Subgroups — C₃₁×#11744707 #42124880

Order 27900.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×C₃⋊(C₃⋊(C₅⋊D₅))	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₅²×C₃₁	C₄	1	1	yes	characteristic, complemented, Hall
3	C₃₁×#4380	—	5	6	no	maximal
4	C₃₁×C₃⋊(C₅⋊D₅)	—	6	2	no	—
5	C₃₁×#1043	—	9	1	no	maximal, Hall
6	C₃₁×C₃⋊D₁₅	—	5	6	no	—
7	C₃×C₅²×C₃₁	—	2	2	no	—
8	C₃₁×C₅⋊D₅	—	9	1	no	—
9	C₃²×C₅×C₃₁	AGL(1,𝔽₅)	1	6	yes	complemented
10	C₃₁×#157	—	25	1	no	Hall
11	D₁₅×C₃₁	—	30	12	no	—
12	#11744707	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₅²×C₃₁	#157	1	1	yes	characteristic, complemented, Hall
14	AGL(1,𝔽₅)×C₃₁	—	45	6	no	—
15	C₃⋊D₃×C₃₁	—	25	1	no	—
16	C₃×C₅×C₃₁	—	2	12	no	—
17	C₃⋊(C₃⋊(C₅⋊D₅))	C₂×C₃₁	1	1	yes	characteristic
18	D₅×C₃₁	—	45	6	no	—
19	C₃²×C₃₁	#1043	1	1	yes	characteristic, complemented, Hall
20	C₃²×C₅²	C₄×C₃₁	1	1	yes	characteristic, complemented, Hall
21	D₃×C₃₁	—	150	2	no	—
22	#4380	—	5	6	no	—
23	C₅×C₃₁	#4380	1	6	yes	complemented
24	C₃⋊(C₅⋊D₅)	—	6	2	no	—
25	C₄×C₃₁	—	225	1	no	Hall
26	#1043	—	9	1	no	Hall
27	C₃×C₃₁	—	2	2	no	—
28	C₃⋊D₁₅	—	5	6	no	—
29	C₃×C₅²	—	2	2	no	—
30	C₂×C₃₁	—	225	1	no	—
31	C₅⋊D₅	—	9	1	no	—
32	C₃²×C₅	AGL(1,𝔽₅)×C₃₁	1	6	yes	complemented
33	#157	—	25	1	no	Hall
34	C₃₁	#11744707	1	1	yes	characteristic, direct product factor, Sylow
35	D₁₅	—	30	12	no	—
36	C₅²	C₃₁×#157	1	1	yes	characteristic, complemented, Sylow
37	AGL(1,𝔽₅)	—	45	6	no	—
38	C₃⋊D₃	—	25	1	no	—
39	C₃×C₅	—	2	12	no	—
40	D₅	—	45	6	no	—
41	C₃²	C₃₁×#1043	1	1	yes	characteristic, complemented, Sylow
42	D₃	—	150	2	no	—
43	C₅	C₃₁×#4380	1	6	yes	complemented
44	C₄	—	225	1	no	Sylow
45	C₃	—	2	2	no	—
46	C₂	—	225	1	no	—

	subgroup	G/H
center, hypercenter	C₃₁	#11744707
commutator, hypocenter	C₃²×C₅²	C₄×C₃₁
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₃₁	C₄
prime radical	C₃₁×C₃⋊(C₃⋊(C₅⋊D₅))	C₂
cosocle	C₃⋊(C₃⋊(C₅⋊D₅))	C₂×C₃₁
LCS	G ▷ C₃²×C₅²
UCS	C₁ ◁ C₃₁

Complements

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

N	H ≅ G/N	type
2. C₃²×C₅²×C₃₁	C₄	semidirect
9. C₃²×C₅×C₃₁	AGL(1,𝔽₅)	semidirect
12. #11744707	C₃₁	direct (N × H)
13. C₅²×C₃₁	#157	semidirect
19. C₃²×C₃₁	#1043	semidirect
20. C₃²×C₅²	C₄×C₃₁	semidirect
23. C₅×C₃₁	#4380	semidirect
32. C₃²×C₅	AGL(1,𝔽₅)×C₃₁	semidirect
34. C₃₁	#11744707	direct (N × H)
36. C₅²	C₃₁×#157	semidirect
41. C₃²	C₃₁×#1043	semidirect
43. C₅	C₃₁×#4380	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×C₃⋊(C₃⋊(C₅⋊D₅))	2
C₃₁×#4380	5
C₃₁×#1043	9
#11744707	31

Subgroups — C31×#11744707 #42124880

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#11744707 #42124880