Group #28356587 — subgroups

Subgroups — C₃×#17987529 #28356587

Order 13851.

992 conjugacy classes of subgroups; only the 274 normal subgroups are shown.

normal subgroups only

This group has 992 conjugacy classes of subgroups — too many to draw. Only its 274 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×#13095333	C₃	1	3	yes	maximal, complemented
2	C₃²×#10587022	C₃	1	1	yes	maximal, characteristic, complemented
3	#17987529	C₃	1	27	yes	maximal, direct product factor
4	C₃×#13095322	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃×#13095320	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃×#13095318	C₃	1	3	yes	maximal, complemented
7	C₃×#13095315	C₃	1	3	yes	maximal, complemented
8	C₃×C₁₉×#733	C₃	1	1	yes	maximal, characteristic, complemented
9	C₃³×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
10	#13095333	—	3	18	no	—
11	#13095333	C₃²	1	27	yes	complemented
12	C₃×#10587025	C₃²	1	3	yes	—
13	C₃×#10587022	—	3	3	no	—
14	C₃×#10587022	—	3	6	no	—
15	C₃×#10587022	—	3	3	no	—
16	C₃×#10587022	C₃²	1	9	yes	complemented
17	C₃×#10587023	—	3	3	no	—
18	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	3	yes	—
19	C₃²×#4294	C₃²	1	1	yes	characteristic
20	C₃²×#4294	C₃²	1	1	yes	characteristic
21	C₃×C₉×C₁₉⋊C₃	—	3	3	no	—
22	#13095322	C₃²	1	9	yes	complemented
23	#13095320	C₃²	1	9	yes	complemented
24	#13095318	C₃²	1	27	yes	complemented
25	#13095315	C₃²	1	27	yes	complemented
26	C₃×C₁₉×3₋¹⁺²	C₃²	1	3	yes	—
27	C₃²×C₉×C₁₉	C₃²	1	1	yes	characteristic
28	C₁₉×#733	C₃²	1	9	yes	complemented
29	C₃×#6913	—	19	1	no	maximal, Sylow
30	C₃³×C₁₉	C₃³	1	1	yes	characteristic
31	C₃²×C₁₉⋊C₃	—	3	3	no	—
32	C₃²×C₁₉⋊C₃	—	3	2	no	—
33	C₃²×C₁₉⋊C₃	—	3	1	no	—
34	C₃²×C₁₉⋊C₃	—	9	3	no	—
35	C₃²×C₁₉⋊C₃	C₃³	1	3	yes	—
36	C₁₉×3₋¹⁺²	—	3	6	no	—
37	C₁₉×3₋¹⁺²	C₃³	1	9	yes	—
38	C₃×C₉×C₁₉	—	3	2	no	—
39	C₃×C₉×C₁₉	—	3	1	no	—
40	C₃×C₉×C₁₉	—	3	3	no	—
41	C₃×C₉×C₁₉	C₃³	1	3	yes	—
42	C₃²⋊(C₁₉⋊C₃)	—	3	6	no	—
43	C₃²⋊(C₁₉⋊C₃)	C₃³	1	9	yes	—
44	#10587025	—	3	6	no	—
45	#10587025	C₃³	1	9	yes	—
46	C₃×#4294	—	3	3	no	—
47	C₃×#4294	—	3	2	no	—
48	C₃×#4294	—	3	1	no	—
49	C₃×#4294	H(3,𝔽₃)	1	3	yes	—
50	C₃×#4294	H(3,𝔽₃)	1	6	yes	—
51	C₃×#4294	C₃³	1	3	yes	—
52	C₃×#4294	C₃³	1	3	yes	—
53	#10587023	—	3	27	no	—
54	#10587022	—	3	27	no	—
55	#10587022	—	3	27	no	—
56	C₉×C₁₉⋊C₃	—	3	27	no	—
57	C₃×#737	—	19	3	no	—
58	C₃²×3₋¹⁺²	—	19	1	no	—
59	#6913	—	19	27	no	—
60	C₃×#733	—	19	1	no	—
61	C₃×#733	C₁₉⋊C₃	1	1	yes	characteristic, complemented
62	C₃×#732	—	19	1	no	—
63	C₃×#731	—	19	3	no	—
64	C₃×AΣL(1,𝔽₂₇)	—	19	3	no	—
65	C₃²×C₁₉	—	3	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	—	3	3	no	—
71	C₃×C₁₉⋊C₃	—	9	6	no	—
72	C₃×C₁₉⋊C₃	—	9	9	no	—
73	C₃×C₁₉⋊C₃	—	9	3	no	—
74	C₃×C₁₉⋊C₃	—	9	3	no	—
75	C₉×C₁₉	—	3	9	no	—
76	C₉×C₁₉	—	3	3	no	—
77	#4294	—	3	9	no	—
78	#4294	—	3	3	no	—
79	#4294	C₃×H(3,𝔽₃)	1	9	yes	—
80	C₃⁴	—	19	1	no	—
81	#737	—	19	27	no	—
82	#737	—	57	18	no	—
83	C₃×3₋¹⁺²	—	19	9	no	—
84	C₃×3₋¹⁺²	—	19	3	no	—
85	C₃×3₋¹⁺²	—	57	6	no	—
86	C₃×3₋¹⁺²	—	57	3	no	—
87	C₃×3₋¹⁺²	—	57	3	no	—
88	C₃×3₋¹⁺²	—	57	3	no	—
89	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	3	yes	—
90	C₃×H(3,𝔽₃)	—	19	3	no	—
91	C₃²×C₉	—	19	1	no	—
92	C₃²×C₉	—	19	1	no	—
93	C₃²×C₉	—	57	3	no	—
94	C₃²×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
95	#733	—	19	9	no	—
96	#733	C₃×C₁₉⋊C₃	1	9	yes	complemented
97	#732	—	19	9	no	—
98	#731	—	19	27	no	—
99	AΣL(1,𝔽₂₇)	—	19	27	no	—
100	C₃×C₁₉	—	3	2	no	—
101	C₃×C₁₉	—	3	1	no	—
102	C₃×C₁₉	#6913	1	3	yes	complemented
103	C₃×C₁₉	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
104	C₁₉⋊C₃	—	9	9	no	—
105	C₁₉⋊C₃	—	9	3	no	—
106	C₃³	—	19	3	no	—
107	C₃³	—	57	2	no	—
108	C₃³	—	57	3	no	—
109	C₃³	—	57	1	no	—
110	C₃³	—	171	3	no	—
111	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
112	3₋¹⁺²	—	3	6	no	—
113	3₋¹⁺²	—	19	9	no	—
114	3₋¹⁺²	—	57	27	no	—
115	3₋¹⁺²	—	57	27	no	—
116	3₋¹⁺²	—	57	27	no	—
117	3₋¹⁺²	—	57	6	no	—
118	3₋¹⁺²	C₃²×C₁₉⋊C₃	1	9	yes	—
119	H(3,𝔽₃)	—	19	9	no	—
120	H(3,𝔽₃)	—	57	6	no	—
121	C₃×C₉	—	3	2	no	—
122	C₃×C₉	—	3	3	no	—
123	C₃×C₉	—	3	1	no	—
124	C₃×C₉	—	19	6	no	—
125	C₃×C₉	—	19	3	no	—
126	C₃×C₉	—	19	3	no	—
127	C₃×C₉	—	19	3	no	—
128	C₃×C₉	—	57	27	no	—
129	C₃×C₉	—	57	2	no	—
130	C₃×C₉	—	57	1	no	—
131	C₃×C₉	—	57	3	no	—
132	C₃×C₉	C₃²×C₁₉⋊C₃	1	3	yes	—
133	C₁₉	C₃×#6913	1	1	yes	characteristic, complemented, Sylow
134	C₃²	—	3	3	no	—
135	C₃²	—	57	9	no	—
136	C₃²	—	57	3	no	—
137	C₃²	—	171	3	no	—
138	C₃²	—	171	6	no	—
139	C₃²	—	171	3	no	—
140	C₃²	—	171	9	no	—
141	C₃²	H(3,𝔽₃)×C₁₉⋊C₃	1	2	yes	—
142	C₃²	H(3,𝔽₃)×C₁₉⋊C₃	1	1	yes	characteristic
143	C₃²	C₃³×C₁₉⋊C₃	1	1	yes	characteristic
144	C₉	—	3	3	no	—
145	C₉	—	3	9	no	—
146	C₉	—	19	9	no	—
147	C₉	—	57	3	no	—
148	C₉	—	57	9	no	—
149	C₃	—	3	2	no	—
150	C₃	—	3	1	no	—
151	C₃	—	171	3	no	—
152	C₃	—	171	9	no	—
153	C₃	#17987529	1	3	yes	direct product factor
154	C₃	C₃×H(3,𝔽₃)×C₁₉⋊C₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#13095333

C₃

yes

maximal, complemented

C₃²×#10587022

C₃

yes

maximal, characteristic, complemented

#17987529

C₃

yes

maximal, direct product factor

C₃×#13095322

C₃

yes

maximal, characteristic, complemented

C₃×#13095320

C₃

yes

maximal, characteristic, complemented

C₃×#13095318

C₃

yes

maximal, complemented

C₃×#13095315

C₃

yes

maximal, complemented

C₃×C₁₉×#733

C₃

yes

maximal, characteristic, complemented

C₃³×C₁₉⋊C₃

C₃²

yes

characteristic

#13095333

—

#13095333

C₃²

yes

complemented

C₃×#10587025

C₃²

yes

—

—

—

—

C₃²

yes

complemented

C₃×#10587023

—

C₃×C₃²⋊(C₁₉⋊C₃)

C₃²

yes

—

C₃²×#4294

C₃²

yes

characteristic

C₃²×#4294

C₃²

yes

characteristic

C₃×C₉×C₁₉⋊C₃

—

#13095322

C₃²

yes

complemented

#13095320

C₃²

yes

complemented

#13095318

C₃²

yes

complemented

#13095315

C₃²

yes

complemented

C₃×C₁₉×3₋¹⁺²

C₃²

yes

—

C₃²×C₉×C₁₉

C₃²

yes

characteristic

C₁₉×#733

C₃²

yes

complemented

C₃×#6913

—

maximal, Sylow

C₃³×C₁₉

C₃³

yes

characteristic

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

C₃³

yes

—

C₁₉×3₋¹⁺²

—

C₁₉×3₋¹⁺²

C₃³

yes

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

C₃³

yes

—

C₃²⋊(C₁₉⋊C₃)

—

C₃²⋊(C₁₉⋊C₃)

C₃³

yes

—

#10587025

—

#10587025

C₃³

yes

—

—

—

—

yes

—

C₃×#4294

H(3,𝔽₃)

yes

—

C₃×#4294

C₃³

yes

—

C₃×#4294

C₃³

yes

—

#10587023

—

#10587022

—

#10587022

—

C₉×C₁₉⋊C₃

—

C₃×#737

—

C₃²×3₋¹⁺²

—

#6913

—

C₃×#733

—

C₃×#733

C₁₉⋊C₃

yes

characteristic, complemented

C₃×#732

—

C₃×#731

—

C₃×AΣL(1,𝔽₂₇)

—

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

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₉×C₁₉

—

C₉×C₁₉

—

#4294

—

#4294

—

#4294

C₃×H(3,𝔽₃)

yes

—

C₃⁴

—

#737

—

#737

—

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)

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃×C₁₉⋊C₃

yes

characteristic

#733

—

#733

C₃×C₁₉⋊C₃

yes

complemented

#732

—

#731

—

AΣL(1,𝔽₂₇)

—

100

C₃×C₁₉

—

101

C₃×C₁₉

—

102

C₃×C₁₉

#6913

yes

complemented

103

C₃×C₁₉

C₃²×H(3,𝔽₃)

yes

characteristic

104

C₁₉⋊C₃

—

105

C₁₉⋊C₃

—

106

C₃³

—

107

C₃³

—

108

C₃³

—

109

C₃³

—

110

C₃³

—

171

—

111

C₃³

C₃²×C₁₉⋊C₃

yes

characteristic

112

3₋¹⁺²

—

113

3₋¹⁺²

—

114

3₋¹⁺²

—

115

3₋¹⁺²

—

116

3₋¹⁺²

—

117

3₋¹⁺²

—

118

3₋¹⁺²

C₃²×C₁₉⋊C₃

yes

—

119

H(3,𝔽₃)

—

120

H(3,𝔽₃)

—

121

C₃×C₉

—

122

C₃×C₉

—

123

C₃×C₉

—

124

C₃×C₉

—

125

C₃×C₉

—

126

C₃×C₉

—

127

C₃×C₉

—

128

C₃×C₉

—

129

C₃×C₉

—

130

C₃×C₉

—

131

C₃×C₉

—

132

C₃×C₉

C₃²×C₁₉⋊C₃

yes

—

133

C₁₉

C₃×#6913

yes

characteristic, complemented, Sylow

134

C₃²

—

135

C₃²

—

136

C₃²

—

137

C₃²

—

171

—

138

C₃²

—

171

—

139

C₃²

—

171

—

140

C₃²

—

171

—

141

C₃²

H(3,𝔽₃)×C₁₉⋊C₃

yes

—

142

C₃²

H(3,𝔽₃)×C₁₉⋊C₃

yes

characteristic

143

C₃²

C₃³×C₁₉⋊C₃

yes

characteristic

144

C₉

—

145

C₉

—

146

C₉

—

147

C₉

—

148

C₉

—

149

C₃

—

150

C₃

—

151

C₃

—

171

—

152

C₃

—

171

—

153

C₃

#17987529

yes

direct product factor

154

C₃

C₃×H(3,𝔽₃)×C₁₉⋊C₃

yes

characteristic

	subgroup	G/H
center	C₃²	H(3,𝔽₃)×C₁₉⋊C₃
commutator, cosocle	C₃²×C₁₉	C₃⁴
Frattini	C₃²	C₃³×C₁₉⋊C₃
Fitting	C₃×C₁₉×#733	C₃
socle	C₃²×C₁₉	C₃×H(3,𝔽₃)
hypercenter	C₃×#733	C₁₉⋊C₃
hypocenter	C₁₉	C₃×#6913
LCS	G ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃×#733

subgroup

G/H

center

C₃²

H(3,𝔽₃)×C₁₉⋊C₃

commutator, cosocle

C₃²×C₁₉

C₃⁴

Frattini

C₃²

C₃³×C₁₉⋊C₃

Fitting

C₃×C₁₉×#733

C₃

socle

C₃²×C₁₉

C₃×H(3,𝔽₃)

C₁₉⋊C₃

C₁₉

G ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₃² ◁ C₃³ ◁ C₃×#733

N	H ≅ G/N	type
1. C₃×#13095333	C₃	semidirect
2. C₃²×#10587022	C₃	semidirect
3. #17987529	C₃	direct (N × H)
4. C₃×#13095322	C₃	semidirect
5. C₃×#13095320	C₃	semidirect
6. C₃×#13095318	C₃	semidirect
7. C₃×#13095315	C₃	semidirect
8. C₃×C₁₉×#733	C₃	semidirect
11. #13095333	C₃²	semidirect
16. C₃×#10587022	C₃²	semidirect
22. #13095322	C₃²	semidirect
23. #13095320	C₃²	semidirect
24. #13095318	C₃²	semidirect
25. #13095315	C₃²	semidirect
28. C₁₉×#733	C₃²	semidirect
61. C₃×#733	C₁₉⋊C₃	semidirect
96. #733	C₃×C₁₉⋊C₃	semidirect
102. C₃×C₁₉	#6913	semidirect
133. C₁₉	C₃×#6913	semidirect
153. C₃	#17987529	direct (N × H)

H ≅ G/N

type

1. C₃×#13095333

C₃

semidirect

2. C₃²×#10587022

C₃

semidirect

3. #17987529

C₃

direct (N × H)

4. C₃×#13095322

C₃

semidirect

5. C₃×#13095320

C₃

semidirect

6. C₃×#13095318

C₃

semidirect

7. C₃×#13095315

C₃

semidirect

8. C₃×C₁₉×#733

C₃

semidirect

11. #13095333

C₃²

semidirect

16. C₃×#10587022

C₃²

semidirect

22. #13095322

C₃²

semidirect

23. #13095320

C₃²

semidirect

24. #13095318

C₃²

semidirect

25. #13095315

C₃²

semidirect

28. C₁₉×#733

C₃²

semidirect

61. C₃×#733

C₁₉⋊C₃

semidirect

96. #733

C₃×C₁₉⋊C₃

semidirect

102. C₃×C₁₉

#6913

semidirect

133. C₁₉

C₃×#6913

semidirect

153. C₃

#17987529

direct (N × H)

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

subgroup	index
C₃×#13095333	3
C₃²×#10587022	3
#17987529	3
C₃×#13095322	3
C₃×#13095320	3
C₃×#13095318	3
C₃×#13095315	3
C₃×C₁₉×#733	3
C₃×#6913	19

subgroup

index

C₃×C₁₉×#733

Subgroups — C₃×#17987529 #28356587

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×#17987529 #28356587

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#17987529 #28356587