Subgroups — #41004987

Order 26973.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582538	C₃	1	1	yes	maximal, characteristic
2	#22582470	C₃	1	2	yes	maximal
3	C₃₇×#6867	C₃	1	1	yes	maximal, characteristic
4	#15688506	C₉	1	3	yes	complemented
5	#15688505	—	3	1	no	—
6	C₃²⋊#66524	—	3	1	no	—
7	C₃²⋊#66524	—	3	2	no	—
8	C₉×#66524	—	3	1	no	—
9	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
10	C₃₇×C₃²⋊C₉	—	3	1	no	—
11	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
12	C₃×C₉×C₃₇	—	3	1	no	—
13	C₃×C₉×C₃₇	—	3	2	no	—
14	C₃×C₉×C₃₇	—	9	1	no	—
15	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
16	C₃×C₉×C₃₇	3₋¹⁺²	1	1	yes	characteristic
17	C₃×C₉×C₃₇	3₋¹⁺²	1	1	yes	characteristic
18	C₃×#66524	—	3	3	no	—
19	C₃×#66524	—	9	2	no	—
20	#10624515	—	37	1	no	maximal, Sylow
21	C₃²×C₃₇	—	9	1	no	—
22	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
23	C₃²×C₃₇	#731	1	1	yes	characteristic
24	C₃²×C₃₇	#731	1	2	yes	—
25	C₉×C₃₇	—	3	1	no	—
26	C₉×C₃₇	—	3	1	no	—
27	C₉×C₃₇	—	3	1	no	—
28	C₉×C₃₇	—	27	1	no	—
29	#66524	—	9	3	no	—
30	#66524	—	27	2	no	—
31	#6887	—	37	1	no	—
32	#6867	—	37	2	no	—
33	#6867	C₃₇⋊C₃	1	1	yes	characteristic
34	C₃×C₃₇	—	3	1	no	—
35	C₃×C₃₇	—	3	2	no	—
36	C₃×C₃₇	—	3	1	no	—
37	C₃×C₃₇	#6867	1	1	yes	characteristic
38	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
39	C₉⋊C₉	—	37	3	no	—
40	C₉⋊C₉	—	111	1	no	—
41	C₃²⋊C₉	—	3	1	no	—
42	C₃²⋊C₉	—	111	2	no	—
43	C₃²⋊C₉	—	111	1	no	—
44	C₉²	—	111	1	no	—
45	C₃₇	#10624515	1	1	yes	characteristic, complemented, Sylow
46	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
47	C₃×C₉	—	3	1	no	—
48	C₃×C₉	—	3	2	no	—
49	C₃×C₉	—	9	1	no	—
50	C₃×C₉	—	333	2	no	—
51	C₃×C₉	—	111	3	no	—
52	C₃×C₉	C₉×C₃₇⋊C₃	1	1	yes	characteristic
53	C₃×C₉	#11758605	1	1	yes	characteristic
54	C₃×C₉	#11758606	1	1	yes	characteristic
55	C₃²	—	9	1	no	—
56	C₃²	#15688502	1	1	yes	characteristic
57	C₃²	#15688515	1	1	yes	characteristic
58	C₃²	#15688517	1	2	yes	—
59	C₉	—	3	1	no	—
60	C₉	—	3	1	no	—
61	C₉	—	3	1	no	—
62	C₉	—	27	1	no	—
63	C₉	—	333	3	no	—
64	C₉	—	999	2	no	—
65	C₃	—	3	1	no	—
66	C₃	—	3	1	no	—
67	C₃	—	3	2	no	—
68	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×#6867	C₃
socle	C₃×C₃₇	#6867
prime radical	C₃³×C₃₇	H(3,𝔽₃)
cosocle	C₃²×C₉×C₃₇	C₃²
hypercenter	#6867	C₃₇⋊C₃
hypocenter	C₃₇	#10624515
LCS	G ▷ C₃×C₉×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃²×C₉ ◁ #6867

Complements

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

N	H ≅ G/N	type
4. #15688506	C₉	semidirect
45. C₃₇	#10624515	semidirect

Maximal subgroups

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

subgroup	index
#22582538	3
#22582470	3
C₃₇×#6867	3
#10624515	37