Subgroups — #1043×#66526 #45686181

Order 33300.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅⋊D₅×#66526	C₂	1	1	yes	maximal, characteristic
2	#1043×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
3	C₅²×#66526	C₄	1	1	yes	characteristic, complemented, Hall
4	AGL(1,𝔽₅)×#66526	—	5	6	no	maximal
5	C₅⋊D₅×C₃₇⋊C₃	C₂×C₃	1	1	yes	characteristic
6	C₃₇×#1043	C₉	1	1	yes	characteristic, complemented, Hall
7	D₅×#66526	—	5	6	no	—
8	C₅²×C₃₇⋊C₃	C₃×C₄	1	1	yes	characteristic
9	AGL(1,𝔽₅)×C₃₇⋊C₃	—	5	6	no	—
10	C₃₇×C₅⋊D₅	C₂×C₉	1	1	yes	characteristic
11	C₅×#66526	AGL(1,𝔽₅)	1	6	yes	complemented
12	C₄×#66526	—	25	1	no	Hall
13	D₅×C₃₇⋊C₃	—	5	6	no	—
14	C₅²×C₃₇	C₄×C₉	1	1	yes	characteristic, complemented, Hall
15	C₉×#1043	—	37	1	no	maximal, Hall
16	AGL(1,𝔽₅)×C₃₇	—	5	6	no	—
17	C₂×#66526	—	25	1	no	—
18	C₅×C₃₇⋊C₃	C₃×AGL(1,𝔽₅)	1	6	yes	—
19	C₉×C₅⋊D₅	—	37	1	no	—
20	C₄×C₃₇⋊C₃	—	25	1	no	—
21	D₅×C₃₇	—	5	6	no	—
22	#66526	#1043	1	1	yes	characteristic, direct product factor, Hall
23	C₃×#1043	—	37	1	no	—
24	C₅²×C₉	—	37	1	no	Hall
25	C₂×C₃₇⋊C₃	—	25	1	no	—
26	C₅×C₃₇	C₉×AGL(1,𝔽₅)	1	6	yes	complemented
27	C₉×AGL(1,𝔽₅)	—	185	6	no	—
28	C₃×C₅⋊D₅	—	37	1	no	—
29	C₄×C₃₇	—	25	1	no	Hall
30	C₃₇⋊C₃	C₃×#1043	1	1	yes	characteristic
31	#1043	#66526	1	1	yes	characteristic, direct product factor, Hall
32	C₉×D₅	—	185	6	no	—
33	C₃×C₅²	—	37	1	no	—
34	C₂×C₃₇	—	25	1	no	—
35	C₃×AGL(1,𝔽₅)	—	185	6	no	—
36	C₅⋊D₅	C₂×#66526	1	1	yes	characteristic
37	C₅×C₉	—	37	6	no	—
38	C₃₇	C₉×#1043	1	1	yes	characteristic, complemented, Sylow
39	C₄×C₉	—	925	1	no	Hall
40	C₃×D₅	—	185	6	no	—
41	C₅²	C₄×#66526	1	1	yes	characteristic, complemented, Sylow
42	AGL(1,𝔽₅)	—	5	6	no	—
43	C₂×C₉	—	925	1	no	—
44	C₃×C₅	—	37	6	no	—
45	C₃×C₄	—	925	1	no	—
46	D₅	—	5	6	no	—
47	C₉	—	37	1	no	Sylow
48	C₂×C₃	—	925	1	no	—
49	C₅	AGL(1,𝔽₅)×#66526	1	6	yes	complemented
50	C₄	—	25	1	no	Sylow
51	C₃	—	37	1	no	—
52	C₂	—	25	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅²×C₃₇	C₄×C₉
prime radical, cosocle	C₅⋊D₅×C₃₇⋊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
6. C₃₇×#1043	C₉	semidirect
11. C₅×#66526	AGL(1,𝔽₅)	semidirect
14. C₅²×C₃₇	C₄×C₉	semidirect
22. #66526	#1043	direct (N × H)
26. C₅×C₃₇	C₉×AGL(1,𝔽₅)	semidirect
31. #1043	#66526	direct (N × H)
38. C₃₇	C₉×#1043	semidirect
41. C₅²	C₄×#66526	semidirect
49. C₅	AGL(1,𝔽₅)×#66526	semidirect

Maximal subgroups

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

subgroup	index
C₅⋊D₅×#66526	2
#1043×C₃₇⋊C₃	3
AGL(1,𝔽₅)×#66526	5
C₉×#1043	37