Subgroups — C₂²×(C₆₁⋊C₃)⋊C₂₅ #32912902

Order 18300.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×(C₆₁⋊C₃)⋊C₂₅	C₂	1	3	yes	maximal, direct product factor
2	C₂²×C₆₁⋊C₂₅	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	(C₆₁⋊C₃)⋊C₂₅	C₂²	1	1	yes	characteristic, direct product factor, Hall
4	C₂²×C₅×C₆₁⋊C₃	C₅	1	1	yes	maximal, characteristic
5	C₂×C₆₁⋊C₂₅	C₂×C₃	1	3	yes	complemented
6	C₂×C₅×C₆₁⋊C₃	C₂×C₅	1	3	yes	—
7	C₆₁⋊C₂₅	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
8	C₂²×C₅×C₆₁	C₃×C₅	1	1	yes	characteristic
9	C₅×C₆₁⋊C₃	C₂²×C₅	1	1	yes	characteristic
10	C₂²×C₆₁⋊C₃	C₂₅	1	1	yes	characteristic, complemented, Hall
11	C₂×C₅×C₆₁	C₂×C₃×C₅	1	3	yes	—
12	C₂×C₆₁⋊C₃	C₂×C₂₅	1	3	yes	complemented
13	C₅×C₆₁	C₂²×C₃×C₅	1	1	yes	characteristic
14	C₂²×C₃×C₂₅	—	61	1	no	maximal, Hall
15	C₂²×C₆₁	C₃×C₂₅	1	1	yes	characteristic, complemented, Hall
16	C₆₁⋊C₃	C₂²×C₂₅	1	1	yes	characteristic, complemented, Hall
17	C₂×C₃×C₂₅	—	61	3	no	—
18	C₂×C₆₁	C₂×C₃×C₂₅	1	3	yes	complemented
19	C₂²×C₂₅	—	61	1	no	Hall
20	C₃×C₂₅	—	61	1	no	Hall
21	C₆₁	C₂²×C₃×C₂₅	1	1	yes	characteristic, complemented, Sylow
22	C₂²×C₃×C₅	—	305	1	no	—
23	C₂×C₂₅	—	61	3	no	—
24	C₂×C₃×C₅	—	305	3	no	—
25	C₂₅	—	61	1	no	Sylow
26	C₂²×C₅	(C₆₁⋊C₃)⋊C₅	1	1	yes	characteristic
27	C₃×C₅	—	305	1	no	—
28	C₂²×C₃	—	305	1	no	Hall
29	C₂×C₅	C₂×(C₆₁⋊C₃)⋊C₅	1	3	yes	—
30	C₂×C₃	—	305	3	no	—
31	C₅	C₂²×(C₆₁⋊C₃)⋊C₅	1	1	yes	characteristic
32	C₂²	(C₆₁⋊C₃)⋊C₂₅	1	1	yes	characteristic, direct product factor, Sylow
33	C₃	—	305	1	no	Sylow
34	C₂	C₂×(C₆₁⋊C₃)⋊C₂₅	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₅	(C₆₁⋊C₃)⋊C₅
commutator, hypocenter	C₆₁	C₂²×C₃×C₂₅
Frattini	C₅	C₂²×(C₆₁⋊C₃)⋊C₅
Fitting, socle	C₂²×C₅×C₆₁	C₃×C₅
prime radical	C₂²×C₅×C₆₁⋊C₃	C₅
cosocle	C₅×C₆₁	C₂²×C₃×C₅
LCS	G ▷ C₆₁
UCS	C₁ ◁ C₂²×C₅

Complements

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

N	H ≅ G/N	type
1. C₂×(C₆₁⋊C₃)⋊C₂₅	C₂	direct (N × H)
2. C₂²×C₆₁⋊C₂₅	C₃	semidirect
3. (C₆₁⋊C₃)⋊C₂₅	C₂²	direct (N × H)
5. C₂×C₆₁⋊C₂₅	C₂×C₃	semidirect
7. C₆₁⋊C₂₅	C₂²×C₃	semidirect
10. C₂²×C₆₁⋊C₃	C₂₅	semidirect
12. C₂×C₆₁⋊C₃	C₂×C₂₅	semidirect
15. C₂²×C₆₁	C₃×C₂₅	semidirect
16. C₆₁⋊C₃	C₂²×C₂₅	semidirect
18. C₂×C₆₁	C₂×C₃×C₂₅	semidirect
21. C₆₁	C₂²×C₃×C₂₅	semidirect
32. C₂²	(C₆₁⋊C₃)⋊C₂₅	direct (N × H)
34. C₂	C₂×(C₆₁⋊C₃)⋊C₂₅	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×(C₆₁⋊C₃)⋊C₂₅	2
C₂²×C₆₁⋊C₂₅	3
C₂²×C₅×C₆₁⋊C₃	5
C₂²×C₃×C₂₅	61

Subgroups — C22×(C61⋊C3)⋊C25 #32912902

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×(C₆₁⋊C₃)⋊C₂₅ #32912902