Subgroups — C₂×C₃²×C₅×D₃₁⋊C₅ #42124822

Order 27900.

480 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, complemented
2	C₃²×C₅×D₃₁⋊C₅	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃×C₅×D₃₁⋊C₅	C₃	1	4	yes	maximal, direct product factor
4	C₃²×C₅×ΓL(1,𝔽₃₂)	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×C₃²×C₅×D₃₁	C₅	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃²×D₃₁⋊C₅	C₅	1	5	yes	maximal, direct product factor
7	C₂×C₃×C₅×ΓL(1,𝔽₃₂)	C₂×C₃	1	4	yes	complemented
8	C₃×C₅×D₃₁⋊C₅	C₂×C₃	1	8	yes	direct product factor
9	C₂×C₅×D₃₁⋊C₅	C₃²	1	1	yes	characteristic, direct product factor, Hall
10	C₃²×C₅×D₃₁	C₂×C₅	1	2	yes	complemented
11	C₂×C₃²×ΓL(1,𝔽₃₂)	C₂×C₅	1	5	yes	complemented
12	C₃²×D₃₁⋊C₅	C₂×C₅	1	10	yes	direct product factor
13	C₂×C₃²×C₅×C₃₁	C₂×C₅	1	1	yes	characteristic, complemented
14	C₃×C₅×ΓL(1,𝔽₃₂)	C₂²×C₃	1	4	yes	complemented
15	C₂×C₃×C₅×D₃₁	C₃×C₅	1	4	yes	complemented
16	C₂×C₃×D₃₁⋊C₅	C₃×C₅	1	20	yes	direct product factor
17	C₂×C₅×ΓL(1,𝔽₃₂)	C₂×C₃²	1	1	yes	characteristic, complemented
18	C₅×D₃₁⋊C₅	C₂×C₃²	1	2	yes	direct product factor
19	C₃²×C₅×C₃₁	C₂²×C₅	1	1	yes	characteristic, complemented
20	C₃²×ΓL(1,𝔽₃₂)	C₂²×C₅	1	5	yes	complemented
21	C₂×C₃²×D₃₁	C₅²	1	1	yes	characteristic, complemented, Hall
22	C₂×C₃×C₅×C₃₁	C₂×C₃×C₅	1	4	yes	complemented
23	C₃×C₅×D₃₁	C₂×C₃×C₅	1	8	yes	complemented
24	C₂×C₃×ΓL(1,𝔽₃₂)	C₂×C₃×C₅	1	20	yes	complemented
25	C₃×D₃₁⋊C₅	C₂×C₃×C₅	1	40	yes	direct product factor
26	C₂²×C₃²×C₅²	—	31	1	no	maximal, Hall
27	C₅×ΓL(1,𝔽₃₂)	C₂²×C₃²	1	1	yes	characteristic, complemented, Hall
28	C₂×C₅×D₃₁	C₃²×C₅	1	1	yes	characteristic, complemented
29	C₂×D₃₁⋊C₅	C₃²×C₅	1	5	yes	direct product factor
30	C₂×C₃²×C₃₁	C₂×C₅²	1	1	yes	characteristic, complemented
31	C₃²×D₃₁	C₂×C₅²	1	2	yes	complemented
32	C₃×C₅×C₃₁	C₂²×C₃×C₅	1	4	yes	complemented
33	C₃×ΓL(1,𝔽₃₂)	C₂²×C₃×C₅	1	20	yes	complemented
34	C₂×C₃²×C₅²	—	31	2	no	—
35	C₂×C₃²×C₅²	—	31	1	no	—
36	C₂×C₃×D₃₁	C₃×C₅²	1	4	yes	complemented
37	C₂×C₅×C₃₁	C₂×C₃²×C₅	1	1	yes	characteristic, complemented
38	C₅×D₃₁	C₂×C₃²×C₅	1	2	yes	complemented
39	C₂×ΓL(1,𝔽₃₂)	C₂×C₃²×C₅	1	5	yes	complemented
40	D₃₁⋊C₅	C₂×C₃²×C₅	1	10	yes	direct product factor
41	C₂²×C₃×C₅²	—	31	4	no	—
42	C₃²×C₃₁	C₂²×C₅²	1	1	yes	characteristic, complemented, Hall
43	C₃²×C₅²	—	31	1	no	Hall
44	C₂×C₃×C₃₁	C₂×C₃×C₅²	1	4	yes	complemented
45	C₃×D₃₁	C₂×C₃×C₅²	1	8	yes	complemented
46	C₂²×C₃²×C₅	—	31	5	no	—
47	C₂²×C₃²×C₅	—	31	1	no	—
48	C₅×C₃₁	C₂²×C₃²×C₅	1	1	yes	characteristic, complemented
49	ΓL(1,𝔽₃₂)	C₂²×C₃²×C₅	1	5	yes	complemented
50	C₂×C₃×C₅²	—	31	8	no	—
51	C₂×C₃×C₅²	—	31	4	no	—
52	C₂×D₃₁	C₃²×C₅²	1	1	yes	characteristic, complemented, Hall
53	C₂²×C₅²	—	31	1	no	Hall
54	C₃×C₃₁	C₂²×C₃×C₅²	1	4	yes	complemented
55	C₂×C₃²×C₅	—	31	10	no	—
56	C₂×C₃²×C₅	—	31	2	no	—
57	C₂×C₃²×C₅	—	31	5	no	—
58	C₂×C₃²×C₅	D₃₁⋊C₅	1	1	yes	characteristic, direct product factor
59	C₃×C₅²	—	31	4	no	—
60	C₂×C₃₁	C₂×C₃²×C₅²	1	1	yes	characteristic, complemented
61	D₃₁	C₂×C₃²×C₅²	1	2	yes	complemented
62	C₂²×C₃×C₅	—	31	20	no	—
63	C₂²×C₃×C₅	—	31	4	no	—
64	C₂×C₅²	—	31	2	no	—
65	C₂×C₅²	—	31	1	no	—
66	C₃²×C₅	—	31	5	no	—
67	C₃²×C₅	C₂×D₃₁⋊C₅	1	1	yes	characteristic, direct product factor
68	C₂²×C₃²	—	31	1	no	Hall
69	C₃₁	C₂²×C₃²×C₅²	1	1	yes	characteristic, complemented, Sylow
70	C₂×C₃×C₅	—	31	40	no	—
71	C₂×C₃×C₅	—	31	8	no	—
72	C₂×C₃×C₅	—	31	20	no	—
73	C₂×C₃×C₅	C₃×D₃₁⋊C₅	1	4	yes	direct product factor
74	C₅²	—	31	1	no	Sylow
75	C₂²×C₅	—	31	5	no	—
76	C₂²×C₅	—	31	1	no	—
77	C₂×C₃²	—	31	2	no	—
78	C₂×C₃²	C₅×D₃₁⋊C₅	1	1	yes	characteristic, direct product factor
79	C₃×C₅	—	31	20	no	—
80	C₃×C₅	C₂×C₃×D₃₁⋊C₅	1	4	yes	direct product factor
81	C₂²×C₃	—	31	4	no	—
82	C₂×C₅	—	31	10	no	—
83	C₂×C₅	—	31	2	no	—
84	C₂×C₅	—	31	5	no	—
85	C₂×C₅	C₃²×D₃₁⋊C₅	1	1	yes	characteristic, direct product factor
86	C₃²	C₂×C₅×D₃₁⋊C₅	1	1	yes	characteristic, direct product factor, Sylow
87	C₂×C₃	—	31	8	no	—
88	C₂×C₃	C₃×C₅×D₃₁⋊C₅	1	4	yes	direct product factor
89	C₅	—	31	5	no	—
90	C₅	C₂×C₃²×D₃₁⋊C₅	1	1	yes	characteristic, direct product factor
91	C₂²	—	31	1	no	Sylow
92	C₃	C₂×C₃×C₅×D₃₁⋊C₅	1	4	yes	direct product factor
93	C₂	—	31	2	no	—
94	C₂	C₃²×C₅×D₃₁⋊C₅	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃²×C₅	D₃₁⋊C₅
commutator, cosocle, hypocenter	C₃₁	C₂²×C₃²×C₅²
Frattini	C₁	G
Fitting, socle	C₂×C₃²×C₅×C₃₁	C₂×C₅
LCS	G ▷ C₃₁
UCS	C₁ ◁ C₂×C₃²×C₅

Complements

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

N	H ≅ G/N	type
1. C₂×C₃²×C₅×ΓL(1,𝔽₃₂)	C₂	semidirect
2. C₃²×C₅×D₃₁⋊C₅	C₂	direct (N × H)
3. C₂×C₃×C₅×D₃₁⋊C₅	C₃	direct (N × H)
4. C₃²×C₅×ΓL(1,𝔽₃₂)	C₂²	semidirect
5. C₂×C₃²×C₅×D₃₁	C₅	semidirect
6. C₂×C₃²×D₃₁⋊C₅	C₅	direct (N × H)
7. C₂×C₃×C₅×ΓL(1,𝔽₃₂)	C₂×C₃	semidirect
8. C₃×C₅×D₃₁⋊C₅	C₂×C₃	direct (N × H)
9. C₂×C₅×D₃₁⋊C₅	C₃²	direct (N × H)
10. C₃²×C₅×D₃₁	C₂×C₅	semidirect
11. C₂×C₃²×ΓL(1,𝔽₃₂)	C₂×C₅	semidirect
12. C₃²×D₃₁⋊C₅	C₂×C₅	direct (N × H)
13. C₂×C₃²×C₅×C₃₁	C₂×C₅	semidirect
14. C₃×C₅×ΓL(1,𝔽₃₂)	C₂²×C₃	semidirect
15. C₂×C₃×C₅×D₃₁	C₃×C₅	semidirect
16. C₂×C₃×D₃₁⋊C₅	C₃×C₅	direct (N × H)
17. C₂×C₅×ΓL(1,𝔽₃₂)	C₂×C₃²	semidirect
18. C₅×D₃₁⋊C₅	C₂×C₃²	direct (N × H)
19. C₃²×C₅×C₃₁	C₂²×C₅	semidirect
20. C₃²×ΓL(1,𝔽₃₂)	C₂²×C₅	semidirect
21. C₂×C₃²×D₃₁	C₅²	semidirect
22. C₂×C₃×C₅×C₃₁	C₂×C₃×C₅	semidirect
23. C₃×C₅×D₃₁	C₂×C₃×C₅	semidirect
24. C₂×C₃×ΓL(1,𝔽₃₂)	C₂×C₃×C₅	semidirect
25. C₃×D₃₁⋊C₅	C₂×C₃×C₅	direct (N × H)
27. C₅×ΓL(1,𝔽₃₂)	C₂²×C₃²	semidirect
28. C₂×C₅×D₃₁	C₃²×C₅	semidirect
29. C₂×D₃₁⋊C₅	C₃²×C₅	direct (N × H)
30. C₂×C₃²×C₃₁	C₂×C₅²	semidirect
31. C₃²×D₃₁	C₂×C₅²	semidirect
32. C₃×C₅×C₃₁	C₂²×C₃×C₅	semidirect
33. C₃×ΓL(1,𝔽₃₂)	C₂²×C₃×C₅	semidirect
36. C₂×C₃×D₃₁	C₃×C₅²	semidirect
37. C₂×C₅×C₃₁	C₂×C₃²×C₅	semidirect
38. C₅×D₃₁	C₂×C₃²×C₅	semidirect
39. C₂×ΓL(1,𝔽₃₂)	C₂×C₃²×C₅	semidirect
40. D₃₁⋊C₅	C₂×C₃²×C₅	direct (N × H)
42. C₃²×C₃₁	C₂²×C₅²	semidirect
44. C₂×C₃×C₃₁	C₂×C₃×C₅²	semidirect
45. C₃×D₃₁	C₂×C₃×C₅²	semidirect
48. C₅×C₃₁	C₂²×C₃²×C₅	semidirect
49. ΓL(1,𝔽₃₂)	C₂²×C₃²×C₅	semidirect
52. C₂×D₃₁	C₃²×C₅²	semidirect
54. C₃×C₃₁	C₂²×C₃×C₅²	semidirect
58. C₂×C₃²×C₅	D₃₁⋊C₅	direct (N × H)
60. C₂×C₃₁	C₂×C₃²×C₅²	semidirect
61. D₃₁	C₂×C₃²×C₅²	semidirect
67. C₃²×C₅	C₂×D₃₁⋊C₅	direct (N × H)
69. C₃₁	C₂²×C₃²×C₅²	semidirect
73. C₂×C₃×C₅	C₃×D₃₁⋊C₅	direct (N × H)
78. C₂×C₃²	C₅×D₃₁⋊C₅	direct (N × H)
80. C₃×C₅	C₂×C₃×D₃₁⋊C₅	direct (N × H)
85. C₂×C₅	C₃²×D₃₁⋊C₅	direct (N × H)
86. C₃²	C₂×C₅×D₃₁⋊C₅	direct (N × H)
88. C₂×C₃	C₃×C₅×D₃₁⋊C₅	direct (N × H)
90. C₅	C₂×C₃²×D₃₁⋊C₅	direct (N × H)
92. C₃	C₂×C₃×C₅×D₃₁⋊C₅	direct (N × H)
94. C₂	C₃²×C₅×D₃₁⋊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
C₃²×C₅×D₃₁⋊C₅	2
C₂×C₃×C₅×D₃₁⋊C₅	3
C₂×C₃²×C₅×D₃₁	5
C₂×C₃²×D₃₁⋊C₅	5
C₂²×C₃²×C₅²	31

Subgroups — C2×C32×C5×D31⋊C5 #42124822

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×C₅×D₃₁⋊C₅ #42124822