Subgroups — #45686183

Order 33300.

96 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	#16814882⋊C₃	C₃	1	1	yes	maximal, characteristic
3	C₅²×#66526	C₄	1	1	yes	characteristic, complemented, Hall
4	#20269084	—	5	2	no	maximal
5	C₅⋊D₅×C₃₇⋊C₃	C₂×C₃	1	1	yes	characteristic
6	#16814882	C₉	1	1	yes	characteristic, complemented, Hall
7	D₅×#66526	—	5	2	no	—
8	D₅×#66526	—	10	2	no	—
9	C₅²×C₃₇⋊C₃	C₃×C₄	1	1	yes	characteristic
10	#10625193⋊C₃	—	5	2	no	—
11	C₃₇×C₅⋊D₅	C₂×C₉	1	1	yes	characteristic
12	C₅×#66526	—	2	2	no	—
13	C₅×#66526	AGL(1,𝔽₅)	1	2	yes	complemented
14	#13050625	—	25	1	no	Hall
15	D₅×C₃₇⋊C₃	—	5	2	no	—
16	D₅×C₃₇⋊C₃	—	10	2	no	—
17	C₅²×C₃₇	C₄×C₉	1	1	yes	characteristic, complemented, Hall
18	C₉×#1044	—	37	1	no	maximal, Hall
19	#10625193	—	5	2	no	—
20	C₂×#66526	—	25	1	no	—
21	C₅×C₃₇⋊C₃	—	2	2	no	—
22	C₅×C₃₇⋊C₃	C₃×AGL(1,𝔽₅)	1	2	yes	—
23	C₉×C₅⋊D₅	—	37	1	no	—
24	Dic₃₇⋊C₃	—	25	1	no	—
25	D₅×C₃₇	—	5	2	no	—
26	D₅×C₃₇	—	10	2	no	—
27	#66526	#1044	1	1	yes	characteristic, complemented, Hall
28	C₃×#1044	—	37	1	no	—
29	C₅²×C₉	—	37	1	no	Hall
30	C₂×C₃₇⋊C₃	—	25	1	no	—
31	C₅×C₃₇	—	2	2	no	—
32	C₅×C₃₇	C₉×AGL(1,𝔽₅)	1	2	yes	complemented
33	C₉×AGL(1,𝔽₅)	—	185	2	no	—
34	C₃×C₅⋊D₅	—	37	1	no	—
35	Dic₃₇	—	25	1	no	Hall
36	C₃₇⋊C₃	C₃×#1044	1	1	yes	characteristic
37	#1044	—	37	1	no	Hall
38	C₉×D₅	—	370	2	no	—
39	C₉×D₅	—	185	2	no	—
40	C₃×C₅²	—	37	1	no	—
41	C₂×C₃₇	—	25	1	no	—
42	C₃×AGL(1,𝔽₅)	—	185	2	no	—
43	C₅⋊D₅	#66526⋊C₂	1	1	yes	characteristic
44	C₅×C₉	—	37	2	no	—
45	C₅×C₉	—	74	2	no	—
46	C₃₇	C₉×#1044	1	1	yes	characteristic, complemented, Sylow
47	C₄×C₉	—	925	1	no	Hall
48	C₃×D₅	—	370	2	no	—
49	C₃×D₅	—	185	2	no	—
50	C₅²	#13050625	1	1	yes	characteristic, complemented, Sylow
51	AGL(1,𝔽₅)	—	185	2	no	—
52	C₂×C₉	—	925	1	no	—
53	C₃×C₅	—	37	2	no	—
54	C₃×C₅	—	74	2	no	—
55	C₃×C₄	—	925	1	no	—
56	D₅	—	5	2	no	—
57	D₅	—	10	2	no	—
58	C₉	—	37	1	no	Sylow
59	C₂×C₃	—	925	1	no	—
60	C₅	—	2	2	no	—
61	C₅	#20269084	1	2	yes	complemented
62	C₄	—	925	1	no	Sylow
63	C₃	—	37	1	no	—
64	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. #16814882	C₉	semidirect
13. C₅×#66526	AGL(1,𝔽₅)	semidirect
17. C₅²×C₃₇	C₄×C₉	semidirect
27. #66526	#1044	semidirect
32. C₅×C₃₇	C₉×AGL(1,𝔽₅)	semidirect
46. C₃₇	C₉×#1044	semidirect
50. C₅²	#13050625	semidirect
61. C₅	#20269084	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
#16814882⋊C₃	3
#20269084	5
C₉×#1044	37