Subgroups — #41008393

Order 26973.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582693	C₃	1	3	yes	maximal
2	C₉²×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
3	C₃²⋊#66526	—	3	3	no	—
4	C₃×C₉×C₃₇⋊C₃	—	3	1	no	—
5	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
6	C₉×#66524	—	3	2	no	—
7	C₉×#66524	C₉	1	2	yes	complemented
8	C₉²×C₃₇	C₉	1	1	yes	characteristic, complemented
9	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
10	C₃×#66526	—	9	3	no	—
11	C₃×C₉×C₃₇	—	3	1	no	—
12	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
13	C₃×#66524	—	3	2	no	—
14	C₃×#66524	3₋¹⁺²	1	2	yes	—
15	C₉×C₃₇⋊C₃	—	3	3	no	—
16	C₉×C₃₇⋊C₃	—	3	6	no	—
17	C₉×C₃₇⋊C₃	—	3	1	no	—
18	C₉×C₃₇⋊C₃	—	3	2	no	—
19	#10624519	—	37	1	no	maximal, Sylow
20	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
21	C₃×C₃₇⋊C₃	—	3	1	no	—
22	C₃×C₃₇⋊C₃	—	3	2	no	—
23	C₃×C₃₇⋊C₃	#732	1	1	yes	characteristic
24	C₃×C₃₇⋊C₃	#733	1	2	yes	—
25	#66526	—	27	3	no	—
26	C₉×C₃₇	—	3	3	no	—
27	C₉×C₃₇	—	3	1	no	—
28	#66524	—	3	6	no	—
29	#66524	—	3	2	no	—
30	C₃×C₉²	—	37	1	no	—
31	#6868	—	37	3	no	—
32	C₃×C₃₇	—	3	1	no	—
33	C₃×C₃₇	#6868	1	1	yes	characteristic
34	C₃₇⋊C₃	—	3	2	no	—
35	C₃₇⋊C₃	#6879	1	1	yes	characteristic
36	C₃₇⋊C₃	#6880	1	2	yes	—
37	C₃²×C₉	—	37	1	no	—
38	C₃²×C₉	—	111	1	no	—
39	C₃²⋊C₉	—	111	3	no	—
40	C₉²	—	37	2	no	—
41	C₉²	—	111	2	no	—
42	C₉²	#66526	1	1	yes	characteristic, complemented
43	C₃₇	#10624519	1	1	yes	characteristic, complemented, Sylow
44	C₃³	—	37	1	no	—
45	C₃×C₉	—	3	1	no	—
46	C₃×C₉	—	37	2	no	—
47	C₃×C₉	—	333	3	no	—
48	C₃×C₉	—	111	1	no	—
49	C₃×C₉	—	111	2	no	—
50	C₃×C₉	—	111	3	no	—
51	C₃×C₉	—	111	6	no	—
52	C₃×C₉	—	111	2	no	—
53	C₃×C₉	C₃×#66526	1	1	yes	characteristic
54	C₃²	—	37	1	no	—
55	C₃²	—	37	2	no	—
56	C₃²	—	111	1	no	—
57	C₃²	—	111	2	no	—
58	C₃²	C₃²⋊#66526	1	1	yes	characteristic
59	C₉	—	3	3	no	—
60	C₉	—	3	1	no	—
61	C₉	—	111	6	no	—
62	C₉	—	111	2	no	—
63	C₉	—	999	3	no	—
64	C₃	—	3	1	no	—
65	C₃	—	37	1	no	—
66	C₃	—	37	2	no	—
67	C₃	—	111	2	no	—
68	C₃	#22582693	1	1	yes	characteristic

	subgroup	G/H
center	C₃	#22582693
commutator	C₃×C₉×C₃₇	C₃×C₉
Frattini	C₃×C₉	C₃×#66526
Fitting	C₉²×C₃₇	C₉
socle	C₃×C₃₇	#6868
prime radical	C₃²×C₃₇⋊C₃	H(3,𝔽₃)
cosocle	C₃×C₉×C₃₇⋊C₃	C₃²
hypercenter	C₉²	#66526
hypocenter	C₃₇	#10624519
LCS	G ▷ C₃×C₉×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃² ◁ C₃×C₉ ◁ C₉²

Complements

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

N	H ≅ G/N	type
7. C₉×#66524	C₉	semidirect
8. C₉²×C₃₇	C₉	semidirect
42. C₉²	#66526	semidirect
43. C₃₇	#10624519	semidirect

Maximal subgroups

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

subgroup	index
#22582693	3
C₉²×C₃₇⋊C₃	3
#10624519	37