Subgroups — C₂×C₅⋊(C₅⋊(#66526⋊C₂)) #45686196

Order 33300.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²×#66526	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅⋊(C₅⋊(#66526⋊C₂))	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₅⋊(C₅⋊(D₃₇⋊C₃))	C₃	1	1	yes	maximal, characteristic
4	C₅²×#66526	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×C₅⋊(#66526⋊C₂)	—	5	6	no	maximal
6	C₂×C₅²×C₃₇⋊C₃	C₂×C₃	1	1	yes	characteristic
7	C₅⋊(C₅⋊(D₃₇⋊C₃))	C₂×C₃	1	2	yes	—
8	C₂×C₅⋊D₁₈₅	C₉	1	1	yes	characteristic, complemented, Hall
9	C₂×C₅×#66526	D₅	1	6	yes	complemented
10	C₅⋊(#66526⋊C₂)	—	5	12	no	—
11	C₅²×C₃₇⋊C₃	C₂²×C₃	1	1	yes	characteristic
12	C₂×C₅⋊(D₃₇⋊C₃)	—	5	6	no	—
13	C₂×C₅²×C₃₇	C₂×C₉	1	1	yes	characteristic, complemented
14	C₅⋊D₁₈₅	C₂×C₉	1	2	yes	complemented
15	C₅×#66526	C₂×D₅	1	6	yes	complemented
16	C₂×#66526⋊C₂	—	25	1	no	Hall
17	C₂×C₅×C₃₇⋊C₃	C₃×D₅	1	6	yes	—
18	C₅⋊(D₃₇⋊C₃)	—	5	12	no	—
19	C₅²×C₃₇	C₂²×C₉	1	1	yes	characteristic, complemented, Hall
20	C₂×C₉×C₅⋊D₅	—	37	1	no	maximal, Hall
21	C₂×D₁₈₅	—	5	6	no	—
22	C₂×#66526	C₅⋊D₅	1	1	yes	characteristic, complemented
23	#66526⋊C₂	—	25	2	no	—
24	C₅×C₃₇⋊C₃	C₂×C₃×D₅	1	6	yes	—
25	C₂×C₅²×C₉	—	37	1	no	—
26	C₉×C₅⋊D₅	—	37	2	no	—
27	C₂×D₃₇⋊C₃	—	25	1	no	—
28	C₂×C₅×C₃₇	C₉×D₅	1	6	yes	complemented
29	D₁₈₅	—	5	12	no	—
30	#66526	C₂×C₅⋊D₅	1	1	yes	characteristic, complemented, Hall
31	C₂×C₃×C₅⋊D₅	—	37	1	no	—
32	C₅²×C₉	—	37	1	no	Hall
33	C₂×C₃₇⋊C₃	C₃×C₅⋊D₅	1	1	yes	characteristic
34	D₃₇⋊C₃	—	25	2	no	—
35	C₅×C₃₇	C₂×C₉×D₅	1	6	yes	complemented
36	C₂×C₉×D₅	—	185	6	no	—
37	C₂×C₃×C₅²	—	37	1	no	—
38	C₃×C₅⋊D₅	—	37	2	no	—
39	C₂×D₃₇	—	25	1	no	Hall
40	C₃₇⋊C₃	C₂×C₃×C₅⋊D₅	1	1	yes	characteristic
41	C₂×C₅⋊D₅	—	37	1	no	Hall
42	C₂×C₅×C₉	—	37	6	no	—
43	C₉×D₅	—	185	12	no	—
44	C₃×C₅²	—	37	1	no	—
45	C₂×C₃₇	C₉×C₅⋊D₅	1	1	yes	characteristic, complemented
46	D₃₇	—	25	2	no	—
47	C₂×C₃×D₅	—	185	6	no	—
48	C₂×C₅²	#66526⋊C₂	1	1	yes	characteristic, complemented
49	C₅⋊D₅	—	37	2	no	—
50	C₅×C₉	—	37	6	no	—
51	C₃₇	C₂×C₉×C₅⋊D₅	1	1	yes	characteristic, complemented, Sylow
52	C₂²×C₉	—	925	1	no	Hall
53	C₂×C₃×C₅	—	37	6	no	—
54	C₃×D₅	—	185	12	no	—
55	C₅²	C₂×#66526⋊C₂	1	1	yes	characteristic, complemented, Sylow
56	C₂×D₅	—	185	6	no	—
57	C₂×C₉	—	37	1	no	—
58	C₂×C₉	—	925	2	no	—
59	C₃×C₅	—	37	6	no	—
60	C₂²×C₃	—	925	1	no	—
61	C₂×C₅	C₅⋊(#66526⋊C₂)	1	6	yes	complemented
62	D₅	—	185	12	no	—
63	C₉	—	37	1	no	Sylow
64	C₂×C₃	—	37	1	no	—
65	C₂×C₃	—	925	2	no	—
66	C₅	C₂×C₅⋊(#66526⋊C₂)	1	6	yes	complemented
67	C₂²	—	925	1	no	Sylow
68	C₃	—	37	1	no	—
69	C₂	—	925	2	no	—
70	C₂	C₅⋊(C₅⋊(#66526⋊C₂))	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂	C₅⋊(C₅⋊(#66526⋊C₂))
commutator, hypocenter	C₅²×C₃₇	C₂²×C₉
Frattini	C₁	G
Fitting, socle	C₂×C₅²×C₃₇	C₂×C₉
prime radical	C₂×C₅⋊(C₅⋊(D₃₇⋊C₃))	C₃
cosocle	C₅²×C₃₇⋊C₃	C₂²×C₃
LCS	G ▷ C₅²×C₃₇
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. C₂×C₅²×#66526	C₂	semidirect
2. C₅⋊(C₅⋊(#66526⋊C₂))	C₂	direct (N × H)
4. C₅²×#66526	C₂²	semidirect
8. C₂×C₅⋊D₁₈₅	C₉	semidirect
9. C₂×C₅×#66526	D₅	semidirect
13. C₂×C₅²×C₃₇	C₂×C₉	semidirect
14. C₅⋊D₁₈₅	C₂×C₉	semidirect
15. C₅×#66526	C₂×D₅	semidirect
19. C₅²×C₃₇	C₂²×C₉	semidirect
22. C₂×#66526	C₅⋊D₅	semidirect
28. C₂×C₅×C₃₇	C₉×D₅	semidirect
30. #66526	C₂×C₅⋊D₅	semidirect
35. C₅×C₃₇	C₂×C₉×D₅	semidirect
45. C₂×C₃₇	C₉×C₅⋊D₅	semidirect
48. C₂×C₅²	#66526⋊C₂	semidirect
51. C₃₇	C₂×C₉×C₅⋊D₅	semidirect
55. C₅²	C₂×#66526⋊C₂	semidirect
61. C₂×C₅	C₅⋊(#66526⋊C₂)	semidirect
66. C₅	C₂×C₅⋊(#66526⋊C₂)	semidirect
70. C₂	C₅⋊(C₅⋊(#66526⋊C₂))	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅²×#66526	2
C₅⋊(C₅⋊(#66526⋊C₂))	2
C₂×C₅⋊(C₅⋊(D₃₇⋊C₃))	3
C₂×C₅⋊(#66526⋊C₂)	5
C₂×C₉×C₅⋊D₅	37

Subgroups — C2×C5⋊(C5⋊(#66526⋊C2)) #45686196

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅⋊(C₅⋊(#66526⋊C₂)) #45686196