Subgroups — #41006498

Order 26973.

518 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
2	C₃×#15688520	C₃	1	9	yes	maximal, complemented
3	#22582554	C₃	1	2	yes	maximal
4	C₃₇×#6891	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	#15688520	—	3	27	no	—
10	C₃²⋊#66524	—	3	6	no	—
11	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
12	C₃×3₋¹⁺²×C₃₇	C₃²	1	3	yes	—
13	C₃₇×C₃²⋊C₉	—	3	3	no	—
14	C₃³×C₃₇	—	3	9	no	—
15	C₃³×C₃₇	—	3	3	no	—
16	C₃³×C₃₇	H(3,𝔽₃)	1	3	yes	—
17	C₃³×C₃₇	C₃³	1	1	yes	characteristic
18	3₋¹⁺²×C₃₇	—	3	9	no	—
19	C₃×C₉×C₃₇	—	3	3	no	—
20	#11758608	—	3	18	no	—
21	C₃×#66524	—	3	27	no	—
22	C₃×#66524	—	3	6	no	—
23	C₃×#66524	—	3	3	no	—
24	C₃×#66524	—	3	6	no	—
25	C₃×#66524	H(3,𝔽₃)	1	9	yes	—
26	#10624733	—	37	1	no	maximal, Sylow
27	C₃²×C₃₇	—	3	27	no	—
28	C₃²×C₃₇	—	3	3	no	—
29	C₃²×C₃₇	—	3	6	no	—
30	C₃²×C₃₇	—	3	3	no	—
31	C₃²×C₃₇	#733	1	9	yes	complemented
32	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	3	yes	—
33	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
34	C₉×C₃₇	—	9	3	no	—
35	#66524	—	3	9	no	—
36	#66524	—	9	6	no	—
37	C₃³×C₉	—	37	1	no	—
38	C₃×#733	—	37	9	no	—
39	#6891	—	37	2	no	—
40	#6891	C₃₇⋊C₃	1	1	yes	characteristic
41	C₃×C₃₇	—	3	9	no	—
42	C₃×C₃₇	—	3	3	no	—
43	C₃×C₃₇	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
44	C₃×C₃₇	C₃×#733	1	3	yes	—
45	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
46	C₃×3₋¹⁺²	—	37	6	no	—
47	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	3	yes	—
48	C₃²×C₉	—	37	3	no	—
49	C₃²×C₉	—	111	9	no	—
50	C₃²×C₉	—	111	3	no	—
51	#733	—	111	27	no	—
52	C₃²⋊C₉	—	3	3	no	—
53	C₃²⋊C₉	—	111	6	no	—
54	C₃₇	#10624733	1	1	yes	characteristic, complemented, Sylow
55	C₃³	—	3	9	no	—
56	C₃³	—	3	3	no	—
57	C₃³	C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
58	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
59	3₋¹⁺²	—	3	9	no	—
60	3₋¹⁺²	—	111	18	no	—
61	C₃×C₉	—	3	3	no	—
62	C₃×C₉	—	37	9	no	—
63	C₃×C₉	—	111	6	no	—
64	C₃×C₉	—	111	27	no	—
65	C₃×C₉	—	111	6	no	—
66	C₃×C₉	—	111	3	no	—
67	C₃²	—	3	27	no	—
68	C₃²	—	3	3	no	—
69	C₃²	—	3	6	no	—
70	C₃²	—	3	3	no	—
71	C₃²	#15688520	1	9	yes	complemented
72	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	3	yes	—
73	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
74	C₉	—	9	3	no	—
75	C₉	—	333	6	no	—
76	C₉	—	111	9	no	—
77	C₃	—	3	9	no	—
78	C₃	—	3	3	no	—
79	C₃	(C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)	1	1	yes	characteristic
80	C₃	C₃×#15688520	1	3	yes	—

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6891	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical	C₃⁴×C₃₇	C₃²
hypercenter	#6891	C₃₇⋊C₃
hypocenter	C₃₇	#10624733
LCS	G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ #6891

Complements

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

N	H ≅ G/N	type
2. C₃×#15688520	C₃	semidirect
31. C₃²×C₃₇	#733	semidirect
54. C₃₇	#10624733	semidirect
71. C₃²	#15688520	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×#15688520	3
#22582554	3
C₃₇×#6891	3
#10624733	37