Group #44535392 — subgroups

Subgroups — C₃²×3₋¹⁺²×C₄₃⋊C₃ #44535392

Order 31347.

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×C₄₃⋊C₃	C₃	1	1	yes	maximal, characteristic
2	C₃²×#11923958	C₃	1	2	yes	maximal, complemented
3	C₃²×#11923957	C₃	1	3	yes	maximal, complemented
4	C₃²×#11923956	C₃	1	3	yes	maximal, complemented
5	C₃×3₋¹⁺²×C₄₃⋊C₃	C₃	1	108	yes	maximal, direct product factor
6	C₃²×C₉×C₄₃⋊C₃	C₃	1	3	yes	maximal, complemented
7	C₃²×3₋¹⁺²×C₄₃	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃³×C₄₃⋊C₃	—	3	27	no	—
9	C₃³×C₄₃⋊C₃	C₃²	1	36	yes	—
10	C₃³×C₄₃⋊C₃	C₃²	1	1	yes	characteristic
11	C₃³×C₄₃⋊C₃	C₃²	1	1	yes	characteristic
12	C₃³×C₄₃⋊C₃	C₃²	1	1	yes	characteristic
13	C₃×#11923958	C₃²	1	72	yes	complemented
14	C₃×#11923957	C₃²	1	108	yes	complemented
15	C₃×#11923956	C₃²	1	108	yes	complemented
16	3₋¹⁺²×C₄₃⋊C₃	C₃²	1	729	yes	direct product factor
17	C₃²×C₄₃⋊C₉	C₃²	1	6	yes	complemented
18	C₃×C₉×C₄₃⋊C₃	C₃²	1	108	yes	complemented
19	C₃⁴×C₄₃	C₃²	1	1	yes	characteristic
20	C₃×3₋¹⁺²×C₄₃	C₃²	1	36	yes	complemented
21	C₃²×C₉×C₄₃	C₃²	1	3	yes	complemented
22	C₃²×C₄₃⋊C₃	—	3	9	no	—
23	C₃²×C₄₃⋊C₃	—	3	9	no	—
24	C₃²×C₄₃⋊C₃	—	3	324	no	—
25	C₃²×C₄₃⋊C₃	3₋¹⁺²	1	27	yes	direct product factor
26	C₃²×C₄₃⋊C₃	C₃³	1	81	yes	—
27	C₃²×C₄₃⋊C₃	C₃³	1	12	yes	—
28	C₃²×C₄₃⋊C₃	C₃³	1	12	yes	—
29	C₃²×C₄₃⋊C₃	C₃³	1	12	yes	—
30	#11923958	C₃³	1	162	yes	complemented
31	#11923957	C₃³	1	243	yes	complemented
32	#11923956	C₃³	1	243	yes	complemented
33	C₃×C₄₃⋊C₉	C₃³	1	72	yes	complemented
34	C₉×C₄₃⋊C₃	C₃³	1	243	yes	complemented
35	C₃³×C₄₃	—	3	9	no	—
36	C₃³×C₄₃	C₃³	1	12	yes	—
37	C₃³×C₄₃	C₃³	1	1	yes	characteristic
38	3₋¹⁺²×C₄₃	C₃³	1	81	yes	complemented
39	C₃×C₉×C₄₃	C₃³	1	36	yes	complemented
40	C₃³×3₋¹⁺²	—	43	1	no	maximal, Sylow
41	C₃²×C₄₃	—	3	36	no	—
42	C₃²×C₄₃	C₃×3₋¹⁺²	1	9	yes	complemented
43	C₃²×C₄₃	C₃⁴	1	9	yes	—
44	C₃²×C₄₃	C₃⁴	1	4	yes	—
45	C₃×C₄₃⋊C₃	—	3	36	no	—
46	C₃×C₄₃⋊C₃	—	3	36	no	—
47	C₃×C₄₃⋊C₃	—	3	243	no	—
48	C₃×C₄₃⋊C₃	C₃×3₋¹⁺²	1	108	yes	direct product factor
49	C₃×C₄₃⋊C₃	C₃⁴	1	9	yes	—
50	C₃×C₄₃⋊C₃	C₃⁴	1	9	yes	—
51	C₃×C₄₃⋊C₃	C₃⁴	1	9	yes	—
52	C₉×C₄₃	C₃⁴	1	27	yes	complemented
53	C₄₃⋊C₉	C₃⁴	1	54	yes	complemented
54	C₃⁵	—	43	1	no	—
55	C₃²×3₋¹⁺²	—	43	2	no	—
56	C₃²×3₋¹⁺²	—	43	108	no	—
57	C₃²×3₋¹⁺²	—	43	3	no	—
58	C₃²×3₋¹⁺²	—	43	3	no	—
59	C₃²×3₋¹⁺²	C₄₃⋊C₃	1	1	yes	characteristic, direct product factor
60	C₃³×C₉	—	43	3	no	—
61	C₃×C₄₃	—	3	9	no	—
62	C₃×C₄₃	C₃²×3₋¹⁺²	1	12	yes	complemented
63	C₃×C₄₃	C₃⁵	1	1	yes	characteristic
64	C₄₃⋊C₃	—	3	9	no	—
65	C₄₃⋊C₃	—	3	9	no	—
66	C₄₃⋊C₃	C₃²×3₋¹⁺²	1	27	yes	direct product factor
67	C₃⁴	—	43	1	no	—
68	C₃⁴	—	43	36	no	—
69	C₃⁴	—	43	1	no	—
70	C₃⁴	—	43	1	no	—
71	C₃⁴	—	129	27	no	—
72	C₃⁴	C₃×C₄₃⋊C₃	1	1	yes	characteristic
73	C₃×3₋¹⁺²	—	43	72	no	—
74	C₃×3₋¹⁺²	—	43	729	no	—
75	C₃×3₋¹⁺²	—	43	108	no	—
76	C₃×3₋¹⁺²	—	43	108	no	—
77	C₃×3₋¹⁺²	C₃×C₄₃⋊C₃	1	36	yes	direct product factor
78	C₃²×C₉	—	43	6	no	—
79	C₃²×C₉	—	43	108	no	—
80	C₃²×C₉	C₃×C₄₃⋊C₃	1	3	yes	complemented
81	C₄₃	C₃³×3₋¹⁺²	1	1	yes	characteristic, complemented, Sylow
82	C₃³	—	3	9	no	—
83	C₃³	—	43	27	no	—
84	C₃³	—	43	12	no	—
85	C₃³	—	43	81	no	—
86	C₃³	—	43	12	no	—
87	C₃³	—	43	12	no	—
88	C₃³	—	129	324	no	—
89	C₃³	—	129	9	no	—
90	C₃³	—	129	9	no	—
91	C₃³	C₃²×C₄₃⋊C₃	1	12	yes	—
92	C₃³	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
93	3₋¹⁺²	—	43	162	no	—
94	3₋¹⁺²	—	43	243	no	—
95	3₋¹⁺²	—	43	243	no	—
96	3₋¹⁺²	C₃²×C₄₃⋊C₃	1	81	yes	direct product factor
97	C₃×C₉	—	43	72	no	—
98	C₃×C₉	—	43	243	no	—
99	C₃×C₉	C₃²×C₄₃⋊C₃	1	36	yes	complemented
100	C₃²	—	3	36	no	—
101	C₃²	—	43	108	no	—
102	C₃²	—	43	9	no	—
103	C₃²	—	43	9	no	—
104	C₃²	—	43	9	no	—
105	C₃²	—	129	243	no	—
106	C₃²	—	129	36	no	—
107	C₃²	—	129	36	no	—
108	C₃²	3₋¹⁺²×C₄₃⋊C₃	1	9	yes	direct product factor
109	C₃²	C₃³×C₄₃⋊C₃	1	9	yes	—
110	C₃²	C₃³×C₄₃⋊C₃	1	4	yes	—
111	C₉	—	43	54	no	—
112	C₉	C₃³×C₄₃⋊C₃	1	27	yes	complemented
113	C₃	—	3	9	no	—
114	C₃	—	43	27	no	—
115	C₃	—	129	9	no	—
116	C₃	—	129	9	no	—
117	C₃	C₃×3₋¹⁺²×C₄₃⋊C₃	1	12	yes	direct product factor
118	C₃	C₃⁴×C₄₃⋊C₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃⁴×C₄₃⋊C₃

C₃

yes

maximal, characteristic

C₃²×#11923958

C₃

yes

maximal, complemented

C₃²×#11923957

C₃

yes

maximal, complemented

C₃²×#11923956

C₃

yes

maximal, complemented

C₃×3₋¹⁺²×C₄₃⋊C₃

C₃

108

yes

maximal, direct product factor

C₃²×C₉×C₄₃⋊C₃

C₃

yes

maximal, complemented

C₃²×3₋¹⁺²×C₄₃

C₃

yes

maximal, characteristic, complemented

C₃³×C₄₃⋊C₃

—

C₃³×C₄₃⋊C₃

C₃²

yes

—

C₃³×C₄₃⋊C₃

C₃²

yes

characteristic

C₃³×C₄₃⋊C₃

C₃²

yes

characteristic

C₃³×C₄₃⋊C₃

C₃²

yes

characteristic

C₃×#11923958

C₃²

yes

complemented

C₃×#11923957

C₃²

108

yes

complemented

C₃×#11923956

C₃²

108

yes

complemented

3₋¹⁺²×C₄₃⋊C₃

C₃²

729

yes

direct product factor

C₃²×C₄₃⋊C₉

C₃²

yes

complemented

C₃×C₉×C₄₃⋊C₃

C₃²

108

yes

complemented

C₃⁴×C₄₃

C₃²

yes

characteristic

C₃×3₋¹⁺²×C₄₃

C₃²

yes

complemented

C₃²×C₉×C₄₃

C₃²

yes

complemented

C₃²×C₄₃⋊C₃

—

C₃²×C₄₃⋊C₃

—

C₃²×C₄₃⋊C₃

—

324

—

C₃²×C₄₃⋊C₃

3₋¹⁺²

yes

direct product factor

C₃²×C₄₃⋊C₃

C₃³

yes

—

C₃²×C₄₃⋊C₃

C₃³

yes

—

C₃²×C₄₃⋊C₃

C₃³

yes

—

C₃²×C₄₃⋊C₃

C₃³

yes

—

#11923958

C₃³

162

yes

complemented

#11923957

C₃³

243

yes

complemented

#11923956

C₃³

243

yes

complemented

C₃×C₄₃⋊C₉

C₃³

yes

complemented

C₉×C₄₃⋊C₃

C₃³

243

yes

complemented

C₃³×C₄₃

—

C₃³×C₄₃

C₃³

yes

—

C₃³×C₄₃

C₃³

yes

characteristic

3₋¹⁺²×C₄₃

C₃³

yes

complemented

C₃×C₉×C₄₃

C₃³

yes

complemented

C₃³×3₋¹⁺²

—

maximal, Sylow

C₃²×C₄₃

—

C₃²×C₄₃

C₃×3₋¹⁺²

yes

complemented

C₃²×C₄₃

C₃⁴

yes

—

C₃²×C₄₃

C₃⁴

yes

—

C₃×C₄₃⋊C₃

—

C₃×C₄₃⋊C₃

—

C₃×C₄₃⋊C₃

—

243

—

C₃×C₄₃⋊C₃

C₃×3₋¹⁺²

108

yes

direct product factor

C₃×C₄₃⋊C₃

C₃⁴

yes

—

C₃×C₄₃⋊C₃

C₃⁴

yes

—

C₃×C₄₃⋊C₃

C₃⁴

yes

—

C₉×C₄₃

C₃⁴

yes

complemented

C₄₃⋊C₉

C₃⁴

yes

complemented

C₃⁵

—

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

—

108

—

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

C₄₃⋊C₃

yes

characteristic, direct product factor

C₃³×C₉

—

C₃×C₄₃

—

C₃×C₄₃

C₃²×3₋¹⁺²

yes

complemented

C₃×C₄₃

C₃⁵

yes

characteristic

C₄₃⋊C₃

—

C₄₃⋊C₃

—

C₄₃⋊C₃

C₃²×3₋¹⁺²

yes

direct product factor

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

129

—

C₃⁴

C₃×C₄₃⋊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×C₄₃⋊C₃

yes

direct product factor

C₃²×C₉

—

C₃²×C₉

—

108

—

C₃²×C₉

C₃×C₄₃⋊C₃

yes

complemented

C₄₃

C₃³×3₋¹⁺²

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

129

324

—

C₃³

—

129

—

C₃³

—

129

—

C₃³

C₃²×C₄₃⋊C₃

yes

—

C₃³

C₃²×C₄₃⋊C₃

yes

characteristic

3₋¹⁺²

—

162

—

3₋¹⁺²

—

243

—

3₋¹⁺²

—

243

—

3₋¹⁺²

C₃²×C₄₃⋊C₃

yes

direct product factor

C₃×C₉

—

C₃×C₉

—

243

—

C₃×C₉

C₃²×C₄₃⋊C₃

yes

complemented

100

C₃²

—

101

C₃²

—

108

—

102

C₃²

—

103

C₃²

—

104

C₃²

—

105

C₃²

—

129

243

—

106

C₃²

—

129

—

107

C₃²

—

129

—

108

C₃²

3₋¹⁺²×C₄₃⋊C₃

yes

direct product factor

109

C₃²

C₃³×C₄₃⋊C₃

yes

—

110

C₃²

C₃³×C₄₃⋊C₃

yes

—

111

C₉

—

112

C₉

C₃³×C₄₃⋊C₃

yes

complemented

113

C₃

—

114

C₃

—

115

C₃

—

129

—

116

C₃

—

129

—

117

C₃

C₃×3₋¹⁺²×C₄₃⋊C₃

yes

direct product factor

118

C₃

C₃⁴×C₄₃⋊C₃

yes

characteristic

	subgroup	G/H
center	C₃³	C₃²×C₄₃⋊C₃
commutator, cosocle	C₃×C₄₃	C₃⁵
Frattini	C₃	C₃⁴×C₄₃⋊C₃
Fitting	C₃²×3₋¹⁺²×C₄₃	C₃
socle	C₃³×C₄₃	C₃³
prime radical	C₃⁴×C₄₃⋊C₃	C₃
hypercenter	C₃²×3₋¹⁺²	C₄₃⋊C₃
hypocenter	C₄₃	C₃³×3₋¹⁺²
LCS	G ▷ C₃×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₃³ ◁ C₃²×3₋¹⁺²

subgroup

G/H

center

C₃³

C₃²×C₄₃⋊C₃

commutator, cosocle

C₃×C₄₃

C₃⁵

Frattini

C₃

C₃⁴×C₄₃⋊C₃

Fitting

C₃²×3₋¹⁺²×C₄₃

C₃

socle

C₃³×C₄₃

C₃³

prime radical

C₃⁴×C₄₃⋊C₃

C₃

hypercenter

C₃²×3₋¹⁺²

C₄₃⋊C₃

hypocenter

C₄₃

C₃³×3₋¹⁺²

LCS

G ▷ C₃×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₃³ ◁ C₃²×3₋¹⁺²

N	H ≅ G/N	type
2. C₃²×#11923958	C₃	semidirect
3. C₃²×#11923957	C₃	semidirect
4. C₃²×#11923956	C₃	semidirect
5. C₃×3₋¹⁺²×C₄₃⋊C₃	C₃	direct (N × H)
6. C₃²×C₉×C₄₃⋊C₃	C₃	semidirect
7. C₃²×3₋¹⁺²×C₄₃	C₃	semidirect
13. C₃×#11923958	C₃²	semidirect
14. C₃×#11923957	C₃²	semidirect
15. C₃×#11923956	C₃²	semidirect
16. 3₋¹⁺²×C₄₃⋊C₃	C₃²	direct (N × H)
17. C₃²×C₄₃⋊C₉	C₃²	semidirect
18. C₃×C₉×C₄₃⋊C₃	C₃²	semidirect
20. C₃×3₋¹⁺²×C₄₃	C₃²	semidirect
21. C₃²×C₉×C₄₃	C₃²	semidirect
25. C₃²×C₄₃⋊C₃	3₋¹⁺²	direct (N × H)
30. #11923958	C₃³	semidirect
31. #11923957	C₃³	semidirect
32. #11923956	C₃³	semidirect
33. C₃×C₄₃⋊C₉	C₃³	semidirect
34. C₉×C₄₃⋊C₃	C₃³	semidirect
38. 3₋¹⁺²×C₄₃	C₃³	semidirect
39. C₃×C₉×C₄₃	C₃³	semidirect
42. C₃²×C₄₃	C₃×3₋¹⁺²	semidirect
48. C₃×C₄₃⋊C₃	C₃×3₋¹⁺²	direct (N × H)
52. C₉×C₄₃	C₃⁴	semidirect
53. C₄₃⋊C₉	C₃⁴	semidirect
59. C₃²×3₋¹⁺²	C₄₃⋊C₃	direct (N × H)
62. C₃×C₄₃	C₃²×3₋¹⁺²	semidirect
66. C₄₃⋊C₃	C₃²×3₋¹⁺²	direct (N × H)
77. C₃×3₋¹⁺²	C₃×C₄₃⋊C₃	direct (N × H)
80. C₃²×C₉	C₃×C₄₃⋊C₃	semidirect
81. C₄₃	C₃³×3₋¹⁺²	semidirect
96. 3₋¹⁺²	C₃²×C₄₃⋊C₃	direct (N × H)
99. C₃×C₉	C₃²×C₄₃⋊C₃	semidirect
108. C₃²	3₋¹⁺²×C₄₃⋊C₃	direct (N × H)
112. C₉	C₃³×C₄₃⋊C₃	semidirect
117. C₃	C₃×3₋¹⁺²×C₄₃⋊C₃	direct (N × H)

H ≅ G/N

type

2. C₃²×#11923958

C₃

semidirect

3. C₃²×#11923957

C₃

semidirect

4. C₃²×#11923956

C₃

semidirect

5. C₃×3₋¹⁺²×C₄₃⋊C₃

C₃

direct (N × H)

6. C₃²×C₉×C₄₃⋊C₃

C₃

semidirect

7. C₃²×3₋¹⁺²×C₄₃

C₃

semidirect

13. C₃×#11923958

C₃²

semidirect

14. C₃×#11923957

C₃²

semidirect

15. C₃×#11923956

C₃²

semidirect

16. 3₋¹⁺²×C₄₃⋊C₃

C₃²

direct (N × H)

17. C₃²×C₄₃⋊C₉

C₃²

semidirect

18. C₃×C₉×C₄₃⋊C₃

C₃²

semidirect

20. C₃×3₋¹⁺²×C₄₃

C₃²

semidirect

21. C₃²×C₉×C₄₃

C₃²

semidirect

25. C₃²×C₄₃⋊C₃

3₋¹⁺²

direct (N × H)

30. #11923958

C₃³

semidirect

31. #11923957

C₃³

semidirect

32. #11923956

C₃³

semidirect

33. C₃×C₄₃⋊C₉

C₃³

semidirect

34. C₉×C₄₃⋊C₃

C₃³

semidirect

38. 3₋¹⁺²×C₄₃

C₃³

semidirect

39. C₃×C₉×C₄₃

C₃³

semidirect

42. C₃²×C₄₃

C₃×3₋¹⁺²

semidirect

48. C₃×C₄₃⋊C₃

C₃×3₋¹⁺²

direct (N × H)

52. C₉×C₄₃

C₃⁴

semidirect

53. C₄₃⋊C₉

C₃⁴

semidirect

59. C₃²×3₋¹⁺²

C₄₃⋊C₃

direct (N × H)

62. C₃×C₄₃

C₃²×3₋¹⁺²

semidirect

66. C₄₃⋊C₃

C₃²×3₋¹⁺²

direct (N × H)

77. C₃×3₋¹⁺²

C₃×C₄₃⋊C₃

direct (N × H)

80. C₃²×C₉

C₃×C₄₃⋊C₃

semidirect

81. C₄₃

C₃³×3₋¹⁺²

semidirect

96. 3₋¹⁺²

C₃²×C₄₃⋊C₃

direct (N × H)

99. C₃×C₉

C₃²×C₄₃⋊C₃

semidirect

108. C₃²

3₋¹⁺²×C₄₃⋊C₃

direct (N × H)

112. C₉

C₃³×C₄₃⋊C₃

semidirect

117. C₃

C₃×3₋¹⁺²×C₄₃⋊C₃

direct (N × H)

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

subgroup	index
C₃⁴×C₄₃⋊C₃	3
C₃²×#11923958	3
C₃²×#11923957	3
C₃²×#11923956	3
C₃×3₋¹⁺²×C₄₃⋊C₃	3
C₃²×C₉×C₄₃⋊C₃	3
C₃²×3₋¹⁺²×C₄₃	3
C₃³×3₋¹⁺²	43

subgroup

index

C₃⁴×C₄₃⋊C₃

C₃²×#11923958

C₃²×#11923957

C₃²×#11923956

C₃×3₋¹⁺²×C₄₃⋊C₃

C₃²×C₉×C₄₃⋊C₃

C₃²×3₋¹⁺²×C₄₃

C₃³×3₋¹⁺²

Subgroups — C₃²×3₋¹⁺²×C₄₃⋊C₃ #44535392

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C32×3−1+2×C43⋊C3 #44535392

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×3₋¹⁺²×C₄₃⋊C₃ #44535392