Group #44534656 — subgroups

Subgroups — C₃×3₋¹⁺²×C₄₃⋊C₉ #44534656

Order 31347.

1148 conjugacy classes of subgroups; only the 670 normal subgroups are shown.

normal subgroups only

This group has 1148 conjugacy classes of subgroups — too many to draw. Only its 670 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	C₃³×C₄₃⋊C₉	C₃	1	1	yes	maximal, characteristic
2	3₋¹⁺²×C₄₃⋊C₉	C₃	1	27	yes	maximal, direct product factor
3	C₃×#16812500	C₃	1	3	yes	maximal, complemented
4	C₃×#16812499	C₃	1	3	yes	maximal, complemented
5	C₃×C₃²⋊(C₄₃⋊C₉)	C₃	1	2	yes	maximal
6	C₃×C₉×C₄₃⋊C₉	C₃	1	3	yes	maximal, complemented
7	C₃²×3₋¹⁺²×C₄₃	C₃	1	1	yes	maximal, characteristic
8	C₃²×C₄₃⋊C₉	—	3	9	no	—
9	C₃²×C₄₃⋊C₉	C₃²	1	6	yes	—
10	C₃²×C₄₃⋊C₉	C₃²	1	1	yes	characteristic
11	C₃²×C₄₃⋊C₉	C₃²	1	1	yes	characteristic
12	C₃²×C₄₃⋊C₉	C₃²	1	9	yes	—
13	C₃²×C₄₃⋊C₉	C₃²	1	1	yes	characteristic
14	#16812500	C₃²	1	27	yes	complemented
15	#16812499	C₃²	1	27	yes	complemented
16	C₃²⋊(C₄₃⋊C₉)	C₃²	1	18	yes	—
17	C₉×C₄₃⋊C₉	C₃²	1	27	yes	complemented
18	C₃⁴×C₄₃	C₃²	1	1	yes	characteristic
19	C₃×3₋¹⁺²×C₄₃	C₉	1	27	yes	complemented
20	C₃×3₋¹⁺²×C₄₃	C₃²	1	9	yes	—
21	C₃²×C₉×C₄₃	C₃²	1	3	yes	—
22	C₃×C₄₃⋊C₉	—	3	3	no	—
23	C₃×C₄₃⋊C₉	—	3	3	no	—
24	C₃×C₄₃⋊C₉	—	3	18	no	—
25	C₃×C₄₃⋊C₉	—	3	27	no	—
26	C₃×C₄₃⋊C₉	3₋¹⁺²	1	9	yes	direct product factor
27	C₃×C₄₃⋊C₉	C₃³	1	18	yes	—
28	C₃×C₄₃⋊C₉	C₃³	1	3	yes	—
29	C₃×C₄₃⋊C₉	C₃³	1	3	yes	—
30	C₃×C₄₃⋊C₉	C₃³	1	3	yes	—
31	C₃³×C₄₃	—	3	3	no	—
32	C₃³×C₄₃	—	3	6	no	—
33	C₃³×C₄₃	C₃×C₉	1	9	yes	—
34	C₃³×C₄₃	C₃³	1	3	yes	—
35	C₃³×C₄₃	C₃³	1	1	yes	characteristic
36	3₋¹⁺²×C₄₃	C₃×C₉	1	81	yes	complemented
37	C₃×C₉×C₄₃	C₃×C₉	1	27	yes	complemented
38	C₃×C₉×C₄₃	C₃³	1	9	yes	—
39	C₃×C₉×3₋¹⁺²	—	43	1	no	maximal, Sylow
40	C₃²×C₄₃	—	3	3	no	—
41	C₃²×C₄₃	—	3	6	no	—
42	C₃²×C₄₃	—	3	27	no	—
43	C₃²×C₄₃	C₃²×C₉	1	9	yes	—
44	C₃²×C₄₃	C₃²×C₉	1	3	yes	—
45	C₃²×C₄₃	C₃×3₋¹⁺²	1	3	yes	—
46	C₃²×C₄₃	#737	1	6	yes	—
47	C₃²×C₄₃	C₃⁴	1	1	yes	characteristic
48	C₉×C₄₃	C₃²×C₉	1	27	yes	complemented
49	C₄₃⋊C₉	—	3	3	no	—
50	C₄₃⋊C₉	—	3	3	no	—
51	C₄₃⋊C₉	—	3	18	no	—
52	C₄₃⋊C₉	C₃×3₋¹⁺²	1	9	yes	direct product factor
53	C₃²×3₋¹⁺²	C₄₃⋊C₃	1	1	yes	characteristic
54	C₃³×C₉	—	43	1	no	—
55	C₉×3₋¹⁺²	—	43	27	no	—
56	C₃×C₉⋊C₉	—	43	3	no	—
57	C₃×C₉⋊C₉	—	43	3	no	—
58	C₃×C₃²⋊C₉	—	43	2	no	—
59	C₃×C₉²	—	43	3	no	—
60	C₃×C₄₃	—	3	9	no	—
61	C₃×C₄₃	C₉×3₋¹⁺²	1	9	yes	complemented
62	C₃×C₄₃	C₃³×C₉	1	1	yes	characteristic
63	C₃×C₄₃	C₃²×3₋¹⁺²	1	1	yes	characteristic
64	C₃×C₄₃	C₃×#737	1	2	yes	—
65	C₃⁴	C₃×C₄₃⋊C₃	1	1	yes	characteristic
66	C₃×3₋¹⁺²	C₄₃⋊C₉	1	27	yes	direct product factor
67	C₃×3₋¹⁺²	C₃×C₄₃⋊C₃	1	9	yes	—
68	C₃²×C₉	—	43	1	no	—
69	C₃²×C₉	—	43	9	no	—
70	C₃²×C₉	—	43	1	no	—
71	C₃²×C₉	—	43	1	no	—
72	C₃²×C₉	—	43	6	no	—
73	C₃²×C₉	—	129	9	no	—
74	C₃²×C₉	C₃×C₄₃⋊C₃	1	3	yes	—
75	C₉⋊C₉	—	43	27	no	—
76	C₉⋊C₉	—	43	27	no	—
77	C₃²⋊C₉	—	43	18	no	—
78	C₉²	—	43	27	no	—
79	C₄₃	C₃×C₉×3₋¹⁺²	1	1	yes	characteristic, complemented, Sylow
80	C₃³	—	3	3	no	—
81	C₃³	—	3	6	no	—
82	C₃³	C₃×C₄₃⋊C₉	1	9	yes	—
83	C₃³	C₃²×C₄₃⋊C₃	1	3	yes	—
84	C₃³	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
85	3₋¹⁺²	C₃×C₄₃⋊C₉	1	81	yes	direct product factor
86	C₃×C₉	—	43	9	no	—
87	C₃×C₉	—	43	3	no	—
88	C₃×C₉	—	43	18	no	—
89	C₃×C₉	—	43	3	no	—
90	C₃×C₉	—	43	3	no	—
91	C₃×C₉	—	129	27	no	—
92	C₃×C₉	—	129	3	no	—
93	C₃×C₉	—	129	3	no	—
94	C₃×C₉	—	129	18	no	—
95	C₃×C₉	C₃×C₄₃⋊C₉	1	27	yes	complemented
96	C₃×C₉	C₃²×C₄₃⋊C₃	1	9	yes	—
97	C₃²	—	3	3	no	—
98	C₃²	—	3	6	no	—
99	C₃²	—	3	27	no	—
100	C₃²	C₃²×C₄₃⋊C₉	1	9	yes	—
101	C₃²	C₃²×C₄₃⋊C₉	1	3	yes	—
102	C₃²	3₋¹⁺²×C₄₃⋊C₃	1	3	yes	—
103	C₃²	#16812526	1	6	yes	—
104	C₃²	C₃³×C₄₃⋊C₃	1	1	yes	characteristic
105	C₉	—	43	9	no	—
106	C₉	—	129	3	no	—
107	C₉	—	129	3	no	—
108	C₉	—	129	18	no	—
109	C₉	C₃²×C₄₃⋊C₉	1	27	yes	complemented
110	C₃	—	3	9	no	—
111	C₃	3₋¹⁺²×C₄₃⋊C₉	1	9	yes	direct product factor
112	C₃	C₃³×C₄₃⋊C₉	1	1	yes	characteristic
113	C₃	C₃×3₋¹⁺²×C₄₃⋊C₃	1	1	yes	characteristic
114	C₃	C₃×#16812526	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃³×C₄₃⋊C₉

C₃

yes

maximal, characteristic

3₋¹⁺²×C₄₃⋊C₉

C₃

yes

maximal, direct product factor

C₃×#16812500

C₃

yes

maximal, complemented

C₃×#16812499

C₃

yes

maximal, complemented

C₃×C₃²⋊(C₄₃⋊C₉)

C₃

yes

maximal

C₃×C₉×C₄₃⋊C₉

C₃

yes

maximal, complemented

C₃²×3₋¹⁺²×C₄₃

C₃

yes

maximal, characteristic

C₃²×C₄₃⋊C₉

—

C₃²×C₄₃⋊C₉

C₃²

yes

—

C₃²×C₄₃⋊C₉

C₃²

yes

characteristic

C₃²×C₄₃⋊C₉

C₃²

yes

characteristic

C₃²×C₄₃⋊C₉

C₃²

yes

—

C₃²×C₄₃⋊C₉

C₃²

yes

characteristic

#16812500

C₃²

yes

complemented

#16812499

C₃²

yes

complemented

C₃²⋊(C₄₃⋊C₉)

C₃²

yes

—

C₉×C₄₃⋊C₉

C₃²

yes

complemented

C₃⁴×C₄₃

C₃²

yes

characteristic

C₃×3₋¹⁺²×C₄₃

C₉

yes

complemented

C₃×3₋¹⁺²×C₄₃

C₃²

yes

—

C₃²×C₉×C₄₃

C₃²

yes

—

C₃×C₄₃⋊C₉

—

C₃×C₄₃⋊C₉

—

C₃×C₄₃⋊C₉

—

C₃×C₄₃⋊C₉

—

C₃×C₄₃⋊C₉

3₋¹⁺²

yes

direct product factor

C₃×C₄₃⋊C₉

C₃³

yes

—

C₃×C₄₃⋊C₉

C₃³

yes

—

C₃×C₄₃⋊C₉

C₃³

yes

—

C₃×C₄₃⋊C₉

C₃³

yes

—

C₃³×C₄₃

—

C₃³×C₄₃

—

C₃³×C₄₃

C₃×C₉

yes

—

C₃³×C₄₃

C₃³

yes

—

C₃³×C₄₃

C₃³

yes

characteristic

3₋¹⁺²×C₄₃

C₃×C₉

yes

complemented

C₃×C₉×C₄₃

C₃×C₉

yes

complemented

C₃×C₉×C₄₃

C₃³

yes

—

C₃×C₉×3₋¹⁺²

—

maximal, Sylow

C₃²×C₄₃

—

C₃²×C₄₃

—

C₃²×C₄₃

—

C₃²×C₄₃

C₃²×C₉

yes

—

C₃²×C₄₃

C₃²×C₉

yes

—

C₃²×C₄₃

C₃×3₋¹⁺²

yes

—

C₃²×C₄₃

#737

yes

—

C₃²×C₄₃

C₃⁴

yes

characteristic

C₉×C₄₃

C₃²×C₉

yes

complemented

C₄₃⋊C₉

—

C₄₃⋊C₉

—

C₄₃⋊C₉

—

C₄₃⋊C₉

C₃×3₋¹⁺²

yes

direct product factor

C₃²×3₋¹⁺²

C₄₃⋊C₃

yes

characteristic

C₃³×C₉

—

C₉×3₋¹⁺²

—

C₃×C₉⋊C₉

—

C₃×C₉⋊C₉

—

C₃×C₃²⋊C₉

—

C₃×C₉²

—

C₃×C₄₃

—

C₃×C₄₃

C₉×3₋¹⁺²

yes

complemented

C₃×C₄₃

C₃³×C₉

yes

characteristic

C₃×C₄₃

C₃²×3₋¹⁺²

yes

characteristic

C₃×C₄₃

C₃×#737

yes

—

C₃⁴

C₃×C₄₃⋊C₃

yes

characteristic

C₃×3₋¹⁺²

C₄₃⋊C₉

yes

direct product factor

C₃×3₋¹⁺²

C₃×C₄₃⋊C₃

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

129

—

C₃²×C₉

C₃×C₄₃⋊C₃

yes

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₉²

—

C₄₃

C₃×C₉×3₋¹⁺²

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

C₃×C₄₃⋊C₉

yes

—

C₃³

C₃²×C₄₃⋊C₃

yes

—

C₃³

C₃²×C₄₃⋊C₃

yes

characteristic

3₋¹⁺²

C₃×C₄₃⋊C₉

yes

direct product factor

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

129

—

C₃×C₉

—

129

—

C₃×C₉

—

129

—

C₃×C₉

—

129

—

C₃×C₉

C₃×C₄₃⋊C₉

yes

complemented

C₃×C₉

C₃²×C₄₃⋊C₃

yes

—

C₃²

—

C₃²

—

C₃²

—

100

C₃²

C₃²×C₄₃⋊C₉

yes

—

101

C₃²

C₃²×C₄₃⋊C₉

yes

—

102

C₃²

3₋¹⁺²×C₄₃⋊C₃

yes

—

103

C₃²

#16812526

yes

—

104

C₃²

C₃³×C₄₃⋊C₃

yes

characteristic

105

C₉

—

106

C₉

—

129

—

107

C₉

—

129

—

108

C₉

—

129

—

109

C₉

C₃²×C₄₃⋊C₉

yes

complemented

110

C₃

—

111

C₃

3₋¹⁺²×C₄₃⋊C₉

yes

direct product factor

112

C₃

C₃³×C₄₃⋊C₉

yes

characteristic

113

C₃

C₃×3₋¹⁺²×C₄₃⋊C₃

yes

characteristic

114

C₃

C₃×#16812526

yes

—

	subgroup	G/H
center	C₃³	C₃²×C₄₃⋊C₃
commutator	C₃×C₄₃	C₃³×C₉
Frattini	C₃²	C₃³×C₄₃⋊C₃
Fitting	C₃²×3₋¹⁺²×C₄₃	C₃
socle	C₃³×C₄₃	C₃³
prime radical	C₃⁴×C₄₃	C₃²
cosocle	C₃²×C₄₃	C₃⁴
hypercenter	C₃²×3₋¹⁺²	C₄₃⋊C₃
hypocenter	C₄₃	C₃×C₉×3₋¹⁺²
LCS	G ▷ C₃×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₃³ ◁ C₃²×3₋¹⁺²

subgroup

G/H

center

C₃³

C₃²×C₄₃⋊C₃

commutator

C₃×C₄₃

C₃³×C₉

Frattini

C₃²

C₃³×C₄₃⋊C₃

Fitting

C₃²×3₋¹⁺²×C₄₃

C₃

socle

C₃³×C₄₃

C₃³

prime radical

C₃⁴×C₄₃

C₃²

cosocle

C₃²×C₄₃

C₃⁴

hypercenter

C₃²×3₋¹⁺²

C₄₃⋊C₃

hypocenter

C₄₃

C₃×C₉×3₋¹⁺²

LCS

G ▷ C₃×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₃³ ◁ C₃²×3₋¹⁺²

N	H ≅ G/N	type
2. 3₋¹⁺²×C₄₃⋊C₉	C₃	direct (N × H)
3. C₃×#16812500	C₃	semidirect
4. C₃×#16812499	C₃	semidirect
6. C₃×C₉×C₄₃⋊C₉	C₃	semidirect
14. #16812500	C₃²	semidirect
15. #16812499	C₃²	semidirect
17. C₉×C₄₃⋊C₉	C₃²	semidirect
19. C₃×3₋¹⁺²×C₄₃	C₉	semidirect
26. C₃×C₄₃⋊C₉	3₋¹⁺²	direct (N × H)
36. 3₋¹⁺²×C₄₃	C₃×C₉	semidirect
37. C₃×C₉×C₄₃	C₃×C₉	semidirect
48. C₉×C₄₃	C₃²×C₉	semidirect
52. C₄₃⋊C₉	C₃×3₋¹⁺²	direct (N × H)
61. C₃×C₄₃	C₉×3₋¹⁺²	semidirect
66. C₃×3₋¹⁺²	C₄₃⋊C₉	direct (N × H)
79. C₄₃	C₃×C₉×3₋¹⁺²	semidirect
85. 3₋¹⁺²	C₃×C₄₃⋊C₉	direct (N × H)
95. C₃×C₉	C₃×C₄₃⋊C₉	semidirect
109. C₉	C₃²×C₄₃⋊C₉	semidirect
111. C₃	3₋¹⁺²×C₄₃⋊C₉	direct (N × H)

H ≅ G/N

type

2. 3₋¹⁺²×C₄₃⋊C₉

C₃

direct (N × H)

3. C₃×#16812500

C₃

semidirect

4. C₃×#16812499

C₃

semidirect

6. C₃×C₉×C₄₃⋊C₉

C₃

semidirect

14. #16812500

C₃²

semidirect

15. #16812499

C₃²

semidirect

17. C₉×C₄₃⋊C₉

C₃²

semidirect

19. C₃×3₋¹⁺²×C₄₃

C₉

semidirect

26. C₃×C₄₃⋊C₉

3₋¹⁺²

direct (N × H)

36. 3₋¹⁺²×C₄₃

C₃×C₉

semidirect

37. C₃×C₉×C₄₃

C₃×C₉

semidirect

48. C₉×C₄₃

C₃²×C₉

semidirect

52. C₄₃⋊C₉

C₃×3₋¹⁺²

direct (N × H)

61. C₃×C₄₃

C₉×3₋¹⁺²

semidirect

66. C₃×3₋¹⁺²

C₄₃⋊C₉

direct (N × H)

79. C₄₃

C₃×C₉×3₋¹⁺²

semidirect

85. 3₋¹⁺²

C₃×C₄₃⋊C₉

direct (N × H)

95. C₃×C₉

C₃×C₄₃⋊C₉

semidirect

109. C₉

C₃²×C₄₃⋊C₉

semidirect

111. C₃

3₋¹⁺²×C₄₃⋊C₉

direct (N × H)

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

subgroup	index
C₃³×C₄₃⋊C₉	3
3₋¹⁺²×C₄₃⋊C₉	3
C₃×#16812500	3
C₃×#16812499	3
C₃×C₃²⋊(C₄₃⋊C₉)	3
C₃×C₉×C₄₃⋊C₉	3
C₃²×3₋¹⁺²×C₄₃	3
C₃×C₉×3₋¹⁺²	43

subgroup

index

C₃³×C₄₃⋊C₉

3₋¹⁺²×C₄₃⋊C₉

C₃×#16812500

C₃×#16812499

C₃×C₃²⋊(C₄₃⋊C₉)

C₃×C₉×C₄₃⋊C₉

C₃²×3₋¹⁺²×C₄₃

C₃×C₉×3₋¹⁺²

Subgroups — C₃×3₋¹⁺²×C₄₃⋊C₉ #44534656

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×3−1+2×C43⋊C9 #44534656

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×3₋¹⁺²×C₄₃⋊C₉ #44534656