Subgroups — #41004906

Order 26973.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#15688521	C₃	1	1	yes	maximal, characteristic
2	#22582472	C₃	1	2	yes	maximal
3	C₃₇×#6855	C₃	1	1	yes	maximal, characteristic
4	C₃×#11758608	—	3	1	no	—
5	#15688521	—	3	3	no	—
6	C₃²⋊#66524	—	3	2	no	—
7	C₃²×C₉×C₃₇	—	3	1	no	—
8	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
9	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
10	C₃×C₉×C₃₇	—	3	3	no	—
11	C₃×C₉×C₃₇	—	3	1	no	—
12	C₃×C₉×C₃₇	—	3	2	no	—
13	C₃×C₉×C₃₇	—	3	1	no	—
14	C₃×C₉×C₃₇	—	9	1	no	—
15	C₃×C₉×C₃₇	—	9	1	no	—
16	C₃×C₉×C₃₇	—	9	1	no	—
17	#11758608	—	3	3	no	—
18	#11758608	—	9	2	no	—
19	C₃×#66524	—	9	1	no	—
20	C₃×#66524	—	9	2	no	—
21	#10624491	—	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	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
26	C₃²×C₃₇	#731	1	2	yes	—
27	C₃²×C₃₇	#732	1	1	yes	characteristic
28	C₉×C₃₇	—	9	3	no	—
29	C₉×C₃₇	—	9	1	no	—
30	#66524	—	9	3	no	—
31	#66524	—	27	2	no	—
32	C₃×#733	—	37	1	no	—
33	#6868	—	37	2	no	—
34	#6855	C₃₇⋊C₃	1	1	yes	characteristic
35	C₃×C₃₇	—	3	1	no	—
36	C₃×C₃₇	—	3	2	no	—
37	C₃×C₃₇	#6859	1	1	yes	characteristic
38	C₃×C₃₇	#6880	1	1	yes	characteristic
39	C₃×C₃₇	#6881	1	1	yes	characteristic
40	C₃×C₃₇	#6882	1	1	yes	characteristic
41	C₃×3₋¹⁺²	—	111	1	no	—
42	C₃²×C₉	—	3	1	no	—
43	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
44	#733	—	111	3	no	—
45	C₃²⋊C₉	—	111	2	no	—
46	C₃₇	#10624491	1	1	yes	characteristic, complemented, Sylow
47	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
48	3₋¹⁺²	—	333	2	no	—
49	3₋¹⁺²	—	111	3	no	—
50	C₃×C₉	—	3	3	no	—
51	C₃×C₉	—	3	1	no	—
52	C₃×C₉	—	3	2	no	—
53	C₃×C₉	—	3	1	no	—
54	C₃×C₉	—	9	1	no	—
55	C₃×C₉	—	9	1	no	—
56	C₃×C₉	—	9	1	no	—
57	C₃×C₉	—	333	1	no	—
58	C₃×C₉	—	333	2	no	—
59	C₃²	—	3	1	no	—
60	C₃²	—	3	1	no	—
61	C₃²	—	3	1	no	—
62	C₃²	#15688512	1	1	yes	characteristic
63	C₃²	#15688517	1	2	yes	—
64	C₃²	#15688519	1	1	yes	characteristic
65	C₉	—	9	3	no	—
66	C₉	—	9	1	no	—
67	C₉	—	333	3	no	—
68	C₉	—	999	2	no	—
69	C₃	—	3	1	no	—
70	C₃	—	3	2	no	—
71	C₃	#22582452	1	1	yes	characteristic
72	C₃	#22582510	1	1	yes	characteristic
73	C₃	#22582513	1	1	yes	characteristic
74	C₃	#22582518	1	1	yes	characteristic

	subgroup	G/H
center	C₃²	#15688519
commutator, cosocle	C₃²×C₉×C₃₇	C₃²
Frattini	C₃²×C₉	C₃×C₃₇⋊C₃
Fitting	C₃₇×#6855	C₃
socle	C₃²×C₃₇	#732
prime radical	C₃³×C₃₇	H(3,𝔽₃)
hypercenter	#6855	C₃₇⋊C₃
hypocenter	C₃₇	#10624491
LCS	G ▷ C₃²×C₉×C₃₇ ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃²×C₉ ◁ #6855

Complements

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

N	H ≅ G/N	type
46. C₃₇	#10624491	semidirect

Maximal subgroups

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

subgroup	index
C₃×#15688521	3
#22582472	3
C₃₇×#6855	3
#10624491	37