Subgroups — #44534788

Order 31347.

1718 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₃²⋊(C₄₃⋊C₃)	C₃	1	3	yes	maximal, complemented
2	#23782794	C₃	1	9	yes	maximal, complemented
3	(C₃×C₄₃⋊C₃)⋊H(3,𝔽₃)	C₃	1	27	yes	maximal, complemented
4	C₃²×H(3,𝔽₃)×C₄₃	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃³×C₄₃⋊C₃	C₃²	1	9	yes	complemented
6	C₃×C₃²⋊(C₄₃⋊C₃)	—	3	27	no	—
7	C₃×C₃²⋊(C₄₃⋊C₃)	C₃²	1	18	yes	complemented
8	C₃×C₃²⋊(C₄₃⋊C₃)	C₃²	1	9	yes	complemented
9	C₃×C₃²⋊(C₄₃⋊C₃)	C₃²	1	81	yes	complemented
10	C₃⁴×C₄₃	C₃²	1	1	yes	characteristic
11	C₃⁴×C₄₃	C₃²	1	3	yes	complemented
12	C₃×H(3,𝔽₃)×C₄₃	—	3	9	no	—
13	C₃×H(3,𝔽₃)×C₄₃	C₃²	1	9	yes	complemented
14	C₃²×C₄₃⋊C₃	—	3	27	no	—
15	C₃²×C₄₃⋊C₃	—	9	27	no	—
16	C₃²×C₄₃⋊C₃	C₃³	1	27	yes	—
17	C₃²⋊(C₄₃⋊C₃)	—	3	54	no	—
18	C₃²⋊(C₄₃⋊C₃)	—	3	27	no	—
19	C₃²⋊(C₄₃⋊C₃)	—	9	81	no	—
20	C₃³×C₄₃	—	3	9	no	—
21	C₃³×C₄₃	—	9	3	no	—
22	C₃³×C₄₃	—	9	9	no	—
23	C₃³×C₄₃	H(3,𝔽₃)	1	9	yes	complemented
24	C₃³×C₄₃	C₃³	1	9	yes	—
25	C₃³×C₄₃	C₃³	1	3	yes	—
26	C₃³×C₄₃	C₃³	1	1	yes	characteristic
27	H(3,𝔽₃)×C₄₃	—	3	27	no	—
28	#10624880	—	43	1	no	maximal, Sylow
29	C₃²×C₄₃	—	3	3	no	—
30	C₃²×C₄₃	—	3	9	no	—
31	C₃²×C₄₃	—	3	9	no	—
32	C₃²×C₄₃	—	3	27	no	—
33	C₃²×C₄₃	—	9	9	no	—
34	C₃²×C₄₃	—	9	27	no	—
35	C₃²×C₄₃	C₃×H(3,𝔽₃)	1	9	yes	—
36	C₃²×C₄₃	C₃×H(3,𝔽₃)	1	3	yes	—
37	C₃²×C₄₃	C₃⁴	1	1	yes	characteristic
38	C₃×C₄₃⋊C₃	—	3	27	no	—
39	C₃×C₄₃⋊C₃	—	3	81	no	—
40	C₃×C₄₃⋊C₃	—	9	81	no	—
41	C₃²×H(3,𝔽₃)	—	43	3	no	—
42	C₃²×H(3,𝔽₃)	C₄₃⋊C₃	1	1	yes	characteristic, complemented
43	C₃²⋊H(3,𝔽₃)	—	43	9	no	—
44	C₃²⋊H(3,𝔽₃)	—	43	27	no	—
45	C₃×C₄₃	—	3	9	no	—
46	C₃×C₄₃	—	3	3	no	—
47	C₃×C₄₃	—	9	9	no	—
48	C₃×C₄₃	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
49	C₃×C₄₃	H(5,𝔽₃)	1	3	yes	—
50	C₄₃⋊C₃	—	9	27	no	—
51	C₃⁴	—	43	9	no	—
52	C₃⁴	C₃×C₄₃⋊C₃	1	1	yes	characteristic
53	C₃⁴	C₃×C₄₃⋊C₃	1	3	yes	complemented
54	C₃×H(3,𝔽₃)	—	3	9	no	—
55	C₃×H(3,𝔽₃)	—	43	9	no	—
56	C₃×H(3,𝔽₃)	—	43	18	no	—
57	C₃×H(3,𝔽₃)	—	43	81	no	—
58	C₃×H(3,𝔽₃)	—	129	27	no	—
59	C₃×H(3,𝔽₃)	C₃×C₄₃⋊C₃	1	9	yes	complemented
60	C₄₃	#10624880	1	1	yes	characteristic, complemented, Sylow
61	C₃³	—	3	9	no	—
62	C₃³	—	9	3	no	—
63	C₃³	—	9	9	no	—
64	C₃³	—	43	27	no	—
65	C₃³	—	129	27	no	—
66	C₃³	—	387	27	no	—
67	C₃³	C₃²⋊(C₄₃⋊C₃)	1	9	yes	complemented
68	C₃³	C₃²×C₄₃⋊C₃	1	9	yes	—
69	C₃³	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
70	C₃³	C₃²×C₄₃⋊C₃	1	3	yes	—
71	H(3,𝔽₃)	—	3	27	no	—
72	H(3,𝔽₃)	—	129	54	no	—
73	H(3,𝔽₃)	—	129	27	no	—
74	H(3,𝔽₃)	—	387	81	no	—
75	C₃²	—	3	3	no	—
76	C₃²	—	3	9	no	—
77	C₃²	—	3	27	no	—
78	C₃²	—	3	9	no	—
79	C₃²	—	9	9	no	—
80	C₃²	—	9	27	no	—
81	C₃²	—	129	81	no	—
82	C₃²	—	129	27	no	—
83	C₃²	—	387	81	no	—
84	C₃²	C₃×C₃²⋊(C₄₃⋊C₃)	1	9	yes	—
85	C₃²	C₃×C₃²⋊(C₄₃⋊C₃)	1	3	yes	—
86	C₃²	C₃³×C₄₃⋊C₃	1	1	yes	characteristic
87	C₃	—	3	3	no	—
88	C₃	—	3	9	no	—
89	C₃	—	9	9	no	—
90	C₃	—	387	27	no	—
91	C₃	C₃²×C₃²⋊(C₄₃⋊C₃)	1	1	yes	characteristic
92	C₃	C₄₃⋊H(5,𝔽₃)	1	3	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₄₃⋊C₃
commutator, socle, cosocle	C₃²×C₄₃	C₃⁴
Fitting	C₃²×H(3,𝔽₃)×C₄₃	C₃
hypercenter	C₃²×H(3,𝔽₃)	C₄₃⋊C₃
hypocenter	C₄₃	#10624880
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×C₃²⋊(C₄₃⋊C₃)	C₃	semidirect
2. #23782794	C₃	semidirect
3. (C₃×C₄₃⋊C₃)⋊H(3,𝔽₃)	C₃	semidirect
4. C₃²×H(3,𝔽₃)×C₄₃	C₃	semidirect
5. C₃³×C₄₃⋊C₃	C₃²	semidirect
7. C₃×C₃²⋊(C₄₃⋊C₃)	C₃²	semidirect
8. C₃×C₃²⋊(C₄₃⋊C₃)	C₃²	semidirect
9. C₃×C₃²⋊(C₄₃⋊C₃)	C₃²	semidirect
11. C₃⁴×C₄₃	C₃²	semidirect
13. C₃×H(3,𝔽₃)×C₄₃	C₃²	semidirect
23. C₃³×C₄₃	H(3,𝔽₃)	semidirect
42. C₃²×H(3,𝔽₃)	C₄₃⋊C₃	semidirect
53. C₃⁴	C₃×C₄₃⋊C₃	semidirect
59. C₃×H(3,𝔽₃)	C₃×C₄₃⋊C₃	semidirect
60. C₄₃	#10624880	semidirect
67. C₃³	C₃²⋊(C₄₃⋊C₃)	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₃²⋊(C₄₃⋊C₃)	3
#23782794	3
(C₃×C₄₃⋊C₃)⋊H(3,𝔽₃)	3
C₃²×H(3,𝔽₃)×C₄₃	3
#10624880	43