Subgroups — #44534793

Order 31347.

1124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#11923958	C₃	1	3	yes	maximal, complemented
2	#23782806	C₃	1	27	yes	maximal, complemented
3	#23782798	C₃	1	9	yes	maximal, complemented
4	C₃²×H(3,𝔽₃)×C₄₃	C₃	1	1	yes	maximal, characteristic
5	C₃×#11923958	—	3	27	no	—
6	C₃×#11923958	C₃²	1	9	yes	—
7	C₃×#11923958	C₃²	1	18	yes	complemented
8	C₃²×C₄₃⋊C₉	C₃²	1	9	yes	complemented
9	C₃²⋊(C₄₃⋊C₉)	C₃²	1	81	yes	complemented
10	C₃⁴×C₄₃	C₃²	1	3	yes	—
11	C₃⁴×C₄₃	C₃²	1	1	yes	characteristic
12	C₃×H(3,𝔽₃)×C₄₃	—	3	9	no	—
13	C₃×H(3,𝔽₃)×C₄₃	C₃²	1	9	yes	—
14	#11923958	—	3	27	no	—
15	#11923958	—	3	54	no	—
16	C₃×C₄₃⋊C₉	—	3	27	no	—
17	C₃×C₄₃⋊C₉	C₃³	1	27	yes	—
18	C₃³×C₄₃	—	3	9	no	—
19	C₃³×C₄₃	—	9	3	no	—
20	C₃³×C₄₃	—	9	9	no	—
21	C₃³×C₄₃	H(3,𝔽₃)	1	9	yes	—
22	C₃³×C₄₃	C₃³	1	9	yes	—
23	C₃³×C₄₃	C₃³	1	3	yes	—
24	C₃³×C₄₃	C₃³	1	1	yes	characteristic
25	H(3,𝔽₃)×C₄₃	—	3	27	no	—
26	#10624881	—	43	1	no	maximal, Sylow
27	C₃²×C₄₃	—	3	3	no	—
28	C₃²×C₄₃	—	3	9	no	—
29	C₃²×C₄₃	—	3	9	no	—
30	C₃²×C₄₃	—	3	27	no	—
31	C₃²×C₄₃	—	9	9	no	—
32	C₃²×C₄₃	—	9	27	no	—
33	C₃²×C₄₃	C₃×H(3,𝔽₃)	1	9	yes	—
34	C₃²×C₄₃	C₃×H(3,𝔽₃)	1	3	yes	—
35	C₃²×C₄₃	C₃⁴	1	1	yes	characteristic
36	C₄₃⋊C₉	—	3	27	no	—
37	C₃²×3₋¹⁺²	—	43	3	no	—
38	C₃²×H(3,𝔽₃)	C₄₃⋊C₃	1	1	yes	characteristic
39	#6893	—	43	27	no	—
40	#6891	—	43	9	no	—
41	C₃×C₄₃	—	3	3	no	—
42	C₃×C₄₃	—	3	9	no	—
43	C₃×C₄₃	—	9	9	no	—
44	C₃×C₄₃	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
45	C₃×C₄₃	3₋¹⁺⁴	1	3	yes	—
46	C₃⁴	C₃×C₄₃⋊C₃	1	3	yes	—
47	C₃⁴	C₃×C₄₃⋊C₃	1	1	yes	characteristic
48	C₃×3₋¹⁺²	—	43	9	no	—
49	C₃×3₋¹⁺²	—	43	18	no	—
50	C₃×3₋¹⁺²	—	129	27	no	—
51	C₃×H(3,𝔽₃)	—	3	9	no	—
52	C₃×H(3,𝔽₃)	C₃×C₄₃⋊C₃	1	9	yes	—
53	C₃²×C₉	—	43	9	no	—
54	C₃²⋊C₉	—	43	81	no	—
55	C₄₃	#10624881	1	1	yes	characteristic, complemented, Sylow
56	C₃³	—	3	9	no	—
57	C₃³	—	9	3	no	—
58	C₃³	—	9	9	no	—
59	C₃³	C₃²⋊(C₄₃⋊C₃)	1	9	yes	—
60	C₃³	C₃²×C₄₃⋊C₃	1	9	yes	—
61	C₃³	C₃²×C₄₃⋊C₃	1	3	yes	—
62	C₃³	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
63	3₋¹⁺²	—	129	54	no	—
64	3₋¹⁺²	—	129	27	no	—
65	H(3,𝔽₃)	—	3	27	no	—
66	C₃×C₉	—	43	27	no	—
67	C₃×C₉	—	129	27	no	—
68	C₃²	—	3	3	no	—
69	C₃²	—	3	9	no	—
70	C₃²	—	3	27	no	—
71	C₃²	—	3	9	no	—
72	C₃²	—	9	9	no	—
73	C₃²	—	9	27	no	—
74	C₃²	C₃×C₃²⋊(C₄₃⋊C₃)	1	9	yes	—
75	C₃²	C₃×C₃²⋊(C₄₃⋊C₃)	1	3	yes	—
76	C₃²	C₃³×C₄₃⋊C₃	1	1	yes	characteristic
77	C₉	—	129	27	no	—
78	C₃	—	3	3	no	—
79	C₃	—	3	9	no	—
80	C₃	—	9	9	no	—
81	C₃	C₃²×C₃²⋊(C₄₃⋊C₃)	1	1	yes	characteristic
82	C₃	#23782924	1	3	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₄₃⋊C₃
commutator, socle, cosocle	C₃²×C₄₃	C₃⁴
Fitting, prime radical	C₃²×H(3,𝔽₃)×C₄₃	C₃
hypercenter	C₃²×H(3,𝔽₃)	C₄₃⋊C₃
hypocenter	C₄₃	#10624881
LCS	G ▷ C₃²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₃² ◁ 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. #23782806	C₃	semidirect
3. #23782798	C₃	semidirect
7. C₃×#11923958	C₃²	semidirect
8. C₃²×C₄₃⋊C₉	C₃²	semidirect
9. C₃²⋊(C₄₃⋊C₉)	C₃²	semidirect
55. C₄₃	#10624881	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
#23782806	3
#23782798	3
C₃²×H(3,𝔽₃)×C₄₃	3
#10624881	43