Subgroups — #41006519

Order 26973.

572 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#15688519	C₃	1	9	yes	maximal, complemented
2	#22582550	C₃	1	3	yes	maximal, complemented
3	C₃²×3₋¹⁺²×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	9	no	—
5	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	9	yes	—
6	#15688519	—	3	27	no	—
7	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
8	C₃×3₋¹⁺²×C₃₇	—	3	3	no	—
9	C₃×3₋¹⁺²×C₃₇	—	3	9	no	—
10	C₃²×C₉×C₃₇	C₃²	1	3	yes	—
11	C₃³×C₃₇	—	3	3	no	—
12	C₃³×C₃₇	—	9	3	no	—
13	C₃³×C₃₇	H(3,𝔽₃)	1	3	yes	—
14	C₃³×C₃₇	C₃³	1	1	yes	characteristic
15	C₃²×C₃₇⋊C₃	—	3	9	no	—
16	3₋¹⁺²×C₃₇	—	3	27	no	—
17	C₃×C₉×C₃₇	—	3	6	no	—
18	C₃×C₉×C₃₇	—	3	3	no	—
19	C₃×C₉×C₃₇	H(3,𝔽₃)	1	9	yes	complemented
20	C₃²⋊(C₃₇⋊C₃)	—	3	27	no	—
21	C₃²⋊(C₃₇⋊C₃)	—	9	27	no	—
22	#10624737	—	37	1	no	maximal, Sylow
23	C₃²×C₃₇	—	3	3	no	—
24	C₃²×C₃₇	—	3	3	no	—
25	C₃²×C₃₇	—	3	9	no	—
26	C₃²×C₃₇	—	9	9	no	—
27	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	3	yes	—
28	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
29	C₃×C₃₇⋊C₃	—	9	9	no	—
30	C₃×C₃₇⋊C₃	—	9	27	no	—
31	C₉×C₃₇	—	3	9	no	—
32	C₃²×3₋¹⁺²	C₃₇⋊C₃	1	1	yes	characteristic, complemented
33	C₃×#732	—	37	9	no	—
34	C₃²⋊H(3,𝔽₃)	—	37	3	no	—
35	C₃×C₃₇	—	3	3	no	—
36	C₃×C₃₇	—	9	3	no	—
37	C₃×C₃₇	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
38	C₃×C₃₇	#6911	1	3	yes	—
39	C₃₇⋊C₃	—	27	9	no	—
40	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
41	C₃×3₋¹⁺²	—	3	3	no	—
42	C₃×3₋¹⁺²	—	3	9	no	—
43	C₃×H(3,𝔽₃)	—	37	9	no	—
44	C₃×H(3,𝔽₃)	—	111	9	no	—
45	C₃²×C₉	C₃×C₃₇⋊C₃	1	3	yes	—
46	#732	—	111	27	no	—
47	C₃₇	#10624737	1	1	yes	characteristic, complemented, Sylow
48	C₃³	—	3	3	no	—
49	C₃³	—	9	3	no	—
50	C₃³	—	111	9	no	—
51	C₃³	C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
52	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
53	3₋¹⁺²	—	3	27	no	—
54	H(3,𝔽₃)	—	333	27	no	—
55	H(3,𝔽₃)	—	111	27	no	—
56	C₃×C₉	—	3	6	no	—
57	C₃×C₉	—	3	3	no	—
58	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	9	yes	complemented
59	C₃²	—	3	3	no	—
60	C₃²	—	3	3	no	—
61	C₃²	—	3	9	no	—
62	C₃²	—	9	9	no	—
63	C₃²	—	333	9	no	—
64	C₃²	—	333	27	no	—
65	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
66	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
67	C₉	—	3	9	no	—
68	C₃	—	3	3	no	—
69	C₃	—	9	3	no	—
70	C₃	—	999	9	no	—
71	C₃	#22582550	1	1	yes	characteristic
72	C₃	#22582648	1	3	yes	—

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃²×3₋¹⁺²×C₃₇	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
hypercenter	C₃²×3₋¹⁺²	C₃₇⋊C₃
hypocenter	C₃₇	#10624737
LCS	G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ C₃²×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
1. C₃×#15688519	C₃	semidirect
2. #22582550	C₃	semidirect
3. C₃²×3₋¹⁺²×C₃₇	C₃	semidirect
19. C₃×C₉×C₃₇	H(3,𝔽₃)	semidirect
32. C₃²×3₋¹⁺²	C₃₇⋊C₃	semidirect
47. C₃₇	#10624737	semidirect
58. C₃×C₉	C₃²⋊(C₃₇⋊C₃)	semidirect

Maximal subgroups

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

subgroup	index
C₃×#15688519	3
#22582550	3
C₃²×3₋¹⁺²×C₃₇	3
#10624737	37