Subgroups — #10597048

Order 600.

48 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₂₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₇₅	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₇₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²⋊D₂₅	—	3	1	no	maximal, Hall
5	C₂×C₃×C₂₅	—	2	1	no	—
6	C₂×C₃×C₂₅	C₂²	1	1	yes	characteristic
7	D₇₅	—	2	1	no	—
8	#1220	—	5	1	no	maximal
9	C₂²×C₂₅	D₃	1	1	yes	characteristic, complemented
10	C₂×D₂₅	—	3	1	no	—
11	Dic₂₅	—	3	1	no	—
12	C₃×C₂₅	D₄	1	1	yes	characteristic, complemented, Hall
13	C₂²×C₃×C₅	D₅	1	1	yes	characteristic
14	C₂×D₁₅	—	5	1	no	—
15	Dic₁₅	—	5	1	no	—
16	C₂×C₂₅	—	2	1	no	—
17	C₂×C₂₅	C₂×D₃	1	1	yes	characteristic
18	D₂₅	—	6	1	no	—
19	C₂²⋊D₅	—	15	1	no	—
20	C₂×C₃×C₅	—	2	1	no	—
21	C₂×C₃×C₅	C₂×D₅	1	1	yes	characteristic
22	D₁₅	—	10	1	no	—
23	C₂₅	#67	1	1	yes	characteristic, complemented, Sylow
24	#67	—	25	1	no	Hall
25	C₂²×C₅	D₁₅	1	1	yes	characteristic
26	C₂×D₅	—	15	1	no	—
27	Dic₅	—	15	1	no	—
28	C₃×C₅	C₂²⋊D₅	1	1	yes	characteristic
29	C₂²×C₃	D₂₅	1	1	yes	characteristic, complemented
30	C₂×D₃	—	25	1	no	—
31	Dic₃	—	25	1	no	—
32	C₂×C₅	—	2	1	no	—
33	C₂×C₅	C₂×D₁₅	1	1	yes	characteristic
34	D₅	—	30	1	no	—
35	D₄	—	75	1	no	Sylow
36	C₂×C₃	—	2	1	no	—
37	C₂×C₃	C₂×D₂₅	1	1	yes	characteristic
38	D₃	—	50	1	no	—
39	C₅	#1220	1	1	yes	characteristic
40	C₂²	—	75	1	no	—
41	C₂²	D₇₅	1	1	yes	characteristic, complemented
42	C₄	—	75	1	no	—
43	C₃	C₂²⋊D₂₅	1	1	yes	characteristic, complemented, Sylow
44	C₂	—	2	1	no	—
45	C₂	—	150	1	no	—
46	C₂	C₂×D₇₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₇₅
commutator, cosocle	C₂×C₃×C₂₅	C₂²
Frattini	C₂×C₅	C₂×D₁₅
Fitting	C₂²×C₃×C₂₅	C₂
socle	C₂×C₃×C₅	C₂×D₅
hypercenter	C₂²	D₇₅
hypocenter	C₃×C₂₅	D₄
LCS	G ▷ C₂×C₃×C₂₅ ▷ C₃×C₂₅
UCS	C₁ ◁ C₂ ◁ C₂²

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×C₂₅	C₂	semidirect
2. C₂×D₇₅	C₂	semidirect
3. Dic₇₅	C₂	semidirect
9. C₂²×C₂₅	D₃	semidirect
12. C₃×C₂₅	D₄	semidirect
23. C₂₅	#67	semidirect
29. C₂²×C₃	D₂₅	semidirect
41. C₂²	D₇₅	semidirect
43. C₃	C₂²⋊D₂₅	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₂₅	2
C₂×D₇₅	2
Dic₇₅	2
C₂²⋊D₂₅	3
#1220	5