Group #28356663 — subgroups

Subgroups — C₃×#17987545 #28356663

Order 13851.

5588 conjugacy classes of subgroups; only the 2884 normal subgroups are shown.

normal subgroups only

This group has 5588 conjugacy classes of subgroups — too many to draw. Only its 2884 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	#17987545	C₃	1	81	yes	maximal, direct product factor
2	C₃×#13095333	C₃	1	16	yes	maximal, complemented
3	C₃×#13095332	C₃	1	8	yes	maximal, complemented
4	C₃×#13095331	C₃	1	2	yes	maximal, complemented
5	C₃²×#10587022	C₃	1	4	yes	maximal, complemented
6	C₃×3₋¹⁺²×C₁₉⋊C₃	C₃	1	8	yes	maximal, complemented
7	C₃×H(3,𝔽₃)×C₁₉⋊C₃	C₃	1	1	yes	maximal, characteristic
8	C₃×C₁₉×#737	C₃	1	1	yes	maximal, characteristic, complemented
9	C₃³×C₁₉⋊C₃	C₃²	1	4	yes	—
10	#13095333	C₃²	1	432	yes	complemented
11	#13095332	C₃²	1	216	yes	complemented
12	#13095331	C₃²	1	54	yes	complemented
13	C₃×#10587025	C₃²	1	16	yes	complemented
14	C₃×#10587022	C₃²	1	8	yes	—
15	C₃×#10587022	C₃²	1	1	yes	characteristic
16	C₃×#10587022	C₃²	1	108	yes	complemented
17	C₃×#10587022	C₃²	1	8	yes	complemented
18	C₃×#10587022	C₃²	1	16	yes	complemented
19	C₃×#10587023	C₃²	1	24	yes	complemented
20	3₋¹⁺²×C₁₉⋊C₃	C₃²	1	216	yes	complemented
21	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	8	yes	—
22	H(3,𝔽₃)×C₁₉⋊C₃	C₃²	1	27	yes	—
23	C₃²×#4294	C₃²	1	8	yes	complemented
24	C₃×C₉×C₁₉⋊C₃	C₃²	1	24	yes	complemented
25	C₁₉×#737	C₃²	1	27	yes	complemented
26	C₃×C₁₉×3₋¹⁺²	C₃²	1	8	yes	complemented
27	C₃×C₁₉×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
28	C₃²×C₉×C₁₉	C₃²	1	4	yes	complemented
29	C₃×3₋¹⁺⁴	—	19	1	no	maximal, Sylow
30	C₃³×C₁₉	C₃³	1	4	yes	—
31	C₃²×C₁₉⋊C₃	—	9	12	no	—
32	C₃²×C₁₉⋊C₃	C₃³	1	8	yes	—
33	C₃²×C₁₉⋊C₃	C₃³	1	1	yes	characteristic
34	C₃²×C₁₉⋊C₃	C₃³	1	36	yes	—
35	C₁₉×3₋¹⁺²	C₃³	1	72	yes	complemented
36	C₁₉×H(3,𝔽₃)	C₃³	1	9	yes	—
37	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
38	C₃×C₉×C₁₉	C₃³	1	8	yes	—
39	C₃×C₉×C₁₉	C₃³	1	36	yes	complemented
40	C₃²⋊(C₁₉⋊C₃)	C₃³	1	72	yes	—
41	#10587025	C₃³	1	144	yes	complemented
42	C₃×#4294	C₃³	1	16	yes	—
43	C₃×#4294	C₃³	1	2	yes	—
44	C₃×#4294	C₃³	1	72	yes	complemented
45	#10587023	C₃³	1	216	yes	complemented
46	#10587022	C₃³	1	72	yes	—
47	#10587022	C₃³	1	9	yes	—
48	#10587022	C₃³	1	72	yes	complemented
49	#10587022	C₃³	1	144	yes	complemented
50	C₉×C₁₉⋊C₃	C₃³	1	216	yes	complemented
51	3₋¹⁺⁴	—	19	81	no	—
52	C₃×#737	—	19	16	no	—
53	C₃×#737	—	19	2	no	—
54	C₃×#737	—	19	8	no	—
55	C₃×#737	C₁₉⋊C₃	1	1	yes	characteristic, complemented
56	C₃²×3₋¹⁺²	—	19	8	no	—
57	C₃²×3₋¹⁺²	—	19	4	no	—
58	C₃²×H(3,𝔽₃)	—	19	1	no	—
59	C₃²×C₁₉	—	3	12	no	—
60	C₃²×C₁₉	C₃⁴	1	12	yes	—
61	C₃²×C₁₉	C₃⁴	1	1	yes	characteristic
62	C₃×C₁₉⋊C₃	—	3	3	no	—
63	C₃×C₁₉⋊C₃	—	3	24	no	—
64	C₃×C₁₉⋊C₃	—	9	36	no	—
65	C₃×C₁₉⋊C₃	C₃⁴	1	24	yes	—
66	C₃×C₁₉⋊C₃	C₃⁴	1	3	yes	—
67	C₉×C₁₉	C₃⁴	1	24	yes	—
68	C₉×C₁₉	C₃⁴	1	3	yes	—
69	#4294	C₃⁴	1	48	yes	—
70	#4294	C₃⁴	1	6	yes	—
71	C₃⁴	—	19	4	no	—
72	#737	—	19	432	no	—
73	#737	—	19	54	no	—
74	#737	—	19	216	no	—
75	#737	C₃×C₁₉⋊C₃	1	27	yes	complemented
76	C₃×3₋¹⁺²	—	19	16	no	—
77	C₃×3₋¹⁺²	—	19	216	no	—
78	C₃×3₋¹⁺²	—	19	8	no	—
79	C₃×3₋¹⁺²	—	19	1	no	—
80	C₃×3₋¹⁺²	—	19	24	no	—
81	C₃×3₋¹⁺²	—	19	16	no	—
82	C₃×3₋¹⁺²	—	19	8	no	—
83	C₃×3₋¹⁺²	—	19	108	no	—
84	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	8	yes	complemented
85	C₃×H(3,𝔽₃)	—	19	8	no	—
86	C₃×H(3,𝔽₃)	—	19	27	no	—
87	C₃×H(3,𝔽₃)	C₃×C₁₉⋊C₃	1	1	yes	characteristic
88	C₃²×C₉	—	19	8	no	—
89	C₃²×C₉	—	19	24	no	—
90	C₃²×C₉	C₃×C₁₉⋊C₃	1	4	yes	complemented
91	C₃×C₁₉	—	3	12	no	—
92	C₃×C₁₉	3₋¹⁺⁴	1	3	yes	complemented
93	C₃×C₁₉	C₃⁵	1	1	yes	characteristic
94	C₁₉⋊C₃	—	3	3	no	—
95	C₁₉⋊C₃	—	3	24	no	—
96	C₃³	—	19	8	no	—
97	C₃³	—	19	1	no	—
98	C₃³	—	19	36	no	—
99	C₃³	—	171	12	no	—
100	C₃³	C₃²×C₁₉⋊C₃	1	4	yes	—
101	3₋¹⁺²	—	19	144	no	—
102	3₋¹⁺²	—	19	72	no	—
103	3₋¹⁺²	—	19	9	no	—
104	3₋¹⁺²	—	19	144	no	—
105	3₋¹⁺²	—	19	72	no	—
106	3₋¹⁺²	—	19	216	no	—
107	3₋¹⁺²	C₃²×C₁₉⋊C₃	1	72	yes	complemented
108	H(3,𝔽₃)	—	19	72	no	—
109	H(3,𝔽₃)	C₃²×C₁₉⋊C₃	1	9	yes	—
110	C₃×C₉	—	19	72	no	—
111	C₃×C₉	—	19	16	no	—
112	C₃×C₉	—	19	2	no	—
113	C₃×C₉	—	19	216	no	—
114	C₃×C₉	C₃²×C₁₉⋊C₃	1	8	yes	—
115	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
116	C₃×C₉	C₃²×C₁₉⋊C₃	1	36	yes	complemented
117	C₁₉	C₃×3₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
118	C₃²	—	3	12	no	—
119	C₃²	—	19	24	no	—
120	C₃²	—	19	3	no	—
121	C₃²	—	57	3	no	—
122	C₃²	—	57	24	no	—
123	C₃²	—	171	36	no	—
124	C₃²	C₃³×C₁₉⋊C₃	1	12	yes	—
125	C₃²	C₃³×C₁₉⋊C₃	1	1	yes	characteristic
126	C₉	—	19	6	no	—
127	C₉	—	19	48	no	—
128	C₉	C₃³×C₁₉⋊C₃	1	24	yes	—
129	C₉	C₃³×C₁₉⋊C₃	1	3	yes	—
130	C₃	—	3	12	no	—
131	C₃	—	57	3	no	—
132	C₃	—	57	24	no	—
133	C₃	#17987545	1	3	yes	direct product factor
134	C₃	C₃⁴×C₁₉⋊C₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17987545

C₃

yes

maximal, direct product factor

C₃×#13095333

C₃

yes

maximal, complemented

C₃×#13095332

C₃

yes

maximal, complemented

C₃×#13095331

C₃

yes

maximal, complemented

C₃²×#10587022

C₃

yes

maximal, complemented

C₃×3₋¹⁺²×C₁₉⋊C₃

C₃

yes

maximal, complemented

C₃×H(3,𝔽₃)×C₁₉⋊C₃

C₃

yes

maximal, characteristic

C₃×C₁₉×#737

C₃

yes

maximal, characteristic, complemented

C₃³×C₁₉⋊C₃

C₃²

yes

—

#13095333

C₃²

432

yes

complemented

#13095332

C₃²

216

yes

complemented

#13095331

C₃²

yes

complemented

C₃×#10587025

C₃²

yes

complemented

C₃×#10587022

C₃²

yes

—

C₃×#10587022

C₃²

yes

characteristic

C₃×#10587022

C₃²

108

yes

complemented

C₃×#10587022

C₃²

yes

complemented

C₃×#10587022

C₃²

yes

complemented

C₃×#10587023

C₃²

yes

complemented

3₋¹⁺²×C₁₉⋊C₃

C₃²

216

yes

complemented

C₃×C₃²⋊(C₁₉⋊C₃)

C₃²

yes

—

H(3,𝔽₃)×C₁₉⋊C₃

C₃²

yes

—

C₃²×#4294

C₃²

yes

complemented

C₃×C₉×C₁₉⋊C₃

C₃²

yes

complemented

C₁₉×#737

C₃²

yes

complemented

C₃×C₁₉×3₋¹⁺²

C₃²

yes

complemented

C₃×C₁₉×H(3,𝔽₃)

C₃²

yes

characteristic

C₃²×C₉×C₁₉

C₃²

yes

complemented

C₃×3₋¹⁺⁴

—

maximal, Sylow

C₃³×C₁₉

C₃³

yes

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

C₃³

yes

—

C₃²×C₁₉⋊C₃

C₃³

yes

characteristic

C₃²×C₁₉⋊C₃

C₃³

yes

—

C₁₉×3₋¹⁺²

C₃³

yes

complemented

C₁₉×H(3,𝔽₃)

C₃³

yes

—

C₃×C₉×C₁₉

C₃³

yes

characteristic

C₃×C₉×C₁₉

C₃³

yes

—

C₃×C₉×C₁₉

C₃³

yes

complemented

C₃²⋊(C₁₉⋊C₃)

C₃³

yes

—

#10587025

C₃³

144

yes

complemented

C₃×#4294

C₃³

yes

—

C₃×#4294

C₃³

yes

—

C₃×#4294

C₃³

yes

complemented

#10587023

C₃³

216

yes

complemented

#10587022

C₃³

yes

—

#10587022

C₃³

yes

—

#10587022

C₃³

yes

complemented

#10587022

C₃³

144

yes

complemented

C₉×C₁₉⋊C₃

C₃³

216

yes

complemented

3₋¹⁺⁴

—

—

—

—

C₁₉⋊C₃

yes

characteristic, complemented

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

—

C₃²×H(3,𝔽₃)

—

C₃²×C₁₉

—

C₃²×C₁₉

C₃⁴

yes

—

C₃²×C₁₉

C₃⁴

yes

characteristic

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

C₃⁴

yes

—

C₃×C₁₉⋊C₃

C₃⁴

yes

—

C₉×C₁₉

C₃⁴

yes

—

C₉×C₁₉

C₃⁴

yes

—

#4294

C₃⁴

yes

—

#4294

C₃⁴

yes

—

C₃⁴

—

#737

—

432

—

#737

—

#737

—

216

—

#737

C₃×C₁₉⋊C₃

yes

complemented

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

216

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

108

—

C₃×3₋¹⁺²

C₃×C₁₉⋊C₃

yes

complemented

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₃×C₁₉⋊C₃

yes

characteristic

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃×C₁₉⋊C₃

yes

complemented

C₃×C₁₉

—

C₃×C₁₉

3₋¹⁺⁴

yes

complemented

C₃×C₁₉

C₃⁵

yes

characteristic

C₁₉⋊C₃

—

C₁₉⋊C₃

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

171

—

100

C₃³

C₃²×C₁₉⋊C₃

yes

—

101

3₋¹⁺²

—

144

—

102

3₋¹⁺²

—

103

3₋¹⁺²

—

104

3₋¹⁺²

—

144

—

105

3₋¹⁺²

—

106

3₋¹⁺²

—

216

—

107

3₋¹⁺²

C₃²×C₁₉⋊C₃

yes

complemented

108

H(3,𝔽₃)

—

109

H(3,𝔽₃)

C₃²×C₁₉⋊C₃

yes

—

110

C₃×C₉

—

111

C₃×C₉

—

112

C₃×C₉

—

113

C₃×C₉

—

216

—

114

C₃×C₉

C₃²×C₁₉⋊C₃

yes

—

115

C₃×C₉

C₃²×C₁₉⋊C₃

yes

characteristic

116

C₃×C₉

C₃²×C₁₉⋊C₃

yes

complemented

117

C₁₉

C₃×3₋¹⁺⁴

yes

characteristic, complemented, Sylow

118

C₃²

—

119

C₃²

—

120

C₃²

—

121

C₃²

—

122

C₃²

—

123

C₃²

—

171

—

124

C₃²

C₃³×C₁₉⋊C₃

yes

—

125

C₃²

C₃³×C₁₉⋊C₃

yes

characteristic

126

C₉

—

127

C₉

—

128

C₉

C₃³×C₁₉⋊C₃

yes

—

129

C₉

C₃³×C₁₉⋊C₃

yes

—

130

C₃

—

131

C₃

—

132

C₃

—

133

C₃

#17987545

yes

direct product factor

134

C₃

C₃⁴×C₁₉⋊C₃

yes

characteristic

	subgroup	G/H
center	C₃²	C₃³×C₁₉⋊C₃
commutator, cosocle	C₃×C₁₉	C₃⁵
Frattini	C₃	C₃⁴×C₁₉⋊C₃
Fitting	C₃×C₁₉×#737	C₃
socle	C₃²×C₁₉	C₃⁴
prime radical	C₃×H(3,𝔽₃)×C₁₉⋊C₃	C₃
hypercenter	C₃×#737	C₁₉⋊C₃
hypocenter	C₁₉	C₃×3₋¹⁺⁴
LCS	G ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ C₃×#737

subgroup

G/H

center

C₃²

C₃³×C₁₉⋊C₃

commutator, cosocle

C₃×C₁₉

C₃⁵

Frattini

C₃

C₃⁴×C₁₉⋊C₃

Fitting

C₃×C₁₉×#737

C₃

socle

C₃²×C₁₉

C₃⁴

prime radical

C₃×H(3,𝔽₃)×C₁₉⋊C₃

C₃

hypercenter

C₃×#737

C₁₉⋊C₃

hypocenter

C₁₉

C₃×3₋¹⁺⁴

LCS

G ▷ C₃×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₃² ◁ C₃×#737

N	H ≅ G/N	type
1. #17987545	C₃	direct (N × H)
2. C₃×#13095333	C₃	semidirect
3. C₃×#13095332	C₃	semidirect
4. C₃×#13095331	C₃	semidirect
5. C₃²×#10587022	C₃	semidirect
6. C₃×3₋¹⁺²×C₁₉⋊C₃	C₃	semidirect
8. C₃×C₁₉×#737	C₃	semidirect
10. #13095333	C₃²	semidirect
11. #13095332	C₃²	semidirect
12. #13095331	C₃²	semidirect
13. C₃×#10587025	C₃²	semidirect
16. C₃×#10587022	C₃²	semidirect
17. C₃×#10587022	C₃²	semidirect
18. C₃×#10587022	C₃²	semidirect
19. C₃×#10587023	C₃²	semidirect
20. 3₋¹⁺²×C₁₉⋊C₃	C₃²	semidirect
23. C₃²×#4294	C₃²	semidirect
24. C₃×C₉×C₁₉⋊C₃	C₃²	semidirect
25. C₁₉×#737	C₃²	semidirect
26. C₃×C₁₉×3₋¹⁺²	C₃²	semidirect
28. C₃²×C₉×C₁₉	C₃²	semidirect
35. C₁₉×3₋¹⁺²	C₃³	semidirect
39. C₃×C₉×C₁₉	C₃³	semidirect
41. #10587025	C₃³	semidirect
44. C₃×#4294	C₃³	semidirect
45. #10587023	C₃³	semidirect
48. #10587022	C₃³	semidirect
49. #10587022	C₃³	semidirect
50. C₉×C₁₉⋊C₃	C₃³	semidirect
55. C₃×#737	C₁₉⋊C₃	semidirect
75. #737	C₃×C₁₉⋊C₃	semidirect
84. C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	semidirect
90. C₃²×C₉	C₃×C₁₉⋊C₃	semidirect
92. C₃×C₁₉	3₋¹⁺⁴	semidirect
107. 3₋¹⁺²	C₃²×C₁₉⋊C₃	semidirect
116. C₃×C₉	C₃²×C₁₉⋊C₃	semidirect
117. C₁₉	C₃×3₋¹⁺⁴	semidirect
133. C₃	#17987545	direct (N × H)

H ≅ G/N

type

1. #17987545

C₃

direct (N × H)

2. C₃×#13095333

C₃

semidirect

3. C₃×#13095332

C₃

semidirect

4. C₃×#13095331

C₃

semidirect

5. C₃²×#10587022

C₃

semidirect

6. C₃×3₋¹⁺²×C₁₉⋊C₃

C₃

semidirect

8. C₃×C₁₉×#737

C₃

semidirect

10. #13095333

C₃²

semidirect

11. #13095332

C₃²

semidirect

12. #13095331

C₃²

semidirect

13. C₃×#10587025

C₃²

semidirect

16. C₃×#10587022

C₃²

semidirect

17. C₃×#10587022

C₃²

semidirect

18. C₃×#10587022

C₃²

semidirect

19. C₃×#10587023

C₃²

semidirect

20. 3₋¹⁺²×C₁₉⋊C₃

C₃²

semidirect

23. C₃²×#4294

C₃²

semidirect

24. C₃×C₉×C₁₉⋊C₃

C₃²

semidirect

25. C₁₉×#737

C₃²

semidirect

26. C₃×C₁₉×3₋¹⁺²

C₃²

semidirect

28. C₃²×C₉×C₁₉

C₃²

semidirect

35. C₁₉×3₋¹⁺²

C₃³

semidirect

39. C₃×C₉×C₁₉

C₃³

semidirect

41. #10587025

C₃³

semidirect

44. C₃×#4294

C₃³

semidirect

45. #10587023

C₃³

semidirect

48. #10587022

C₃³

semidirect

49. #10587022

C₃³

semidirect

50. C₉×C₁₉⋊C₃

C₃³

semidirect

55. C₃×#737

C₁₉⋊C₃

semidirect

75. #737

C₃×C₁₉⋊C₃

semidirect

84. C₃×3₋¹⁺²

C₃×C₁₉⋊C₃

semidirect

90. C₃²×C₉

C₃×C₁₉⋊C₃

semidirect

92. C₃×C₁₉

3₋¹⁺⁴

semidirect

107. 3₋¹⁺²

C₃²×C₁₉⋊C₃

semidirect

116. C₃×C₉

C₃²×C₁₉⋊C₃

semidirect

117. C₁₉

C₃×3₋¹⁺⁴

semidirect

133. C₃

#17987545

direct (N × H)

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

subgroup	index
#17987545	3
C₃×#13095333	3
C₃×#13095332	3
C₃×#13095331	3
C₃²×#10587022	3
C₃×3₋¹⁺²×C₁₉⋊C₃	3
C₃×H(3,𝔽₃)×C₁₉⋊C₃	3
C₃×C₁₉×#737	3
C₃×3₋¹⁺⁴	19

subgroup

index

C₃×3₋¹⁺²×C₁₉⋊C₃

C₃×H(3,𝔽₃)×C₁₉⋊C₃

C₃×C₁₉×#737

C₃×3₋¹⁺⁴

Subgroups — C₃×#17987545 #28356663

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×#17987545 #28356663

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#17987545 #28356663