Subgroups — #41004989

Order 26973.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#15688506	C₃	1	1	yes	maximal, characteristic
2	#22582474	C₃	1	2	yes	maximal
3	C₃₇×#6869	C₃	1	1	yes	maximal, characteristic
4	C₃²×#66524	—	3	1	no	—
5	#15688511	—	3	2	no	—
6	#15688506	—	3	1	no	—
7	#15688506	—	3	1	no	—
8	#15688506	C₉	1	3	yes	—
9	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
10	C₃₇×C₂₇⋊C₃	—	3	1	no	—
11	C₃³×C₃₇	3₋¹⁺²	1	1	yes	characteristic
12	C₃×C₉×C₃₇	—	3	1	no	—
13	C₃×C₉×C₃₇	—	3	2	no	—
14	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
15	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
16	C₃×C₉×C₃₇	3₋¹⁺²	1	1	yes	characteristic
17	C₃×#66524	—	3	3	no	—
18	C₃×#66524	—	9	3	no	—
19	C₂₇×C₃₇	—	9	1	no	—
20	#11758601	—	9	2	no	—
21	#10624516	—	37	1	no	maximal, Sylow
22	C₃²×C₃₇	—	3	3	no	—
23	C₃²×C₃₇	—	3	1	no	—
24	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
25	C₉×C₃₇	—	3	1	no	—
26	C₉×C₃₇	—	3	1	no	—
27	C₉×C₃₇	#731	1	1	yes	characteristic
28	C₉×C₃₇	#731	1	2	yes	—
29	#66524	—	9	3	no	—
30	C₃×C₉⋊C₉	—	37	1	no	—
31	#6869	—	37	2	no	—
32	#6869	C₃₇⋊C₃	1	1	yes	characteristic
33	C₃×C₃₇	—	3	1	no	—
34	C₃×C₃₇	—	3	3	no	—
35	C₃×C₃₇	#6867	1	1	yes	characteristic
36	C₃²×C₉	—	111	1	no	—
37	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
38	C₂₇⋊C₃	—	3	1	no	—
39	C₂₇⋊C₃	—	111	2	no	—
40	C₉⋊C₉	—	37	3	no	—
41	C₉⋊C₉	—	111	1	no	—
42	C₉⋊C₉	—	111	1	no	—
43	C₃₇	#10624516	1	1	yes	characteristic, complemented, Sylow
44	C₃³	#11758605	1	1	yes	characteristic
45	C₃×C₉	—	3	1	no	—
46	C₃×C₉	—	3	2	no	—
47	C₃×C₉	—	333	3	no	—
48	C₃×C₉	—	111	3	no	—
49	C₃×C₉	C₉×C₃₇⋊C₃	1	1	yes	characteristic
50	C₃×C₉	#11758606	1	1	yes	characteristic
51	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
52	C₂₇	—	9	1	no	—
53	C₂₇	—	333	2	no	—
54	C₃²	—	3	3	no	—
55	C₃²	—	3	1	no	—
56	C₃²	#15688502	1	1	yes	characteristic
57	C₉	—	3	1	no	—
58	C₉	—	3	1	no	—
59	C₉	—	333	3	no	—
60	C₉	#15688515	1	1	yes	characteristic
61	C₉	#15688517	1	2	yes	—
62	C₃	—	3	1	no	—
63	C₃	—	3	3	no	—
64	C₃	#22582469	1	1	yes	characteristic

	subgroup	G/H
center	C₃	#22582469
commutator	C₃×C₉×C₃₇	C₃×C₉
Frattini	C₃²×C₉	C₃×C₃₇⋊C₃
Fitting	C₃₇×#6869	C₃
socle	C₃×C₃₇	#6867
prime radical	C₃³×C₃₇	3₋¹⁺²
cosocle	C₃²×C₉×C₃₇	C₃²
hypercenter	#6869	C₃₇⋊C₃
hypocenter	C₃₇	#10624516
LCS	G ▷ C₃×C₉×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃×C₉ ◁ C₃²×C₉ ◁ #6869

Complements

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

N	H ≅ G/N	type
43. C₃₇	#10624516	semidirect

Maximal subgroups

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

subgroup	index
C₃×#15688506	3
#22582474	3
C₃₇×#6869	3
#10624516	37