Subgroups — #41006514

Order 26973.

488 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#15688516	C₃	1	9	yes	maximal, complemented
2	#22582554	C₃	1	2	yes	maximal, complemented
3	#22582550	C₃	1	1	yes	maximal, characteristic
4	C₃²×3₋¹⁺²×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃×#11758608	C₃²	1	6	yes	—
6	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
7	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	3	yes	—
8	#15688516	—	3	27	no	—
9	C₃²⋊#66524	—	3	6	no	—
10	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
11	C₃×3₋¹⁺²×C₃₇	—	3	3	no	—
12	C₃×3₋¹⁺²×C₃₇	—	3	9	no	—
13	C₃²×C₉×C₃₇	C₃²	1	3	yes	—
14	C₃³×C₃₇	—	3	3	no	—
15	C₃³×C₃₇	—	9	3	no	—
16	C₃³×C₃₇	H(3,𝔽₃)	1	3	yes	—
17	C₃³×C₃₇	C₃³	1	1	yes	characteristic
18	C₃²×C₃₇⋊C₃	—	3	3	no	—
19	3₋¹⁺²×C₃₇	—	3	27	no	—
20	C₃×C₉×C₃₇	—	3	6	no	—
21	C₃×C₉×C₃₇	—	3	3	no	—
22	C₃×C₉×C₃₇	H(3,𝔽₃)	1	9	yes	complemented
23	C₃²⋊(C₃₇⋊C₃)	—	3	9	no	—
24	C₃²⋊(C₃₇⋊C₃)	—	9	9	no	—
25	#11758608	—	3	18	no	—
26	C₃×#66524	—	3	6	no	—
27	#10624736	—	37	1	no	maximal, Sylow
28	C₃²×C₃₇	—	3	3	no	—
29	C₃²×C₃₇	—	3	3	no	—
30	C₃²×C₃₇	—	3	3	no	—
31	C₃²×C₃₇	—	3	6	no	—
32	C₃²×C₃₇	—	9	9	no	—
33	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	3	yes	—
34	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
35	C₃×C₃₇⋊C₃	—	9	3	no	—
36	C₃×C₃₇⋊C₃	—	9	9	no	—
37	C₉×C₃₇	—	3	9	no	—
38	#66524	—	9	6	no	—
39	C₃²×3₋¹⁺²	C₃₇⋊C₃	1	1	yes	characteristic, complemented
40	C₃×#731	—	37	9	no	—
41	#6891	—	37	2	no	—
42	C₃²⋊H(3,𝔽₃)	—	37	1	no	—
43	C₃×C₃₇	—	3	3	no	—
44	C₃×C₃₇	—	9	3	no	—
45	C₃×C₃₇	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
46	C₃×C₃₇	#6910	1	3	yes	—
47	C₃₇⋊C₃	—	27	3	no	—
48	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
49	C₃×3₋¹⁺²	—	3	3	no	—
50	C₃×3₋¹⁺²	—	3	9	no	—
51	C₃×3₋¹⁺²	—	37	6	no	—
52	C₃×H(3,𝔽₃)	—	37	3	no	—
53	C₃×H(3,𝔽₃)	—	111	3	no	—
54	C₃²×C₉	C₃×C₃₇⋊C₃	1	3	yes	—
55	#731	—	111	27	no	—
56	C₃²⋊C₉	—	111	6	no	—
57	C₃₇	#10624736	1	1	yes	characteristic, complemented, Sylow
58	C₃³	—	3	3	no	—
59	C₃³	—	9	3	no	—
60	C₃³	—	111	3	no	—
61	C₃³	C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
62	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
63	3₋¹⁺²	—	3	27	no	—
64	3₋¹⁺²	—	111	18	no	—
65	H(3,𝔽₃)	—	333	9	no	—
66	H(3,𝔽₃)	—	111	9	no	—
67	C₃×C₉	—	3	6	no	—
68	C₃×C₉	—	3	3	no	—
69	C₃×C₉	—	111	6	no	—
70	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	9	yes	complemented
71	C₃²	—	3	3	no	—
72	C₃²	—	3	3	no	—
73	C₃²	—	3	3	no	—
74	C₃²	—	3	6	no	—
75	C₃²	—	9	9	no	—
76	C₃²	—	333	3	no	—
77	C₃²	—	333	9	no	—
78	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
79	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
80	C₉	—	3	9	no	—
81	C₉	—	333	6	no	—
82	C₃	—	3	3	no	—
83	C₃	—	9	3	no	—
84	C₃	—	999	3	no	—
85	C₃	#22582550	1	1	yes	characteristic
86	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	#22582550	C₃
hypercenter	C₃²×3₋¹⁺²	C₃₇⋊C₃
hypocenter	C₃₇	#10624736
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
1. C₃×#15688516	C₃	semidirect
2. #22582554	C₃	semidirect
4. C₃²×3₋¹⁺²×C₃₇	C₃	semidirect
22. C₃×C₉×C₃₇	H(3,𝔽₃)	semidirect
39. C₃²×3₋¹⁺²	C₃₇⋊C₃	semidirect
57. C₃₇	#10624736	semidirect
70. C₃×C₉	C₃²⋊(C₃₇⋊C₃)	semidirect

Maximal subgroups

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

subgroup	index
C₃×#15688516	3
#22582554	3
#22582550	3
C₃²×3₋¹⁺²×C₃₇	3
#10624736	37