Subgroups — C₃₁×C₂²⋊(C₅²⋊C₉) #42124576

Order 27900.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅²×C₃₁	C₃	1	1	yes	maximal, characteristic
2	C₃₁×C₅²⋊C₉	—	4	1	no	maximal, Hall
3	C₂×C₃×C₅²×C₃₁	—	3	1	no	—
4	C₂²×C₅²×C₃₁	C₉	1	1	yes	characteristic, complemented, Hall
5	C₃×C₅²×C₃₁	A₄	1	1	yes	characteristic
6	C₂²×C₃×C₅×C₃₁	—	3	2	no	—
7	C₂×C₅²×C₃₁	—	3	1	no	—
8	C₃₁×C₂²⋊C₉	—	25	1	no	maximal, Hall
9	C₂×C₃×C₅×C₃₁	—	3	6	no	—
10	C₂²⋊(C₅²⋊C₉)	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₅²×C₃₁	C₂²⋊C₉	1	1	yes	characteristic, complemented, Hall
12	C₂²×C₅×C₃₁	—	3	2	no	—
13	C₃×C₅×C₃₁	—	3	2	no	—
14	C₂²×C₃×C₃₁	C₅²⋊C₃	1	1	yes	characteristic
15	C₂×C₅×C₃₁	—	3	6	no	—
16	C₂²×C₃×C₅²	C₃×C₃₁	1	1	yes	characteristic
17	C₉×C₃₁	—	100	1	no	Hall
18	C₅²⋊C₉	—	4	1	no	Hall
19	C₂×C₃×C₃₁	—	3	1	no	—
20	C₅×C₃₁	—	3	2	no	—
21	C₂×C₃×C₅²	—	3	1	no	—
22	C₂²×C₃₁	C₅²⋊C₉	1	1	yes	characteristic, complemented, Hall
23	C₂²×C₅²	C₉×C₃₁	1	1	yes	characteristic, complemented, Hall
24	C₃×C₃₁	C₂²⋊(C₅²⋊C₃)	1	1	yes	characteristic
25	C₃×C₅²	A₄×C₃₁	1	1	yes	characteristic
26	C₂×C₃₁	—	3	1	no	—
27	C₂²×C₃×C₅	—	3	2	no	—
28	C₂×C₅²	—	3	1	no	—
29	C₂²⋊C₉	—	25	1	no	Hall
30	C₃₁	C₂²⋊(C₅²⋊C₉)	1	1	yes	characteristic, direct product factor, Sylow
31	C₂×C₃×C₅	—	3	6	no	—
32	C₅²	C₃₁×C₂²⋊C₉	1	1	yes	characteristic, complemented, Sylow
33	C₂²×C₅	—	3	2	no	—
34	C₃×C₅	—	3	2	no	—
35	C₂²×C₃	C₃₁×C₅²⋊C₃	1	1	yes	characteristic
36	C₂×C₅	—	3	6	no	—
37	C₉	—	100	1	no	Sylow
38	C₂×C₃	—	3	1	no	—
39	C₅	—	3	2	no	—
40	C₂²	C₃₁×C₅²⋊C₉	1	1	yes	characteristic, complemented, Sylow
41	C₃	C₃₁×C₂²⋊(C₅²⋊C₃)	1	1	yes	characteristic
42	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₃×C₃₁	C₂²⋊(C₅²⋊C₃)
commutator, hypocenter	C₂²×C₅²	C₉×C₃₁
Frattini	C₃	C₃₁×C₂²⋊(C₅²⋊C₃)
Fitting, socle, prime radical	C₂²×C₃×C₅²×C₃₁	C₃
cosocle	C₂²×C₃×C₅²	C₃×C₃₁
LCS	G ▷ C₂²×C₅²
UCS	C₁ ◁ C₃×C₃₁

Complements

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

N	H ≅ G/N	type
4. C₂²×C₅²×C₃₁	C₉	semidirect
10. C₂²⋊(C₅²⋊C₉)	C₃₁	direct (N × H)
11. C₅²×C₃₁	C₂²⋊C₉	semidirect
22. C₂²×C₃₁	C₅²⋊C₉	semidirect
23. C₂²×C₅²	C₉×C₃₁	semidirect
30. C₃₁	C₂²⋊(C₅²⋊C₉)	direct (N × H)
32. C₅²	C₃₁×C₂²⋊C₉	semidirect
40. C₂²	C₃₁×C₅²⋊C₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₅²×C₃₁	3
C₃₁×C₅²⋊C₉	4
C₃₁×C₂²⋊C₉	25
C₂²⋊(C₅²⋊C₉)	31

Subgroups — C31×C22⋊(C52⋊C9) #42124576

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×C₂²⋊(C₅²⋊C₉) #42124576