Subgroups — #45686140

Order 33300.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30659275	C₂	1	1	yes	maximal, characteristic
2	C₅²×#3866⋊C₃	C₃	1	1	yes	maximal, characteristic
3	C₅²⋊#66526	C₄	1	1	yes	characteristic, complemented, Hall
4	C₅²×D₃₇⋊C₃	C₂×C₃	1	1	yes	characteristic
5	C₅²×#3866	C₉	1	1	yes	characteristic, complemented, Hall
6	C₅²×C₃₇⋊C₃	C₃×C₄	1	1	yes	characteristic
7	C₅×#3866⋊C₃	—	3	2	no	—
8	C₅²×D₃₇	C₂×C₉	1	1	yes	characteristic
9	AGL(1,𝔽₃₇)	—	25	1	no	maximal, Hall
10	C₅×D₃₇⋊C₃	—	3	2	no	—
11	C₅²×C₃₇	C₄×C₉	1	1	yes	characteristic, complemented, Hall
12	C₄×C₅²⋊C₉	—	37	1	no	maximal, Hall
13	C₅×#3866	—	3	2	no	—
14	#66526⋊C₂	—	25	1	no	—
15	C₅×C₃₇⋊C₃	—	3	2	no	—
16	C₂×C₅²⋊C₉	—	37	1	no	—
17	#3866⋊C₃	C₅²⋊C₃	1	1	yes	characteristic
18	C₅×D₃₇	—	3	2	no	—
19	#66526	—	25	1	no	Hall
20	C₃×C₄×C₅²	—	37	1	no	—
21	C₅²⋊C₉	—	37	1	no	Hall
22	D₃₇⋊C₃	C₂×C₅²⋊C₃	1	1	yes	characteristic
23	C₅×C₃₇	—	3	2	no	—
24	C₂×C₃×C₅²	—	37	1	no	—
25	#3866	C₅²⋊C₉	1	1	yes	characteristic, complemented, Hall
26	C₃₇⋊C₃	C₄×C₅²⋊C₃	1	1	yes	characteristic
27	C₄×C₅²	—	37	1	no	Hall
28	C₃×C₅²	—	37	1	no	—
29	D₃₇	C₂×C₅²⋊C₉	1	1	yes	characteristic
30	C₃×C₄×C₅	—	111	2	no	—
31	C₂×C₅²	—	37	1	no	—
32	C₃₇	C₄×C₅²⋊C₉	1	1	yes	characteristic, complemented, Sylow
33	C₄×C₉	—	925	1	no	Hall
34	C₂×C₃×C₅	—	111	2	no	—
35	C₅²	AGL(1,𝔽₃₇)	1	1	yes	characteristic, complemented, Sylow
36	C₄×C₅	—	111	2	no	—
37	C₂×C₉	—	925	1	no	—
38	C₃×C₅	—	111	2	no	—
39	C₃×C₄	—	37	1	no	—
40	C₂×C₅	—	111	2	no	—
41	C₉	—	925	1	no	Sylow
42	C₂×C₃	—	37	1	no	—
43	C₅	—	3	2	no	—
44	C₄	—	37	1	no	Sylow
45	C₃	—	37	1	no	—
46	C₂	—	37	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅²×C₃₇	C₄×C₉
prime radical, cosocle	C₅²×D₃₇⋊C₃	C₂×C₃
LCS	G ▷ C₅²×C₃₇
UCS	C₁

Complements

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

N	H ≅ G/N	type
3. C₅²⋊#66526	C₄	semidirect
5. C₅²×#3866	C₉	semidirect
11. C₅²×C₃₇	C₄×C₉	semidirect
25. #3866	C₅²⋊C₉	semidirect
32. C₃₇	C₄×C₅²⋊C₉	semidirect
35. C₅²	AGL(1,𝔽₃₇)	semidirect

Maximal subgroups

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

subgroup	index
#30659275	2
C₅²×#3866⋊C₃	3
AGL(1,𝔽₃₇)	25
C₄×C₅²⋊C₉	37