—	3	1	no	—
165	C₉	—	3	1	no	—
166	C₉	—	9	3	no	—
167	C₉	—	57	3	no	—
168	C₉	—	57	3	no	—
169	C₉	—	171	3	no	—
170	C₉	—	171	3	no	—
171	C₃	—	3	1	no	—
172	C₃	—	3	1	no	—
173	C₃	—	3	1	no	—
174	C₃	—	3	1	no	—
175	C₃	—	171	3	no	—
176	C₃	(C₃×C₁₉⋊C₃)⋊H(3,𝔽₃)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#13095332

C₃

yes

maximal, characteristic

#17987519

C₃

yes

maximal, complemented

#17987518

C₃

yes

maximal, complemented

#17987511

C₃

yes

maximal

C₃×#13095322

C₃

yes

maximal, characteristic, complemented

C₃×#13095322

C₃

yes

maximal, characteristic, complemented

C₃×C₁₉×#733

C₃

yes

maximal, characteristic, complemented

—

—

—

C₃²

yes

—

C₃×#10587025

C₃²

yes

—

C₃×#10587025

C₃²

yes

characteristic

C₃×#10587025

C₃²

yes

characteristic

C₃×#10587022

—

C₃×#10587023

—

C₃×C₃²⋊(C₁₉⋊C₃)

C₃²

yes

characteristic

C₃×C₉×C₁₉⋊C₃

—

—

—

—

—

—

—

—

—

—

C₃×C₁₉×3₋¹⁺²

C₃²

yes

—

C₃²×C₉×C₁₉

C₃²

yes

characteristic

C₁₉×#733

—

C₁₉×#733

—

C₁₉×#733

—

#10624758

—

maximal, Sylow

C₃³×C₁₉

C₃³

yes

characteristic

C₃²×C₁₉⋊C₃

—

C₁₉×3₋¹⁺²

—

C₁₉×3₋¹⁺²

—

C₁₉×3₋¹⁺²

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

yes

C₃×C₉×C₁₉

yes

C₃×C₉×C₁₉

yes

C₃²⋊(C₁₉⋊C₃)

—

C₃²⋊(C₁₉⋊C₃)

—

C₃²⋊(C₁₉⋊C₃)

H(3,𝔽₃)

yes

—

—

—

—

—

—

—

—

—

—

—

yes

—

#10587025

H(3,𝔽₃)

yes

—

—

—

—

—

—

—

C₉×C₁₉⋊C₃

—

—

—

—

—

—

—

C₁₉⋊C₃

yes

characteristic, complemented

C₃²×C₁₉

—

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

characteristic

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₉×C₁₉

—

C₉×C₁₉

—

C₉×C₁₉

—

C₉×C₁₉

—

—

—

—

—

—

—

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃×C₁₉⋊C₃

yes

—

100

C₃×H(3,𝔽₃)

—

101

C₃²×C₉

—

102

C₃²×C₉

C₃×C₁₉⋊C₃

yes

characteristic

103

#733

—

104

#733

—

105

#733

—

106

#733

—

107

#733

—

108

#733

—

109

#733

—

110

#733

—

111

#733

—

112

#733

—

113

#733

—

114

#731

—

115

C₃×C₁₉

—

116

C₃×C₁₉

—

117

C₃×C₁₉

—

118

C₃×C₁₉

—

119

C₃×C₁₉

C₃²⋊H(3,𝔽₃)

yes

characteristic

120

C₁₉⋊C₃

—

121

C₃³

—

122

C₃³

C₃²×C₁₉⋊C₃

yes

characteristic

123

3₋¹⁺²

—

124

3₋¹⁺²

—

125

3₋¹⁺²

—

126

3₋¹⁺²

—

127

3₋¹⁺²

—

128

3₋¹⁺²

—

129

3₋¹⁺²

—

130

3₋¹⁺²

—

131

3₋¹⁺²

—

132

3₋¹⁺²

—

133

3₋¹⁺²

—

134

3₋¹⁺²

—

135

3₋¹⁺²

—

136

3₋¹⁺²

—

137

3₋¹⁺²

—

138

3₋¹⁺²

—

139

3₋¹⁺²

—

140

H(3,𝔽₃)

—

141

H(3,𝔽₃)

—

142

H(3,𝔽₃)

—

143

C₃×C₉

—

144

C₃×C₉

—

145

C₃×C₉

—

146

C₃×C₉

—

147

C₃×C₉

—

148

C₃×C₉

—

149

C₃×C₉

—

150

C₃×C₉

—

151

C₃×C₉

—

152

C₃×C₉

C₃²⋊(C₁₉⋊C₃)

yes

characteristic

153

C₃×C₉

C₃²⋊(C₁₉⋊C₃)

yes

characteristic

154

C₃×C₉

C₃²⋊(C₁₉⋊C₃)

yes

characteristic

155

C₁₉

#10624758

yes

characteristic, complemented, Sylow

156

C₃²

—

157

C₃²

—

158

C₃²

—

171

—

159

C₃²

H(3,𝔽₃)×C₁₉⋊C₃

yes

characteristic

160

C₃²

H(3,𝔽₃)×C₁₉⋊C₃

yes

characteristic

161

C₃²

H(3,𝔽₃)×C₁₉⋊C₃

yes

characteristic

162

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

yes

characteristic

163

C₉

—

164

C₉

—

165

C₉

—

166

C₉

—

167

C₉

—

168

C₉

—

169

C₉

—

171

—

170

C₉

—

171

—

171

C₃

—

172

C₃

—

173

C₃

—

174

C₃

—

175

C₃

—

171

—

176

C₃

(C₃×C₁₉⋊C₃)⋊H(3,𝔽₃)

yes

characteristic

	subgroup	G/H
center	C₃	(C₃×C₁₉⋊C₃)⋊H(3,𝔽₃)
commutator, cosocle	C₃³×C₁₉	C₃³
Frattini	C₃³	C₃²×C₁₉⋊C₃
Fitting	C₃×C₁₉×#733	C₃
socle	C₃×C₁₉	C₃²⋊H(3,𝔽₃)
prime radical	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²
hypercenter	C₃×#733	C₁₉⋊C₃
hypocenter	C₁₉	#10624758
LCS	G ▷ C₃³×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃×#733

subgroup

G/H

center

C₃

(C₃×C₁₉⋊C₃)⋊H(3,𝔽₃)

commutator, cosocle

C₃³×C₁₉

C₃³

Frattini

C₃³

C₃²×C₁₉⋊C₃

Fitting

C₃×C₁₉×#733

C₃

socle

C₃×C₁₉

C₃²⋊H(3,𝔽₃)

prime radical

C₃×C₃²⋊(C₁₉⋊C₃)

C₃²

C₁₉⋊C₃

C₁₉

G ▷ C₃³×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₃ ◁ C₃³ ◁ C₃×#733

N	H ≅ G/N	type
2. #17987519	C₃	semidirect
3. #17987518	C₃	semidirect
5. C₃×#13095322	C₃	semidirect
6. C₃×#13095322	C₃	semidirect
7. C₃×C₁₉×#733	C₃	semidirect
74. C₃×#733	C₁₉⋊C₃	semidirect
155. C₁₉	#10624758	semidirect

H ≅ G/N

type

2. #17987519

C₃

semidirect

3. #17987518

C₃

semidirect

5. C₃×#13095322

C₃

semidirect

6. C₃×#13095322

C₃

semidirect

7. C₃×C₁₉×#733

C₃

semidirect

74. C₃×#733

C₁₉⋊C₃

semidirect

155. C₁₉

#10624758

semidirect

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

subgroup	index
C₃×#13095332	3
#17987519	3
#17987518	3
#17987511	3
C₃×#13095322	3
C₃×C₁₉×#733	3
#10624758	19

subgroup

index

C₃×C₁₉×#733

Subgroups — #28355064

Conjugacy classes of subgroups

Complements

Maximal subgroups