Subgroups — #41006738

Order 26973.

356 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582554	C₃	1	3	yes	maximal
2	#22582470	C₃	1	9	yes	maximal, complemented
3	C₃²×3₋¹⁺²×C₃₇	C₃	1	1	yes	maximal, characteristic
4	C₃×#11758608	—	3	3	no	—
5	C₃²⋊#66524	—	3	6	no	—
6	C₃²⋊#66524	C₃²	1	9	yes	—
7	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
8	C₃×3₋¹⁺²×C₃₇	—	3	2	no	—
9	C₃×3₋¹⁺²×C₃₇	—	3	1	no	—
10	C₃×3₋¹⁺²×C₃₇	—	3	9	no	—
11	C₃²×C₉×C₃₇	C₃²	1	3	yes	—
12	C₃³×C₃₇	—	3	3	no	—
13	C₃³×C₃₇	—	9	3	no	—
14	C₃³×C₃₇	H(3,𝔽₃)	1	2	yes	—
15	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
16	C₃³×C₃₇	C₃³	1	1	yes	characteristic
17	3₋¹⁺²×C₃₇	—	3	18	no	—
18	3₋¹⁺²×C₃₇	—	3	9	no	—
19	C₃×C₉×C₃₇	—	3	3	no	—
20	C₃×C₉×C₃₇	—	3	6	no	—
21	C₃×C₉×C₃₇	3₋¹⁺²	1	9	yes	complemented
22	#11758608	—	9	9	no	—
23	C₃×#66524	—	3	9	no	—
24	#10624771	—	37	1	no	maximal, Sylow
25	C₃²×C₃₇	—	3	3	no	—
26	C₃²×C₃₇	—	3	2	no	—
27	C₃²×C₃₇	—	3	1	no	—
28	C₃²×C₃₇	—	3	3	no	—
29	C₃²×C₃₇	—	3	6	no	—
30	C₃²×C₃₇	—	9	9	no	—
31	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
32	C₃²×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
33	C₉×C₃₇	—	3	9	no	—
34	#66524	—	9	9	no	—
35	C₃²×3₋¹⁺²	C₃₇⋊C₃	1	1	yes	characteristic
36	#6891	—	37	3	no	—
37	#6867	—	37	9	no	—
38	C₃×C₃₇	—	3	3	no	—
39	C₃×C₃₇	—	9	1	no	—
40	C₃×C₃₇	—	9	2	no	—
41	C₃×C₃₇	#6891	1	1	yes	characteristic
42	C₃×C₃₇	#6910	1	3	yes	—
43	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
44	C₃×3₋¹⁺²	—	3	2	no	—
45	C₃×3₋¹⁺²	—	3	1	no	—
46	C₃×3₋¹⁺²	—	3	9	no	—
47	C₃×3₋¹⁺²	—	111	3	no	—
48	C₃²×C₉	C₃×C₃₇⋊C₃	1	3	yes	—
49	C₃²⋊C₉	—	37	9	no	—
50	C₃²⋊C₉	—	111	6	no	—
51	C₃₇	#10624771	1	1	yes	characteristic, complemented, Sylow
52	C₃³	—	3	3	no	—
53	C₃³	—	9	3	no	—
54	C₃³	C₃²⋊(C₃₇⋊C₃)	1	2	yes	—
55	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
56	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
57	3₋¹⁺²	—	3	18	no	—
58	3₋¹⁺²	—	3	9	no	—
59	3₋¹⁺²	—	333	9	no	—
60	C₃×C₉	—	3	3	no	—
61	C₃×C₉	—	3	6	no	—
62	C₃×C₉	—	111	9	no	—
63	C₃×C₉	#11758608	1	9	yes	complemented
64	C₃²	—	3	3	no	—
65	C₃²	—	3	2	no	—
66	C₃²	—	3	1	no	—
67	C₃²	—	3	3	no	—
68	C₃²	—	3	6	no	—
69	C₃²	—	9	9	no	—
70	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
71	C₃²	C₃×#11758608	1	3	yes	—
72	C₉	—	3	9	no	—
73	C₉	—	333	9	no	—
74	C₃	—	3	3	no	—
75	C₃	—	9	1	no	—
76	C₃	—	9	2	no	—
77	C₃	#22582554	1	1	yes	characteristic
78	C₃	#22582643	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)
prime radical	C₃⁴×C₃₇	C₃²
hypercenter	C₃²×3₋¹⁺²	C₃₇⋊C₃
hypocenter	C₃₇	#10624771
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
2. #22582470	C₃	semidirect
21. C₃×C₉×C₃₇	3₋¹⁺²	semidirect
51. C₃₇	#10624771	semidirect
63. C₃×C₉	#11758608	semidirect

Maximal subgroups

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

subgroup	index
#22582554	3
#22582470	3
C₃²×3₋¹⁺²×C₃₇	3
#10624771	37