Subgroups — #41004896

Order 26973.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582472	C₃	1	1	yes	maximal, characteristic
2	#22582472	C₃	1	1	yes	maximal, characteristic
3	#22582441	C₃	1	1	yes	maximal, characteristic
4	C₃×C₃₇×#733	C₃	1	1	yes	maximal, characteristic
5	C₃²×#66524	—	3	1	no	—
6	C₃²⋊#66524	—	3	1	no	—
7	C₃²⋊#66524	—	3	1	no	—
8	C₃×3₋¹⁺²×C₃₇	—	3	1	no	—
9	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
10	C₃₇×#733	—	3	3	no	—
11	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
12	3₋¹⁺²×C₃₇	—	3	3	no	—
13	3₋¹⁺²×C₃₇	—	9	2	no	—
14	C₃×C₉×C₃₇	—	3	1	no	—
15	C₃×C₉×C₃₇	—	3	2	no	—
16	C₃×C₉×C₃₇	—	3	1	no	—
17	C₃×C₉×C₃₇	—	9	1	no	—
18	C₃×#66524	—	3	3	no	—
19	C₃×#66524	—	9	1	no	—
20	C₃×#66524	—	9	1	no	—
21	C₃×#66524	—	9	1	no	—
22	C₃×#66524	—	9	1	no	—
23	C₃×#66524	—	9	1	no	—
24	#10624490	—	37	1	no	maximal, Sylow
25	C₃²×C₃₇	—	3	1	no	—
26	C₃²×C₃₇	—	3	1	no	—
27	C₃²×C₃₇	—	3	1	no	—
28	C₃²×C₃₇	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
29	C₃²×C₃₇	#731	1	2	yes	—
30	C₃²×C₃₇	#732	1	1	yes	characteristic
31	C₉×C₃₇	—	9	1	no	—
32	C₉×C₃₇	—	9	3	no	—
33	#66524	—	9	3	no	—
34	#66524	—	27	1	no	—
35	#66524	—	27	1	no	—
36	C₃×#733	C₃₇⋊C₃	1	1	yes	characteristic
37	#6868	—	37	1	no	—
38	#6868	—	37	1	no	—
39	#6855	—	37	1	no	—
40	C₃×C₃₇	—	3	1	no	—
41	C₃×C₃₇	—	3	2	no	—
42	C₃×C₃₇	#6859	1	1	yes	characteristic
43	C₃×C₃₇	#6878	1	1	yes	characteristic
44	C₃×C₃₇	#6882	1	1	yes	characteristic
45	C₃×C₃₇	#6883	1	1	yes	characteristic
46	C₃×3₋¹⁺²	—	3	1	no	—
47	C₃²×C₉	—	111	1	no	—
48	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
49	#733	—	3	3	no	—
50	C₃²⋊C₉	—	111	1	no	—
51	C₃²⋊C₉	—	111	1	no	—
52	C₃₇	#10624490	1	1	yes	characteristic, complemented, Sylow
53	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
54	3₋¹⁺²	—	3	3	no	—
55	3₋¹⁺²	—	9	2	no	—
56	C₃×C₉	—	3	1	no	—
57	C₃×C₉	—	3	2	no	—
58	C₃×C₉	—	3	1	no	—
59	C₃×C₉	—	9	1	no	—
60	C₃×C₉	—	333	1	no	—
61	C₃×C₉	—	333	1	no	—
62	C₃×C₉	—	333	1	no	—
63	C₃×C₉	—	333	1	no	—
64	C₃×C₉	—	333	1	no	—
65	C₃×C₉	—	111	3	no	—
66	C₃²	—	3	1	no	—
67	C₃²	—	3	1	no	—
68	C₃²	—	3	1	no	—
69	C₃²	#15688513	1	1	yes	characteristic
70	C₃²	#15688515	1	2	yes	—
71	C₃²	#15688518	1	1	yes	characteristic
72	C₉	—	9	1	no	—
73	C₉	—	9	3	no	—
74	C₉	—	333	3	no	—
75	C₉	—	999	1	no	—
76	C₉	—	999	1	no	—
77	C₃	—	3	1	no	—
78	C₃	—	3	2	no	—
79	C₃	#22582451	1	1	yes	characteristic
80	C₃	#22582505	1	1	yes	characteristic
81	C₃	#22582516	1	1	yes	characteristic
82	C₃	#22582522	1	1	yes	characteristic

	subgroup	G/H
center	C₃²	#15688518
commutator, cosocle	C₃²×C₉×C₃₇	C₃²
Frattini	C₃²×C₉	C₃×C₃₇⋊C₃
Fitting	C₃×C₃₇×#733	C₃
socle	C₃²×C₃₇	#732
prime radical	C₃³×C₃₇	H(3,𝔽₃)
hypercenter	C₃×#733	C₃₇⋊C₃
hypocenter	C₃₇	#10624490
LCS	G ▷ C₃²×C₉×C₃₇ ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃²×C₉ ◁ C₃×#733

Complements

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

N	H ≅ G/N	type
52. C₃₇	#10624490	semidirect

Maximal subgroups

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

subgroup	index
#22582472	3
#22582441	3
C₃×C₃₇×#733	3
#10624490	37