Subgroups — #41007674

Order 26973.

716 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582580	C₃	1	27	yes	maximal, complemented
2	#22582578	C₃	1	6	yes	maximal, complemented
3	#22582554	C₃	1	3	yes	maximal
4	#22582538	C₃	1	3	yes	maximal, complemented
5	C₃²×3₋¹⁺²×C₃₇	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	9	yes	—
11	C₃²⋊#66524	C₃²	1	18	yes	—
12	C₉×#66524	C₃²	1	9	yes	complemented
13	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
14	C₃×3₋¹⁺²×C₃₇	—	3	9	no	—
15	C₃×3₋¹⁺²×C₃₇	C₃²	1	9	yes	—
16	C₃²×C₉×C₃₇	C₃²	1	2	yes	—
17	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
18	C₃³×C₃₇	—	3	3	no	—
19	C₃³×C₃₇	—	9	3	no	—
20	C₃³×C₃₇	C₃³	1	3	yes	—
21	C₃³×C₃₇	C₃³	1	1	yes	characteristic
22	3₋¹⁺²×C₃₇	—	3	27	no	—
23	C₃×C₉×C₃₇	—	3	6	no	—
24	C₃×C₉×C₃₇	3₋¹⁺²	1	9	yes	complemented
25	C₃×C₉×C₃₇	C₃³	1	6	yes	—
26	C₃×C₉×C₃₇	C₃³	1	3	yes	—
27	#11758608	—	9	9	no	—
28	C₃×#66524	C₃³	1	27	yes	—
29	#10624885	—	37	1	no	maximal, Sylow
30	C₃²×C₃₇	—	3	3	no	—
31	C₃²×C₃₇	—	3	3	no	—
32	C₃²×C₃₇	—	3	3	no	—
33	C₃²×C₃₇	—	3	6	no	—
34	C₃²×C₃₇	—	9	9	no	—
35	C₃²×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
36	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
37	C₉×C₃₇	—	3	6	no	—
38	C₉×C₃₇	C₃×3₋¹⁺²	1	9	yes	—
39	#66524	—	3	27	no	—
40	C₃²×3₋¹⁺²	C₃₇⋊C₃	1	1	yes	characteristic
41	#6896	—	37	27	no	—
42	#6895	—	37	6	no	—
43	#6891	—	37	3	no	—
44	#6887	—	37	3	no	—
45	C₃×C₃₇	—	3	3	no	—
46	C₃×C₃₇	—	9	3	no	—
47	C₃×C₃₇	C₃²×3₋¹⁺²	1	1	yes	characteristic
48	C₃×C₃₇	3₋¹⁺⁴	1	3	yes	—
49	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
50	C₃×3₋¹⁺²	—	3	9	no	—
51	C₃×3₋¹⁺²	—	37	9	no	—
52	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	9	yes	—
53	C₃²×C₉	C₃×C₃₇⋊C₃	1	2	yes	—
54	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
55	C₉⋊C₉	—	37	54	no	—
56	C₉⋊C₉	—	37	9	no	—
57	C₉⋊C₉	—	37	9	no	—
58	C₃²⋊C₉	—	37	9	no	—
59	C₃²⋊C₉	—	37	18	no	—
60	C₉²	—	37	9	no	—
61	C₃₇	#10624885	1	1	yes	characteristic, complemented, Sylow
62	C₃³	—	3	3	no	—
63	C₃³	—	9	3	no	—
64	C₃³	C₃²×C₃₇⋊C₃	1	3	yes	—
65	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
66	3₋¹⁺²	—	3	27	no	—
67	3₋¹⁺²	—	333	9	no	—
68	C₃×C₉	—	3	6	no	—
69	C₃×C₉	—	37	27	no	—
70	C₃×C₉	#11758608	1	9	yes	complemented
71	C₃×C₉	C₃²×C₃₇⋊C₃	1	6	yes	—
72	C₃×C₉	C₃²×C₃₇⋊C₃	1	3	yes	—
73	C₃²	—	3	3	no	—
74	C₃²	—	3	3	no	—
75	C₃²	—	3	3	no	—
76	C₃²	—	3	6	no	—
77	C₃²	—	9	9	no	—
78	C₃²	C₃×#11758608	1	3	yes	—
79	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
80	C₉	—	3	6	no	—
81	C₉	—	111	27	no	—
82	C₉	C₃×#11758608	1	9	yes	—
83	C₃	—	3	3	no	—
84	C₃	—	9	3	no	—
85	C₃	C₃²×#11758608	1	1	yes	characteristic
86	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	#10624885
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. #22582580	C₃	semidirect
2. #22582578	C₃	semidirect
4. #22582538	C₃	semidirect
7. #15688506	C₃²	semidirect
8. #15688505	C₃²	semidirect
9. #15688504	C₃²	semidirect
12. C₉×#66524	C₃²	semidirect
24. C₃×C₉×C₃₇	3₋¹⁺²	semidirect
61. C₃₇	#10624885	semidirect
70. 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
#22582580	3
#22582578	3
#22582554	3
#22582538	3
C₃²×3₋¹⁺²×C₃₇	3
#10624885	37