Subgroups — #41007522

Order 26973.

944 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	3₋¹⁺²×#66524	C₃	1	27	yes	maximal, complemented
2	#22582538	C₃	1	9	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	#15688506	C₃²	1	27	yes	complemented
7	#15688505	C₃²	1	27	yes	complemented
8	C₃²⋊#66524	C₃²	1	9	yes	—
9	C₃²⋊#66524	C₃²	1	18	yes	—
10	C₉×#66524	C₃²	1	27	yes	complemented
11	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
12	C₃×3₋¹⁺²×C₃₇	C₉	1	27	yes	complemented
13	C₃×3₋¹⁺²×C₃₇	C₃²	1	9	yes	—
14	C₃²×C₉×C₃₇	C₃²	1	3	yes	—
15	C₃³×C₃₇	—	9	3	no	—
16	C₃³×C₃₇	C₃×C₉	1	9	yes	—
17	C₃³×C₃₇	C₃³	1	1	yes	characteristic
18	C₃³×C₃₇	C₃³	1	3	yes	—
19	3₋¹⁺²×C₃₇	C₃×C₉	1	81	yes	complemented
20	C₃×C₉×C₃₇	C₃×C₉	1	27	yes	complemented
21	C₃×C₉×C₃₇	C₃³	1	9	yes	—
22	C₃×#66524	—	9	9	no	—
23	C₃×#66524	C₃³	1	27	yes	—
24	#10624863	—	37	1	no	maximal, Sylow
25	C₃²×C₃₇	—	3	3	no	—
26	C₃²×C₃₇	—	3	3	no	—
27	C₃²×C₃₇	—	3	6	no	—
28	C₃²×C₃₇	—	9	9	no	—
29	C₃²×C₃₇	C₃²×C₉	1	3	yes	—
30	C₃²×C₃₇	C₃²×C₉	1	9	yes	—
31	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
32	C₉×C₃₇	C₃²×C₉	1	27	yes	—
33	#66524	—	3	27	no	—
34	C₃²×3₋¹⁺²	C₃₇⋊C₃	1	1	yes	characteristic
35	C₉×3₋¹⁺²	—	37	27	no	—
36	#6887	—	37	9	no	—
37	C₃×C₃²⋊C₉	—	37	3	no	—
38	C₃×C₃₇	—	3	9	no	—
39	C₃×C₃₇	—	3	3	no	—
40	C₃×C₃₇	C₃³×C₉	1	1	yes	characteristic
41	C₃×C₃₇	3₋¹⁺⁴	1	3	yes	—
42	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
43	C₃×3₋¹⁺²	#66524	1	27	yes	complemented
44	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	9	yes	—
45	C₃²×C₉	—	37	9	no	—
46	C₃²×C₉	C₃×C₃₇⋊C₃	1	3	yes	—
47	C₉⋊C₉	—	37	27	no	—
48	C₉⋊C₉	—	37	27	no	—
49	C₃²⋊C₉	—	37	9	no	—
50	C₃²⋊C₉	—	37	18	no	—
51	C₉²	—	37	27	no	—
52	C₃₇	#10624863	1	1	yes	characteristic, complemented, Sylow
53	C₃³	—	9	3	no	—
54	C₃³	C₃×#66524	1	9	yes	—
55	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
56	C₃³	C₃²×C₃₇⋊C₃	1	3	yes	—
57	3₋¹⁺²	C₃×#66524	1	81	yes	complemented
58	C₃×C₉	—	37	27	no	—
59	C₃×C₉	—	333	9	no	—
60	C₃×C₉	C₃×#66524	1	27	yes	complemented
61	C₃×C₉	C₃²×C₃₇⋊C₃	1	9	yes	—
62	C₃²	—	3	3	no	—
63	C₃²	—	3	3	no	—
64	C₃²	—	3	6	no	—
65	C₃²	—	9	9	no	—
66	C₃²	C₃²×#66524	1	3	yes	—
67	C₃²	C₃²×#66524	1	9	yes	—
68	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
69	C₉	—	111	27	no	—
70	C₉	C₃²×#66524	1	27	yes	—
71	C₃	—	3	9	no	—
72	C₃	—	3	3	no	—
73	C₃	C₃³×#66524	1	1	yes	characteristic
74	C₃	#22582679	1	3	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₃₇⋊C₃
commutator	C₃×C₃₇	C₃³×C₉
Fitting	C₃²×3₋¹⁺²×C₃₇	C₃
socle, cosocle	C₃²×C₃₇	C₃⁴
prime radical	C₃⁴×C₃₇	C₃²
hypercenter	C₃²×3₋¹⁺²	C₃₇⋊C₃
hypocenter	C₃₇	#10624863
LCS	G ▷ C₃×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. 3₋¹⁺²×#66524	C₃	semidirect
2. #22582538	C₃	semidirect
6. #15688506	C₃²	semidirect
7. #15688505	C₃²	semidirect
10. C₉×#66524	C₃²	semidirect
12. C₃×3₋¹⁺²×C₃₇	C₉	semidirect
19. 3₋¹⁺²×C₃₇	C₃×C₉	semidirect
20. C₃×C₉×C₃₇	C₃×C₉	semidirect
43. C₃×3₋¹⁺²	#66524	semidirect
52. C₃₇	#10624863	semidirect
57. 3₋¹⁺²	C₃×#66524	semidirect
60. C₃×C₉	C₃×#66524	semidirect

Maximal subgroups

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

subgroup	index
3₋¹⁺²×#66524	3
#22582538	3
C₃×C₃²⋊#66524	3
C₃²×3₋¹⁺²×C₃₇	3
#10624863	37