Subgroups — #41007680

Order 26973.

728 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582578	C₃	1	9	yes	maximal, complemented
2	#22582568	C₃	1	27	yes	maximal, complemented
3	C₃×C₃²⋊#66524	C₃	1	3	yes	maximal
4	C₃²×3₋¹⁺²×C₃₇	C₃	1	1	yes	maximal, characteristic
5	C₃²×#66524	C₃²	1	9	yes	—
6	#15688504	C₃²	1	81	yes	complemented
7	C₃²⋊#66524	C₃²	1	18	yes	—
8	C₃²⋊#66524	C₃²	1	9	yes	—
9	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
10	C₃×3₋¹⁺²×C₃₇	—	3	9	no	—
11	C₃×3₋¹⁺²×C₃₇	C₃²	1	9	yes	—
12	C₃²×C₉×C₃₇	C₃²	1	3	yes	—
13	C₃³×C₃₇	—	9	3	no	—
14	C₃³×C₃₇	3₋¹⁺²	1	9	yes	—
15	C₃³×C₃₇	C₃³	1	1	yes	characteristic
16	C₃³×C₃₇	C₃³	1	3	yes	—
17	3₋¹⁺²×C₃₇	—	3	27	no	—
18	C₃×C₉×C₃₇	—	3	9	no	—
19	C₃×C₉×C₃₇	C₃³	1	9	yes	—
20	C₃×#66524	—	9	9	no	—
21	C₃×#66524	C₃³	1	27	yes	—
22	#10624886	—	37	1	no	maximal, Sylow
23	C₃²×C₃₇	—	3	3	no	—
24	C₃²×C₃₇	—	3	3	no	—
25	C₃²×C₃₇	—	3	6	no	—
26	C₃²×C₃₇	—	9	9	no	—
27	C₃²×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
28	C₃²×C₃₇	C₃×3₋¹⁺²	1	9	yes	—
29	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
30	C₉×C₃₇	—	3	9	no	—
31	#66524	—	3	27	no	—
32	C₃²×3₋¹⁺²	C₃₇⋊C₃	1	1	yes	characteristic
33	#6895	—	37	9	no	—
34	#6894	—	37	27	no	—
35	C₃×C₃²⋊C₉	—	37	3	no	—
36	C₃×C₃₇	—	3	9	no	—
37	C₃×C₃₇	—	3	3	no	—
38	C₃×C₃₇	C₃²×3₋¹⁺²	1	1	yes	characteristic
39	C₃×C₃₇	3₋¹⁺⁴	1	3	yes	—
40	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
41	C₃×3₋¹⁺²	—	3	9	no	—
42	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	9	yes	—
43	C₃²×C₉	—	37	9	no	—
44	C₃²×C₉	C₃×C₃₇⋊C₃	1	3	yes	—
45	C₉⋊C₉	—	37	81	no	—
46	C₃²⋊C₉	—	37	18	no	—
47	C₃²⋊C₉	—	37	9	no	—
48	C₃₇	#10624886	1	1	yes	characteristic, complemented, Sylow
49	C₃³	—	9	3	no	—
50	C₃³	#11758608	1	9	yes	—
51	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
52	C₃³	C₃²×C₃₇⋊C₃	1	3	yes	—
53	3₋¹⁺²	—	3	27	no	—
54	C₃×C₉	—	3	9	no	—
55	C₃×C₉	—	37	27	no	—
56	C₃×C₉	—	333	9	no	—
57	C₃×C₉	C₃²×C₃₇⋊C₃	1	9	yes	—
58	C₃²	—	3	3	no	—
59	C₃²	—	3	3	no	—
60	C₃²	—	3	6	no	—
61	C₃²	—	9	9	no	—
62	C₃²	C₃×#11758608	1	3	yes	—
63	C₃²	C₃×#11758608	1	9	yes	—
64	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
65	C₉	—	3	9	no	—
66	C₉	—	111	27	no	—
67	C₃	—	3	9	no	—
68	C₃	—	3	3	no	—
69	C₃	C₃²×#11758608	1	1	yes	characteristic
70	C₃	#22582679	1	3	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₃₇⋊C₃
commutator, socle, cosocle	C₃²×C₃₇	C₃⁴
Fitting	C₃²×3₋¹⁺²×C₃₇	C₃
prime radical	C₃⁴×C₃₇	C₃²
hypercenter	C₃²×3₋¹⁺²	C₃₇⋊C₃
hypocenter	C₃₇	#10624886
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃²×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
1. #22582578	C₃	semidirect
2. #22582568	C₃	semidirect
6. #15688504	C₃²	semidirect
48. C₃₇	#10624886	semidirect

Maximal subgroups

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

subgroup	index
#22582578	3
#22582568	3
C₃×C₃²⋊#66524	3
C₃²×3₋¹⁺²×C₃₇	3
#10624886	37