Subgroups — C₅²×C₂²⋊#66526 #45686325

Order 33300.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅²×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
2	C₅²×#66526	—	4	1	no	maximal, Hall
3	C₅×C₂²⋊#66526	C₅	1	6	yes	maximal, direct product factor
4	C₂×C₅²×C₃₇⋊C₃	—	3	1	no	—
5	C₂²×C₅²×C₃₇	C₉	1	1	yes	characteristic, complemented, Hall
6	C₅²×C₃₇⋊C₃	A₄	1	1	yes	characteristic
7	C₂²×C₅×C₃₇⋊C₃	C₃×C₅	1	6	yes	—
8	C₂×C₅²×C₃₇	—	3	1	no	—
9	C₅×#66526	—	4	6	no	—
10	C₂²⋊#66526	C₅²	1	1	yes	characteristic, direct product factor, Hall
11	C₂×C₅×C₃₇⋊C₃	—	3	6	no	—
12	C₅²×C₃₇	C₂²⋊C₉	1	1	yes	characteristic, complemented, Hall
13	C₅²×C₂²⋊C₉	—	37	1	no	maximal, Hall
14	C₂²×C₅×C₃₇	C₅×C₉	1	6	yes	complemented
15	C₅×C₃₇⋊C₃	C₅×A₄	1	6	yes	—
16	C₂²×C₃₇⋊C₃	C₃×C₅²	1	1	yes	characteristic
17	C₂×C₅×C₃₇	—	3	6	no	—
18	#66526	—	4	1	no	Hall
19	C₂²×C₃×C₅²	—	37	1	no	—
20	C₅²×C₉	—	148	1	no	Hall
21	C₂×C₃₇⋊C₃	—	3	1	no	—
22	C₅×C₃₇	C₅×C₂²⋊C₉	1	6	yes	complemented
23	C₅×C₂²⋊C₉	—	37	6	no	—
24	C₂×C₃×C₅²	—	111	1	no	—
25	C₂²×C₃₇	C₅²×C₉	1	1	yes	characteristic, complemented, Hall
26	C₃₇⋊C₃	C₅²×A₄	1	1	yes	characteristic
27	C₂²×C₅²	#66526	1	1	yes	characteristic, complemented, Hall
28	C₃×C₅²	—	37	1	no	—
29	C₂×C₃₇	—	3	1	no	—
30	C₂²×C₃×C₅	—	37	6	no	—
31	C₂×C₅²	—	3	1	no	—
32	C₅×C₉	—	148	6	no	—
33	C₃₇	C₅²×C₂²⋊C₉	1	1	yes	characteristic, complemented, Sylow
34	C₂²⋊C₉	—	37	1	no	Hall
35	C₂×C₃×C₅	—	111	6	no	—
36	C₅²	C₂²⋊#66526	1	1	yes	characteristic, direct product factor, Sylow
37	C₂²×C₅	C₅×#66526	1	6	yes	complemented
38	C₃×C₅	—	37	6	no	—
39	C₂²×C₃	—	37	1	no	—
40	C₂×C₅	—	3	6	no	—
41	C₉	—	148	1	no	Sylow
42	C₂×C₃	—	111	1	no	—
43	C₅	C₅×C₂²⋊#66526	1	6	yes	direct product factor
44	C₂²	C₅²×#66526	1	1	yes	characteristic, complemented, Sylow
45	C₃	—	37	1	no	—
46	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₅²	C₂²⋊#66526
commutator, hypocenter	C₂²×C₃₇	C₅²×C₉
Frattini	C₁	G
Fitting, socle	C₂²×C₅²×C₃₇	C₉
prime radical	C₂²×C₅²×C₃₇⋊C₃	C₃
cosocle	C₂²×C₃₇⋊C₃	C₃×C₅²
LCS	G ▷ C₂²×C₃₇
UCS	C₁ ◁ C₅²

Complements

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

N	H ≅ G/N	type
3. C₅×C₂²⋊#66526	C₅	direct (N × H)
5. C₂²×C₅²×C₃₇	C₉	semidirect
10. C₂²⋊#66526	C₅²	direct (N × H)
12. C₅²×C₃₇	C₂²⋊C₉	semidirect
14. C₂²×C₅×C₃₇	C₅×C₉	semidirect
22. C₅×C₃₇	C₅×C₂²⋊C₉	semidirect
25. C₂²×C₃₇	C₅²×C₉	semidirect
27. C₂²×C₅²	#66526	semidirect
33. C₃₇	C₅²×C₂²⋊C₉	semidirect
36. C₅²	C₂²⋊#66526	direct (N × H)
37. C₂²×C₅	C₅×#66526	semidirect
43. C₅	C₅×C₂²⋊#66526	direct (N × H)
44. C₂²	C₅²×#66526	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₅²×C₃₇⋊C₃	3
C₅²×#66526	4
C₅×C₂²⋊#66526	5
C₅²×C₂²⋊C₉	37

Subgroups — C52×C22⋊#66526 #45686325

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×C₂²⋊#66526 #45686325