Subgroups — C₂×C₃×C₅×C₈×C₃₇ #72187708

Order 8880.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₅×C₃₇	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×C₈×C₃₇	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₅×C₈×C₃₇	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂²×C₃×C₅×C₃₇	C₄	1	1	yes	characteristic
5	C₃×C₄×C₅×C₃₇	C₄	1	1	yes	characteristic
6	C₃×C₄×C₅×C₃₇	C₂²	1	1	yes	characteristic
7	C₂×C₃×C₈×C₃₇	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂×C₄×C₅×C₃₇	C₂×C₃	1	1	yes	characteristic
9	C₅×C₈×C₃₇	C₂×C₃	1	2	yes	direct product factor
10	C₂×C₃×C₅×C₃₇	C₈	1	2	yes	direct product factor
11	C₂×C₃×C₅×C₃₇	C₂×C₄	1	1	yes	characteristic
12	C₂×C₃×C₄×C₃₇	C₂×C₅	1	1	yes	characteristic
13	C₃×C₈×C₃₇	C₂×C₅	1	2	yes	direct product factor
14	C₂²×C₅×C₃₇	C₃×C₄	1	1	yes	characteristic
15	C₄×C₅×C₃₇	C₃×C₄	1	1	yes	characteristic
16	C₄×C₅×C₃₇	C₂²×C₃	1	1	yes	characteristic
17	C₂×C₈×C₃₇	C₃×C₅	1	1	yes	characteristic, direct product factor, Hall
18	C₃×C₅×C₃₇	C₂×C₈	1	1	yes	characteristic, direct product factor, Hall
19	C₂²×C₃×C₃₇	C₄×C₅	1	1	yes	characteristic
20	C₃×C₄×C₃₇	C₄×C₅	1	1	yes	characteristic
21	C₃×C₄×C₃₇	C₂²×C₅	1	1	yes	characteristic
22	C₂×C₅×C₃₇	C₃×C₈	1	2	yes	direct product factor
23	C₂×C₅×C₃₇	C₂×C₃×C₄	1	1	yes	characteristic
24	C₂×C₄×C₃₇	C₂×C₃×C₅	1	1	yes	characteristic
25	C₈×C₃₇	C₂×C₃×C₅	1	2	yes	direct product factor
26	C₂×C₃×C₅×C₈	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Hall
27	C₂×C₃×C₃₇	C₅×C₈	1	2	yes	direct product factor
28	C₂×C₃×C₃₇	C₂×C₄×C₅	1	1	yes	characteristic
29	C₅×C₃₇	C₂×C₃×C₈	1	1	yes	characteristic, direct product factor, Hall
30	C₂²×C₃₇	C₃×C₄×C₅	1	1	yes	characteristic
31	C₄×C₃₇	C₃×C₄×C₅	1	1	yes	characteristic
32	C₄×C₃₇	C₂²×C₃×C₅	1	1	yes	characteristic
33	C₂×C₃×C₄×C₅	C₂×C₃₇	1	1	yes	characteristic
34	C₃×C₅×C₈	C₂×C₃₇	1	2	yes	direct product factor
35	C₃×C₃₇	C₂×C₅×C₈	1	1	yes	characteristic, direct product factor, Hall
36	C₂×C₅×C₈	C₃×C₃₇	1	1	yes	characteristic, direct product factor, Hall
37	C₂×C₃₇	C₃×C₅×C₈	1	2	yes	direct product factor
38	C₂×C₃₇	C₂×C₃×C₄×C₅	1	1	yes	characteristic
39	C₂²×C₃×C₅	C₄×C₃₇	1	1	yes	characteristic
40	C₃×C₄×C₅	C₄×C₃₇	1	1	yes	characteristic
41	C₃×C₄×C₅	C₂²×C₃₇	1	1	yes	characteristic
42	C₂×C₃×C₈	C₅×C₃₇	1	1	yes	characteristic, direct product factor, Hall
43	C₂×C₄×C₅	C₂×C₃×C₃₇	1	1	yes	characteristic
44	C₅×C₈	C₂×C₃×C₃₇	1	2	yes	direct product factor
45	C₃₇	C₂×C₃×C₅×C₈	1	1	yes	characteristic, direct product factor, Sylow
46	C₂×C₃×C₅	C₈×C₃₇	1	2	yes	direct product factor
47	C₂×C₃×C₅	C₂×C₄×C₃₇	1	1	yes	characteristic
48	C₂×C₃×C₄	C₂×C₅×C₃₇	1	1	yes	characteristic
49	C₃×C₈	C₂×C₅×C₃₇	1	2	yes	direct product factor
50	C₂²×C₅	C₃×C₄×C₃₇	1	1	yes	characteristic
51	C₄×C₅	C₃×C₄×C₃₇	1	1	yes	characteristic
52	C₄×C₅	C₂²×C₃×C₃₇	1	1	yes	characteristic
53	C₂×C₈	C₃×C₅×C₃₇	1	1	yes	characteristic, direct product factor, Sylow
54	C₃×C₅	C₂×C₈×C₃₇	1	1	yes	characteristic, direct product factor, Hall
55	C₂²×C₃	C₄×C₅×C₃₇	1	1	yes	characteristic
56	C₃×C₄	C₄×C₅×C₃₇	1	1	yes	characteristic
57	C₃×C₄	C₂²×C₅×C₃₇	1	1	yes	characteristic
58	C₂×C₅	C₃×C₈×C₃₇	1	2	yes	direct product factor
59	C₂×C₅	C₂×C₃×C₄×C₃₇	1	1	yes	characteristic
60	C₂×C₄	C₂×C₃×C₅×C₃₇	1	1	yes	characteristic
61	C₈	C₂×C₃×C₅×C₃₇	1	2	yes	direct product factor
62	C₂×C₃	C₅×C₈×C₃₇	1	2	yes	direct product factor
63	C₂×C₃	C₂×C₄×C₅×C₃₇	1	1	yes	characteristic
64	C₅	C₂×C₃×C₈×C₃₇	1	1	yes	characteristic, direct product factor, Sylow
65	C₂²	C₃×C₄×C₅×C₃₇	1	1	yes	characteristic
66	C₄	C₃×C₄×C₅×C₃₇	1	1	yes	characteristic
67	C₄	C₂²×C₃×C₅×C₃₇	1	1	yes	characteristic
68	C₃	C₂×C₅×C₈×C₃₇	1	1	yes	characteristic, direct product factor, Sylow
69	C₂	C₃×C₅×C₈×C₃₇	1	2	yes	direct product factor
70	C₂	C₂×C₃×C₄×C₅×C₃₇	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄	C₂²×C₃×C₅×C₃₇
socle, prime radical	C₂²×C₃×C₅×C₃₇	C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×C₈×C₃₇	C₂	direct (N × H)
3. C₂×C₅×C₈×C₃₇	C₃	direct (N × H)
7. C₂×C₃×C₈×C₃₇	C₅	direct (N × H)
9. C₅×C₈×C₃₇	C₂×C₃	direct (N × H)
10. C₂×C₃×C₅×C₃₇	C₈	direct (N × H)
13. C₃×C₈×C₃₇	C₂×C₅	direct (N × H)
17. C₂×C₈×C₃₇	C₃×C₅	direct (N × H)
18. C₃×C₅×C₃₇	C₂×C₈	direct (N × H)
22. C₂×C₅×C₃₇	C₃×C₈	direct (N × H)
25. C₈×C₃₇	C₂×C₃×C₅	direct (N × H)
26. C₂×C₃×C₅×C₈	C₃₇	direct (N × H)
27. C₂×C₃×C₃₇	C₅×C₈	direct (N × H)
29. C₅×C₃₇	C₂×C₃×C₈	direct (N × H)
34. C₃×C₅×C₈	C₂×C₃₇	direct (N × H)
35. C₃×C₃₇	C₂×C₅×C₈	direct (N × H)
36. C₂×C₅×C₈	C₃×C₃₇	direct (N × H)
37. C₂×C₃₇	C₃×C₅×C₈	direct (N × H)
42. C₂×C₃×C₈	C₅×C₃₇	direct (N × H)
44. C₅×C₈	C₂×C₃×C₃₇	direct (N × H)
45. C₃₇	C₂×C₃×C₅×C₈	direct (N × H)
46. C₂×C₃×C₅	C₈×C₃₇	direct (N × H)
49. C₃×C₈	C₂×C₅×C₃₇	direct (N × H)
53. C₂×C₈	C₃×C₅×C₃₇	direct (N × H)
54. C₃×C₅	C₂×C₈×C₃₇	direct (N × H)
58. C₂×C₅	C₃×C₈×C₃₇	direct (N × H)
61. C₈	C₂×C₃×C₅×C₃₇	direct (N × H)
62. C₂×C₃	C₅×C₈×C₃₇	direct (N × H)
64. C₅	C₂×C₃×C₈×C₃₇	direct (N × H)
68. C₃	C₂×C₅×C₈×C₃₇	direct (N × H)
69. C₂	C₃×C₅×C₈×C₃₇	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×C₃×C₄×C₅×C₃₇	2
C₃×C₅×C₈×C₃₇	2
C₂×C₅×C₈×C₃₇	3
C₂×C₃×C₈×C₃₇	5
C₂×C₃×C₅×C₈	37

Subgroups — C2×C3×C5×C8×C37 #72187708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅×C₈×C₃₇ #72187708