Group #28356651 — subgroups

Subgroups — C₃²×3₋¹⁺²×C₁₉⋊C₃ #28356651

Order 13851.

7208 conjugacy classes of subgroups; only the 3082 normal subgroups are shown.

normal subgroups only

This group has 7208 conjugacy classes of subgroups — too many to draw. Only its 3082 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	C₃⁴×C₁₉⋊C₃	C₃	1	1	yes	maximal, characteristic
2	C₃²×#10587025	C₃	1	2	yes	maximal, complemented
3	C₃²×#10587022	C₃	1	3	yes	maximal, complemented
4	C₃²×#10587023	C₃	1	3	yes	maximal, complemented
5	C₃×3₋¹⁺²×C₁₉⋊C₃	C₃	1	108	yes	maximal, direct product factor
6	C₃²×C₉×C₁₉⋊C₃	C₃	1	3	yes	maximal, complemented
7	C₃²×C₁₉×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃³×C₁₉⋊C₃	—	3	27	no	—
9	C₃³×C₁₉⋊C₃	C₃²	1	36	yes	—
10	C₃³×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
11	C₃³×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
12	C₃³×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
13	C₃×#10587025	C₃²	1	72	yes	complemented
14	C₃×#10587022	C₃²	1	108	yes	complemented
15	C₃×#10587023	C₃²	1	108	yes	complemented
16	3₋¹⁺²×C₁₉⋊C₃	C₃²	1	729	yes	direct product factor
17	C₃²×#4294	C₃²	1	6	yes	complemented
18	C₃×C₉×C₁₉⋊C₃	C₃²	1	108	yes	complemented
19	C₃⁴×C₁₉	C₃²	1	1	yes	characteristic
20	C₃×C₁₉×3₋¹⁺²	C₃²	1	36	yes	complemented
21	C₃²×C₉×C₁₉	C₃²	1	3	yes	complemented
22	C₃³×3₋¹⁺²	—	19	1	no	maximal, Sylow
23	C₃³×C₁₉	—	3	9	no	—
24	C₃³×C₁₉	C₃³	1	12	yes	—
25	C₃³×C₁₉	C₃³	1	1	yes	characteristic
26	C₃²×C₁₉⋊C₃	—	3	9	no	—
27	C₃²×C₁₉⋊C₃	—	3	9	no	—
28	C₃²×C₁₉⋊C₃	—	3	324	no	—
29	C₃²×C₁₉⋊C₃	3₋¹⁺²	1	27	yes	direct product factor
30	C₃²×C₁₉⋊C₃	C₃³	1	81	yes	—
31	C₃²×C₁₉⋊C₃	C₃³	1	12	yes	—
32	C₃²×C₁₉⋊C₃	C₃³	1	12	yes	—
33	C₃²×C₁₉⋊C₃	C₃³	1	12	yes	—
34	C₁₉×3₋¹⁺²	C₃³	1	81	yes	complemented
35	C₃×C₉×C₁₉	C₃³	1	36	yes	complemented
36	#10587025	C₃³	1	162	yes	complemented
37	C₃×#4294	C₃³	1	72	yes	complemented
38	#10587023	C₃³	1	243	yes	complemented
39	#10587022	C₃³	1	243	yes	complemented
40	C₉×C₁₉⋊C₃	C₃³	1	243	yes	complemented
41	C₃⁵	—	19	1	no	—
42	C₃²×3₋¹⁺²	—	19	2	no	—
43	C₃²×3₋¹⁺²	—	19	108	no	—
44	C₃²×3₋¹⁺²	—	19	3	no	—
45	C₃²×3₋¹⁺²	—	19	3	no	—
46	C₃²×3₋¹⁺²	C₁₉⋊C₃	1	1	yes	characteristic, direct product factor
47	C₃³×C₉	—	19	3	no	—
48	C₃²×C₁₉	—	3	36	no	—
49	C₃²×C₁₉	C₃×3₋¹⁺²	1	9	yes	complemented
50	C₃²×C₁₉	C₃⁴	1	9	yes	—
51	C₃²×C₁₉	C₃⁴	1	4	yes	—
52	C₃×C₁₉⋊C₃	—	3	36	no	—
53	C₃×C₁₉⋊C₃	—	3	36	no	—
54	C₃×C₁₉⋊C₃	—	3	243	no	—
55	C₃×C₁₉⋊C₃	C₃×3₋¹⁺²	1	108	yes	direct product factor
56	C₃×C₁₉⋊C₃	C₃⁴	1	9	yes	—
57	C₃×C₁₉⋊C₃	C₃⁴	1	9	yes	—
58	C₃×C₁₉⋊C₃	C₃⁴	1	9	yes	—
59	C₉×C₁₉	C₃⁴	1	27	yes	complemented
60	#4294	C₃⁴	1	54	yes	complemented
61	C₃⁴	—	19	1	no	—
62	C₃⁴	—	19	36	no	—
63	C₃⁴	—	19	1	no	—
64	C₃⁴	—	19	1	no	—
65	C₃⁴	—	57	27	no	—
66	C₃⁴	C₃×C₁₉⋊C₃	1	1	yes	characteristic
67	C₃×3₋¹⁺²	—	19	72	no	—
68	C₃×3₋¹⁺²	—	19	729	no	—
69	C₃×3₋¹⁺²	—	19	108	no	—
70	C₃×3₋¹⁺²	—	19	108	no	—
71	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	36	yes	direct product factor
72	C₃²×C₉	—	19	6	no	—
73	C₃²×C₉	—	19	108	no	—
74	C₃²×C₉	C₃×C₁₉⋊C₃	1	3	yes	complemented
75	C₃×C₁₉	—	3	9	no	—
76	C₃×C₁₉	C₃²×3₋¹⁺²	1	12	yes	complemented
77	C₃×C₁₉	C₃⁵	1	1	yes	characteristic
78	C₁₉⋊C₃	—	3	9	no	—
79	C₁₉⋊C₃	—	3	9	no	—
80	C₁₉⋊C₃	C₃²×3₋¹⁺²	1	27	yes	direct product factor
81	C₃³	—	3	9	no	—
82	C₃³	—	19	27	no	—
83	C₃³	—	19	12	no	—
84	C₃³	—	19	81	no	—
85	C₃³	—	19	12	no	—
86	C₃³	—	19	12	no	—
87	C₃³	—	57	324	no	—
88	C₃³	—	57	9	no	—
89	C₃³	—	57	9	no	—
90	C₃³	C₃²×C₁₉⋊C₃	1	12	yes	—
91	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
92	3₋¹⁺²	—	19	162	no	—
93	3₋¹⁺²	—	19	243	no	—
94	3₋¹⁺²	—	19	243	no	—
95	3₋¹⁺²	C₃²×C₁₉⋊C₃	1	81	yes	direct product factor
96	C₃×C₉	—	19	72	no	—
97	C₃×C₉	—	19	243	no	—
98	C₃×C₉	C₃²×C₁₉⋊C₃	1	36	yes	complemented
99	C₁₉	C₃³×3₋¹⁺²	1	1	yes	characteristic, complemented, Sylow
100	C₃²	—	3	36	no	—
101	C₃²	—	19	108	no	—
102	C₃²	—	19	9	no	—
103	C₃²	—	19	9	no	—
104	C₃²	—	19	9	no	—
105	C₃²	—	57	243	no	—
106	C₃²	—	57	36	no	—
107	C₃²	—	57	36	no	—
108	C₃²	3₋¹⁺²×C₁₉⋊C₃	1	9	yes	direct product factor
109	C₃²	C₃³×C₁₉⋊C₃	1	9	yes	—
110	C₃²	C₃³×C₁₉⋊C₃	1	4	yes	—
111	C₉	—	19	54	no	—
112	C₉	C₃³×C₁₉⋊C₃	1	27	yes	complemented
113	C₃	—	3	9	no	—
114	C₃	—	19	27	no	—
115	C₃	—	57	9	no	—
116	C₃	—	57	9	no	—
117	C₃	C₃×3₋¹⁺²×C₁₉⋊C₃	1	12	yes	direct product factor
118	C₃	C₃⁴×C₁₉⋊C₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃⁴×C₁₉⋊C₃

C₃

yes

maximal, characteristic

C₃²×#10587025

C₃

yes

maximal, complemented

C₃²×#10587022

C₃

yes

maximal, complemented

C₃²×#10587023

C₃

yes

maximal, complemented

C₃×3₋¹⁺²×C₁₉⋊C₃

C₃

108

yes

maximal, direct product factor

C₃²×C₉×C₁₉⋊C₃

C₃

yes

maximal, complemented

C₃²×C₁₉×3₋¹⁺²

C₃

yes

maximal, characteristic, complemented

C₃³×C₁₉⋊C₃

—

C₃³×C₁₉⋊C₃

C₃²

yes

—

C₃³×C₁₉⋊C₃

C₃²

yes

characteristic

C₃³×C₁₉⋊C₃

C₃²

yes

characteristic

C₃³×C₁₉⋊C₃

C₃²

yes

characteristic

C₃×#10587025

C₃²

yes

complemented

C₃×#10587022

C₃²

108

yes

complemented

C₃×#10587023

C₃²

108

yes

complemented

3₋¹⁺²×C₁₉⋊C₃

C₃²

729

yes

direct product factor

C₃²×#4294

C₃²

yes

complemented

C₃×C₉×C₁₉⋊C₃

C₃²

108

yes

complemented

C₃⁴×C₁₉

C₃²

yes

characteristic

C₃×C₁₉×3₋¹⁺²

C₃²

yes

complemented

C₃²×C₉×C₁₉

C₃²

yes

complemented

C₃³×3₋¹⁺²

—

maximal, Sylow

C₃³×C₁₉

—

C₃³×C₁₉

C₃³

yes

—

C₃³×C₁₉

C₃³

yes

characteristic

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

324

—

C₃²×C₁₉⋊C₃

3₋¹⁺²

yes

direct product factor

C₃²×C₁₉⋊C₃

C₃³

yes

—

C₃²×C₁₉⋊C₃

C₃³

yes

—

C₃²×C₁₉⋊C₃

C₃³

yes

—

C₃²×C₁₉⋊C₃

C₃³

yes

—

C₁₉×3₋¹⁺²

C₃³

yes

complemented

C₃×C₉×C₁₉

C₃³

yes

complemented

#10587025

C₃³

162

yes

complemented

C₃×#4294

C₃³

yes

complemented

#10587023

C₃³

243

yes

complemented

#10587022

C₃³

243

yes

complemented

C₉×C₁₉⋊C₃

C₃³

243

yes

complemented

C₃⁵

—

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

—

108

—

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

C₁₉⋊C₃

yes

characteristic, direct product factor

C₃³×C₉

—

C₃²×C₁₉

—

C₃²×C₁₉

C₃×3₋¹⁺²

yes

complemented

C₃²×C₁₉

C₃⁴

yes

—

C₃²×C₁₉

C₃⁴

yes

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

243

—

C₃×C₁₉⋊C₃

C₃×3₋¹⁺²

108

yes

direct product factor

C₃×C₁₉⋊C₃

C₃⁴

yes

—

C₃×C₁₉⋊C₃

C₃⁴

yes

—

C₃×C₁₉⋊C₃

C₃⁴

yes

—

C₉×C₁₉

C₃⁴

yes

complemented

#4294

C₃⁴

yes

complemented

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

C₃×C₁₉⋊C₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

729

—

C₃×3₋¹⁺²

—

108

—

C₃×3₋¹⁺²

—

108

—

C₃×3₋¹⁺²

C₃×C₁₉⋊C₃

yes

direct product factor

C₃²×C₉

—

C₃²×C₉

—

108

—

C₃²×C₉

C₃×C₁₉⋊C₃

yes

complemented

C₃×C₁₉

—

C₃×C₁₉

C₃²×3₋¹⁺²

yes

complemented

C₃×C₁₉

C₃⁵

yes

characteristic

C₁₉⋊C₃

—

C₁₉⋊C₃

—

C₁₉⋊C₃

C₃²×3₋¹⁺²

yes

direct product factor

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

324

—

C₃³

—

C₃³

—

C₃³

C₃²×C₁₉⋊C₃

yes

—

C₃³

C₃²×C₁₉⋊C₃

yes

characteristic

3₋¹⁺²

—

162

—

3₋¹⁺²

—

243

—

3₋¹⁺²

—

243

—

3₋¹⁺²

C₃²×C₁₉⋊C₃

yes

direct product factor

C₃×C₉

—

C₃×C₉

—

243

—

C₃×C₉

C₃²×C₁₉⋊C₃

yes

complemented

C₁₉

C₃³×3₋¹⁺²

yes

characteristic, complemented, Sylow

100

C₃²

—

101

C₃²

—

108

—

102

C₃²

—

103

C₃²

—

104

C₃²

—

105

C₃²

—

243

—

106

C₃²

—

107

C₃²

—

108

C₃²

3₋¹⁺²×C₁₉⋊C₃

yes

direct product factor

109

C₃²

C₃³×C₁₉⋊C₃

yes

—

110

C₃²

C₃³×C₁₉⋊C₃

yes

—

111

C₉

—

112

C₉

C₃³×C₁₉⋊C₃

yes

complemented

113

C₃

—

114

C₃

—

115

C₃

—

116

C₃

—

117

C₃

C₃×3₋¹⁺²×C₁₉⋊C₃

yes

direct product factor

118

C₃

C₃⁴×C₁₉⋊C₃

yes

characteristic

	subgroup	G/H
center	C₃³	C₃²×C₁₉⋊C₃
commutator, cosocle	C₃×C₁₉	C₃⁵
Frattini	C₃	C₃⁴×C₁₉⋊C₃
Fitting	C₃²×C₁₉×3₋¹⁺²	C₃
socle	C₃³×C₁₉	C₃³
prime radical	C₃⁴×C₁₉⋊C₃	C₃
hypercenter	C₃²×3₋¹⁺²	C₁₉⋊C₃
hypocenter	C₁₉	C₃³×3₋¹⁺²
LCS	G ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃³ ◁ C₃²×3₋¹⁺²

subgroup

G/H

center

C₃³

C₃²×C₁₉⋊C₃

commutator, cosocle

C₃×C₁₉

C₃⁵

Frattini

C₃

C₃⁴×C₁₉⋊C₃

Fitting

C₃²×C₁₉×3₋¹⁺²

C₃

socle

C₃³×C₁₉

C₃³

prime radical

C₃⁴×C₁₉⋊C₃

C₃

hypercenter

C₃²×3₋¹⁺²

C₁₉⋊C₃

hypocenter

C₁₉

C₃³×3₋¹⁺²

LCS

G ▷ C₃×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₃³ ◁ C₃²×3₋¹⁺²

N	H ≅ G/N	type
2. C₃²×#10587025	C₃	semidirect
3. C₃²×#10587022	C₃	semidirect
4. C₃²×#10587023	C₃	semidirect
5. C₃×3₋¹⁺²×C₁₉⋊C₃	C₃	direct (N × H)
6. C₃²×C₉×C₁₉⋊C₃	C₃	semidirect
7. C₃²×C₁₉×3₋¹⁺²	C₃	semidirect
13. C₃×#10587025	C₃²	semidirect
14. C₃×#10587022	C₃²	semidirect
15. C₃×#10587023	C₃²	semidirect
16. 3₋¹⁺²×C₁₉⋊C₃	C₃²	direct (N × H)
17. C₃²×#4294	C₃²	semidirect
18. C₃×C₉×C₁₉⋊C₃	C₃²	semidirect
20. C₃×C₁₉×3₋¹⁺²	C₃²	semidirect
21. C₃²×C₉×C₁₉	C₃²	semidirect
29. C₃²×C₁₉⋊C₃	3₋¹⁺²	direct (N × H)
34. C₁₉×3₋¹⁺²	C₃³	semidirect
35. C₃×C₉×C₁₉	C₃³	semidirect
36. #10587025	C₃³	semidirect
37. C₃×#4294	C₃³	semidirect
38. #10587023	C₃³	semidirect
39. #10587022	C₃³	semidirect
40. C₉×C₁₉⋊C₃	C₃³	semidirect
46. C₃²×3₋¹⁺²	C₁₉⋊C₃	direct (N × H)
49. C₃²×C₁₉	C₃×3₋¹⁺²	semidirect
55. C₃×C₁₉⋊C₃	C₃×3₋¹⁺²	direct (N × H)
59. C₉×C₁₉	C₃⁴	semidirect
60. #4294	C₃⁴	semidirect
71. C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	direct (N × H)
74. C₃²×C₉	C₃×C₁₉⋊C₃	semidirect
76. C₃×C₁₉	C₃²×3₋¹⁺²	semidirect
80. C₁₉⋊C₃	C₃²×3₋¹⁺²	direct (N × H)
95. 3₋¹⁺²	C₃²×C₁₉⋊C₃	direct (N × H)
98. C₃×C₉	C₃²×C₁₉⋊C₃	semidirect
99. C₁₉	C₃³×3₋¹⁺²	semidirect
108. C₃²	3₋¹⁺²×C₁₉⋊C₃	direct (N × H)
112. C₉	C₃³×C₁₉⋊C₃	semidirect
117. C₃	C₃×3₋¹⁺²×C₁₉⋊C₃	direct (N × H)

H ≅ G/N

type

2. C₃²×#10587025

C₃

semidirect

3. C₃²×#10587022

C₃

semidirect

4. C₃²×#10587023

C₃

semidirect

5. C₃×3₋¹⁺²×C₁₉⋊C₃

C₃

direct (N × H)

6. C₃²×C₉×C₁₉⋊C₃

C₃

semidirect

7. C₃²×C₁₉×3₋¹⁺²

C₃

semidirect

13. C₃×#10587025

C₃²

semidirect

14. C₃×#10587022

C₃²

semidirect

15. C₃×#10587023

C₃²

semidirect

16. 3₋¹⁺²×C₁₉⋊C₃

C₃²

direct (N × H)

17. C₃²×#4294

C₃²

semidirect

18. C₃×C₉×C₁₉⋊C₃

C₃²

semidirect

20. C₃×C₁₉×3₋¹⁺²

C₃²

semidirect

21. C₃²×C₉×C₁₉

C₃²

semidirect

29. C₃²×C₁₉⋊C₃

3₋¹⁺²

direct (N × H)

34. C₁₉×3₋¹⁺²

C₃³

semidirect

35. C₃×C₉×C₁₉

C₃³

semidirect

36. #10587025

C₃³

semidirect

37. C₃×#4294

C₃³

semidirect

38. #10587023

C₃³

semidirect

39. #10587022

C₃³

semidirect

40. C₉×C₁₉⋊C₃

C₃³

semidirect

46. C₃²×3₋¹⁺²

C₁₉⋊C₃

direct (N × H)

49. C₃²×C₁₉

C₃×3₋¹⁺²

semidirect

55. C₃×C₁₉⋊C₃

C₃×3₋¹⁺²

direct (N × H)

59. C₉×C₁₉

C₃⁴

semidirect

60. #4294

C₃⁴

semidirect

71. C₃×3₋¹⁺²

C₃×C₁₉⋊C₃

direct (N × H)

74. C₃²×C₉

C₃×C₁₉⋊C₃

semidirect

76. C₃×C₁₉

C₃²×3₋¹⁺²

semidirect

80. C₁₉⋊C₃

C₃²×3₋¹⁺²

direct (N × H)

95. 3₋¹⁺²

C₃²×C₁₉⋊C₃

direct (N × H)

98. C₃×C₉

C₃²×C₁₉⋊C₃

semidirect

99. C₁₉

C₃³×3₋¹⁺²

semidirect

108. C₃²

3₋¹⁺²×C₁₉⋊C₃

direct (N × H)

112. C₉

C₃³×C₁₉⋊C₃

semidirect

117. C₃

C₃×3₋¹⁺²×C₁₉⋊C₃

direct (N × H)

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

subgroup	index
C₃⁴×C₁₉⋊C₃	3
C₃²×#10587025	3
C₃²×#10587022	3
C₃²×#10587023	3
C₃×3₋¹⁺²×C₁₉⋊C₃	3
C₃²×C₉×C₁₉⋊C₃	3
C₃²×C₁₉×3₋¹⁺²	3
C₃³×3₋¹⁺²	19

subgroup

index

C₃⁴×C₁₉⋊C₃

C₃²×#10587025

C₃²×#10587022

C₃²×#10587023

C₃×3₋¹⁺²×C₁₉⋊C₃

C₃²×C₉×C₁₉⋊C₃

C₃²×C₁₉×3₋¹⁺²

C₃³×3₋¹⁺²

Subgroups — C₃²×3₋¹⁺²×C₁₉⋊C₃ #28356651

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C32×3−1+2×C19⋊C3 #28356651

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×3₋¹⁺²×C₁₉⋊C₃ #28356651