Subgroups — #44533661

Order 31347.

488 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#11923958	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×#16812509	C₃	1	9	yes	maximal, complemented
3	#23782798	C₃	1	2	yes	maximal, complemented
4	C₄₃×C₃²⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
5	C₃×#11923958	—	3	9	no	—
6	C₃×#11923958	—	3	3	no	—
7	C₃×#11923958	C₃²	1	6	yes	—
8	C₃²×C₄₃⋊C₉	C₃²	1	3	yes	—
9	#16812509	—	3	27	no	—
10	C₃²⋊(C₄₃⋊C₉)	—	3	6	no	—
11	C₃⁴×C₄₃	C₃²	1	1	yes	characteristic
12	C₃×H(3,𝔽₃)×C₄₃	—	3	3	no	—
13	C₃×H(3,𝔽₃)×C₄₃	C₃²	1	3	yes	—
14	#11923958	—	3	27	no	—
15	#11923958	—	3	18	no	—
16	C₃×C₄₃⋊C₉	—	3	6	no	—
17	C₃×C₄₃⋊C₉	—	3	3	no	—
18	C₃×C₄₃⋊C₉	—	3	6	no	—
19	C₃×C₄₃⋊C₉	H(3,𝔽₃)	1	9	yes	complemented
20	C₃³×C₄₃	—	3	3	no	—
21	C₃³×C₄₃	—	3	3	no	—
22	C₃³×C₄₃	—	9	3	no	—
23	C₃³×C₄₃	H(3,𝔽₃)	1	3	yes	—
24	C₃³×C₄₃	C₃³	1	1	yes	characteristic
25	H(3,𝔽₃)×C₄₃	—	3	9	no	—
26	H(3,𝔽₃)×C₄₃	—	9	9	no	—
27	#10624736	—	43	1	no	maximal, Sylow
28	C₃²×C₄₃	—	3	3	no	—
29	C₃²×C₄₃	—	3	3	no	—
30	C₃²×C₄₃	—	3	3	no	—
31	C₃²×C₄₃	—	3	6	no	—
32	C₃²×C₄₃	—	9	9	no	—
33	C₃²×C₄₃	—	9	3	no	—
34	C₃²×C₄₃	—	9	9	no	—
35	C₃²×C₄₃	C₃×H(3,𝔽₃)	1	3	yes	—
36	C₃²×C₄₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
37	C₄₃⋊C₉	—	3	9	no	—
38	C₄₃⋊C₉	—	9	6	no	—
39	C₃²×3₋¹⁺²	—	43	1	no	—
40	C₃×#731	—	43	9	no	—
41	#6891	—	43	2	no	—
42	C₃²⋊H(3,𝔽₃)	C₄₃⋊C₃	1	1	yes	characteristic
43	C₃×C₄₃	—	3	3	no	—
44	C₃×C₄₃	—	9	3	no	—
45	C₃×C₄₃	—	27	3	no	—
46	C₃×C₄₃	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
47	C₃×C₄₃	#6910	1	3	yes	—
48	C₃⁴	C₃×C₄₃⋊C₃	1	1	yes	characteristic
49	C₃×3₋¹⁺²	—	43	6	no	—
50	C₃×3₋¹⁺²	—	129	9	no	—
51	C₃×3₋¹⁺²	—	129	3	no	—
52	C₃×H(3,𝔽₃)	—	3	3	no	—
53	C₃×H(3,𝔽₃)	C₃×C₄₃⋊C₃	1	3	yes	—
54	C₃²×C₉	—	43	3	no	—
55	#731	—	129	27	no	—
56	C₃²⋊C₉	—	129	6	no	—
57	C₄₃	#10624736	1	1	yes	characteristic, complemented, Sylow
58	C₃³	—	3	3	no	—
59	C₃³	—	3	3	no	—
60	C₃³	—	9	3	no	—
61	C₃³	C₃²⋊(C₄₃⋊C₃)	1	3	yes	—
62	C₃³	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
63	3₋¹⁺²	—	129	27	no	—
64	3₋¹⁺²	—	129	18	no	—
65	H(3,𝔽₃)	—	3	9	no	—
66	H(3,𝔽₃)	—	9	9	no	—
67	C₃×C₉	—	43	9	no	—
68	C₃×C₉	—	129	6	no	—
69	C₃×C₉	—	129	3	no	—
70	C₃×C₉	—	129	6	no	—
71	C₃²	—	3	3	no	—
72	C₃²	—	3	3	no	—
73	C₃²	—	3	3	no	—
74	C₃²	—	3	6	no	—
75	C₃²	—	9	9	no	—
76	C₃²	—	9	3	no	—
77	C₃²	—	9	9	no	—
78	C₃²	H(3,𝔽₃)×C₄₃⋊C₃	1	3	yes	—
79	C₃²	C₃×C₃²⋊(C₄₃⋊C₃)	1	1	yes	characteristic
80	C₉	—	129	9	no	—
81	C₉	—	387	6	no	—
82	C₃	—	3	3	no	—
83	C₃	—	9	3	no	—
84	C₃	—	27	3	no	—
85	C₃	(C₃×C₄₃⋊C₃)⋊H(3,𝔽₃)	1	1	yes	characteristic
86	C₃	#23782886	1	3	yes	—

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₄₃⋊C₃)
commutator, cosocle	C₃³×C₄₃	C₃³
Frattini	C₃³	C₃²×C₄₃⋊C₃
Fitting, prime radical	C₄₃×C₃²⋊H(3,𝔽₃)	C₃
socle	C₃²×C₄₃	C₃×H(3,𝔽₃)
hypercenter	C₃²⋊H(3,𝔽₃)	C₄₃⋊C₃
hypocenter	C₄₃	#10624736
LCS	G ▷ C₃³×C₄₃ ▷ C₃×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ C₃²⋊H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. C₃²×#11923958	C₃	semidirect
2. C₃×#16812509	C₃	semidirect
3. #23782798	C₃	semidirect
19. C₃×C₄₃⋊C₉	H(3,𝔽₃)	semidirect
57. C₄₃	#10624736	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#11923958	3
C₃×#16812509	3
#23782798	3
C₄₃×C₃²⋊H(3,𝔽₃)	3
#10624736	43