Subgroups — C₅²×#41⋊C₃ #11925671

Order 1200.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅²×#41	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₃×C₄×C₅²	—	4	1	no	maximal
4	C₅×#41⋊C₃	C₅	1	6	yes	maximal, direct product factor
5	C₅²×Q₈	C₂×C₃	1	1	yes	characteristic
6	C₅²×D₄	—	3	1	no	—
7	C₂×C₄×C₅²	—	3	1	no	—
8	C₂×C₃×C₅²	—	4	1	no	—
9	C₅×SL(2,𝔽₃)	C₂×C₅	1	6	yes	complemented
10	C₂²×C₅²	—	3	1	no	—
11	C₄×C₅²	—	3	1	no	—
12	C₄×C₅²	A₄	1	1	yes	characteristic
13	C₅×#41	C₃×C₅	1	6	yes	complemented
14	C₃×C₅²	—	4	1	no	Hall
15	C₃×C₄×C₅	—	4	6	no	—
16	C₂×C₅²	—	6	1	no	—
17	C₂×C₅²	C₂×A₄	1	1	yes	characteristic
18	#41⋊C₃	C₅²	1	1	yes	characteristic, direct product factor, Hall
19	C₅×Q₈	C₂×C₃×C₅	1	6	yes	—
20	C₅×D₄	—	3	6	no	—
21	C₂×C₄×C₅	—	3	6	no	—
22	C₂×C₃×C₅	—	4	6	no	—
23	C₅²	#41⋊C₃	1	1	yes	characteristic, direct product factor, Sylow
24	SL(2,𝔽₃)	C₂×C₅²	1	1	yes	characteristic, complemented
25	C₂²×C₅	—	3	6	no	—
26	C₄×C₅	—	3	6	no	—
27	C₄×C₅	C₅×A₄	1	6	yes	—
28	#41	C₃×C₅²	1	1	yes	characteristic, complemented, Sylow
29	C₃×C₅	—	4	6	no	—
30	C₃×C₄	—	4	1	no	—
31	C₂×C₅	—	6	6	no	—
32	C₂×C₅	C₂×C₅×A₄	1	6	yes	—
33	Q₈	C₂×C₃×C₅²	1	1	yes	characteristic
34	D₄	—	3	1	no	—
35	C₂×C₄	—	3	1	no	—
36	C₂×C₃	—	4	1	no	—
37	C₅	C₅×#41⋊C₃	1	6	yes	direct product factor
38	C₂²	—	3	1	no	—
39	C₄	—	3	1	no	—
40	C₄	C₅²×A₄	1	1	yes	characteristic
41	C₃	—	4	1	no	Sylow
42	C₂	—	6	1	no	—
43	C₂	C₂×C₅²×A₄	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₄×C₅²	A₄
commutator, cosocle, hypocenter	Q₈	C₂×C₃×C₅²
Frattini	C₂	C₂×C₅²×A₄
Fitting	C₅²×#41	C₃
socle	C₂×C₅²	C₂×A₄
LCS	G ▷ Q₈
UCS	C₁ ◁ C₄×C₅²

Complements

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

N	H ≅ G/N	type
1. C₅²×SL(2,𝔽₃)	C₂	semidirect
2. C₅²×#41	C₃	semidirect
4. C₅×#41⋊C₃	C₅	direct (N × H)
9. C₅×SL(2,𝔽₃)	C₂×C₅	semidirect
13. C₅×#41	C₃×C₅	semidirect
18. #41⋊C₃	C₅²	direct (N × H)
23. C₅²	#41⋊C₃	direct (N × H)
24. SL(2,𝔽₃)	C₂×C₅²	semidirect
28. #41	C₃×C₅²	semidirect
37. C₅	C₅×#41⋊C₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅²×SL(2,𝔽₃)	2
C₅²×#41	3
C₃×C₄×C₅²	4
C₅×#41⋊C₃	5

Subgroups — C52×#41⋊C3 #11925671

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×#41⋊C₃ #11925671