Subgroups — C₃×#22582510 #41007331

Order 26973.

224 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#15688521	C₃	1	3	yes	maximal
2	#22582510	C₃	1	9	yes	maximal, direct product factor
3	C₃×C₉²×C₃₇	C₃	1	1	yes	maximal, characteristic
4	C₃×#11758608	—	3	3	no	—
5	#15688521	—	3	6	no	—
6	#15688521	C₃²	1	9	yes	—
7	C₃²×C₉×C₃₇	—	3	1	no	—
8	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
9	C₉²×C₃₇	—	3	2	no	—
10	C₉²×C₃₇	C₃²	1	3	yes	—
11	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
12	C₃×C₉×C₃₇	—	3	9	no	—
13	C₃×C₉×C₃₇	—	3	2	no	—
14	C₃×C₉×C₃₇	—	3	1	no	—
15	C₃×C₉×C₃₇	—	3	3	no	—
16	C₃×C₉×C₃₇	H(3,𝔽₃)	1	2	yes	—
17	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
18	#11758608	—	3	9	no	—
19	#11758608	—	9	6	no	—
20	C₃×#66524	—	9	3	no	—
21	C₃×#6880	—	37	1	no	maximal, Sylow
22	C₃²×C₃₇	—	3	3	no	—
23	C₃²×C₃₇	#732	1	2	yes	—
24	C₃²×C₃₇	#732	1	1	yes	characteristic
25	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
26	C₉×C₃₇	—	3	9	no	—
27	C₉×C₃₇	—	3	2	no	—
28	C₉×C₃₇	—	3	1	no	—
29	#66524	—	9	9	no	—
30	C₃×#733	—	37	3	no	—
31	C₃×C₉²	C₃₇⋊C₃	1	1	yes	characteristic
32	#6880	—	37	9	no	—
33	C₃×C₃₇	—	3	1	no	—
34	C₃×C₃₇	—	3	2	no	—
35	C₃×C₃₇	#6880	1	3	yes	complemented
36	C₃×C₃₇	C₃×#732	1	1	yes	characteristic
37	C₃×3₋¹⁺²	—	111	3	no	—
38	C₃²×C₉	—	3	1	no	—
39	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
40	#733	—	37	9	no	—
41	#733	—	111	6	no	—
42	C₉²	—	3	2	no	—
43	C₉²	C₃×C₃₇⋊C₃	1	3	yes	—
44	C₃₇	C₃×#6880	1	1	yes	characteristic, complemented, Sylow
45	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
46	3₋¹⁺²	—	333	6	no	—
47	3₋¹⁺²	—	111	9	no	—
48	C₃×C₉	—	3	9	no	—
49	C₃×C₉	—	3	2	no	—
50	C₃×C₉	—	3	1	no	—
51	C₃×C₉	—	3	3	no	—
52	C₃×C₉	—	333	3	no	—
53	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	2	yes	—
54	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
55	C₃²	—	3	3	no	—
56	C₃²	#15688519	1	2	yes	—
57	C₃²	#15688519	1	1	yes	characteristic
58	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
59	C₉	—	3	9	no	—
60	C₉	—	3	2	no	—
61	C₉	—	3	1	no	—
62	C₉	—	333	9	no	—
63	C₃	—	3	1	no	—
64	C₃	—	3	2	no	—
65	C₃	#22582510	1	3	yes	direct product factor
66	C₃	C₃×#15688519	1	1	yes	characteristic

	subgroup	G/H
center	C₃²	#15688519
commutator, cosocle	C₃×C₉×C₃₇	C₃³
Frattini	C₃×C₉	C₃²×C₃₇⋊C₃
Fitting	C₃×C₉²×C₃₇	C₃
socle	C₃²×C₃₇	#732
prime radical	C₃³×C₃₇	H(3,𝔽₃)
hypercenter	C₃×C₉²	C₃₇⋊C₃
hypocenter	C₃₇	C₃×#6880
LCS	G ▷ C₃×C₉×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃²×C₉ ◁ C₃×C₉²

Complements

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

N	H ≅ G/N	type
2. #22582510	C₃	direct (N × H)
35. C₃×C₃₇	#6880	semidirect
44. C₃₇	C₃×#6880	semidirect
65. C₃	#22582510	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#15688521	3
#22582510	3
C₃×C₉²×C₃₇	3
C₃×#6880	37

Subgroups — C3×#22582510 #41007331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#22582510 #41007331