Subgroups — #41006491

Order 26973.

560 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×#66524	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×#15688515	C₃	1	9	yes	maximal, complemented
3	#22582554	C₃	1	2	yes	maximal, complemented
4	C₃₇×C₃²⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
5	C₃×#11758608	C₃²	1	6	yes	—
6	C₃²×#66524	—	3	3	no	—
7	C₃²×#66524	—	3	9	no	—
8	C₃²×#66524	C₃²	1	3	yes	—
9	#15688515	—	3	27	no	—
10	C₃²⋊#66524	—	3	6	no	—
11	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
12	C₃×H(3,𝔽₃)×C₃₇	—	3	3	no	—
13	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	3	yes	—
14	C₃³×C₃₇	—	3	9	no	—
15	C₃³×C₃₇	—	3	3	no	—
16	C₃³×C₃₇	—	3	3	no	—
17	C₃³×C₃₇	H(3,𝔽₃)	1	3	yes	—
18	C₃³×C₃₇	C₃³	1	1	yes	characteristic
19	H(3,𝔽₃)×C₃₇	—	3	9	no	—
20	H(3,𝔽₃)×C₃₇	—	9	9	no	—
21	#11758608	—	3	18	no	—
22	C₃×#66524	—	3	27	no	—
23	C₃×#66524	—	3	6	no	—
24	C₃×#66524	—	3	3	no	—
25	C₃×#66524	—	3	6	no	—
26	C₃×#66524	H(3,𝔽₃)	1	9	yes	complemented
27	#10624731	—	37	1	no	maximal, Sylow
28	C₃²×C₃₇	—	3	27	no	—
29	C₃²×C₃₇	—	3	3	no	—
30	C₃²×C₃₇	—	3	6	no	—
31	C₃²×C₃₇	—	3	3	no	—
32	C₃²×C₃₇	—	9	3	no	—
33	C₃²×C₃₇	—	9	9	no	—
34	C₃²×C₃₇	#731	1	9	yes	complemented
35	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	3	yes	—
36	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
37	#66524	—	3	9	no	—
38	#66524	—	9	6	no	—
39	C₃³×C₉	—	37	1	no	—
40	C₃×#731	—	37	9	no	—
41	#6891	—	37	2	no	—
42	C₃²⋊H(3,𝔽₃)	C₃₇⋊C₃	1	1	yes	characteristic
43	C₃×C₃₇	—	3	9	no	—
44	C₃×C₃₇	—	3	3	no	—
45	C₃×C₃₇	—	27	3	no	—
46	C₃×C₃₇	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
47	C₃×C₃₇	C₃×#731	1	3	yes	—
48	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
49	C₃×3₋¹⁺²	—	37	6	no	—
50	C₃×H(3,𝔽₃)	—	3	3	no	—
51	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	3	yes	—
52	C₃²×C₉	—	37	3	no	—
53	C₃²×C₉	—	111	9	no	—
54	C₃²×C₉	—	111	3	no	—
55	#731	—	111	27	no	—
56	C₃²⋊C₉	—	111	6	no	—
57	C₃₇	#10624731	1	1	yes	characteristic, complemented, Sylow
58	C₃³	—	3	9	no	—
59	C₃³	—	3	3	no	—
60	C₃³	—	3	3	no	—
61	C₃³	C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
62	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
63	3₋¹⁺²	—	111	18	no	—
64	H(3,𝔽₃)	—	3	9	no	—
65	H(3,𝔽₃)	—	9	9	no	—
66	C₃×C₉	—	37	9	no	—
67	C₃×C₉	—	111	6	no	—
68	C₃×C₉	—	111	27	no	—
69	C₃×C₉	—	111	6	no	—
70	C₃×C₉	—	111	3	no	—
71	C₃²	—	3	27	no	—
72	C₃²	—	3	3	no	—
73	C₃²	—	3	6	no	—
74	C₃²	—	3	3	no	—
75	C₃²	—	9	3	no	—
76	C₃²	—	9	9	no	—
77	C₃²	#15688515	1	9	yes	complemented
78	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	3	yes	—
79	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
80	C₉	—	333	6	no	—
81	C₉	—	111	9	no	—
82	C₃	—	3	9	no	—
83	C₃	—	3	3	no	—
84	C₃	—	27	3	no	—
85	C₃	(C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)	1	1	yes	characteristic
86	C₃	C₃×#15688515	1	3	yes	—

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting, prime radical	C₃₇×C₃²⋊H(3,𝔽₃)	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
hypercenter	C₃²⋊H(3,𝔽₃)	C₃₇⋊C₃
hypocenter	C₃₇	#10624731
LCS	G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ C₃²⋊H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. C₃³×#66524	C₃	semidirect
2. C₃×#15688515	C₃	semidirect
3. #22582554	C₃	semidirect
26. C₃×#66524	H(3,𝔽₃)	semidirect
34. C₃²×C₃₇	#731	semidirect
57. C₃₇	#10624731	semidirect
77. C₃²	#15688515	semidirect

Maximal subgroups

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

subgroup	index
C₃³×#66524	3
C₃×#15688515	3
#22582554	3
C₃₇×C₃²⋊H(3,𝔽₃)	3
#10624731	37