Subgroups — #44533869

Order 31347.

428 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23782798	C₃	1	3	yes	maximal
2	#23782716	C₃	1	9	yes	maximal, complemented
3	C₃³×C₉×C₄₃	C₃	1	1	yes	maximal, characteristic
4	C₃×#11923958	—	3	3	no	—
5	C₃²⋊(C₄₃⋊C₉)	—	3	6	no	—
6	C₃²⋊(C₄₃⋊C₉)	C₃²	1	9	yes	—
7	C₃⁴×C₄₃	C₃²	1	1	yes	characteristic
8	C₃²×C₉×C₄₃	—	3	1	no	—
9	C₃²×C₉×C₄₃	—	3	2	no	—
10	C₃²×C₉×C₄₃	—	3	9	no	—
11	C₃²×C₉×C₄₃	C₃²	1	3	yes	—
12	#11923958	—	9	9	no	—
13	C₃×C₄₃⋊C₉	—	3	9	no	—
14	C₃³×C₄₃	—	3	3	no	—
15	C₃³×C₄₃	—	3	6	no	—
16	C₃³×C₄₃	—	3	3	no	—
17	C₃³×C₄₃	H(3,𝔽₃)	1	1	yes	characteristic
18	C₃³×C₄₃	H(3,𝔽₃)	1	2	yes	—
19	C₃³×C₄₃	C₃³	1	1	yes	characteristic
20	C₃×C₉×C₄₃	—	3	9	no	—
21	C₃×C₉×C₄₃	—	3	18	no	—
22	C₃×C₉×C₄₃	—	3	3	no	—
23	C₃×C₉×C₄₃	—	3	6	no	—
24	C₃×C₉×C₄₃	3₋¹⁺²	1	9	yes	complemented
25	#10624768	—	43	1	no	maximal, Sylow
26	C₃²×C₄₃	—	3	27	no	—
27	C₃²×C₄₃	—	3	1	no	—
28	C₃²×C₄₃	—	3	2	no	—
29	C₃²×C₄₃	—	3	2	no	—
30	C₃²×C₄₃	—	3	1	no	—
31	C₃²×C₄₃	—	3	2	no	—
32	C₃²×C₄₃	—	3	2	no	—
33	C₃²×C₄₃	—	3	2	no	—
34	C₃²×C₄₃	#732	1	3	yes	—
35	C₃²×C₄₃	#733	1	6	yes	—
36	C₃²×C₄₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
37	C₃²×C₄₃	C₃×3₋¹⁺²	1	3	yes	—
38	C₉×C₄₃	—	3	9	no	—
39	C₄₃⋊C₉	—	9	9	no	—
40	C₃³×C₉	C₄₃⋊C₃	1	1	yes	characteristic
41	#6891	—	43	3	no	—
42	#6868	—	43	9	no	—
43	C₃×C₄₃	—	3	3	no	—
44	C₃×C₄₃	—	3	6	no	—
45	C₃×C₄₃	—	3	3	no	—
46	C₃×C₄₃	#6891	1	1	yes	characteristic
47	C₃×C₄₃	C₃×#732	1	1	yes	characteristic
48	C₃×C₄₃	C₃×#733	1	2	yes	—
49	C₃⁴	C₃×C₄₃⋊C₃	1	1	yes	characteristic
50	C₃×3₋¹⁺²	—	129	3	no	—
51	C₃²×C₉	—	3	1	no	—
52	C₃²×C₉	—	3	2	no	—
53	C₃²×C₉	—	3	9	no	—
54	C₃²×C₉	C₃×C₄₃⋊C₃	1	3	yes	—
55	C₃²⋊C₉	—	43	9	no	—
56	C₃²⋊C₉	—	129	6	no	—
57	C₄₃	#10624768	1	1	yes	characteristic, complemented, Sylow
58	C₃³	—	3	3	no	—
59	C₃³	—	3	6	no	—
60	C₃³	—	3	3	no	—
61	C₃³	C₃²⋊(C₄₃⋊C₃)	1	1	yes	characteristic
62	C₃³	C₃²⋊(C₄₃⋊C₃)	1	2	yes	—
63	C₃³	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
64	3₋¹⁺²	—	387	9	no	—
65	C₃×C₉	—	3	9	no	—
66	C₃×C₉	—	3	18	no	—
67	C₃×C₉	—	3	3	no	—
68	C₃×C₉	—	3	6	no	—
69	C₃×C₉	—	129	9	no	—
70	C₃×C₉	#11923958	1	9	yes	complemented
71	C₃²	—	3	27	no	—
72	C₃²	—	3	1	no	—
73	C₃²	—	3	2	no	—
74	C₃²	—	3	2	no	—
75	C₃²	—	3	1	no	—
76	C₃²	—	3	2	no	—
77	C₃²	—	3	2	no	—
78	C₃²	—	3	2	no	—
79	C₃²	#16812513	1	3	yes	—
80	C₃²	#16812515	1	6	yes	—
81	C₃²	C₃×C₃²⋊(C₄₃⋊C₃)	1	1	yes	characteristic
82	C₃²	C₃×#11923958	1	3	yes	—
83	C₉	—	3	9	no	—
84	C₉	—	387	9	no	—
85	C₃	—	3	3	no	—
86	C₃	—	3	6	no	—
87	C₃	—	3	3	no	—
88	C₃	#23782798	1	1	yes	characteristic
89	C₃	C₃×#16812513	1	1	yes	characteristic
90	C₃	C₃×#16812515	1	2	yes	—

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₄₃⋊C₃)
commutator, cosocle	C₃³×C₄₃	C₃³
Frattini	C₃³	C₃²×C₄₃⋊C₃
Fitting	C₃³×C₉×C₄₃	C₃
socle	C₃²×C₄₃	C₃×H(3,𝔽₃)
prime radical	C₃⁴×C₄₃	C₃²
hypercenter	C₃³×C₉	C₄₃⋊C₃
hypocenter	C₄₃	#10624768
LCS	G ▷ C₃³×C₄₃ ▷ C₃×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ C₃³×C₉

Complements

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

N	H ≅ G/N	type
2. #23782716	C₃	semidirect
24. C₃×C₉×C₄₃	3₋¹⁺²	semidirect
57. C₄₃	#10624768	semidirect
70. C₃×C₉	#11923958	semidirect

Maximal subgroups

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

subgroup	index
#23782798	3
#23782716	3
C₃³×C₉×C₄₃	3
#10624768	43