Subgroups — C₅²×C₃₁×#157 #42124870

Order 27900.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×C₃⋊D₃×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₅²×C₃₁	C₄	1	1	yes	characteristic, complemented, Hall
3	C₅×C₃₁×#157	C₅	1	6	yes	maximal, direct product factor
4	C₅²×D₃×C₃₁	—	6	2	no	—
5	C₄×C₅²×C₃₁	—	9	1	no	maximal, Hall
6	C₅×C₃⋊D₃×C₃₁	C₂×C₅	1	6	yes	—
7	C₃×C₅²×C₃₁	—	2	2	no	—
8	C₂×C₅²×C₃₁	—	9	1	no	—
9	C₃²×C₅×C₃₁	C₄×C₅	1	6	yes	complemented
10	C₃₁×#157	C₅²	1	1	yes	characteristic, direct product factor, Hall
11	C₅×D₃×C₃₁	—	6	12	no	—
12	C₅²×#157	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₅²×C₃₁	#157	1	1	yes	characteristic, direct product factor, Hall
14	C₄×C₅×C₃₁	—	9	6	no	—
15	C₃⋊D₃×C₃₁	C₂×C₅²	1	1	yes	characteristic
16	C₃×C₅×C₃₁	—	2	12	no	—
17	C₅²×C₃⋊D₃	C₂×C₃₁	1	1	yes	characteristic
18	C₂×C₅×C₃₁	—	9	6	no	—
19	C₃²×C₃₁	C₄×C₅²	1	1	yes	characteristic, complemented, Hall
20	C₃²×C₅²	C₄×C₃₁	1	1	yes	characteristic, complemented, Hall
21	D₃×C₃₁	—	6	2	no	—
22	C₅×#157	C₅×C₃₁	1	6	yes	direct product factor
23	C₅×C₃₁	C₅×#157	1	6	yes	direct product factor
24	C₅²×D₃	—	6	2	no	—
25	C₄×C₃₁	—	9	1	no	Hall
26	C₄×C₅²	—	9	1	no	Hall
27	C₃×C₃₁	—	2	2	no	—
28	C₅×C₃⋊D₃	C₂×C₅×C₃₁	1	6	yes	—
29	C₃×C₅²	—	2	2	no	—
30	C₂×C₃₁	—	9	1	no	—
31	C₂×C₅²	—	9	1	no	—
32	C₃²×C₅	C₄×C₅×C₃₁	1	6	yes	complemented
33	#157	C₅²×C₃₁	1	1	yes	characteristic, direct product factor, Hall
34	C₃₁	C₅²×#157	1	1	yes	characteristic, direct product factor, Sylow
35	C₅×D₃	—	6	12	no	—
36	C₅²	C₃₁×#157	1	1	yes	characteristic, direct product factor, Sylow
37	C₄×C₅	—	9	6	no	—
38	C₃⋊D₃	C₂×C₅²×C₃₁	1	1	yes	characteristic
39	C₃×C₅	—	2	12	no	—
40	C₂×C₅	—	9	6	no	—
41	C₃²	C₄×C₅²×C₃₁	1	1	yes	characteristic, complemented, Sylow
42	D₃	—	6	2	no	—
43	C₅	C₅×C₃₁×#157	1	6	yes	direct product factor
44	C₄	—	9	1	no	Sylow
45	C₃	—	2	2	no	—
46	C₂	—	9	1	no	—

	subgroup	G/H
center, hypercenter	C₅²×C₃₁	#157
commutator, hypocenter	C₃²	C₄×C₅²×C₃₁
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₃₁	C₄
prime radical	C₅²×C₃⋊D₃×C₃₁	C₂
cosocle	C₃⋊D₃	C₂×C₅²×C₃₁
LCS	G ▷ C₃²
UCS	C₁ ◁ C₅²×C₃₁

Complements

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

N	H ≅ G/N	type
2. C₃²×C₅²×C₃₁	C₄	semidirect
3. C₅×C₃₁×#157	C₅	direct (N × H)
9. C₃²×C₅×C₃₁	C₄×C₅	semidirect
10. C₃₁×#157	C₅²	direct (N × H)
12. C₅²×#157	C₃₁	direct (N × H)
13. C₅²×C₃₁	#157	direct (N × H)
19. C₃²×C₃₁	C₄×C₅²	semidirect
20. C₃²×C₅²	C₄×C₃₁	semidirect
22. C₅×#157	C₅×C₃₁	direct (N × H)
23. C₅×C₃₁	C₅×#157	direct (N × H)
32. C₃²×C₅	C₄×C₅×C₃₁	semidirect
33. #157	C₅²×C₃₁	direct (N × H)
34. C₃₁	C₅²×#157	direct (N × H)
36. C₅²	C₃₁×#157	direct (N × H)
41. C₃²	C₄×C₅²×C₃₁	semidirect
43. C₅	C₅×C₃₁×#157	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×C₃⋊D₃×C₃₁	2
C₅×C₃₁×#157	5
C₄×C₅²×C₃₁	9
C₅²×#157	31

Subgroups — C52×C31×#157 #42124870

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×C₃₁×#157 #42124870