Subgroups — #44535199

Order 31347.

740 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23782806	C₃	1	9	yes	maximal, complemented
2	#23782800	C₃	1	27	yes	maximal, complemented
3	#23782798	C₃	1	3	yes	maximal, complemented
4	C₄₃×C₃²⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
5	C₃×#11923958	C₃²	1	9	yes	—
6	C₃×#11923958	C₃²	1	18	yes	complemented
7	C₃²×C₄₃⋊C₉	C₃²	1	9	yes	complemented
8	C₃²⋊(C₄₃⋊C₉)	C₃²	1	18	yes	—
9	C₃²⋊(C₄₃⋊C₉)	C₃²	1	9	yes	—
10	C₃²⋊(C₄₃⋊C₉)	C₃²	1	54	yes	complemented
11	C₃⁴×C₄₃	C₃²	1	1	yes	characteristic
12	C₃×H(3,𝔽₃)×C₄₃	C₃²	1	9	yes	—
13	C₃×H(3,𝔽₃)×C₄₃	C₃²	1	3	yes	—
14	#11923958	—	9	9	no	—
15	#11923958	—	9	18	no	—
16	C₃×C₄₃⋊C₉	—	9	9	no	—
17	C₃×C₄₃⋊C₉	C₃³	1	27	yes	—
18	C₃³×C₄₃	—	9	1	no	—
19	C₃³×C₄₃	—	9	3	no	—
20	C₃³×C₄₃	C₃³	1	9	yes	—
21	C₃³×C₄₃	C₃³	1	1	yes	characteristic
22	C₃³×C₄₃	C₃³	1	3	yes	—
23	H(3,𝔽₃)×C₄₃	—	9	3	no	—
24	H(3,𝔽₃)×C₄₃	—	9	9	no	—
25	#10624910	—	43	1	no	maximal, Sylow
26	C₃²×C₄₃	—	3	1	no	—
27	C₃²×C₄₃	—	3	9	no	—
28	C₃²×C₄₃	—	3	3	no	—
29	C₃²×C₄₃	—	3	3	no	—
30	C₃²×C₄₃	—	3	27	no	—
31	C₃²×C₄₃	—	3	3	no	—
32	C₃²×C₄₃	—	3	6	no	—
33	C₃²×C₄₃	—	9	9	no	—
34	C₃²×C₄₃	C₃⁴	1	1	yes	characteristic
35	C₄₃⋊C₉	—	3	27	no	—
36	#6893	—	43	9	no	—
37	C₉⋊H(3,𝔽₃)	—	43	27	no	—
38	#6891	—	43	3	no	—
39	C₃²⋊H(3,𝔽₃)	C₄₃⋊C₃	1	1	yes	characteristic
40	C₃×C₄₃	—	9	9	no	—
41	C₃×C₄₃	—	9	1	no	—
42	C₃×C₄₃	—	9	3	no	—
43	C₃×C₄₃	H(5,𝔽₃)	1	1	yes	characteristic
44	C₃×C₄₃	3₋¹⁺⁴	1	3	yes	—
45	C₃⁴	C₃×C₄₃⋊C₃	1	1	yes	characteristic
46	C₃×3₋¹⁺²	—	43	9	no	—
47	C₃×3₋¹⁺²	—	43	18	no	—
48	C₃×H(3,𝔽₃)	C₃×C₄₃⋊C₃	1	9	yes	—
49	C₃×H(3,𝔽₃)	C₃×C₄₃⋊C₃	1	3	yes	—
50	C₃²×C₉	—	43	9	no	—
51	C₃²⋊C₉	—	43	18	no	—
52	C₃²⋊C₉	—	43	9	no	—
53	C₃²⋊C₉	—	43	54	no	—
54	C₄₃	#10624910	1	1	yes	characteristic, complemented, Sylow
55	C₃³	—	9	1	no	—
56	C₃³	—	9	3	no	—
57	C₃³	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
58	C₃³	C₃²×C₄₃⋊C₃	1	9	yes	—
59	C₃³	C₃²×C₄₃⋊C₃	1	3	yes	—
60	3₋¹⁺²	—	387	18	no	—
61	3₋¹⁺²	—	387	9	no	—
62	H(3,𝔽₃)	—	9	3	no	—
63	H(3,𝔽₃)	—	9	9	no	—
64	C₃×C₉	—	43	27	no	—
65	C₃×C₉	—	387	9	no	—
66	C₃²	—	3	9	no	—
67	C₃²	—	3	1	no	—
68	C₃²	—	3	3	no	—
69	C₃²	—	3	3	no	—
70	C₃²	—	3	27	no	—
71	C₃²	—	3	3	no	—
72	C₃²	—	3	6	no	—
73	C₃²	—	9	9	no	—
74	C₃²	C₃³×C₄₃⋊C₃	1	1	yes	characteristic
75	C₉	—	129	27	no	—
76	C₃	—	9	9	no	—
77	C₃	—	9	1	no	—
78	C₃	—	9	3	no	—
79	C₃	C₄₃⋊H(5,𝔽₃)	1	1	yes	characteristic
80	C₃	#23782924	1	3	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₄₃⋊C₃
commutator, socle, cosocle	C₃²×C₄₃	C₃⁴
Fitting, prime radical	C₄₃×C₃²⋊H(3,𝔽₃)	C₃
hypercenter	C₃²⋊H(3,𝔽₃)	C₄₃⋊C₃
hypocenter	C₄₃	#10624910
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. #23782806	C₃	semidirect
2. #23782800	C₃	semidirect
3. #23782798	C₃	semidirect
6. C₃×#11923958	C₃²	semidirect
7. C₃²×C₄₃⋊C₉	C₃²	semidirect
10. C₃²⋊(C₄₃⋊C₉)	C₃²	semidirect
54. C₄₃	#10624910	semidirect

Maximal subgroups

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

subgroup	index
#23782806	3
#23782800	3
#23782798	3
C₄₃×C₃²⋊H(3,𝔽₃)	3
#10624910	43