Group #28353851 — subgroups

Subgroups — #28353851

Order 13851.

494 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 55 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	C₃×3₋¹⁺²×C₁₉⋊C₃	C₃	1	2	yes	maximal, complemented
2	#17987419	C₃	1	1	yes	maximal, characteristic, complemented
3	(C₃×C₁₉⋊C₃)⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
4	#17987313	C₃	1	2	yes	maximal, complemented
5	#17987312	C₃	1	2	yes	maximal, complemented
6	#17987311	C₃	1	4	yes	maximal, complemented
7	C₁₉×#6857	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃³×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
9	C₃×#10587025	C₃²	1	4	yes	—
10	C₃×#10587022	—	3	2	no	—
11	C₃×#10587023	—	3	2	no	—
12	C₃×#10587023	—	3	1	no	—
13	3₋¹⁺²×C₁₉⋊C₃	—	3	18	no	—
14	C₃×C₃²⋊(C₁₉⋊C₃)	—	3	1	no	—
15	C₃×C₃²⋊(C₁₉⋊C₃)	—	3	2	no	—
16	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	2	yes	—
17	C₃×C₉×C₁₉⋊C₃	—	3	2	no	—
18	C₃²⋊#4294	C₃²	1	2	yes	—
19	#13095303	—	3	2	no	—
20	C₃×C₁₉×3₋¹⁺²	C₃²	1	2	yes	—
21	C₃×C₁₉×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
22	C₁₉×C₃²⋊C₉	C₃²	1	1	yes	characteristic
23	#10624608	—	19	1	no	maximal, Sylow
24	C₃³×C₁₉	—	3	1	no	—
25	C₃³×C₁₉	C₃³	1	1	yes	characteristic
26	C₃²×C₁₉⋊C₃	—	3	6	no	—
27	C₃²×C₁₉⋊C₃	—	3	2	no	—
28	C₃²×C₁₉⋊C₃	—	9	1	no	—
29	C₃²×C₁₉⋊C₃	—	9	1	no	—
30	C₃²×C₁₉⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
31	C₃²×C₁₉⋊C₃	H(3,𝔽₃)	1	2	yes	—
32	C₁₉×3₋¹⁺²	—	3	6	no	—
33	C₁₉×H(3,𝔽₃)	—	9	1	no	—
34	C₃×C₉×C₁₉	—	3	2	no	—
35	C₃×C₉×C₁₉	—	3	1	no	—
36	C₃²⋊(C₁₉⋊C₃)	—	9	3	no	—
37	C₃²⋊(C₁₉⋊C₃)	—	9	6	no	—
38	C₃²⋊(C₁₉⋊C₃)	—	9	2	no	—
39	#10587025	—	3	12	no	—
40	C₃×#4294	—	3	4	no	—
41	C₃×#4294	—	3	2	no	—
42	#10587023	—	3	18	no	—
43	#10587023	—	9	3	no	—
44	#10587022	—	3	18	no	—
45	C₉×C₁₉⋊C₃	—	3	18	no	—
46	C₃²×3₋¹⁺²	—	19	2	no	—
47	#6891	—	19	1	no	—
48	C₃²⋊H(3,𝔽₃)	—	19	1	no	—
49	#6857	—	19	4	no	—
50	#6857	—	19	2	no	—
51	#6857	—	19	2	no	—
52	#6857	C₁₉⋊C₃	1	1	yes	characteristic, complemented
53	C₃²×C₁₉	—	3	2	no	—
54	C₃²×C₁₉	—	3	1	no	—
55	C₃²×C₁₉	—	3	1	no	—
56	C₃²×C₁₉	—	9	2	no	—
57	C₃²×C₁₉	—	9	1	no	—
58	C₃²×C₁₉	—	9	1	no	—
59	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
60	C₃×C₁₉⋊C₃	—	3	2	no	—
61	C₃×C₁₉⋊C₃	—	3	2	no	—
62	C₃×C₁₉⋊C₃	—	3	2	no	—
63	C₃×C₁₉⋊C₃	—	3	2	no	—
64	C₃×C₁₉⋊C₃	—	3	1	no	—
65	C₃×C₁₉⋊C₃	—	3	1	no	—
66	C₃×C₁₉⋊C₃	—	9	9	no	—
67	C₃×C₁₉⋊C₃	—	9	4	no	—
68	C₃×C₁₉⋊C₃	—	9	2	no	—
69	C₃×C₁₉⋊C₃	—	9	2	no	—
70	C₃×C₁₉⋊C₃	AΣL(1,𝔽₂₇)	1	6	yes	—
71	C₉×C₁₉	—	3	6	no	—
72	C₉×C₁₉	—	9	1	no	—
73	#4294	—	3	12	no	—
74	#4294	—	9	2	no	—
75	C₃⁴	—	19	1	no	—
76	C₃×3₋¹⁺²	—	19	4	no	—
77	C₃×3₋¹⁺²	—	57	2	no	—
78	C₃×3₋¹⁺²	—	57	2	no	—
79	C₃×3₋¹⁺²	—	57	1	no	—
80	C₃×3₋¹⁺²	—	57	18	no	—
81	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	2	yes	—
82	C₃×H(3,𝔽₃)	—	19	2	no	—
83	C₃×H(3,𝔽₃)	—	57	2	no	—
84	C₃×H(3,𝔽₃)	—	57	1	no	—
85	C₃×H(3,𝔽₃)	C₃×C₁₉⋊C₃	1	1	yes	characteristic
86	C₃²×C₉	—	57	2	no	—
87	C₃²⋊C₉	—	19	2	no	—
88	C₃²⋊C₉	—	57	2	no	—
89	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
90	C₃×C₁₉	—	9	1	no	—
91	C₃×C₁₉	—	27	1	no	—
92	C₃×C₁₉	C₃×AΣL(1,𝔽₂₇)	1	2	yes	—
93	C₃×C₁₉	#6909	1	1	yes	characteristic
94	C₃×C₁₉	#6910	1	1	yes	characteristic
95	C₁₉⋊C₃	—	3	6	no	—
96	C₁₉⋊C₃	—	9	1	no	—
97	C₁₉⋊C₃	—	27	2	no	—
98	C₃³	—	3	1	no	—
99	C₃³	—	19	2	no	—
100	C₃³	—	19	1	no	—
101	C₃³	—	57	6	no	—
102	C₃³	—	57	2	no	—
103	C₃³	—	171	1	no	—
104	C₃³	—	171	1	no	—
105	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
106	3₋¹⁺²	—	3	6	no	—
107	3₋¹⁺²	—	57	18	no	—
108	3₋¹⁺²	—	57	18	no	—
109	3₋¹⁺²	—	57	12	no	—
110	3₋¹⁺²	—	171	3	no	—
111	H(3,𝔽₃)	—	9	1	no	—
112	H(3,𝔽₃)	—	171	2	no	—
113	H(3,𝔽₃)	—	171	6	no	—
114	H(3,𝔽₃)	—	171	3	no	—
115	C₃×C₉	—	3	2	no	—
116	C₃×C₉	—	3	1	no	—
117	C₃×C₉	—	57	18	no	—
118	C₃×C₉	—	57	4	no	—
119	C₃×C₉	—	57	2	no	—
120	C₁₉	#10624608	1	1	yes	characteristic, complemented, Sylow
121	C₃²	—	3	2	no	—
122	C₃²	—	3	1	no	—
123	C₃²	—	3	1	no	—
124	C₃²	—	9	2	no	—
125	C₃²	—	9	1	no	—
126	C₃²	—	9	1	no	—
127	C₃²	—	19	6	no	—
128	C₃²	—	57	2	no	—
129	C₃²	—	57	2	no	—
130	C₃²	—	57	2	no	—
131	C₃²	—	57	2	no	—
132	C₃²	—	57	1	no	—
133	C₃²	—	57	1	no	—
134	C₃²	—	171	4	no	—
135	C₃²	—	171	2	no	—
136	C₃²	—	171	2	no	—
137	C₃²	—	171	9	no	—
138	C₃²	H(3,𝔽₃)×C₁₉⋊C₃	1	1	yes	characteristic
139	C₉	—	3	6	no	—
140	C₉	—	9	1	no	—
141	C₉	—	57	12	no	—
142	C₉	—	171	2	no	—
143	C₃	—	513	2	no	—
144	C₃	—	9	1	no	—
145	C₃	—	27	1	no	—
146	C₃	—	57	6	no	—
147	C₃	—	171	1	no	—
148	C₃	C₁₉⋊C₃×AΣL(1,𝔽₂₇)	1	2	yes	—
149	C₃	#17987503	1	1	yes	characteristic
150	C₃	#17987507	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×3₋¹⁺²×C₁₉⋊C₃

C₃

yes

maximal, complemented

#17987419

C₃

yes

maximal, characteristic, complemented

(C₃×C₁₉⋊C₃)⋊H(3,𝔽₃)

C₃

yes

maximal, characteristic

#17987313

C₃

yes

maximal, complemented

#17987312

C₃

yes

maximal, complemented

#17987311

C₃

yes

maximal, complemented

C₁₉×#6857

C₃

yes

maximal, characteristic, complemented

C₃³×C₁₉⋊C₃

C₃²

yes

characteristic

C₃×#10587025

C₃²

yes

—

C₃×#10587022

—

C₃×#10587023

—

C₃×#10587023

—

3₋¹⁺²×C₁₉⋊C₃

—

C₃×C₃²⋊(C₁₉⋊C₃)

—

C₃×C₃²⋊(C₁₉⋊C₃)

—

C₃×C₃²⋊(C₁₉⋊C₃)

C₃²

yes

—

C₃×C₉×C₁₉⋊C₃

—

C₃²⋊#4294

C₃²

yes

—

#13095303

—

C₃×C₁₉×3₋¹⁺²

C₃²

yes

—

C₃×C₁₉×H(3,𝔽₃)

C₃²

yes

characteristic

C₁₉×C₃²⋊C₉

C₃²

yes

characteristic

#10624608

—

maximal, Sylow

C₃³×C₁₉

—

C₃³×C₁₉

C₃³

yes

characteristic

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

H(3,𝔽₃)

yes

characteristic

C₃²×C₁₉⋊C₃

H(3,𝔽₃)

yes

—

C₁₉×3₋¹⁺²

—

C₁₉×H(3,𝔽₃)

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃²⋊(C₁₉⋊C₃)

—

C₃²⋊(C₁₉⋊C₃)

—

C₃²⋊(C₁₉⋊C₃)

—

—

—

—

—

—

—

C₉×C₁₉⋊C₃

—

C₃²×3₋¹⁺²

—

#6891

—

C₃²⋊H(3,𝔽₃)

—

—

—

—

C₁₉⋊C₃

yes

characteristic, complemented

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

characteristic

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

AΣL(1,𝔽₂₇)

yes

—

C₉×C₁₉

—

C₉×C₁₉

—

#4294

—

#4294

—

C₃⁴

—

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×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₃×C₁₉⋊C₃

yes

characteristic

C₃²×C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×C₁₉⋊C₃

yes

characteristic

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

C₃×AΣL(1,𝔽₂₇)

yes

—

C₃×C₁₉

yes

C₃×C₁₉

yes

C₁₉⋊C₃

—

C₁₉⋊C₃

—

C₁₉⋊C₃

—

C₃³

—

C₃³

—

100

C₃³

—

101

C₃³

—

102

C₃³

—

103

C₃³

—

171

—

104

C₃³

—

171

—

105

C₃³

C₃²×C₁₉⋊C₃

yes

characteristic

106

3₋¹⁺²

—

107

3₋¹⁺²

—

108

3₋¹⁺²

—

109

3₋¹⁺²

—

110

3₋¹⁺²

—

171

—

111

H(3,𝔽₃)

—

112

H(3,𝔽₃)

—

171

—

113

H(3,𝔽₃)

—

171

—

114

H(3,𝔽₃)

—

171

—

115

C₃×C₉

—

116

C₃×C₉

—

117

C₃×C₉

—

118

C₃×C₉

—

119

C₃×C₉

—

120

C₁₉

#10624608

yes

characteristic, complemented, Sylow

121

C₃²

—

122

C₃²

—

123

C₃²

—

124

C₃²

—

125

C₃²

—

126

C₃²

—

127

C₃²

—

128

C₃²

—

129

C₃²

—

130

C₃²

—

131

C₃²

—

132

C₃²

—

133

C₃²

—

134

C₃²

—

171

—

135

C₃²

—

171

—

136

C₃²

—

171

—

137

C₃²

—

171

—

138

C₃²

H(3,𝔽₃)×C₁₉⋊C₃

yes

characteristic

139

C₉

—

140

C₉

—

141

C₉

—

142

C₉

—

171

—

143

C₃

—

513

—

144

C₃

—

145

C₃

—

146

C₃

—

147

C₃

—

171

—

148

C₃

C₁₉⋊C₃×AΣL(1,𝔽₂₇)

yes

—

149

C₃

#17987503

yes

characteristic

150

C₃

#17987507

yes

characteristic

	subgroup	G/H
center	C₃²	H(3,𝔽₃)×C₁₉⋊C₃
commutator, cosocle	C₃³×C₁₉	C₃³
Frattini	C₃³	C₃²×C₁₉⋊C₃
Fitting	C₁₉×#6857	C₃
socle	C₃²×C₁₉	C₃×H(3,𝔽₃)
prime radical	(C₃×C₁₉⋊C₃)⋊H(3,𝔽₃)	C₃
hypercenter	#6857	C₁₉⋊C₃
hypocenter	C₁₉	#10624608
LCS	G ▷ C₃³×C₁₉ ▷ C₃²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ #6857

subgroup

G/H

center

C₃²

H(3,𝔽₃)×C₁₉⋊C₃

commutator, cosocle

C₃³×C₁₉

C₃³

Frattini

C₃³

C₃²×C₁₉⋊C₃

Fitting

C₁₉×#6857

C₃

socle

C₃²×C₁₉

C₃×H(3,𝔽₃)

prime radical

(C₃×C₁₉⋊C₃)⋊H(3,𝔽₃)

C₃

C₁₉⋊C₃

C₁₉

G ▷ C₃³×C₁₉ ▷ C₃²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₃² ◁ C₃³ ◁ #6857

N	H ≅ G/N	type
1. C₃×3₋¹⁺²×C₁₉⋊C₃	C₃	semidirect
2. #17987419	C₃	semidirect
4. #17987313	C₃	semidirect
5. #17987312	C₃	semidirect
6. #17987311	C₃	semidirect
7. C₁₉×#6857	C₃	semidirect
52. #6857	C₁₉⋊C₃	semidirect
120. C₁₉	#10624608	semidirect

H ≅ G/N

type

1. C₃×3₋¹⁺²×C₁₉⋊C₃

C₃

semidirect

2. #17987419

C₃

semidirect

4. #17987313

C₃

semidirect

5. #17987312

C₃

semidirect

6. #17987311

C₃

semidirect

7. C₁₉×#6857

C₃

semidirect

52. #6857

C₁₉⋊C₃

semidirect

120. C₁₉

#10624608

semidirect

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

subgroup	index
C₃×3₋¹⁺²×C₁₉⋊C₃	3
#17987419	3
(C₃×C₁₉⋊C₃)⋊H(3,𝔽₃)	3
#17987313	3
#17987312	3
#17987311	3
C₁₉×#6857	3
#10624608	19

subgroup

index

C₃×3₋¹⁺²×C₁₉⋊C₃

(C₃×C₁₉⋊C₃)⋊H(3,𝔽₃)

C₁₉×#6857

Subgroups — #28353851

Conjugacy classes of subgroups

Complements

Maximal subgroups