Subgroups — #44533391

Order 31347.

404 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₃²⋊(C₄₃⋊C₉)	C₃	1	3	yes	maximal
2	#23782716	C₃	1	9	yes	maximal, complemented
3	C₃²×3₋¹⁺²×C₄₃	C₃	1	1	yes	maximal, characteristic
4	C₃²×C₄₃⋊C₉	—	3	3	no	—
5	C₃²⋊(C₄₃⋊C₉)	—	3	6	no	—
6	C₃²⋊(C₄₃⋊C₉)	C₃²	1	9	yes	—
7	C₃⁴×C₄₃	C₃²	1	1	yes	characteristic
8	C₃×3₋¹⁺²×C₄₃	—	3	3	no	—
9	C₃×3₋¹⁺²×C₄₃	—	3	6	no	—
10	C₃×3₋¹⁺²×C₄₃	C₉	1	9	yes	complemented
11	C₃²×C₉×C₄₃	C₃²	1	3	yes	—
12	C₃×C₄₃⋊C₉	—	3	9	no	—
13	C₃×C₄₃⋊C₉	—	9	9	no	—
14	C₃³×C₄₃	—	9	2	no	—
15	C₃³×C₄₃	—	9	1	no	—
16	C₃³×C₄₃	C₃×C₉	1	3	yes	—
17	C₃³×C₄₃	H(3,𝔽₃)	1	3	yes	—
18	C₃³×C₄₃	3₋¹⁺²	1	6	yes	—
19	C₃³×C₄₃	C₃³	1	1	yes	characteristic
20	3₋¹⁺²×C₄₃	—	3	27	no	—
21	C₃×C₉×C₄₃	—	3	3	no	—
22	C₃×C₉×C₄₃	—	3	6	no	—
23	C₃×C₉×C₄₃	C₃×C₉	1	9	yes	complemented
24	#10624691	—	43	1	no	maximal, Sylow
25	C₃²×C₄₃	—	3	6	no	—
26	C₃²×C₄₃	—	3	2	no	—
27	C₃²×C₄₃	—	3	3	no	—
28	C₃²×C₄₃	—	3	1	no	—
29	C₃²×C₄₃	—	9	3	no	—
30	C₃²×C₄₃	—	9	6	no	—
31	C₃²×C₄₃	C₃²⋊C₉	1	9	yes	—
32	C₃²×C₄₃	C₃²×C₉	1	1	yes	characteristic
33	C₃²×C₄₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
34	C₃²×C₄₃	C₃×3₋¹⁺²	1	2	yes	—
35	C₉×C₄₃	—	3	9	no	—
36	C₄₃⋊C₉	—	9	9	no	—
37	C₃²×3₋¹⁺²	C₄₃⋊C₃	1	1	yes	characteristic
38	C₃×C₃²⋊C₉	—	43	3	no	—
39	#6868	—	43	9	no	—
40	C₃×C₄₃	—	3	9	no	—
41	C₃×C₄₃	—	3	1	no	—
42	C₃×C₄₃	—	3	2	no	—
43	C₃×C₄₃	C₃×C₃²⋊C₉	1	1	yes	characteristic
44	C₃×C₄₃	#6911	1	1	yes	characteristic
45	C₃×C₄₃	#6912	1	2	yes	—
46	C₃⁴	C₃×C₄₃⋊C₃	1	1	yes	characteristic
47	C₃×3₋¹⁺²	—	3	3	no	—
48	C₃×3₋¹⁺²	—	3	6	no	—
49	C₃×3₋¹⁺²	C₄₃⋊C₉	1	9	yes	complemented
50	C₃²×C₉	—	129	3	no	—
51	C₃²×C₉	C₃×C₄₃⋊C₃	1	3	yes	—
52	C₃²⋊C₉	—	43	9	no	—
53	C₃²⋊C₉	—	129	6	no	—
54	C₄₃	#10624691	1	1	yes	characteristic, complemented, Sylow
55	C₃³	—	9	1	no	—
56	C₃³	—	9	2	no	—
57	C₃³	C₃×C₄₃⋊C₉	1	3	yes	—
58	C₃³	C₃²⋊(C₄₃⋊C₃)	1	3	yes	—
59	C₃³	#11923958	1	6	yes	—
60	C₃³	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
61	3₋¹⁺²	—	3	27	no	—
62	C₃×C₉	—	3	3	no	—
63	C₃×C₉	—	3	6	no	—
64	C₃×C₉	—	129	9	no	—
65	C₃×C₉	—	387	9	no	—
66	C₃×C₉	C₃×C₄₃⋊C₉	1	9	yes	complemented
67	C₃²	—	3	3	no	—
68	C₃²	—	3	1	no	—
69	C₃²	—	3	6	no	—
70	C₃²	—	3	2	no	—
71	C₃²	—	9	3	no	—
72	C₃²	—	9	6	no	—
73	C₃²	C₃²⋊(C₄₃⋊C₉)	1	9	yes	—
74	C₃²	C₃²×C₄₃⋊C₉	1	1	yes	characteristic
75	C₃²	C₃×C₃²⋊(C₄₃⋊C₃)	1	1	yes	characteristic
76	C₃²	C₃×#11923958	1	2	yes	—
77	C₉	—	3	9	no	—
78	C₉	—	387	9	no	—
79	C₃	—	3	9	no	—
80	C₃	—	3	1	no	—
81	C₃	—	3	2	no	—
82	C₃	C₃×C₃²⋊(C₄₃⋊C₉)	1	1	yes	characteristic
83	C₃	#23782892	1	1	yes	characteristic
84	C₃	#23782896	1	2	yes	—

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₄₃⋊C₃)
commutator	C₃²×C₄₃	C₃²×C₉
Frattini	C₃³	C₃²×C₄₃⋊C₃
Fitting	C₃²×3₋¹⁺²×C₄₃	C₃
socle	C₃²×C₄₃	C₃×H(3,𝔽₃)
prime radical	C₃⁴×C₄₃	C₃²
cosocle	C₃³×C₄₃	C₃³
hypercenter	C₃²×3₋¹⁺²	C₄₃⋊C₃
hypocenter	C₄₃	#10624691
LCS	G ▷ C₃²×C₄₃ ▷ C₃×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ C₃²×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
2. #23782716	C₃	semidirect
10. C₃×3₋¹⁺²×C₄₃	C₉	semidirect
23. C₃×C₉×C₄₃	C₃×C₉	semidirect
49. C₃×3₋¹⁺²	C₄₃⋊C₉	semidirect
54. C₄₃	#10624691	semidirect
66. 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
C₃×C₃²⋊(C₄₃⋊C₉)	3
#23782716	3
C₃²×3₋¹⁺²×C₄₃	3
#10624691	43