Subgroups — #45686176

Order 33300.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×#66526⋊C₂	C₂	1	1	yes	maximal, characteristic
2	#26012827	C₃	1	1	yes	maximal, characteristic
3	C₅²×#66526	C₄	1	1	yes	characteristic, complemented, Hall
4	#20269086	—	5	6	no	maximal
5	C₅²×D₃₇⋊C₃	C₂×C₃	1	1	yes	characteristic
6	#16814887	C₉	1	1	yes	characteristic, complemented, Hall
7	C₅×#66526⋊C₂	D₅	1	6	yes	—
8	C₅²×C₃₇⋊C₃	C₃×C₄	1	1	yes	characteristic
9	#10625194⋊C₃	—	5	6	no	—
10	C₅²×D₃₇	C₂×C₉	1	1	yes	characteristic
11	C₅×#66526	Dic₅	1	6	yes	complemented
12	AGL(1,𝔽₃₇)	—	25	1	no	Hall
13	C₅×D₃₇⋊C₃	C₃×D₅	1	6	yes	—
14	C₅²×C₃₇	C₄×C₉	1	1	yes	characteristic, complemented, Hall
15	C₉×#1039	—	37	1	no	maximal, Hall
16	#10625194	—	5	6	no	—
17	#66526⋊C₂	C₅⋊D₅	1	1	yes	characteristic
18	C₅×C₃₇⋊C₃	C₃×Dic₅	1	6	yes	—
19	C₂×C₅²×C₉	—	37	1	no	—
20	#3866⋊C₃	—	25	1	no	—
21	C₅×D₃₇	C₉×D₅	1	6	yes	—
22	#66526	#1039	1	1	yes	characteristic, complemented, Hall
23	C₃×#1039	—	37	1	no	—
24	C₅²×C₉	—	37	1	no	Hall
25	D₃₇⋊C₃	C₃×C₅⋊D₅	1	1	yes	characteristic
26	C₅×C₃₇	C₉×Dic₅	1	6	yes	complemented
27	C₉×Dic₅	—	185	6	no	—
28	C₂×C₃×C₅²	—	37	1	no	—
29	#3866	—	25	1	no	Hall
30	C₃₇⋊C₃	C₃×#1039	1	1	yes	characteristic
31	#1039	—	37	1	no	Hall
32	C₂×C₅×C₉	—	37	6	no	—
33	C₃×C₅²	—	37	1	no	—
34	D₃₇	C₉×C₅⋊D₅	1	1	yes	characteristic
35	C₃×Dic₅	—	185	6	no	—
36	C₂×C₅²	—	37	1	no	—
37	C₅×C₉	—	37	6	no	—
38	C₃₇	C₉×#1039	1	1	yes	characteristic, complemented, Sylow
39	C₄×C₉	—	925	1	no	Hall
40	C₂×C₃×C₅	—	37	6	no	—
41	C₅²	AGL(1,𝔽₃₇)	1	1	yes	characteristic, complemented, Sylow
42	Dic₅	—	185	6	no	—
43	C₂×C₉	—	37	1	no	—
44	C₃×C₅	—	37	6	no	—
45	C₃×C₄	—	925	1	no	—
46	C₂×C₅	—	37	6	no	—
47	C₉	—	37	1	no	Sylow
48	C₂×C₃	—	37	1	no	—
49	C₅	#20269086	1	6	yes	complemented
50	C₄	—	925	1	no	Sylow
51	C₃	—	37	1	no	—
52	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
6. #16814887	C₉	semidirect
11. C₅×#66526	Dic₅	semidirect
14. C₅²×C₃₇	C₄×C₉	semidirect
22. #66526	#1039	semidirect
26. C₅×C₃₇	C₉×Dic₅	semidirect
38. C₃₇	C₉×#1039	semidirect
41. C₅²	AGL(1,𝔽₃₇)	semidirect
49. C₅	#20269086	semidirect

Maximal subgroups

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

subgroup	index
C₅²×#66526⋊C₂	2
#26012827	3
#20269086	5
C₉×#1039	37