Subgroups — #44533820

Order 31347.

476 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23782886	C₃	1	8	yes	maximal, complemented
2	C₃×#16812506	C₃	1	4	yes	maximal, complemented
3	C₄₃×H(5,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃³×C₄₃⋊C₃	C₃²	1	1	yes	characteristic, complemented
5	C₃×#11923958	C₃²	1	8	yes	complemented
6	#16812509	—	3	24	no	—
7	#16812506	—	3	12	no	—
8	C₃×H(3,𝔽₃)×C₄₃	—	3	12	no	—
9	C₃×H(3,𝔽₃)×C₄₃	C₃²	1	4	yes	—
10	C₃²×C₄₃⋊C₃	—	3	4	no	—
11	C₃²×C₄₃⋊C₃	—	9	3	no	—
12	#11923958	—	3	8	no	—
13	#11923958	—	3	16	no	—
14	C₃×C₄₃⋊C₉	—	3	8	no	—
15	C₃³×C₄₃	—	3	9	no	—
16	C₃³×C₄₃	—	3	4	no	—
17	C₃³×C₄₃	C₃³	1	1	yes	characteristic
18	H(3,𝔽₃)×C₄₃	—	3	18	no	—
19	H(3,𝔽₃)×C₄₃	—	3	8	no	—
20	H(3,𝔽₃)×C₄₃	H(3,𝔽₃)	1	12	yes	—
21	#10624761	—	43	1	no	maximal, Sylow
22	C₃²×C₄₃	—	3	12	no	—
23	C₃²×C₄₃	—	9	1	no	—
24	C₃²×C₄₃	—	27	4	no	—
25	C₃²×C₄₃	—	27	9	no	—
26	C₃²×C₄₃	C₃×H(3,𝔽₃)	1	4	yes	—
27	C₃×C₄₃⋊C₃	—	9	1	no	—
28	C₃×C₄₃⋊C₃	—	9	12	no	—
29	C₄₃⋊C₉	—	9	8	no	—
30	H(5,𝔽₃)	C₄₃⋊C₃	1	1	yes	characteristic, complemented
31	#6910	—	43	8	no	—
32	C₃×AΣL(1,𝔽₂₇)	—	43	4	no	—
33	C₃×C₄₃	—	3	4	no	—
34	C₃×C₄₃	—	9	12	no	—
35	C₃×C₄₃	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
36	C₄₃⋊C₃	—	9	3	no	—
37	C₃⁴	—	43	1	no	—
38	C₃×3₋¹⁺²	—	43	8	no	—
39	C₃×H(3,𝔽₃)	—	3	12	no	—
40	C₃×H(3,𝔽₃)	C₃×C₄₃⋊C₃	1	4	yes	—
41	#731	—	129	24	no	—
42	AΣL(1,𝔽₂₇)	—	129	12	no	—
43	C₄₃	#10624761	1	1	yes	characteristic, complemented, Sylow
44	C₃³	—	3	9	no	—
45	C₃³	—	3	4	no	—
46	C₃³	—	129	4	no	—
47	C₃³	—	387	3	no	—
48	C₃³	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
49	3₋¹⁺²	—	129	8	no	—
50	3₋¹⁺²	—	129	16	no	—
51	H(3,𝔽₃)	—	3	18	no	—
52	H(3,𝔽₃)	—	3	8	no	—
53	H(3,𝔽₃)	C₃²⋊(C₄₃⋊C₃)	1	12	yes	—
54	C₃×C₉	—	129	8	no	—
55	C₃²	—	3	12	no	—
56	C₃²	—	9	1	no	—
57	C₃²	—	27	4	no	—
58	C₃²	—	27	9	no	—
59	C₃²	—	387	12	no	—
60	C₃²	—	387	1	no	—
61	C₃²	C₃×C₃²⋊(C₄₃⋊C₃)	1	4	yes	—
62	C₉	—	387	8	no	—
63	C₃	—	3	4	no	—
64	C₃	—	9	12	no	—
65	C₃	—	387	3	no	—
66	C₃	#23782794	1	1	yes	characteristic

	subgroup	G/H
center	C₃	#23782794
commutator, cosocle	C₃³×C₄₃	C₃³
Frattini	C₃³	C₃²×C₄₃⋊C₃
Fitting	C₄₃×H(5,𝔽₃)	C₃
socle	C₃×C₄₃	C₃²⋊H(3,𝔽₃)
hypercenter	H(5,𝔽₃)	C₄₃⋊C₃
hypocenter	C₄₃	#10624761
LCS	G ▷ C₃³×C₄₃ ▷ C₃×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ H(5,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. #23782886	C₃	semidirect
2. C₃×#16812506	C₃	semidirect
3. C₄₃×H(5,𝔽₃)	C₃	semidirect
4. C₃³×C₄₃⋊C₃	C₃²	semidirect
5. C₃×#11923958	C₃²	semidirect
30. H(5,𝔽₃)	C₄₃⋊C₃	semidirect
43. C₄₃	#10624761	semidirect

Maximal subgroups

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

subgroup	index
#23782886	3
C₃×#16812506	3
C₄₃×H(5,𝔽₃)	3
#10624761	43