Subgroups — #41008369

Order 26973.

242 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582691	C₃	1	3	yes	maximal
2	C₃²×#11758606	C₃	1	1	yes	maximal, characteristic
3	C₃²⋊#66526	—	3	3	no	—
4	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
5	C₃×#11758606	—	3	1	no	—
6	C₃×#11758606	—	3	2	no	—
7	C₃×#11758606	—	3	9	no	—
8	C₃²×#66524	C₉	1	2	yes	complemented
9	C₃²×C₉×C₃₇	C₉	1	1	yes	characteristic, complemented
10	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
11	C₃²×C₃₇⋊C₃	—	3	3	no	—
12	C₃²×C₃₇⋊C₃	—	9	3	no	—
13	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
14	C₃²×C₃₇⋊C₃	3₋¹⁺²	1	2	yes	—
15	C₃×#66526	—	9	3	no	—
16	C₃×C₉×C₃₇	—	3	1	no	—
17	C₃×C₉×C₃₇	—	3	3	no	—
18	C₃×#66524	—	3	2	no	—
19	C₃×#66524	—	3	6	no	—
20	#11758606	—	3	18	no	—
21	#11758606	—	3	9	no	—
22	#10624506	—	37	1	no	maximal, Sylow
23	C₃²×C₃₇	—	3	3	no	—
24	C₃²×C₃₇	—	3	1	no	—
25	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
26	C₃×C₃₇⋊C₃	—	3	2	no	—
27	C₃×C₃₇⋊C₃	—	9	1	no	—
28	C₃×C₃₇⋊C₃	—	9	9	no	—
29	C₃×C₃₇⋊C₃	—	9	2	no	—
30	C₃×C₃₇⋊C₃	AΣL(1,𝔽₂₇)	1	3	yes	—
31	#66526	—	27	3	no	—
32	C₉×C₃₇	—	3	3	no	—
33	#66524	—	3	6	no	—
34	C₃²×3₋¹⁺²	—	37	1	no	—
35	#6866	—	37	3	no	—
36	C₃×C₃₇	—	3	3	no	—
37	C₃×C₃₇	—	3	1	no	—
38	C₃×C₃₇	#6866	1	1	yes	characteristic
39	C₃₇⋊C₃	—	3	3	no	—
40	C₃₇⋊C₃	—	9	2	no	—
41	C₃⁴	—	37	1	no	—
42	C₃×3₋¹⁺²	—	111	1	no	—
43	C₃×3₋¹⁺²	—	111	2	no	—
44	C₃×3₋¹⁺²	—	111	9	no	—
45	C₃²×C₉	—	37	2	no	—
46	C₃²×C₉	#66526	1	1	yes	characteristic, complemented
47	C₃²⋊C₉	—	111	3	no	—
48	C₃₇	#10624506	1	1	yes	characteristic, complemented, Sylow
49	C₃³	—	37	1	no	—
50	C₃³	—	37	2	no	—
51	C₃³	—	333	3	no	—
52	C₃³	—	111	3	no	—
53	C₃³	C₃×#66526	1	1	yes	characteristic
54	3₋¹⁺²	—	111	9	no	—
55	3₋¹⁺²	—	111	18	no	—
56	C₃×C₉	—	3	1	no	—
57	C₃×C₉	—	3	3	no	—
58	C₃×C₉	—	333	3	no	—
59	C₃×C₉	—	111	2	no	—
60	C₃×C₉	—	111	6	no	—
61	C₃²	—	3	3	no	—
62	C₃²	—	3	1	no	—
63	C₃²	—	37	3	no	—
64	C₃²	—	333	2	no	—
65	C₃²	—	333	9	no	—
66	C₃²	—	333	1	no	—
67	C₃²	—	111	2	no	—
68	C₃²	C₃²⋊#66526	1	1	yes	characteristic
69	C₉	—	3	3	no	—
70	C₉	—	111	6	no	—
71	C₉	—	999	3	no	—
72	C₃	—	3	1	no	—
73	C₃	—	3	3	no	—
74	C₃	—	333	2	no	—
75	C₃	—	111	3	no	—
76	C₃	#22582691	1	1	yes	characteristic

	subgroup	G/H
center	C₃	#22582691
commutator	C₃³×C₃₇	C₃×C₉
Frattini	C₃³	C₃×#66526
Fitting	C₃²×C₉×C₃₇	C₉
socle	C₃×C₃₇	#6866
prime radical, cosocle	C₃³×C₃₇⋊C₃	C₃²
hypercenter	C₃²×C₉	#66526
hypocenter	C₃₇	#10624506
LCS	G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃² ◁ C₃³ ◁ C₃²×C₉

Complements

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

N	H ≅ G/N	type
8. C₃²×#66524	C₉	semidirect
9. C₃²×C₉×C₃₇	C₉	semidirect
46. C₃²×C₉	#66526	semidirect
48. C₃₇	#10624506	semidirect

Maximal subgroups

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

subgroup	index
#22582691	3
C₃²×#11758606	3
#10624506	37