Subgroups — C₆₁×C₂²⋊(C₅²⋊C₃) #32913147

Order 18300.

40 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅²×C₆₁	C₃	1	1	yes	maximal, characteristic, complemented, Hall
2	C₆₁×C₅²⋊C₃	—	4	1	no	maximal, Hall
3	C₂×C₅²×C₆₁	—	3	1	no	—
4	C₅²×C₆₁	A₄	1	1	yes	characteristic, complemented, Hall
5	C₂²×C₅×C₆₁	—	3	2	no	—
6	A₄×C₆₁	—	25	1	no	maximal, Hall
7	C₂×C₅×C₆₁	—	3	6	no	—
8	C₅×C₆₁	—	3	2	no	—
9	C₂²⋊(C₅²⋊C₃)	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂²×C₆₁	C₅²⋊C₃	1	1	yes	characteristic, complemented, Hall
11	C₃×C₆₁	—	100	1	no	Hall
12	C₂×C₆₁	—	3	1	no	—
13	C₂²×C₅²	C₃×C₆₁	1	1	yes	characteristic, complemented, Hall
14	C₅²⋊C₃	—	4	1	no	Hall
15	C₆₁	C₂²⋊(C₅²⋊C₃)	1	1	yes	characteristic, direct product factor, Sylow
16	C₂×C₅²	—	3	1	no	—
17	C₅²	A₄×C₆₁	1	1	yes	characteristic, complemented, Sylow
18	C₂²×C₅	—	3	2	no	—
19	A₄	—	25	1	no	Hall
20	C₂×C₅	—	3	6	no	—
21	C₅	—	3	2	no	—
22	C₂²	C₆₁×C₅²⋊C₃	1	1	yes	characteristic, complemented, Sylow
23	C₃	—	100	1	no	Sylow
24	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₆₁	C₂²⋊(C₅²⋊C₃)
commutator, cosocle, hypocenter	C₂²×C₅²	C₃×C₆₁
Frattini	C₁	G
Fitting, socle	C₂²×C₅²×C₆₁	C₃
LCS	G ▷ C₂²×C₅²
UCS	C₁ ◁ C₆₁

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

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