Subgroups — C₃²×C₅×Dic₃₁⋊C₅ #42124607

Order 27900.

288 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₅×ΓL(1,𝔽₃₂)	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×Dic₃₁⋊C₅	C₃	1	4	yes	maximal, direct product factor
3	C₃²×C₅×ΓL(1,𝔽₃₂)	C₄	1	1	yes	characteristic, complemented, Hall
4	C₃²×C₅×Dic₃₁	C₅	1	1	yes	maximal, characteristic, complemented
5	C₃²×Dic₃₁⋊C₅	C₅	1	5	yes	maximal, direct product factor
6	C₂×C₃×C₅×ΓL(1,𝔽₃₂)	C₂×C₃	1	4	yes	—
7	C₅×Dic₃₁⋊C₅	C₃²	1	1	yes	characteristic, direct product factor, Hall
8	C₂×C₃²×ΓL(1,𝔽₃₂)	C₂×C₅	1	5	yes	—
9	C₂×C₃²×C₅×C₃₁	C₂×C₅	1	1	yes	characteristic
10	C₃×C₅×ΓL(1,𝔽₃₂)	C₃×C₄	1	4	yes	complemented
11	C₃×C₅×Dic₃₁	C₃×C₅	1	4	yes	complemented
12	C₃×Dic₃₁⋊C₅	C₃×C₅	1	20	yes	direct product factor
13	C₂×C₅×ΓL(1,𝔽₃₂)	C₂×C₃²	1	1	yes	characteristic
14	C₃²×C₅×C₃₁	C₄×C₅	1	1	yes	characteristic, complemented
15	C₃²×ΓL(1,𝔽₃₂)	C₄×C₅	1	5	yes	complemented
16	C₃²×Dic₃₁	C₅²	1	1	yes	characteristic, complemented, Hall
17	C₂×C₃×C₅×C₃₁	C₂×C₃×C₅	1	4	yes	—
18	C₂×C₃×ΓL(1,𝔽₃₂)	C₂×C₃×C₅	1	20	yes	—
19	C₃²×C₄×C₅²	—	31	1	no	maximal, Hall
20	C₅×ΓL(1,𝔽₃₂)	C₃²×C₄	1	1	yes	characteristic, complemented, Hall
21	C₅×Dic₃₁	C₃²×C₅	1	1	yes	characteristic, complemented
22	Dic₃₁⋊C₅	C₃²×C₅	1	5	yes	direct product factor
23	C₂×C₃²×C₃₁	C₂×C₅²	1	1	yes	characteristic
24	C₃×C₅×C₃₁	C₃×C₄×C₅	1	4	yes	complemented
25	C₃×ΓL(1,𝔽₃₂)	C₃×C₄×C₅	1	20	yes	complemented
26	C₂×C₃²×C₅²	—	31	1	no	—
27	C₃×Dic₃₁	C₃×C₅²	1	4	yes	complemented
28	C₂×C₅×C₃₁	C₂×C₃²×C₅	1	1	yes	characteristic
29	C₂×ΓL(1,𝔽₃₂)	C₂×C₃²×C₅	1	5	yes	—
30	C₃×C₄×C₅²	—	31	4	no	—
31	C₃²×C₃₁	C₄×C₅²	1	1	yes	characteristic, complemented, Hall
32	C₃²×C₅²	—	31	1	no	Hall
33	C₂×C₃×C₃₁	C₂×C₃×C₅²	1	4	yes	—
34	C₃²×C₄×C₅	—	31	5	no	—
35	C₃²×C₄×C₅	—	31	1	no	—
36	C₅×C₃₁	C₃²×C₄×C₅	1	1	yes	characteristic, complemented
37	ΓL(1,𝔽₃₂)	C₃²×C₄×C₅	1	5	yes	complemented
38	C₂×C₃×C₅²	—	31	4	no	—
39	Dic₃₁	C₃²×C₅²	1	1	yes	characteristic, complemented, Hall
40	C₄×C₅²	—	31	1	no	Hall
41	C₃×C₃₁	C₃×C₄×C₅²	1	4	yes	complemented
42	C₂×C₃²×C₅	—	31	5	no	—
43	C₂×C₃²×C₅	D₃₁⋊C₅	1	1	yes	characteristic
44	C₃×C₅²	—	31	4	no	—
45	C₂×C₃₁	C₂×C₃²×C₅²	1	1	yes	characteristic
46	C₃×C₄×C₅	—	31	20	no	—
47	C₃×C₄×C₅	—	31	4	no	—
48	C₂×C₅²	—	31	1	no	—
49	C₃²×C₅	—	31	5	no	—
50	C₃²×C₅	Dic₃₁⋊C₅	1	1	yes	characteristic, direct product factor
51	C₃²×C₄	—	31	1	no	Hall
52	C₃₁	C₃²×C₄×C₅²	1	1	yes	characteristic, complemented, Sylow
53	C₂×C₃×C₅	—	31	20	no	—
54	C₂×C₃×C₅	C₃×D₃₁⋊C₅	1	4	yes	—
55	C₅²	—	31	1	no	Sylow
56	C₄×C₅	—	31	5	no	—
57	C₄×C₅	—	31	1	no	—
58	C₂×C₃²	C₅×D₃₁⋊C₅	1	1	yes	characteristic
59	C₃×C₅	—	31	20	no	—
60	C₃×C₅	C₃×Dic₃₁⋊C₅	1	4	yes	direct product factor
61	C₃×C₄	—	31	4	no	—
62	C₂×C₅	—	31	5	no	—
63	C₂×C₅	C₃²×D₃₁⋊C₅	1	1	yes	characteristic
64	C₃²	C₅×Dic₃₁⋊C₅	1	1	yes	characteristic, direct product factor, Sylow
65	C₂×C₃	C₃×C₅×D₃₁⋊C₅	1	4	yes	—
66	C₅	—	31	5	no	—
67	C₅	C₃²×Dic₃₁⋊C₅	1	1	yes	characteristic, direct product factor
68	C₄	—	31	1	no	Sylow
69	C₃	C₃×C₅×Dic₃₁⋊C₅	1	4	yes	direct product factor
70	C₂	C₃²×C₅×D₃₁⋊C₅	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃²×C₅	D₃₁⋊C₅
commutator, hypocenter	C₃₁	C₃²×C₄×C₅²
Frattini	C₂	C₃²×C₅×D₃₁⋊C₅
Fitting, socle	C₂×C₃²×C₅×C₃₁	C₂×C₅
prime radical	C₂×C₃²×C₅×ΓL(1,𝔽₃₂)	C₂
cosocle	C₂×C₃₁	C₂×C₃²×C₅²
LCS	G ▷ C₃₁
UCS	C₁ ◁ C₂×C₃²×C₅

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×Dic₃₁⋊C₅	C₃	direct (N × H)
3. C₃²×C₅×ΓL(1,𝔽₃₂)	C₄	semidirect
4. C₃²×C₅×Dic₃₁	C₅	semidirect
5. C₃²×Dic₃₁⋊C₅	C₅	direct (N × H)
7. C₅×Dic₃₁⋊C₅	C₃²	direct (N × H)
10. C₃×C₅×ΓL(1,𝔽₃₂)	C₃×C₄	semidirect
11. C₃×C₅×Dic₃₁	C₃×C₅	semidirect
12. C₃×Dic₃₁⋊C₅	C₃×C₅	direct (N × H)
14. C₃²×C₅×C₃₁	C₄×C₅	semidirect
15. C₃²×ΓL(1,𝔽₃₂)	C₄×C₅	semidirect
16. C₃²×Dic₃₁	C₅²	semidirect
20. C₅×ΓL(1,𝔽₃₂)	C₃²×C₄	semidirect
21. C₅×Dic₃₁	C₃²×C₅	semidirect
22. Dic₃₁⋊C₅	C₃²×C₅	direct (N × H)
24. C₃×C₅×C₃₁	C₃×C₄×C₅	semidirect
25. C₃×ΓL(1,𝔽₃₂)	C₃×C₄×C₅	semidirect
27. C₃×Dic₃₁	C₃×C₅²	semidirect
31. C₃²×C₃₁	C₄×C₅²	semidirect
36. C₅×C₃₁	C₃²×C₄×C₅	semidirect
37. ΓL(1,𝔽₃₂)	C₃²×C₄×C₅	semidirect
39. Dic₃₁	C₃²×C₅²	semidirect
41. C₃×C₃₁	C₃×C₄×C₅²	semidirect
50. C₃²×C₅	Dic₃₁⋊C₅	direct (N × H)
52. C₃₁	C₃²×C₄×C₅²	semidirect
60. C₃×C₅	C₃×Dic₃₁⋊C₅	direct (N × H)
64. C₃²	C₅×Dic₃₁⋊C₅	direct (N × H)
67. C₅	C₃²×Dic₃₁⋊C₅	direct (N × H)
69. C₃	C₃×C₅×Dic₃₁⋊C₅	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₅×ΓL(1,𝔽₃₂)	2
C₃×C₅×Dic₃₁⋊C₅	3
C₃²×C₅×Dic₃₁	5
C₃²×Dic₃₁⋊C₅	5
C₃²×C₄×C₅²	31

Subgroups — C32×C5×Dic31⋊C5 #42124607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₅×Dic₃₁⋊C₅ #42124607