Subgroups — #44534039

Order 31347.

256 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#16812513	C₃	1	1	yes	maximal, characteristic
2	#23782802	C₃	1	3	yes	maximal, complemented
3	#23782721	C₃	1	6	yes	maximal, complemented
4	#23782716	C₃	1	2	yes	maximal, complemented
5	C₃×C₄₃×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃×#11923957	—	3	3	no	—
7	C₃×C₃²⋊(C₄₃⋊C₃)	C₃²	1	3	yes	—
8	#16812513	—	3	3	no	—
9	C₃²⋊(C₄₃⋊C₉)	C₃²	1	6	yes	—
10	(C₃×C₄₃⋊C₃)⋊C₉	—	3	6	no	—
11	C₃²×C₉×C₄₃	C₃²	1	1	yes	characteristic
12	C₃²×C₉×C₄₃	C₃²	1	2	yes	—
13	C₃²×C₉×C₄₃	C₃²	1	1	yes	characteristic
14	C₄₃×C₉⋊C₉	—	3	3	no	—
15	C₃²×C₄₃⋊C₃	—	3	3	no	—
16	#11923957	—	9	9	no	—
17	C₃²⋊(C₄₃⋊C₃)	—	3	3	no	—
18	C₃²⋊(C₄₃⋊C₃)	—	3	6	no	—
19	C₃×C₄₃⋊C₉	—	3	6	no	—
20	C₃³×C₄₃	C₃³	1	1	yes	characteristic
21	C₃×C₉×C₄₃	—	3	1	no	—
22	C₃×C₉×C₄₃	—	3	2	no	—
23	C₃×C₉×C₄₃	—	3	2	no	—
24	C₃×C₉×C₄₃	—	9	1	no	—
25	C₃×C₉×C₄₃	—	9	2	no	—
26	C₃×C₉×C₄₃	H(3,𝔽₃)	1	3	yes	—
27	C₃×C₉×C₄₃	3₋¹⁺²	1	3	yes	complemented
28	#10624790	—	43	1	no	maximal, Sylow
29	C₃²×C₄₃	—	3	2	no	—
30	C₃²×C₄₃	—	3	1	no	—
31	C₃²×C₄₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
32	C₃²×C₄₃	C₃×3₋¹⁺²	1	1	yes	characteristic
33	C₃²×C₄₃	#737	1	2	yes	—
34	C₃×C₄₃⋊C₃	—	9	6	no	—
35	C₃×C₄₃⋊C₃	—	9	3	no	—
36	C₃×C₄₃⋊C₃	—	9	3	no	—
37	C₉×C₄₃	—	3	3	no	—
38	C₉×C₄₃	—	3	3	no	—
39	C₉×C₄₃	—	9	2	no	—
40	C₄₃⋊C₉	—	9	6	no	—
41	C₃×#732	—	43	1	no	—
42	C₉⋊H(3,𝔽₃)	—	43	3	no	—
43	C₃×C₉⋊C₉	C₄₃⋊C₃	1	1	yes	characteristic, complemented
44	H(3,𝔽₃)⋊C₉	—	43	6	no	—
45	#6868	—	43	2	no	—
46	C₃×C₄₃	—	3	1	no	—
47	C₃×C₄₃	—	3	2	no	—
48	C₃×C₄₃	C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
49	C₃×C₄₃	#6911	1	1	yes	characteristic
50	C₃×C₄₃	#6913	1	2	yes	—
51	C₄₃⋊C₃	—	27	3	no	—
52	C₃×3₋¹⁺²	—	129	3	no	—
53	C₃×H(3,𝔽₃)	—	43	3	no	—
54	C₃²×C₉	C₃×C₄₃⋊C₃	1	1	yes	characteristic
55	C₃²×C₉	C₃×C₄₃⋊C₃	1	2	yes	—
56	C₃²×C₉	C₃×C₄₃⋊C₃	1	1	yes	characteristic
57	#732	—	129	3	no	—
58	C₉⋊C₉	—	3	3	no	—
59	C₃²⋊C₉	—	43	6	no	—
60	C₃²⋊C₉	—	129	6	no	—
61	C₄₃	#10624790	1	1	yes	characteristic, complemented, Sylow
62	C₃³	—	129	3	no	—
63	C₃³	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
64	3₋¹⁺²	—	387	9	no	—
65	H(3,𝔽₃)	—	129	3	no	—
66	H(3,𝔽₃)	—	129	6	no	—
67	C₃×C₉	—	3	1	no	—
68	C₃×C₉	—	3	2	no	—
69	C₃×C₉	—	3	2	no	—
70	C₃×C₉	—	9	1	no	—
71	C₃×C₉	—	9	2	no	—
72	C₃×C₉	—	129	6	no	—
73	C₃×C₉	C₃²⋊(C₄₃⋊C₃)	1	3	yes	—
74	C₃×C₉	#11923957	1	3	yes	complemented
75	C₃²	—	3	1	no	—
76	C₃²	—	3	2	no	—
77	C₃²	—	387	3	no	—
78	C₃²	—	387	3	no	—
79	C₃²	—	387	6	no	—
80	C₃²	C₃×C₃²⋊(C₄₃⋊C₃)	1	1	yes	characteristic
81	C₃²	C₃×#11923957	1	1	yes	characteristic
82	C₃²	#16812525	1	2	yes	—
83	C₉	—	3	3	no	—
84	C₉	—	3	3	no	—
85	C₉	—	9	2	no	—
86	C₉	—	387	6	no	—
87	C₃	—	3	1	no	—
88	C₃	—	3	2	no	—
89	C₃	—	1161	3	no	—
90	C₃	#23782802	1	1	yes	characteristic
91	C₃	#23782892	1	1	yes	characteristic
92	C₃	#23782905	1	2	yes	—

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₄₃⋊C₃)
commutator, cosocle	C₃³×C₄₃	C₃³
Frattini	C₃³	C₃²×C₄₃⋊C₃
Fitting	C₃×C₄₃×C₉⋊C₉	C₃
socle	C₃²×C₄₃	C₃×H(3,𝔽₃)
prime radical	C₃×#16812513	C₃
hypercenter	C₃×C₉⋊C₉	C₄₃⋊C₃
hypocenter	C₄₃	#10624790
LCS	G ▷ C₃³×C₄₃ ▷ C₃×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₃² ◁ C₃²×C₉ ◁ C₃×C₉⋊C₉

Complements

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

N	H ≅ G/N	type
2. #23782802	C₃	semidirect
3. #23782721	C₃	semidirect
4. #23782716	C₃	semidirect
5. C₃×C₄₃×C₉⋊C₉	C₃	semidirect
27. C₃×C₉×C₄₃	3₋¹⁺²	semidirect
43. C₃×C₉⋊C₉	C₄₃⋊C₃	semidirect
61. C₄₃	#10624790	semidirect
74. C₃×C₉	#11923957	semidirect

Maximal subgroups

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

subgroup	index
C₃×#16812513	3
#23782802	3
#23782721	3
#23782716	3
C₃×C₄₃×C₉⋊C₉	3
#10624790	43