Subgroups — #41005004

Order 26973.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582472	C₃	1	2	yes	maximal
2	#22582472	C₃	1	1	yes	maximal, characteristic
3	C₃×C₃₇×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic
4	C₃²⋊#66524	—	3	2	no	—
5	C₃²⋊#66524	—	3	1	no	—
6	C₃²×C₉×C₃₇	—	3	1	no	—
7	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
8	C₃₇×C₉⋊C₉	—	3	2	no	—
9	C₃₇×C₉⋊C₉	C₉	1	3	yes	complemented
10	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
11	C₃×C₉×C₃₇	—	3	1	no	—
12	C₃×C₉×C₃₇	—	3	2	no	—
13	C₃×C₉×C₃₇	—	3	3	no	—
14	C₃×C₉×C₃₇	—	9	1	no	—
15	C₃×C₉×C₃₇	—	9	1	no	—
16	C₃×C₉×C₃₇	—	9	1	no	—
17	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
18	C₃×C₉×C₃₇	3₋¹⁺²	1	2	yes	—
19	C₃×#66524	—	9	2	no	—
20	C₃×#66524	—	9	1	no	—
21	#10624521	—	37	1	no	maximal, Sylow
22	C₃²×C₃₇	—	3	1	no	—
23	C₃²×C₃₇	—	3	1	no	—
24	C₃²×C₃₇	—	3	1	no	—
25	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
26	C₃²×C₃₇	#732	1	1	yes	characteristic
27	C₃²×C₃₇	#733	1	2	yes	—
28	C₉×C₃₇	—	3	1	no	—
29	C₉×C₃₇	—	3	2	no	—
30	C₉×C₃₇	—	9	3	no	—
31	#66524	—	27	2	no	—
32	#66524	—	27	1	no	—
33	C₃×C₉⋊C₉	C₃₇⋊C₃	1	1	yes	characteristic
34	#6868	—	37	2	no	—
35	#6868	—	37	1	no	—
36	C₃×C₃₇	—	3	1	no	—
37	C₃×C₃₇	—	3	2	no	—
38	C₃×C₃₇	#6868	1	1	yes	characteristic
39	C₃×C₃₇	#6881	1	1	yes	characteristic
40	C₃×C₃₇	#6882	1	1	yes	characteristic
41	C₃×C₃₇	#6883	1	1	yes	characteristic
42	C₃²×C₉	—	3	1	no	—
43	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
44	C₉⋊C₉	—	3	2	no	—
45	C₉⋊C₉	#66524	1	3	yes	complemented
46	C₃²⋊C₉	—	111	2	no	—
47	C₃²⋊C₉	—	111	1	no	—
48	C₃₇	#10624521	1	1	yes	characteristic, complemented, Sylow
49	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
50	C₃×C₉	—	3	1	no	—
51	C₃×C₉	—	3	2	no	—
52	C₃×C₉	—	3	3	no	—
53	C₃×C₉	—	9	1	no	—
54	C₃×C₉	—	9	1	no	—
55	C₃×C₉	—	9	1	no	—
56	C₃×C₉	—	333	2	no	—
57	C₃×C₉	—	333	1	no	—
58	C₃×C₉	C₃×#66524	1	1	yes	characteristic
59	C₃×C₉	#11758608	1	2	yes	—
60	C₃²	—	3	1	no	—
61	C₃²	—	3	1	no	—
62	C₃²	—	3	1	no	—
63	C₃²	C₃²⋊#66524	1	1	yes	characteristic
64	C₃²	#15688519	1	1	yes	characteristic
65	C₃²	#15688521	1	2	yes	—
66	C₉	—	3	1	no	—
67	C₉	—	3	2	no	—
68	C₉	—	9	3	no	—
69	C₉	—	999	2	no	—
70	C₉	—	999	1	no	—
71	C₃	—	3	1	no	—
72	C₃	—	3	2	no	—
73	C₃	#22582472	1	1	yes	characteristic
74	C₃	#22582513	1	1	yes	characteristic
75	C₃	#22582518	1	1	yes	characteristic
76	C₃	#22582523	1	1	yes	characteristic

	subgroup	G/H
center	C₃²	#15688519
commutator	C₃×C₉×C₃₇	C₃×C₉
Frattini	C₃²×C₉	C₃×C₃₇⋊C₃
Fitting	C₃×C₃₇×C₉⋊C₉	C₃
socle	C₃²×C₃₇	#732
prime radical	C₃³×C₃₇	H(3,𝔽₃)
cosocle	C₃²×C₉×C₃₇	C₃²
hypercenter	C₃×C₉⋊C₉	C₃₇⋊C₃
hypocenter	C₃₇	#10624521
LCS	G ▷ C₃×C₉×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃²×C₉ ◁ C₃×C₉⋊C₉

Complements

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

N	H ≅ G/N	type
9. C₃₇×C₉⋊C₉	C₉	semidirect
45. C₉⋊C₉	#66524	semidirect
48. C₃₇	#10624521	semidirect

Maximal subgroups

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

subgroup	index
#22582472	3
C₃×C₃₇×C₉⋊C₉	3
#10624521	37