Subgroups — #41008065

Order 26973.

596 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582601	C₃	1	9	yes	maximal, complemented
2	#22582590	C₃	1	9	yes	maximal, complemented
3	#22582585	C₃	1	18	yes	maximal, complemented
4	#22582554	C₃	1	3	yes	maximal
5	C₃₇×#6891	C₃	1	1	yes	maximal, characteristic
6	C₃×#11758608	C₃²	1	9	yes	—
7	#15688506	C₃²	1	9	yes	complemented
8	#15688505	C₃²	1	9	yes	complemented
9	#15688504	C₃²	1	54	yes	complemented
10	C₃²⋊#66524	C₃²	1	18	yes	—
11	C₃²⋊#66524	C₃²	1	9	yes	—
12	C₉×#66524	C₃²	1	9	yes	complemented
13	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
14	C₃×3₋¹⁺²×C₃₇	C₃²	1	3	yes	—
15	C₃₇×C₃²⋊C₉	C₃²	1	6	yes	—
16	C₃₇×C₃²⋊C₉	C₃²	1	3	yes	—
17	C₃³×C₃₇	—	9	3	no	—
18	C₃³×C₃₇	—	9	1	no	—
19	C₃³×C₃₇	C₃³	1	2	yes	—
20	C₃³×C₃₇	C₃³	1	1	yes	characteristic
21	C₃³×C₃₇	C₃³	1	1	yes	characteristic
22	3₋¹⁺²×C₃₇	—	9	3	no	—
23	C₃×C₉×C₃₇	C₃³	1	9	yes	—
24	#11758608	—	9	9	no	—
25	C₃×#66524	C₃³	1	27	yes	—
26	#10624912	—	37	1	no	maximal, Sylow
27	C₃²×C₃₇	—	3	3	no	—
28	C₃²×C₃₇	—	3	1	no	—
29	C₃²×C₃₇	—	3	6	no	—
30	C₃²×C₃₇	—	3	3	no	—
31	C₃²×C₃₇	—	3	2	no	—
32	C₃²×C₃₇	—	3	1	no	—
33	C₃²×C₃₇	—	9	9	no	—
34	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
35	C₉×C₃₇	—	3	9	no	—
36	#66524	—	3	27	no	—
37	#6900	—	37	9	no	—
38	#6898	—	37	9	no	—
39	#6897	—	37	18	no	—
40	#6891	—	37	3	no	—
41	#6891	C₃₇⋊C₃	1	1	yes	characteristic
42	C₃×C₃₇	—	9	1	no	—
43	C₃×C₃₇	—	9	2	no	—
44	C₃×C₃₇	—	9	1	no	—
45	C₃×C₃₇	3₋¹⁺⁴	1	3	yes	—
46	C₃×C₃₇	3₋¹⁺⁴	1	1	yes	characteristic
47	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
48	C₃×3₋¹⁺²	—	37	9	no	—
49	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	3	yes	—
50	C₉⋊C₉	—	37	54	no	—
51	C₉⋊C₉	—	37	9	no	—
52	C₉⋊C₉	—	37	9	no	—
53	C₃²⋊C₉	—	37	18	no	—
54	C₃²⋊C₉	—	37	9	no	—
55	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	6	yes	—
56	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	—
57	C₉²	—	37	9	no	—
58	C₃₇	#10624912	1	1	yes	characteristic, complemented, Sylow
59	C₃³	—	9	3	no	—
60	C₃³	—	9	1	no	—
61	C₃³	C₃²×C₃₇⋊C₃	1	2	yes	—
62	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
63	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
64	3₋¹⁺²	—	9	3	no	—
65	3₋¹⁺²	—	333	9	no	—
66	C₃×C₉	—	37	27	no	—
67	C₃×C₉	C₃²×C₃₇⋊C₃	1	9	yes	—
68	C₃²	—	3	3	no	—
69	C₃²	—	3	6	no	—
70	C₃²	—	3	3	no	—
71	C₃²	—	3	1	no	—
72	C₃²	—	3	2	no	—
73	C₃²	—	3	1	no	—
74	C₃²	—	9	9	no	—
75	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
76	C₉	—	3	9	no	—
77	C₉	—	111	27	no	—
78	C₃	—	9	1	no	—
79	C₃	—	9	2	no	—
80	C₃	—	9	1	no	—
81	C₃	#22582679	1	3	yes	—
82	C₃	#22582680	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₃₇⋊C₃
commutator, socle, cosocle	C₃²×C₃₇	C₃⁴
Fitting	C₃₇×#6891	C₃
prime radical	C₃⁴×C₃₇	C₃²
hypercenter	#6891	C₃₇⋊C₃
hypocenter	C₃₇	#10624912
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ #6891

Complements

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

N	H ≅ G/N	type
1. #22582601	C₃	semidirect
2. #22582590	C₃	semidirect
3. #22582585	C₃	semidirect
7. #15688506	C₃²	semidirect
8. #15688505	C₃²	semidirect
9. #15688504	C₃²	semidirect
12. C₉×#66524	C₃²	semidirect
58. C₃₇	#10624912	semidirect

Maximal subgroups

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

subgroup	index
#22582601	3
#22582590	3
#22582585	3
#22582554	3
C₃₇×#6891	3
#10624912	37