Group #28355017 — subgroups

Subgroups — #28355017

Order 13851.

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

normal subgroups only

This group has 414 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	#17987547	C₃	1	1	yes	maximal, characteristic, complemented
2	#17987518	C₃	1	2	yes	maximal, complemented
3	#17987514	C₃	1	1	yes	maximal, characteristic
4	#17987511	C₃	1	2	yes	maximal, complemented
5	#17987510	C₃	1	2	yes	maximal, complemented
6	#17987509	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃×#13095322	C₃	1	1	yes	maximal, characteristic, complemented
8	#17987431	C₃	1	2	yes	maximal, complemented
9	C₃×C₁₉×#731	C₃	1	1	yes	maximal, characteristic, complemented
10	#13095331	—	3	3	no	—
11	#13095331	—	3	6	no	—
12	C₃×#10587025	—	3	3	no	—
13	C₃×#10587025	C₃²	1	2	yes	—
14	C₃×#10587025	C₃²	1	1	yes	characteristic
15	C₃×#10587025	C₃²	1	1	yes	characteristic
16	C₃×#10587025	C₃²	1	1	yes	characteristic
17	C₃×#10587025	C₃²	1	2	yes	—
18	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	2	yes	—
19	#13095322	—	3	3	no	—
20	#13095321	—	3	6	no	—
21	#13095319	—	3	3	no	—
22	#13095318	—	3	6	no	—
23	#13095317	—	3	6	no	—
24	#13095316	—	3	3	no	—
25	#13095303	—	3	6	no	—
26	C₃×C₁₉×3₋¹⁺²	C₃²	1	2	yes	—
27	C₃×C₁₉×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
28	C₃²×C₉×C₁₉	C₃²	1	1	yes	characteristic
29	C₁₉×#731	—	3	3	no	—
30	#10624751	—	19	1	no	maximal, Sylow
31	C₃³×C₁₉	—	3	1	no	—
32	C₃³×C₁₉	C₃³	1	1	yes	characteristic
33	C₃²×C₁₉⋊C₃	—	3	2	no	—
34	C₁₉×3₋¹⁺²	—	3	2	no	—
35	C₁₉×3₋¹⁺²	—	3	2	no	—
36	C₁₉×3₋¹⁺²	—	3	2	no	—
37	C₁₉×H(3,𝔽₃)	—	3	2	no	—
38	C₁₉×H(3,𝔽₃)	H(3,𝔽₃)	1	3	yes	—
39	C₃×C₉×C₁₉	—	3	1	no	—
40	C₃×C₉×C₁₉	—	3	2	no	—
41	C₃×C₉×C₁₉	—	3	1	no	—
42	C₃×C₉×C₁₉	—	3	2	no	—
43	C₃²⋊(C₁₉⋊C₃)	—	3	2	no	—
44	C₃²⋊(C₁₉⋊C₃)	—	3	2	no	—
45	C₃²⋊(C₁₉⋊C₃)	—	3	2	no	—
46	#10587025	—	3	18	no	—
47	#10587025	—	3	2	no	—
48	#10587025	—	3	2	no	—
49	#10587025	—	3	2	no	—
50	#10587025	—	3	2	no	—
51	#10587025	—	3	2	no	—
52	#10587025	—	3	2	no	—
53	#10587025	—	3	2	no	—
54	#10587025	—	3	2	no	—
55	#10587025	—	3	2	no	—
56	#10587025	—	3	1	no	—
57	#10587025	—	3	1	no	—
58	#10587025	H(3,𝔽₃)	1	3	yes	—
59	C₃×#4294	—	3	9	no	—
60	C₃×#4294	—	3	1	no	—
61	C₃×#4294	—	3	2	no	—
62	C₃×#4294	—	3	1	no	—
63	C₃×#4294	—	3	1	no	—
64	C₃×#4294	H(3,𝔽₃)	1	6	yes	—
65	3₋¹⁺⁴	—	19	1	no	—
66	#6912	—	19	2	no	—
67	#6911	—	19	1	no	—
68	#6910	—	19	2	no	—
69	#6910	—	19	2	no	—
70	#6910	—	19	1	no	—
71	C₃×#733	—	19	1	no	—
72	C₃×#731	C₁₉⋊C₃	1	1	yes	characteristic, complemented
73	#6893	—	19	2	no	—
74	C₃²×C₁₉	—	3	3	no	—
75	C₃²×C₁₉	—	9	1	no	—
76	C₃²×C₁₉	—	27	1	no	—
77	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	2	yes	—
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	—	9	6	no	—
81	C₃×C₁₉⋊C₃	—	9	2	no	—
82	C₉×C₁₉	—	9	1	no	—
83	C₉×C₁₉	—	9	2	no	—
84	#4294	—	3	3	no	—
85	#4294	—	3	6	no	—
86	#4294	—	9	2	no	—
87	#4294	—	9	1	no	—
88	#4294	—	9	1	no	—
89	#737	—	57	6	no	—
90	#737	—	57	3	no	—
91	C₃×3₋¹⁺²	—	19	2	no	—
92	C₃×3₋¹⁺²	—	19	1	no	—
93	C₃×3₋¹⁺²	—	19	1	no	—
94	C₃×3₋¹⁺²	—	19	2	no	—
95	C₃×3₋¹⁺²	—	19	1	no	—
96	C₃×3₋¹⁺²	—	57	3	no	—
97	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	2	yes	—
98	C₃×H(3,𝔽₃)	—	19	2	no	—
99	C₃×H(3,𝔽₃)	C₃×C₁₉⋊C₃	1	1	yes	characteristic
100	C₃²×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
101	#733	—	57	3	no	—
102	#733	—	57	6	no	—
103	#732	—	57	3	no	—
104	#731	—	3	3	no	—
105	#731	—	57	6	no	—
106	#731	—	57	6	no	—
107	#731	—	57	3	no	—
108	C₃²⋊C₉	—	57	6	no	—
109	C₃×C₁₉	—	3	2	no	—
110	C₃×C₁₉	—	3	1	no	—
111	C₃×C₁₉	—	3	1	no	—
112	C₃×C₁₉	—	9	3	no	—
113	C₃×C₁₉	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
114	C₁₉⋊C₃	—	27	2	no	—
115	C₃³	—	3	1	no	—
116	C₃³	—	57	2	no	—
117	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
118	3₋¹⁺²	—	3	2	no	—
119	3₋¹⁺²	—	3	2	no	—
120	3₋¹⁺²	—	3	2	no	—
121	3₋¹⁺²	—	19	3	no	—
122	3₋¹⁺²	—	57	18	no	—
123	3₋¹⁺²	—	57	2	no	—
124	3₋¹⁺²	—	57	2	no	—
125	3₋¹⁺²	—	57	2	no	—
126	3₋¹⁺²	—	57	2	no	—
127	3₋¹⁺²	—	57	2	no	—
128	3₋¹⁺²	—	57	2	no	—
129	3₋¹⁺²	—	57	1	no	—
130	3₋¹⁺²	—	57	1	no	—
131	3₋¹⁺²	—	57	2	no	—
132	3₋¹⁺²	—	57	2	no	—
133	3₋¹⁺²	—	57	2	no	—
134	H(3,𝔽₃)	—	3	2	no	—
135	H(3,𝔽₃)	—	57	2	no	—
136	H(3,𝔽₃)	—	57	2	no	—
137	H(3,𝔽₃)	—	57	2	no	—
138	H(3,𝔽₃)	C₃²⋊(C₁₉⋊C₃)	1	3	yes	—
139	C₃×C₉	—	3	2	no	—
140	C₃×C₉	—	3	2	no	—
141	C₃×C₉	—	3	1	no	—
142	C₃×C₉	—	3	1	no	—
143	C₃×C₉	—	19	6	no	—
144	C₃×C₉	—	57	9	no	—
145	C₃×C₉	—	57	1	no	—
146	C₃×C₉	—	57	2	no	—
147	C₃×C₉	—	57	1	no	—
148	C₃×C₉	—	57	1	no	—
149	C₁₉	#10624751	1	1	yes	characteristic, complemented, Sylow
150	C₃²	—	3	3	no	—
151	C₃²	—	9	1	no	—
152	C₃²	—	27	1	no	—
153	C₃²	—	171	6	no	—
154	C₃²	—	171	2	no	—
155	C₃²	H(3,𝔽₃)×C₁₉⋊C₃	1	1	yes	characteristic
156	C₃²	H(3,𝔽₃)×C₁₉⋊C₃	1	2	yes	—
157	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
158	C₉	—	9	1	no	—
159	C₉	—	9	2	no	—
160	C₉	—	57	6	no	—
161	C₉	—	57	3	no	—
162	C₉	—	171	2	no	—
163	C₉	—	171	1	no	—
164	C₉	—	171	1	no	—
165	C₃	—	513	2	no	—
166	C₃	—	3	1	no	—
167	C₃	—	3	2	no	—
168	C₃	—	3	1	no	—
169	C₃	—	9	3	no	—
170	C₃	(C₃×C₁₉⋊C₃)⋊H(3,𝔽₃)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17987547

C₃

yes

maximal, characteristic, complemented

#17987518

C₃

yes

maximal, complemented

#17987514

C₃

yes

maximal, characteristic

#17987511

C₃

yes

maximal, complemented

#17987510

C₃

yes

maximal, complemented

#17987509

C₃

yes

maximal, characteristic, complemented

C₃×#13095322

C₃

yes

maximal, characteristic, complemented

#17987431

C₃

yes

maximal, complemented

C₃×C₁₉×#731

C₃

yes

maximal, characteristic, complemented

—

—

—

C₃²

yes

—

C₃×#10587025

C₃²

yes

characteristic

C₃×#10587025

C₃²

yes

characteristic

C₃×#10587025

C₃²

yes

characteristic

C₃×#10587025

C₃²

yes

—

C₃×C₃²⋊(C₁₉⋊C₃)

C₃²

yes

—

—

—

—

—

—

—

—

C₃×C₁₉×3₋¹⁺²

C₃²

yes

—

C₃×C₁₉×H(3,𝔽₃)

C₃²

yes

characteristic

C₃²×C₉×C₁₉

C₃²

yes

characteristic

C₁₉×#731

—

#10624751

—

maximal, Sylow

C₃³×C₁₉

—

C₃³×C₁₉

C₃³

yes

characteristic

C₃²×C₁₉⋊C₃

—

C₁₉×3₋¹⁺²

—

C₁₉×3₋¹⁺²

—

C₁₉×3₋¹⁺²

—

C₁₉×H(3,𝔽₃)

—

C₁₉×H(3,𝔽₃)

H(3,𝔽₃)

yes

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃²⋊(C₁₉⋊C₃)

—

C₃²⋊(C₁₉⋊C₃)

—

C₃²⋊(C₁₉⋊C₃)

—

—

—

—

—

—

—

—

—

—

—

—

—

yes

—

—

—

—

—

—

yes

—

3₋¹⁺⁴

—

—

—

—

—

—

—

C₁₉⋊C₃

yes

characteristic, complemented

#6893

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

—

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

—

—

—

—

—

—

—

—

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

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₃×C₁₉⋊C₃

yes

characteristic

100

C₃²×C₉

C₃×C₁₉⋊C₃

yes

characteristic

101

#733

—

102

#733

—

103

#732

—

104

#731

—

105

#731

—

106

#731

—

107

#731

—

108

C₃²⋊C₉

—

109

C₃×C₁₉

—

110

C₃×C₁₉

—

111

C₃×C₁₉

—

112

C₃×C₁₉

—

113

C₃×C₁₉

C₃²⋊H(3,𝔽₃)

yes

characteristic

114

C₁₉⋊C₃

—

115

C₃³

—

116

C₃³

—

117

C₃³

C₃²×C₁₉⋊C₃

yes

characteristic

118

3₋¹⁺²

—

119

3₋¹⁺²

—

120

3₋¹⁺²

—

121

3₋¹⁺²

—

122

3₋¹⁺²

—

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

H(3,𝔽₃)

—

135

H(3,𝔽₃)

—

136

H(3,𝔽₃)

—

137

H(3,𝔽₃)

—

138

H(3,𝔽₃)

C₃²⋊(C₁₉⋊C₃)

yes

—

139

C₃×C₉

—

140

C₃×C₉

—

141

C₃×C₉

—

142

C₃×C₉

—

143

C₃×C₉

—

144

C₃×C₉

—

145

C₃×C₉

—

146

C₃×C₉

—

147

C₃×C₉

—

148

C₃×C₉

—

149

C₁₉

#10624751

yes

characteristic, complemented, Sylow

150

C₃²

—

151

C₃²

—

152

C₃²

—

153

C₃²

—

171

—

154

C₃²

—

171

—

155

C₃²

H(3,𝔽₃)×C₁₉⋊C₃

yes

characteristic

156

C₃²

H(3,𝔽₃)×C₁₉⋊C₃

yes

—

157

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

yes

characteristic

158

C₉

—

159

C₉

—

160

C₉

—

161

C₉

—

162

C₉

—

171

—

163

C₉

—

171

—

164

C₉

—

171

—

165

C₃

—

513

—

166

C₃

—

167

C₃

—

168

C₃

—

169

C₃

—

170

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×#731	C₃
socle	C₃×C₁₉	C₃²⋊H(3,𝔽₃)
prime radical	#17987514	C₃
hypercenter	C₃×#731	C₁₉⋊C₃
hypocenter	C₁₉	#10624751
LCS	G ▷ C₃³×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃×#731

subgroup

G/H

center

C₃

(C₃×C₁₉⋊C₃)⋊H(3,𝔽₃)

commutator, cosocle

C₃³×C₁₉

C₃³

Frattini

C₃³

C₃²×C₁₉⋊C₃

Fitting

C₃×C₁₉×#731

C₃

socle

C₃×C₁₉

C₃²⋊H(3,𝔽₃)

C₃

C₁₉⋊C₃

C₁₉

G ▷ C₃³×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₃ ◁ C₃³ ◁ C₃×#731

N	H ≅ G/N	type
1. #17987547	C₃	semidirect
2. #17987518	C₃	semidirect
4. #17987511	C₃	semidirect
5. #17987510	C₃	semidirect
6. #17987509	C₃	semidirect
7. C₃×#13095322	C₃	semidirect
8. #17987431	C₃	semidirect
9. C₃×C₁₉×#731	C₃	semidirect
72. C₃×#731	C₁₉⋊C₃	semidirect
149. C₁₉	#10624751	semidirect

H ≅ G/N

type

1. #17987547

C₃

semidirect

2. #17987518

C₃

semidirect

4. #17987511

C₃

semidirect

5. #17987510

C₃

semidirect

6. #17987509

C₃

semidirect

7. C₃×#13095322

C₃

semidirect

8. #17987431

C₃

semidirect

9. C₃×C₁₉×#731

C₃

semidirect

72. C₃×#731

C₁₉⋊C₃

semidirect

149. C₁₉

#10624751

semidirect

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

subgroup	index
#17987547	3
#17987518	3
#17987514	3
#17987511	3
#17987510	3
#17987509	3
C₃×#13095322	3
#17987431	3
C₃×C₁₉×#731	3
#10624751	19

subgroup

index

C₃×C₁₉×#731

Subgroups — #28355017

Conjugacy classes of subgroups

Complements

Maximal subgroups