Subgroups — #44534675

Order 31347.

926 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	3₋¹⁺²×C₄₃⋊C₉	C₃	1	24	yes	maximal, complemented
2	H(3,𝔽₃)×C₄₃⋊C₉	C₃	1	3	yes	maximal
3	#23782782	C₃	1	8	yes	maximal, complemented
4	C₃×#16812499	C₃	1	4	yes	maximal, complemented
5	C₃×C₄₃×#737	C₃	1	1	yes	maximal, characteristic
6	C₃²×C₄₃⋊C₉	C₃²	1	12	yes	—
7	#16812500	C₃²	1	24	yes	—
8	#16812499	C₃²	1	1	yes	characteristic
9	#16812499	C₃²	1	8	yes	—
10	#16812499	C₃²	1	24	yes	—
11	C₃²⋊(C₄₃⋊C₉)	C₃²	1	24	yes	—
12	C₉×C₄₃⋊C₉	C₃²	1	24	yes	—
13	C₄₃×#737	C₉	1	27	yes	complemented
14	C₃×3₋¹⁺²×C₄₃	C₃²	1	8	yes	—
15	C₃×H(3,𝔽₃)×C₄₃	C₃²	1	1	yes	characteristic
16	C₃²×C₉×C₄₃	C₃²	1	4	yes	—
17	C₃×C₄₃⋊C₉	—	9	12	no	—
18	C₃×C₄₃⋊C₉	C₃³	1	24	yes	—
19	C₃×C₄₃⋊C₉	C₃³	1	3	yes	—
20	C₃³×C₄₃	C₃³	1	4	yes	—
21	3₋¹⁺²×C₄₃	C₃×C₉	1	72	yes	complemented
22	H(3,𝔽₃)×C₄₃	C₃×C₉	1	9	yes	—
23	C₃×C₉×C₄₃	C₃×C₉	1	36	yes	complemented
24	C₃×C₉×C₄₃	C₃³	1	1	yes	characteristic
25	C₃×C₉×C₄₃	C₃³	1	8	yes	—
26	#10624864	—	43	1	no	maximal, Sylow
27	C₃²×C₄₃	—	3	12	no	—
28	C₃²×C₄₃	C₃²×C₉	1	12	yes	—
29	C₃²×C₄₃	C₃⁴	1	1	yes	characteristic
30	C₉×C₄₃	C₃²×C₉	1	3	yes	—
31	C₉×C₄₃	C₃²×C₉	1	24	yes	—
32	C₄₃⋊C₉	—	3	3	no	—
33	C₄₃⋊C₉	—	3	24	no	—
34	C₃×#737	C₄₃⋊C₃	1	1	yes	characteristic
35	C₉×3₋¹⁺²	—	43	24	no	—
36	C₉×H(3,𝔽₃)	—	43	3	no	—
37	#6887	—	43	8	no	—
38	C₃×C₉⋊C₉	—	43	4	no	—
39	C₃×C₄₃	—	3	12	no	—
40	C₃×C₄₃	C₃³×C₉	1	1	yes	characteristic
41	C₃×C₄₃	3₋¹⁺⁴	1	3	yes	—
42	#737	C₄₃⋊C₉	1	27	yes	complemented
43	C₃×3₋¹⁺²	C₃×C₄₃⋊C₃	1	8	yes	—
44	C₃×H(3,𝔽₃)	C₃×C₄₃⋊C₃	1	1	yes	characteristic
45	C₃²×C₉	—	43	12	no	—
46	C₃²×C₉	C₃×C₄₃⋊C₃	1	4	yes	—
47	C₉⋊C₉	—	43	1	no	—
48	C₉⋊C₉	—	43	8	no	—
49	C₉⋊C₉	—	43	24	no	—
50	C₉⋊C₉	—	43	24	no	—
51	C₃²⋊C₉	—	43	24	no	—
52	C₉²	—	43	24	no	—
53	C₄₃	#10624864	1	1	yes	characteristic, complemented, Sylow
54	C₃³	C₃²×C₄₃⋊C₃	1	4	yes	—
55	3₋¹⁺²	C₃×C₄₃⋊C₉	1	72	yes	complemented
56	H(3,𝔽₃)	C₃×C₄₃⋊C₉	1	9	yes	—
57	C₃×C₉	—	43	24	no	—
58	C₃×C₉	—	43	3	no	—
59	C₃×C₉	—	387	12	no	—
60	C₃×C₉	C₃×C₄₃⋊C₉	1	36	yes	complemented
61	C₃×C₉	C₃²×C₄₃⋊C₃	1	8	yes	—
62	C₃×C₉	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
63	C₃²	—	3	12	no	—
64	C₃²	C₃²×C₄₃⋊C₉	1	12	yes	—
65	C₃²	C₃³×C₄₃⋊C₃	1	1	yes	characteristic
66	C₉	—	129	24	no	—
67	C₉	—	129	3	no	—
68	C₉	C₃²×C₄₃⋊C₉	1	3	yes	—
69	C₉	C₃²×C₄₃⋊C₉	1	24	yes	—
70	C₃	—	3	12	no	—
71	C₃	C₃³×C₄₃⋊C₉	1	1	yes	characteristic
72	C₃	#23782922	1	3	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₄₃⋊C₃
commutator	C₃×C₄₃	C₃³×C₉
Fitting	C₃×C₄₃×#737	C₃
socle, cosocle	C₃²×C₄₃	C₃⁴
prime radical	C₃×H(3,𝔽₃)×C₄₃	C₃²
hypercenter	C₃×#737	C₄₃⋊C₃
hypocenter	C₄₃	#10624864
LCS	G ▷ C₃×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₃² ◁ C₃×#737

Complements

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

N	H ≅ G/N	type
1. 3₋¹⁺²×C₄₃⋊C₉	C₃	semidirect
3. #23782782	C₃	semidirect
4. C₃×#16812499	C₃	semidirect
13. C₄₃×#737	C₉	semidirect
21. 3₋¹⁺²×C₄₃	C₃×C₉	semidirect
23. C₃×C₉×C₄₃	C₃×C₉	semidirect
42. #737	C₄₃⋊C₉	semidirect
53. C₄₃	#10624864	semidirect
55. 3₋¹⁺²	C₃×C₄₃⋊C₉	semidirect
60. C₃×C₉	C₃×C₄₃⋊C₉	semidirect

Maximal subgroups

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

subgroup	index
3₋¹⁺²×C₄₃⋊C₉	3
H(3,𝔽₃)×C₄₃⋊C₉	3
#23782782	3
C₃×#16812499	3
C₃×C₄₃×#737	3
#10624864	43