Group #28353737 — subgroups

Subgroups — #28353737

Order 13851.

560 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 64 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	C₃²×#10587025	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃²×C₃²⋊(C₁₉⋊C₃)	C₃	1	1	yes	maximal, characteristic
3	#17987421	C₃	1	1	yes	maximal, characteristic, complemented
4	#17987313	C₃	1	9	yes	maximal, complemented
5	C₁₉×#6891	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃³×C₁₉⋊C₃	—	3	1	no	—
7	C₃×#10587025	—	3	1	no	—
8	C₃×#10587025	—	3	1	no	—
9	C₃×#10587025	—	3	9	no	—
10	C₃×#10587025	C₃²	1	3	yes	complemented
11	C₃×#10587025	C₃²	1	3	yes	complemented
12	C₃×C₃²⋊(C₁₉⋊C₃)	—	3	1	no	—
13	C₃×C₃²⋊(C₁₉⋊C₃)	—	3	1	no	—
14	C₃×C₃²⋊(C₁₉⋊C₃)	—	3	9	no	—
15	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	3	yes	—
16	C₃²×#4294	—	3	1	no	—
17	C₃²⋊#4294	—	3	1	no	—
18	C₃²⋊#4294	—	3	1	no	—
19	C₃²⋊#4294	—	3	1	no	—
20	C₃⁴×C₁₉	C₃²	1	1	yes	characteristic
21	C₃×C₁₉×3₋¹⁺²	—	3	1	no	—
22	C₁₉×C₃²⋊C₉	—	3	1	no	—
23	C₁₉×C₃²⋊C₉	—	3	1	no	—
24	C₁₉×C₃²⋊C₉	C₃²	1	3	yes	complemented
25	#10624590	—	19	1	no	maximal, Sylow
26	C₃³×C₁₉	—	3	3	no	—
27	C₃³×C₁₉	—	3	3	no	—
28	C₃³×C₁₉	—	9	1	no	—
29	C₃³×C₁₉	—	9	1	no	—
30	C₃³×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
31	C₃³×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
32	C₃³×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
33	C₃³×C₁₉	C₃³	1	1	yes	characteristic
34	C₃²×C₁₉⋊C₃	—	3	9	no	—
35	C₃²×C₁₉⋊C₃	—	3	3	no	—
36	C₃²×C₁₉⋊C₃	—	9	3	no	—
37	C₃²×C₁₉⋊C₃	—	9	3	no	—
38	C₃²×C₁₉⋊C₃	—	9	3	no	—
39	C₁₉×3₋¹⁺²	—	9	3	no	—
40	C₃×C₉×C₁₉	—	3	3	no	—
41	C₃²⋊(C₁₉⋊C₃)	—	3	9	no	—
42	C₃²⋊(C₁₉⋊C₃)	—	3	9	no	—
43	C₃²⋊(C₁₉⋊C₃)	—	3	9	no	—
44	#10587025	—	3	9	no	—
45	#10587025	—	3	9	no	—
46	#10587025	—	3	9	no	—
47	#10587025	—	9	3	no	—
48	C₃×#4294	—	3	9	no	—
49	C₃×#4294	—	3	3	no	—
50	C₃×#4294	—	3	3	no	—
51	C₃²×3₋¹⁺²	—	19	1	no	—
52	C₃²×H(3,𝔽₃)	—	19	1	no	—
53	#6891	—	19	1	no	—
54	#6891	C₁₉⋊C₃	1	1	yes	characteristic, complemented
55	#6857	—	19	9	no	—
56	C₃²×C₁₉	—	3	1	no	—
57	C₃²×C₁₉	—	3	1	no	—
58	C₃²×C₁₉	—	3	1	no	—
59	C₃²×C₁₉	—	3	1	no	—
60	C₃²×C₁₉	—	3	1	no	—
61	C₃²×C₁₉	—	3	1	no	—
62	C₃²×C₁₉	—	3	1	no	—
63	C₃²×C₁₉	—	3	1	no	—
64	C₃²×C₁₉	—	3	1	no	—
65	C₃²×C₁₉	—	3	1	no	—
66	C₃²×C₁₉	—	3	1	no	—
67	C₃²×C₁₉	—	3	1	no	—
68	C₃²×C₁₉	—	3	1	no	—
69	C₃²×C₁₉	—	3	1	no	—
70	C₃²×C₁₉	—	9	9	no	—
71	C₃²×C₁₉	AΣL(1,𝔽₂₇)	1	3	yes	—
72	C₃²×C₁₉	AΣL(1,𝔽₂₇)	1	3	yes	—
73	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
74	C₃×C₁₉⋊C₃	—	3	9	no	—
75	C₃×C₁₉⋊C₃	—	9	27	no	—
76	C₃×C₁₉⋊C₃	—	9	3	no	—
77	C₃×C₁₉⋊C₃	—	9	3	no	—
78	C₃×C₁₉⋊C₃	—	9	3	no	—
79	C₉×C₁₉	—	9	3	no	—
80	#4294	—	3	9	no	—
81	#4294	—	9	3	no	—
82	C₃⁴	—	57	1	no	—
83	C₃⁴	C₃×C₁₉⋊C₃	1	1	yes	characteristic
84	C₃×3₋¹⁺²	—	3	1	no	—
85	C₃×3₋¹⁺²	—	19	3	no	—
86	C₃×3₋¹⁺²	—	19	3	no	—
87	C₃×3₋¹⁺²	—	57	1	no	—
88	C₃×3₋¹⁺²	—	57	1	no	—
89	C₃×3₋¹⁺²	—	57	9	no	—
90	C₃×H(3,𝔽₃)	—	19	3	no	—
91	C₃×H(3,𝔽₃)	—	57	1	no	—
92	C₃×H(3,𝔽₃)	—	57	1	no	—
93	C₃×H(3,𝔽₃)	—	57	9	no	—
94	C₃²×C₉	—	57	1	no	—
95	C₃²⋊C₉	—	3	1	no	—
96	C₃²⋊C₉	—	3	1	no	—
97	C₃²⋊C₉	—	57	1	no	—
98	C₃²⋊C₉	—	57	1	no	—
99	C₃²⋊C₉	—	57	1	no	—
100	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	3	yes	complemented
101	C₃×C₁₉	—	3	3	no	—
102	C₃×C₁₉	—	3	3	no	—
103	C₃×C₁₉	—	9	1	no	—
104	C₃×C₁₉	—	9	1	no	—
105	C₃×C₁₉	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
106	C₃×C₁₉	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
107	C₃×C₁₉	#6909	1	1	yes	characteristic
108	C₃×C₁₉	#6910	1	1	yes	characteristic
109	C₁₉⋊C₃	—	9	9	no	—
110	C₃³	—	3	3	no	—
111	C₃³	—	3	3	no	—
112	C₃³	—	9	1	no	—
113	C₃³	—	9	1	no	—
114	C₃³	—	57	9	no	—
115	C₃³	—	57	3	no	—
116	C₃³	—	171	3	no	—
117	C₃³	—	171	3	no	—
118	C₃³	—	171	3	no	—
119	C₃³	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
120	C₃³	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
121	C₃³	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
122	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
123	3₋¹⁺²	—	9	3	no	—
124	3₋¹⁺²	—	57	9	no	—
125	3₋¹⁺²	—	57	9	no	—
126	3₋¹⁺²	—	57	9	no	—
127	3₋¹⁺²	—	171	3	no	—
128	H(3,𝔽₃)	—	57	9	no	—
129	H(3,𝔽₃)	—	57	9	no	—
130	H(3,𝔽₃)	—	57	9	no	—
131	C₃×C₉	—	3	3	no	—
132	C₃×C₉	—	57	9	no	—
133	C₃×C₉	—	57	3	no	—
134	C₃×C₉	—	57	3	no	—
135	C₁₉	#10624590	1	1	yes	characteristic, complemented, Sylow
136	C₃²	—	3	1	no	—
137	C₃²	—	3	1	no	—
138	C₃²	—	3	1	no	—
139	C₃²	—	3	1	no	—
140	C₃²	—	3	1	no	—
141	C₃²	—	3	1	no	—
142	C₃²	—	3	1	no	—
143	C₃²	—	3	1	no	—
144	C₃²	—	3	1	no	—
145	C₃²	—	3	1	no	—
146	C₃²	—	3	1	no	—
147	C₃²	—	3	1	no	—
148	C₃²	—	3	1	no	—
149	C₃²	—	3	1	no	—
150	C₃²	—	9	9	no	—
151	C₃²	—	57	9	no	—
152	C₃²	—	171	27	no	—
153	C₃²	—	171	3	no	—
154	C₃²	—	171	3	no	—
155	C₃²	—	171	3	no	—
156	C₃²	#13095313	1	3	yes	—
157	C₃²	#13095315	1	3	yes	—
158	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
159	C₉	—	9	3	no	—
160	C₉	—	57	9	no	—
161	C₉	—	171	3	no	—
162	C₃	—	3	3	no	—
163	C₃	—	3	3	no	—
164	C₃	—	9	1	no	—
165	C₃	—	9	1	no	—
166	C₃	—	171	9	no	—
167	C₃	C₃×#13095313	1	1	yes	characteristic
168	C₃	C₃×#13095315	1	1	yes	characteristic
169	C₃	#17987506	1	1	yes	characteristic
170	C₃	#17987510	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#10587025

C₃

yes

maximal, characteristic, complemented

C₃²×C₃²⋊(C₁₉⋊C₃)

C₃

yes

maximal, characteristic

#17987421

C₃

yes

maximal, characteristic, complemented

#17987313

C₃

yes

maximal, complemented

C₁₉×#6891

C₃

yes

maximal, characteristic, complemented

C₃³×C₁₉⋊C₃

—

—

—

—

C₃²

yes

complemented

C₃×#10587025

C₃²

yes

complemented

C₃×C₃²⋊(C₁₉⋊C₃)

—

C₃×C₃²⋊(C₁₉⋊C₃)

—

C₃×C₃²⋊(C₁₉⋊C₃)

—

C₃×C₃²⋊(C₁₉⋊C₃)

C₃²

yes

—

C₃²×#4294

—

C₃²⋊#4294

—

C₃²⋊#4294

—

C₃²⋊#4294

—

C₃⁴×C₁₉

C₃²

yes

characteristic

C₃×C₁₉×3₋¹⁺²

—

C₁₉×C₃²⋊C₉

—

C₁₉×C₃²⋊C₉

—

C₁₉×C₃²⋊C₉

C₃²

yes

complemented

#10624590

—

maximal, Sylow

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

yes

C₃³×C₁₉

yes

C₃³×C₁₉

yes

C₃³×C₁₉

C₃³

yes

characteristic

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₁₉×3₋¹⁺²

—

C₃×C₉×C₁₉

—

C₃²⋊(C₁₉⋊C₃)

—

C₃²⋊(C₁₉⋊C₃)

—

C₃²⋊(C₁₉⋊C₃)

—

—

—

—

—

—

—

—

C₃²×3₋¹⁺²

—

C₃²×H(3,𝔽₃)

—

#6891

—

#6891

C₁₉⋊C₃

yes

characteristic, complemented

#6857

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

AΣL(1,𝔽₂₇)

yes

—

C₃²×C₁₉

AΣL(1,𝔽₂₇)

yes

—

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

characteristic

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₉×C₁₉

—

#4294

—

#4294

—

C₃⁴

—

C₃⁴

C₃×C₁₉⋊C₃

yes

characteristic

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×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃²×C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

100

C₃²⋊C₉

C₃×C₁₉⋊C₃

yes

complemented

101

C₃×C₁₉

—

102

C₃×C₁₉

—

103

C₃×C₁₉

—

104

C₃×C₁₉

—

105

C₃×C₁₉

C₃×AΣL(1,𝔽₂₇)

yes

characteristic

106

C₃×C₁₉

C₃×AΣL(1,𝔽₂₇)

yes

characteristic

107

C₃×C₁₉

#6909

yes

characteristic

108

C₃×C₁₉

#6910

yes

characteristic

109

C₁₉⋊C₃

—

110

C₃³

—

111

C₃³

—

112

C₃³

—

113

C₃³

—

114

C₃³

—

115

C₃³

—

116

C₃³

—

171

—

117

C₃³

—

171

—

118

C₃³

—

171

—

119

C₃³

C₃²⋊(C₁₉⋊C₃)

yes

characteristic

120

C₃³

C₃²⋊(C₁₉⋊C₃)

yes

characteristic

121

C₃³

C₃²⋊(C₁₉⋊C₃)

yes

characteristic

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

—

171

—

128

H(3,𝔽₃)

—

129

H(3,𝔽₃)

—

130

H(3,𝔽₃)

—

131

C₃×C₉

—

132

C₃×C₉

—

133

C₃×C₉

—

134

C₃×C₉

—

135

C₁₉

#10624590

yes

characteristic, complemented, Sylow

136

C₃²

—

137

C₃²

—

138

C₃²

—

139

C₃²

—

140

C₃²

—

141

C₃²

—

142

C₃²

—

143

C₃²

—

144

C₃²

—

145

C₃²

—

146

C₃²

—

147

C₃²

—

148

C₃²

—

149

C₃²

—

150

C₃²

—

151

C₃²

—

152

C₃²

—

171

—

153

C₃²

—

171

—

154

C₃²

—

171

—

155

C₃²

—

171

—

156

C₃²

#13095313

yes

—

157

C₃²

#13095315

yes

—

158

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

yes

characteristic

159

C₉

—

160

C₉

—

161

C₉

—

171

—

162

C₃

—

163

C₃

—

164

C₃

—

165

C₃

—

166

C₃

—

171

—

167

C₃

C₃×#13095313

yes

characteristic

168

C₃

C₃×#13095315

yes

characteristic

169

C₃

#17987506

yes

characteristic

170

C₃

#17987510

yes

characteristic

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)
commutator, cosocle	C₃³×C₁₉	C₃³
Frattini	C₃³	C₃²×C₁₉⋊C₃
Fitting	C₁₉×#6891	C₃
socle	C₃²×C₁₉	C₃×H(3,𝔽₃)
prime radical	C₃²×C₃²⋊(C₁₉⋊C₃)	C₃
hypercenter	#6891	C₁₉⋊C₃
hypocenter	C₁₉	#10624590
LCS	G ▷ C₃³×C₁₉ ▷ C₃²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ #6891

subgroup

G/H

center

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

commutator, cosocle

C₃³×C₁₉

C₃³

Frattini

C₃³

C₃²×C₁₉⋊C₃

Fitting

C₁₉×#6891

C₃

socle

C₃²×C₁₉

C₃×H(3,𝔽₃)

prime radical

C₃²×C₃²⋊(C₁₉⋊C₃)

C₃

C₁₉⋊C₃

C₁₉

G ▷ C₃³×C₁₉ ▷ C₃²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₃² ◁ C₃⁴ ◁ #6891

N	H ≅ G/N	type
1. C₃²×#10587025	C₃	semidirect
3. #17987421	C₃	semidirect
4. #17987313	C₃	semidirect
5. C₁₉×#6891	C₃	semidirect
10. C₃×#10587025	C₃²	semidirect
11. C₃×#10587025	C₃²	semidirect
24. C₁₉×C₃²⋊C₉	C₃²	semidirect
54. #6891	C₁₉⋊C₃	semidirect
100. C₃²⋊C₉	C₃×C₁₉⋊C₃	semidirect
135. C₁₉	#10624590	semidirect

H ≅ G/N

type

1. C₃²×#10587025

C₃

semidirect

3. #17987421

C₃

semidirect

4. #17987313

C₃

semidirect

5. C₁₉×#6891

C₃

semidirect

10. C₃×#10587025

C₃²

semidirect

11. C₃×#10587025

C₃²

semidirect

24. C₁₉×C₃²⋊C₉

C₃²

semidirect

54. #6891

C₁₉⋊C₃

semidirect

100. C₃²⋊C₉

C₃×C₁₉⋊C₃

semidirect

135. C₁₉

#10624590

semidirect

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

subgroup	index
C₃²×#10587025	3
C₃²×C₃²⋊(C₁₉⋊C₃)	3
#17987421	3
#17987313	3
C₁₉×#6891	3
#10624590	19

subgroup

index

C₃²×C₃²⋊(C₁₉⋊C₃)

C₁₉×#6891

Subgroups — #28353737

Conjugacy classes of subgroups

Complements

Maximal subgroups