Subgroups — C₃₁×#11744709 #42124876

Order 27900.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₃₁×C₃⋊D₁₅	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₅²×C₃₁	C₄	1	1	yes	characteristic, complemented, Hall
3	C₃₁×#4379	—	5	1	no	maximal
4	C₃₁×#4380	—	5	1	no	maximal
5	C₅×D₁₅×C₃₁	—	6	2	no	—
6	C₃₁×#1042	—	9	1	no	maximal, Hall
7	C₃₁×C₃⋊D₁₅	D₅	1	1	yes	characteristic
8	C₅×C₃⋊D₃×C₃₁	—	5	1	no	—
9	C₃×C₅²×C₃₁	—	2	2	no	—
10	C₅×D₅×C₃₁	—	9	1	no	—
11	C₃²×C₅×C₃₁	—	4	1	no	—
12	C₃²×C₅×C₃₁	Dic₅	1	1	yes	characteristic, complemented
13	C₃²×C₅×C₃₁	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
14	C₃₁×#157	—	25	1	no	Hall
15	D₁₅×C₃₁	—	6	2	no	—
16	C₅×D₃×C₃₁	—	30	2	no	—
17	#11744709	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₅²×C₃₁	#157	1	1	yes	characteristic, complemented, Hall
19	AGL(1,𝔽₅)×C₃₁	—	45	1	no	—
20	Dic₅×C₃₁	—	45	1	no	—
21	C₃⋊D₃×C₃₁	—	5	1	no	—
22	C₃×C₅×C₃₁	—	2	2	no	—
23	C₃×C₅×C₃₁	—	2	2	no	—
24	C₃×C₅×C₃₁	—	4	4	no	—
25	C₅×C₃⋊D₁₅	C₂×C₃₁	1	1	yes	characteristic
26	C₂×C₅×C₃₁	—	45	1	no	—
27	D₅×C₃₁	—	9	1	no	—
28	C₃²×C₃₁	#1042	1	1	yes	characteristic, complemented, Hall
29	C₃²×C₅²	C₄×C₃₁	1	1	yes	characteristic, complemented, Hall
30	D₃×C₃₁	—	30	2	no	—
31	#4380	—	5	1	no	—
32	#4379	—	5	1	no	—
33	C₅×C₃₁	—	4	1	no	—
34	C₅×C₃₁	#4379	1	1	yes	characteristic, complemented
35	C₅×C₃₁	#4380	1	1	yes	characteristic, complemented
36	C₅×D₁₅	—	6	2	no	—
37	C₄×C₃₁	—	225	1	no	Hall
38	#1042	—	9	1	no	Hall
39	C₃×C₃₁	—	2	2	no	—
40	C₃⋊D₁₅	D₅×C₃₁	1	1	yes	characteristic
41	C₅×C₃⋊D₃	—	5	1	no	—
42	C₃×C₅²	—	2	2	no	—
43	C₂×C₃₁	—	45	1	no	—
44	C₅×D₅	—	9	1	no	—
45	C₃²×C₅	—	4	1	no	—
46	C₃²×C₅	Dic₅×C₃₁	1	1	yes	characteristic, complemented
47	C₃²×C₅	AGL(1,𝔽₅)×C₃₁	1	1	yes	characteristic, complemented
48	#157	—	25	1	no	Hall
49	C₃₁	#11744709	1	1	yes	characteristic, direct product factor, Sylow
50	D₁₅	—	6	2	no	—
51	C₅×D₃	—	30	2	no	—
52	C₅²	C₃₁×#157	1	1	yes	characteristic, complemented, Sylow
53	AGL(1,𝔽₅)	—	45	1	no	—
54	Dic₅	—	45	1	no	—
55	C₃⋊D₃	—	5	1	no	—
56	C₃×C₅	—	2	2	no	—
57	C₃×C₅	—	2	2	no	—
58	C₃×C₅	—	4	4	no	—
59	C₂×C₅	—	45	1	no	—
60	D₅	—	9	1	no	—
61	C₃²	C₃₁×#1042	1	1	yes	characteristic, complemented, Sylow
62	D₃	—	30	2	no	—
63	C₅	—	4	1	no	—
64	C₅	C₃₁×#4380	1	1	yes	characteristic, complemented
65	C₅	C₃₁×#4379	1	1	yes	characteristic, complemented
66	C₄	—	225	1	no	Sylow
67	C₃	—	2	2	no	—
68	C₂	—	45	1	no	—

	subgroup	G/H
center, hypercenter	C₃₁	#11744709
commutator, hypocenter	C₃²×C₅²	C₄×C₃₁
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₃₁	C₄
prime radical	C₅×C₃₁×C₃⋊D₁₅	C₂
cosocle	C₅×C₃⋊D₁₅	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
12. C₃²×C₅×C₃₁	Dic₅	semidirect
13. C₃²×C₅×C₃₁	AGL(1,𝔽₅)	semidirect
17. #11744709	C₃₁	direct (N × H)
18. C₅²×C₃₁	#157	semidirect
28. C₃²×C₃₁	#1042	semidirect
29. C₃²×C₅²	C₄×C₃₁	semidirect
34. C₅×C₃₁	#4379	semidirect
35. C₅×C₃₁	#4380	semidirect
46. C₃²×C₅	Dic₅×C₃₁	semidirect
47. C₃²×C₅	AGL(1,𝔽₅)×C₃₁	semidirect
49. C₃₁	#11744709	direct (N × H)
52. C₅²	C₃₁×#157	semidirect
61. C₃²	C₃₁×#1042	semidirect
64. C₅	C₃₁×#4380	semidirect
65. C₅	C₃₁×#4379	semidirect

Maximal subgroups

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

subgroup	index
C₅×C₃₁×C₃⋊D₁₅	2
C₃₁×#4379	5
C₃₁×#4380	5
C₃₁×#1042	9
#11744709	31

Subgroups — C31×#11744709 #42124876

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#11744709 #42124876