Group #28356645 — subgroups

Subgroups — #28356645

Order 13851.

948 conjugacy classes of subgroups; only the 286 normal subgroups are shown.

normal subgroups only

This group has 948 conjugacy classes of subgroups — too many to draw. Only its 286 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	#17987546	C₃	1	3	yes	maximal, complemented
2	#17987528	C₃	1	18	yes	maximal, complemented
3	#17987510	C₃	1	3	yes	maximal, complemented
4	#17987505	C₃	1	3	yes	maximal, complemented
5	#17987500	C₃	1	9	yes	maximal, complemented
6	C₃×#13095322	C₃	1	2	yes	maximal, complemented
7	C₃×#13095320	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃×C₁₉×#737	C₃	1	1	yes	maximal, characteristic, complemented
9	#13095333	—	3	18	no	—
10	#13095332	C₃²	1	27	yes	complemented
11	C₃×#10587025	C₃²	1	6	yes	—
12	3₋¹⁺²×C₁₉⋊C₃	—	3	9	no	—
13	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	3	yes	—
14	#13095322	C₃²	1	6	yes	complemented
15	#13095322	C₃²	1	12	yes	complemented
16	#13095320	C₃²	1	3	yes	—
17	#13095320	C₃²	1	6	yes	—
18	#13095317	C₃²	1	27	yes	complemented
19	#13095314	C₃²	1	27	yes	complemented
20	C₁₉×#737	—	3	3	no	—
21	C₁₉×#737	—	3	6	no	—
22	C₃×C₁₉×3₋¹⁺²	C₃²	1	1	yes	characteristic
23	C₃×C₁₉×3₋¹⁺²	C₃²	1	1	yes	characteristic
24	C₃×C₁₉×3₋¹⁺²	C₃²	1	6	yes	complemented
25	C₃×C₁₉×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
26	C₃²×C₉×C₁₉	C₃²	1	1	yes	characteristic
27	C₃²×C₉×C₁₉	C₃²	1	3	yes	complemented
28	#10624936	—	19	1	no	maximal, Sylow
29	C₃³×C₁₉	C₃³	1	3	yes	—
30	C₃³×C₁₉	C₃³	1	1	yes	characteristic
31	C₃²×C₁₉⋊C₃	—	3	3	no	—
32	C₁₉×3₋¹⁺²	—	3	2	no	—
33	C₁₉×3₋¹⁺²	—	3	1	no	—
34	C₁₉×3₋¹⁺²	—	3	18	no	—
35	C₁₉×3₋¹⁺²	H(3,𝔽₃)	1	3	yes	—
36	C₁₉×3₋¹⁺²	H(3,𝔽₃)	1	6	yes	—
37	C₁₉×H(3,𝔽₃)	—	3	2	no	—
38	C₁₉×H(3,𝔽₃)	—	3	1	no	—
39	C₃×C₉×C₁₉	—	3	2	no	—
40	C₃×C₉×C₁₉	—	3	1	no	—
41	C₃×C₉×C₁₉	—	3	9	no	—
42	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
43	C₃×C₉×C₁₉	C₃³	1	2	yes	—
44	C₃×C₉×C₁₉	C₃³	1	3	yes	—
45	C₃×C₉×C₁₉	C₃³	1	3	yes	—
46	C₃²⋊(C₁₉⋊C₃)	—	3	6	no	—
47	C₃²⋊(C₁₉⋊C₃)	C₃³	1	9	yes	—
48	#10587025	—	3	6	no	—
49	#10587025	—	3	6	no	—
50	#10587025	C₃³	1	18	yes	—
51	C₃×#4294	—	3	6	no	—
52	#10587023	—	3	27	no	—
53	#10587022	—	3	27	no	—
54	C₉×C₁₉⋊C₃	—	3	27	no	—
55	3₋¹⁺⁴	—	19	3	no	—
56	C₃×#737	C₁₉⋊C₃	1	1	yes	characteristic, complemented
57	#6913	—	19	18	no	—
58	#6910	—	19	3	no	—
59	#6909	—	19	3	no	—
60	#6908	—	19	9	no	—
61	C₃×#733	—	19	2	no	—
62	C₃×#732	—	19	1	no	—
63	C₃²×C₁₉	—	3	3	no	—
64	C₃²×C₁₉	—	9	1	no	—
65	C₃²×C₁₉	—	9	3	no	—
66	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	2	yes	—
67	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
68	C₃²×C₁₉	C₃⁴	1	1	yes	characteristic
69	C₃×C₁₉⋊C₃	—	3	9	no	—
70	C₃×C₁₉⋊C₃	—	27	3	no	—
71	C₉×C₁₉	—	3	1	no	—
72	C₉×C₁₉	—	3	2	no	—
73	C₉×C₁₉	—	3	3	no	—
74	C₉×C₁₉	C₃×H(3,𝔽₃)	1	9	yes	—
75	#4294	—	3	18	no	—
76	#737	—	3	3	no	—
77	#737	—	3	6	no	—
78	#737	—	19	27	no	—
79	#737	—	57	18	no	—
80	C₃×3₋¹⁺²	—	19	6	no	—
81	C₃×3₋¹⁺²	—	57	9	no	—
82	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	1	yes	characteristic
83	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	1	yes	characteristic
84	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	6	yes	complemented
85	C₃×H(3,𝔽₃)	—	19	3	no	—
86	C₃×H(3,𝔽₃)	C₃×C₁₉⋊C₃	1	1	yes	characteristic
87	C₃²×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
88	C₃²×C₉	C₃×C₁₉⋊C₃	1	3	yes	complemented
89	#733	—	19	12	no	—
90	#733	—	19	6	no	—
91	#732	—	19	6	no	—
92	#732	—	19	3	no	—
93	#731	—	19	27	no	—
94	AΣL(1,𝔽₂₇)	—	19	27	no	—
95	C₃×C₁₉	—	3	1	no	—
96	C₃×C₁₉	—	3	1	no	—
97	C₃×C₁₉	—	3	2	no	—
98	C₃×C₁₉	—	9	3	no	—
99	C₃×C₁₉	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
100	C₁₉⋊C₃	—	9	9	no	—
101	C₃³	—	57	3	no	—
102	C₃³	C₃²×C₁₉⋊C₃	1	3	yes	—
103	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
104	3₋¹⁺²	—	3	2	no	—
105	3₋¹⁺²	—	3	1	no	—
106	3₋¹⁺²	—	3	18	no	—
107	3₋¹⁺²	—	19	18	no	—
108	3₋¹⁺²	—	57	6	no	—
109	3₋¹⁺²	—	57	6	no	—
110	3₋¹⁺²	—	57	27	no	—
111	3₋¹⁺²	—	57	27	no	—
112	3₋¹⁺²	C₃²⋊(C₁₉⋊C₃)	1	3	yes	—
113	3₋¹⁺²	C₃²⋊(C₁₉⋊C₃)	1	6	yes	—
114	H(3,𝔽₃)	—	3	2	no	—
115	H(3,𝔽₃)	—	3	1	no	—
116	H(3,𝔽₃)	—	19	9	no	—
117	H(3,𝔽₃)	—	57	6	no	—
118	C₃×C₉	—	3	2	no	—
119	C₃×C₉	—	3	1	no	—
120	C₃×C₉	—	3	9	no	—
121	C₃×C₉	—	57	6	no	—
122	C₃×C₉	—	57	27	no	—
123	C₃×C₉	C₃²×C₁₉⋊C₃	1	3	yes	—
124	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
125	C₃×C₉	C₃²×C₁₉⋊C₃	1	2	yes	—
126	C₃×C₉	C₃²×C₁₉⋊C₃	1	3	yes	—
127	C₁₉	#10624936	1	1	yes	characteristic, complemented, Sylow
128	C₃²	—	513	3	no	—
129	C₃²	—	3	3	no	—
130	C₃²	—	9	1	no	—
131	C₃²	—	9	3	no	—
132	C₃²	—	57	9	no	—
133	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	2	yes	—
134	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
135	C₃²	C₃³×C₁₉⋊C₃	1	1	yes	characteristic
136	C₉	—	3	1	no	—
137	C₉	—	3	2	no	—
138	C₉	—	3	3	no	—
139	C₉	—	57	18	no	—
140	C₉	C₃×C₃²⋊(C₁₉⋊C₃)	1	9	yes	—
141	C₃	—	3	1	no	—
142	C₃	—	3	1	no	—
143	C₃	—	3	2	no	—
144	C₃	—	9	3	no	—
145	C₃	—	171	9	no	—
146	C₃	C₃²×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17987546

C₃

yes

maximal, complemented

#17987528

C₃

yes

maximal, complemented

#17987510

C₃

yes

maximal, complemented

#17987505

C₃

yes

maximal, complemented

#17987500

C₃

yes

maximal, complemented

C₃×#13095322

C₃

yes

maximal, complemented

C₃×#13095320

C₃

yes

maximal, characteristic, complemented

C₃×C₁₉×#737

C₃

yes

maximal, characteristic, complemented

#13095333

—

#13095332

C₃²

yes

complemented

C₃×#10587025

C₃²

yes

—

3₋¹⁺²×C₁₉⋊C₃

—

C₃×C₃²⋊(C₁₉⋊C₃)

C₃²

yes

—

#13095322

C₃²

yes

complemented

#13095322

C₃²

yes

complemented

#13095320

C₃²

yes

—

#13095320

C₃²

yes

—

#13095317

C₃²

yes

complemented

#13095314

C₃²

yes

complemented

C₁₉×#737

—

C₁₉×#737

—

C₃×C₁₉×3₋¹⁺²

C₃²

yes

characteristic

C₃×C₁₉×3₋¹⁺²

C₃²

yes

characteristic

C₃×C₁₉×3₋¹⁺²

C₃²

yes

complemented

C₃×C₁₉×H(3,𝔽₃)

C₃²

yes

characteristic

C₃²×C₉×C₁₉

C₃²

yes

characteristic

C₃²×C₉×C₁₉

C₃²

yes

complemented

#10624936

—

maximal, Sylow

C₃³×C₁₉

C₃³

yes

—

C₃³×C₁₉

C₃³

yes

characteristic

C₃²×C₁₉⋊C₃

—

C₁₉×3₋¹⁺²

—

C₁₉×3₋¹⁺²

—

C₁₉×3₋¹⁺²

—

C₁₉×3₋¹⁺²

H(3,𝔽₃)

yes

—

C₁₉×3₋¹⁺²

H(3,𝔽₃)

yes

—

C₁₉×H(3,𝔽₃)

—

C₁₉×H(3,𝔽₃)

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

C₃³

yes

characteristic

C₃×C₉×C₁₉

C₃³

yes

—

C₃×C₉×C₁₉

C₃³

yes

—

C₃×C₉×C₁₉

C₃³

yes

—

C₃²⋊(C₁₉⋊C₃)

—

C₃²⋊(C₁₉⋊C₃)

C₃³

yes

—

#10587025

—

#10587025

—

#10587025

C₃³

yes

—

C₃×#4294

—

#10587023

—

#10587022

—

C₉×C₁₉⋊C₃

—

3₋¹⁺⁴

—

C₃×#737

C₁₉⋊C₃

yes

characteristic, complemented

—

—

—

—

—

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

—

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₁₉

C₃⁴

yes

characteristic

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₉×C₁₉

—

C₉×C₁₉

—

C₉×C₁₉

—

C₉×C₁₉

C₃×H(3,𝔽₃)

yes

—

—

—

—

—

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃×C₁₉⋊C₃

yes

characteristic

C₃×3₋¹⁺²

C₃×C₁₉⋊C₃

yes

characteristic

C₃×3₋¹⁺²

C₃×C₁₉⋊C₃

yes

complemented

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₃×C₁₉⋊C₃

yes

characteristic

C₃²×C₉

C₃×C₁₉⋊C₃

yes

characteristic

C₃²×C₉

C₃×C₁₉⋊C₃

yes

—

—

—

—

—

AΣL(1,𝔽₂₇)

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

C₃²×H(3,𝔽₃)

yes

characteristic

100

C₁₉⋊C₃

—

101

C₃³

—

102

C₃³

C₃²×C₁₉⋊C₃

yes

—

103

C₃³

C₃²×C₁₉⋊C₃

yes

characteristic

104

3₋¹⁺²

—

105

3₋¹⁺²

—

106

3₋¹⁺²

—

107

3₋¹⁺²

—

108

3₋¹⁺²

—

109

3₋¹⁺²

—

110

3₋¹⁺²

—

111

3₋¹⁺²

—

112

3₋¹⁺²

C₃²⋊(C₁₉⋊C₃)

yes

—

113

3₋¹⁺²

C₃²⋊(C₁₉⋊C₃)

yes

—

114

H(3,𝔽₃)

—

115

H(3,𝔽₃)

—

116

H(3,𝔽₃)

—

117

H(3,𝔽₃)

—

118

C₃×C₉

—

119

C₃×C₉

—

120

C₃×C₉

—

121

C₃×C₉

—

122

C₃×C₉

—

123

C₃×C₉

C₃²×C₁₉⋊C₃

yes

—

124

C₃×C₉

C₃²×C₁₉⋊C₃

yes

characteristic

125

C₃×C₉

C₃²×C₁₉⋊C₃

yes

—

126

C₃×C₉

C₃²×C₁₉⋊C₃

yes

—

127

C₁₉

#10624936

yes

characteristic, complemented, Sylow

128

C₃²

—

513

—

129

C₃²

—

130

C₃²

—

131

C₃²

—

132

C₃²

—

133

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

yes

—

134

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

yes

characteristic

135

C₃²

C₃³×C₁₉⋊C₃

yes

characteristic

136

C₉

—

137

C₉

—

138

C₉

—

139

C₉

—

140

C₉

C₃×C₃²⋊(C₁₉⋊C₃)

yes

—

141

C₃

—

142

C₃

—

143

C₃

—

144

C₃

—

145

C₃

—

171

—

146

C₃

C₃²×C₃²⋊(C₁₉⋊C₃)

yes

characteristic

	subgroup	G/H
center	C₃	C₃²×C₃²⋊(C₁₉⋊C₃)
commutator, cosocle	C₃²×C₁₉	C₃⁴
Frattini	C₃²	C₃³×C₁₉⋊C₃
Fitting	C₃×C₁₉×#737	C₃
socle	C₃×C₁₉	C₃²×H(3,𝔽₃)
hypercenter	C₃×#737	C₁₉⋊C₃
hypocenter	C₁₉	#10624936
LCS	G ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃ ◁ C₃×3₋¹⁺² ◁ C₃×#737

subgroup

G/H

center

C₃

C₃²×C₃²⋊(C₁₉⋊C₃)

commutator, cosocle

C₃²×C₁₉

C₃⁴

Frattini

C₃²

C₃³×C₁₉⋊C₃

Fitting

C₃×C₁₉×#737

C₃

socle

C₃×C₁₉

C₃²×H(3,𝔽₃)

C₁₉⋊C₃

C₁₉

G ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₃ ◁ C₃×3₋¹⁺² ◁ C₃×#737

N	H ≅ G/N	type
1. #17987546	C₃	semidirect
2. #17987528	C₃	semidirect
3. #17987510	C₃	semidirect
4. #17987505	C₃	semidirect
5. #17987500	C₃	semidirect
6. C₃×#13095322	C₃	semidirect
7. C₃×#13095320	C₃	semidirect
8. C₃×C₁₉×#737	C₃	semidirect
10. #13095332	C₃²	semidirect
14. #13095322	C₃²	semidirect
15. #13095322	C₃²	semidirect
18. #13095317	C₃²	semidirect
19. #13095314	C₃²	semidirect
24. C₃×C₁₉×3₋¹⁺²	C₃²	semidirect
27. C₃²×C₉×C₁₉	C₃²	semidirect
56. C₃×#737	C₁₉⋊C₃	semidirect
84. C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	semidirect
88. C₃²×C₉	C₃×C₁₉⋊C₃	semidirect
127. C₁₉	#10624936	semidirect

H ≅ G/N

type

1. #17987546

C₃

semidirect

2. #17987528

C₃

semidirect

3. #17987510

C₃

semidirect

4. #17987505

C₃

semidirect

5. #17987500

C₃

semidirect

6. C₃×#13095322

C₃

semidirect

7. C₃×#13095320

C₃

semidirect

8. C₃×C₁₉×#737

C₃

semidirect

10. #13095332

C₃²

semidirect

14. #13095322

C₃²

semidirect

15. #13095322

C₃²

semidirect

18. #13095317

C₃²

semidirect

19. #13095314

C₃²

semidirect

24. C₃×C₁₉×3₋¹⁺²

C₃²

semidirect

27. C₃²×C₉×C₁₉

C₃²

semidirect

56. C₃×#737

C₁₉⋊C₃

semidirect

84. C₃×3₋¹⁺²

C₃×C₁₉⋊C₃

semidirect

88. C₃²×C₉

C₃×C₁₉⋊C₃

semidirect

127. C₁₉

#10624936

semidirect

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

subgroup	index
#17987546	3
#17987528	3
#17987510	3
#17987505	3
#17987500	3
C₃×#13095322	3
C₃×#13095320	3
C₃×C₁₉×#737	3
#10624936	19

subgroup

index

C₃×C₁₉×#737

Subgroups — #28356645

Conjugacy classes of subgroups

Complements

Maximal subgroups