Subgroups — C₂²×C₈×C₁₉×C₃₇ #72199909

Order 22496.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₁₉×C₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×C₁₉×C₃₇	C₂	1	6	yes	maximal, direct product factor
3	C₂³×C₁₉×C₃₇	C₄	1	1	yes	characteristic
4	C₂×C₄×C₁₉×C₃₇	C₄	1	3	yes	—
5	C₂×C₄×C₁₉×C₃₇	C₂²	1	3	yes	—
6	C₈×C₁₉×C₃₇	C₂²	1	4	yes	direct product factor
7	C₂²×C₁₉×C₃₇	C₈	1	4	yes	direct product factor
8	C₂²×C₁₉×C₃₇	C₂×C₄	1	3	yes	—
9	C₄×C₁₉×C₃₇	C₂×C₄	1	3	yes	—
10	C₄×C₁₉×C₃₇	C₂³	1	1	yes	characteristic
11	C₂×C₁₉×C₃₇	C₂×C₈	1	6	yes	direct product factor
12	C₂×C₁₉×C₃₇	C₂²×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₃₇	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₁₉×C₃₇	C₂²×C₈	1	1	yes	characteristic, direct product factor, Hall
15	C₂²×C₈×C₁₉	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂²×C₄×C₃₇	C₂×C₁₉	1	1	yes	characteristic
17	C₂×C₈×C₃₇	C₂×C₁₉	1	6	yes	direct product factor
18	C₂²×C₄×C₁₉	C₂×C₃₇	1	1	yes	characteristic
19	C₂×C₈×C₁₉	C₂×C₃₇	1	6	yes	direct product factor
20	C₂³×C₃₇	C₄×C₁₉	1	1	yes	characteristic
21	C₂×C₄×C₃₇	C₄×C₁₉	1	3	yes	—
22	C₂×C₄×C₃₇	C₂²×C₁₉	1	3	yes	—
23	C₈×C₃₇	C₂²×C₁₉	1	4	yes	direct product factor
24	C₂³×C₁₉	C₄×C₃₇	1	1	yes	characteristic
25	C₂×C₄×C₁₉	C₄×C₃₇	1	3	yes	—
26	C₂×C₄×C₁₉	C₂²×C₃₇	1	3	yes	—
27	C₈×C₁₉	C₂²×C₃₇	1	4	yes	direct product factor
28	C₂²×C₃₇	C₈×C₁₉	1	4	yes	direct product factor
29	C₂²×C₃₇	C₂×C₄×C₁₉	1	3	yes	—
30	C₄×C₃₇	C₂×C₄×C₁₉	1	3	yes	—
31	C₄×C₃₇	C₂³×C₁₉	1	1	yes	characteristic
32	C₂²×C₁₉	C₈×C₃₇	1	4	yes	direct product factor
33	C₂²×C₁₉	C₂×C₄×C₃₇	1	3	yes	—
34	C₄×C₁₉	C₂×C₄×C₃₇	1	3	yes	—
35	C₄×C₁₉	C₂³×C₃₇	1	1	yes	characteristic
36	C₂×C₃₇	C₂×C₈×C₁₉	1	6	yes	direct product factor
37	C₂×C₃₇	C₂²×C₄×C₁₉	1	1	yes	characteristic
38	C₂×C₁₉	C₂×C₈×C₃₇	1	6	yes	direct product factor
39	C₂×C₁₉	C₂²×C₄×C₃₇	1	1	yes	characteristic
40	C₃₇	C₂²×C₈×C₁₉	1	1	yes	characteristic, direct product factor, Sylow
41	C₂²×C₈	C₁₉×C₃₇	1	1	yes	characteristic, direct product factor, Sylow
42	C₁₉	C₂²×C₈×C₃₇	1	1	yes	characteristic, direct product factor, Sylow
43	C₂²×C₄	C₂×C₁₉×C₃₇	1	1	yes	characteristic
44	C₂×C₈	C₂×C₁₉×C₃₇	1	6	yes	direct product factor
45	C₂³	C₄×C₁₉×C₃₇	1	1	yes	characteristic
46	C₂×C₄	C₄×C₁₉×C₃₇	1	3	yes	—
47	C₂×C₄	C₂²×C₁₉×C₃₇	1	3	yes	—
48	C₈	C₂²×C₁₉×C₃₇	1	4	yes	direct product factor
49	C₂²	C₈×C₁₉×C₃₇	1	4	yes	direct product factor
50	C₂²	C₂×C₄×C₁₉×C₃₇	1	3	yes	—
51	C₄	C₂×C₄×C₁₉×C₃₇	1	3	yes	—
52	C₄	C₂³×C₁₉×C₃₇	1	1	yes	characteristic
53	C₂	C₂×C₈×C₁₉×C₃₇	1	6	yes	direct product factor
54	C₂	C₂²×C₄×C₁₉×C₃₇	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄	C₂³×C₁₉×C₃₇
socle, prime radical	C₂³×C₁₉×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. C₂×C₈×C₁₉×C₃₇	C₂	direct (N × H)
6. C₈×C₁₉×C₃₇	C₂²	direct (N × H)
7. C₂²×C₁₉×C₃₇	C₈	direct (N × H)
11. C₂×C₁₉×C₃₇	C₂×C₈	direct (N × H)
13. C₂²×C₈×C₃₇	C₁₉	direct (N × H)
14. C₁₉×C₃₇	C₂²×C₈	direct (N × H)
15. C₂²×C₈×C₁₉	C₃₇	direct (N × H)
17. C₂×C₈×C₃₇	C₂×C₁₉	direct (N × H)
19. C₂×C₈×C₁₉	C₂×C₃₇	direct (N × H)
23. C₈×C₃₇	C₂²×C₁₉	direct (N × H)
27. C₈×C₁₉	C₂²×C₃₇	direct (N × H)
28. C₂²×C₃₇	C₈×C₁₉	direct (N × H)
32. C₂²×C₁₉	C₈×C₃₇	direct (N × H)
36. C₂×C₃₇	C₂×C₈×C₁₉	direct (N × H)
38. C₂×C₁₉	C₂×C₈×C₃₇	direct (N × H)
40. C₃₇	C₂²×C₈×C₁₉	direct (N × H)
41. C₂²×C₈	C₁₉×C₃₇	direct (N × H)
42. C₁₉	C₂²×C₈×C₃₇	direct (N × H)
44. C₂×C₈	C₂×C₁₉×C₃₇	direct (N × H)
48. C₈	C₂²×C₁₉×C₃₇	direct (N × H)
49. C₂²	C₈×C₁₉×C₃₇	direct (N × H)
53. C₂	C₂×C₈×C₁₉×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
C₂²×C₄×C₁₉×C₃₇	2
C₂×C₈×C₁₉×C₃₇	2
C₂²×C₈×C₃₇	19
C₂²×C₈×C₁₉	37

Subgroups — C22×C8×C19×C37 #72199909

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₈×C₁₉×C₃₇ #72199909