Subgroups — #44535351

Order 31347.

1062 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₃⋊H(5,𝔽₃)	C₃	1	2	yes	maximal, complemented
2	C₃×#16812524	C₃	1	1	yes	maximal, characteristic, complemented
3	#23782891	C₃	1	24	yes	maximal, complemented
4	C₃×#16812512	C₃	1	12	yes	maximal, complemented
5	C₄₃×H(5,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
6	#16812524	—	3	9	no	—
7	C₃×#11923958	C₃²	1	8	yes	complemented
8	C₃×C₃²⋊(C₄₃⋊C₃)	C₃²	1	24	yes	complemented
9	H(3,𝔽₃)×C₄₃⋊C₃	—	3	18	no	—
10	C₃²×C₄₃⋊C₉	C₃²	1	4	yes	complemented
11	#16812512	C₃²	1	9	yes	complemented
12	#16812512	C₃²	1	72	yes	complemented
13	C₃×H(3,𝔽₃)×C₄₃	—	3	9	no	—
14	C₃×H(3,𝔽₃)×C₄₃	C₃²	1	1	yes	characteristic
15	C₃×H(3,𝔽₃)×C₄₃	C₃²	1	12	yes	complemented
16	C₃²×C₄₃⋊C₃	—	3	24	no	—
17	#11923958	—	3	24	no	—
18	C₃²⋊(C₄₃⋊C₃)	—	3	48	no	—
19	C₃²⋊(C₄₃⋊C₃)	C₃³	1	18	yes	—
20	C₃×C₄₃⋊C₉	—	3	12	no	—
21	C₃×C₄₃⋊C₉	C₃³	1	1	yes	characteristic
22	C₃×C₄₃⋊C₉	C₃³	1	8	yes	—
23	C₃³×C₄₃	—	3	12	no	—
24	C₃³×C₄₃	C₃³	1	4	yes	—
25	H(3,𝔽₃)×C₄₃	—	3	24	no	—
26	H(3,𝔽₃)×C₄₃	H(3,𝔽₃)	1	9	yes	—
27	H(3,𝔽₃)×C₄₃	C₃³	1	9	yes	—
28	#10624932	—	43	1	no	maximal, Sylow
29	C₃²×C₄₃	—	3	9	no	—
30	C₃²×C₄₃	—	9	4	no	—
31	C₃²×C₄₃	—	27	12	no	—
32	C₃²×C₄₃	C₃×H(3,𝔽₃)	1	12	yes	—
33	C₃²×C₄₃	C₃⁴	1	1	yes	characteristic
34	C₃×C₄₃⋊C₃	—	3	18	no	—
35	C₃×C₄₃⋊C₃	—	27	24	no	—
36	C₄₃⋊C₉	—	3	1	no	—
37	C₄₃⋊C₉	—	3	8	no	—
38	H(5,𝔽₃)	—	43	2	no	—
39	H(5,𝔽₃)	C₄₃⋊C₃	1	1	yes	characteristic, complemented
40	C₃×#737	—	43	1	no	—
41	#6911	—	43	24	no	—
42	C₃×#732	—	43	12	no	—
43	C₃×C₄₃	—	3	12	no	—
44	C₃×C₄₃	—	3	1	no	—
45	C₃×C₄₃	—	9	9	no	—
46	C₃×C₄₃	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
47	C₄₃⋊C₃	—	9	18	no	—
48	#737	—	129	9	no	—
49	C₃×3₋¹⁺²	—	43	8	no	—
50	C₃×H(3,𝔽₃)	—	3	9	no	—
51	C₃×H(3,𝔽₃)	—	43	24	no	—
52	C₃×H(3,𝔽₃)	—	129	18	no	—
53	C₃×H(3,𝔽₃)	C₃×C₄₃⋊C₃	1	1	yes	characteristic
54	C₃×H(3,𝔽₃)	C₃×C₄₃⋊C₃	1	12	yes	complemented
55	C₃²×C₉	—	43	4	no	—
56	#732	—	43	9	no	—
57	#732	—	43	72	no	—
58	C₄₃	#10624932	1	1	yes	characteristic, complemented, Sylow
59	C₃³	—	3	12	no	—
60	C₃³	—	129	24	no	—
61	C₃³	C₃²×C₄₃⋊C₃	1	4	yes	—
62	3₋¹⁺²	—	129	24	no	—
63	H(3,𝔽₃)	—	3	24	no	—
64	H(3,𝔽₃)	—	43	18	no	—
65	H(3,𝔽₃)	—	129	48	no	—
66	H(3,𝔽₃)	C₃²⋊(C₄₃⋊C₃)	1	9	yes	—
67	H(3,𝔽₃)	C₃²×C₄₃⋊C₃	1	9	yes	—
68	C₃×C₉	—	43	1	no	—
69	C₃×C₉	—	43	8	no	—
70	C₃×C₉	—	129	12	no	—
71	C₃²	—	3	9	no	—
72	C₃²	—	9	4	no	—
73	C₃²	—	27	12	no	—
74	C₃²	—	129	18	no	—
75	C₃²	—	1161	24	no	—
76	C₃²	C₃×C₃²⋊(C₄₃⋊C₃)	1	12	yes	—
77	C₃²	C₃³×C₄₃⋊C₃	1	1	yes	characteristic
78	C₉	—	129	1	no	—
79	C₉	—	129	8	no	—
80	C₃	—	3	1	no	—
81	C₃	—	3	12	no	—
82	C₃	—	9	9	no	—
83	C₃	—	387	18	no	—
84	C₃	C₃²×C₃²⋊(C₄₃⋊C₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₃	C₃²×C₃²⋊(C₄₃⋊C₃)
commutator, cosocle	C₃²×C₄₃	C₃⁴
Frattini	C₃²	C₃³×C₄₃⋊C₃
Fitting	C₄₃×H(5,𝔽₃)	C₃
socle	C₃×C₄₃	C₃²×H(3,𝔽₃)
hypercenter	H(5,𝔽₃)	C₄₃⋊C₃
hypocenter	C₄₃	#10624932
LCS	G ▷ C₃²×C₄₃ ▷ C₃×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₃ ◁ C₃×H(3,𝔽₃) ◁ H(5,𝔽₃)

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₄₃⋊H(5,𝔽₃)	C₃	semidirect
2. C₃×#16812524	C₃	semidirect
3. #23782891	C₃	semidirect
4. C₃×#16812512	C₃	semidirect
5. C₄₃×H(5,𝔽₃)	C₃	semidirect
7. C₃×#11923958	C₃²	semidirect
8. C₃×C₃²⋊(C₄₃⋊C₃)	C₃²	semidirect
10. C₃²×C₄₃⋊C₉	C₃²	semidirect
11. #16812512	C₃²	semidirect
12. #16812512	C₃²	semidirect
15. C₃×H(3,𝔽₃)×C₄₃	C₃²	semidirect
39. H(5,𝔽₃)	C₄₃⋊C₃	semidirect
54. C₃×H(3,𝔽₃)	C₃×C₄₃⋊C₃	semidirect
58. C₄₃	#10624932	semidirect

Maximal subgroups

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

subgroup	index
C₄₃⋊H(5,𝔽₃)	3
C₃×#16812524	3
#23782891	3
C₃×#16812512	3
C₄₃×H(5,𝔽₃)	3
#10624932	43