Subgroups — C₃³×C₅×C₈×C₂₃ #72202113

Order 24840.

448 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₄×C₅×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₅×C₈×C₂₃	C₃	1	13	yes	maximal, direct product factor
3	C₂×C₃³×C₅×C₂₃	C₄	1	1	yes	characteristic
4	C₃³×C₈×C₂₃	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₃³×C₅×C₂₃	C₈	1	1	yes	characteristic, direct product factor, Hall
6	C₃×C₅×C₈×C₂₃	C₃²	1	13	yes	direct product factor
7	C₃³×C₄×C₂₃	C₂×C₅	1	1	yes	characteristic
8	C₃²×C₄×C₅×C₂₃	C₂×C₃	1	13	yes	—
9	C₂×C₃²×C₅×C₂₃	C₃×C₄	1	13	yes	—
10	C₃²×C₈×C₂₃	C₃×C₅	1	13	yes	direct product factor
11	C₃×C₄×C₅×C₂₃	C₂×C₃²	1	13	yes	—
12	C₂×C₃³×C₂₃	C₄×C₅	1	1	yes	characteristic
13	C₃³×C₅×C₈	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₃²×C₅×C₂₃	C₃×C₈	1	13	yes	direct product factor
15	C₅×C₈×C₂₃	C₃³	1	1	yes	characteristic, direct product factor, Hall
16	C₃²×C₄×C₂₃	C₂×C₃×C₅	1	13	yes	—
17	C₂×C₃×C₅×C₂₃	C₃²×C₄	1	13	yes	—
18	C₃³×C₂₃	C₅×C₈	1	1	yes	characteristic, direct product factor, Hall
19	C₃×C₈×C₂₃	C₃²×C₅	1	13	yes	direct product factor
20	C₃³×C₄×C₅	C₂×C₂₃	1	1	yes	characteristic
21	C₄×C₅×C₂₃	C₂×C₃³	1	1	yes	characteristic
22	C₂×C₃²×C₂₃	C₃×C₄×C₅	1	13	yes	—
23	C₃²×C₅×C₈	C₃×C₂₃	1	13	yes	direct product factor
24	C₃×C₅×C₂₃	C₃²×C₈	1	13	yes	direct product factor
25	C₃×C₄×C₂₃	C₂×C₃²×C₅	1	13	yes	—
26	C₂×C₃³×C₅	C₄×C₂₃	1	1	yes	characteristic
27	C₂×C₅×C₂₃	C₃³×C₄	1	1	yes	characteristic
28	C₃³×C₈	C₅×C₂₃	1	1	yes	characteristic, direct product factor, Hall
29	C₃²×C₂₃	C₃×C₅×C₈	1	13	yes	direct product factor
30	C₈×C₂₃	C₃³×C₅	1	1	yes	characteristic, direct product factor, Hall
31	C₃²×C₄×C₅	C₂×C₃×C₂₃	1	13	yes	—
32	C₂×C₃×C₂₃	C₃²×C₄×C₅	1	13	yes	—
33	C₃³×C₅	C₈×C₂₃	1	1	yes	characteristic, direct product factor, Hall
34	C₃×C₅×C₈	C₃²×C₂₃	1	13	yes	direct product factor
35	C₅×C₂₃	C₃³×C₈	1	1	yes	characteristic, direct product factor, Hall
36	C₃³×C₄	C₂×C₅×C₂₃	1	1	yes	characteristic
37	C₄×C₂₃	C₂×C₃³×C₅	1	1	yes	characteristic
38	C₂×C₃²×C₅	C₃×C₄×C₂₃	1	13	yes	—
39	C₃²×C₈	C₃×C₅×C₂₃	1	13	yes	direct product factor
40	C₃×C₂₃	C₃²×C₅×C₈	1	13	yes	direct product factor
41	C₃×C₄×C₅	C₂×C₃²×C₂₃	1	13	yes	—
42	C₂×C₃³	C₄×C₅×C₂₃	1	1	yes	characteristic
43	C₂×C₂₃	C₃³×C₄×C₅	1	1	yes	characteristic
44	C₃²×C₅	C₃×C₈×C₂₃	1	13	yes	direct product factor
45	C₅×C₈	C₃³×C₂₃	1	1	yes	characteristic, direct product factor, Hall
46	C₃²×C₄	C₂×C₃×C₅×C₂₃	1	13	yes	—
47	C₂×C₃×C₅	C₃²×C₄×C₂₃	1	13	yes	—
48	C₃³	C₅×C₈×C₂₃	1	1	yes	characteristic, direct product factor, Sylow
49	C₃×C₈	C₃²×C₅×C₂₃	1	13	yes	direct product factor
50	C₂₃	C₃³×C₅×C₈	1	1	yes	characteristic, direct product factor, Sylow
51	C₄×C₅	C₂×C₃³×C₂₃	1	1	yes	characteristic
52	C₂×C₃²	C₃×C₄×C₅×C₂₃	1	13	yes	—
53	C₃×C₅	C₃²×C₈×C₂₃	1	13	yes	direct product factor
54	C₃×C₄	C₂×C₃²×C₅×C₂₃	1	13	yes	—
55	C₂×C₅	C₃³×C₄×C₂₃	1	1	yes	characteristic
56	C₃²	C₃×C₅×C₈×C₂₃	1	13	yes	direct product factor
57	C₈	C₃³×C₅×C₂₃	1	1	yes	characteristic, direct product factor, Sylow
58	C₂×C₃	C₃²×C₄×C₅×C₂₃	1	13	yes	—
59	C₅	C₃³×C₈×C₂₃	1	1	yes	characteristic, direct product factor, Sylow
60	C₄	C₂×C₃³×C₅×C₂₃	1	1	yes	characteristic
61	C₃	C₃²×C₅×C₈×C₂₃	1	13	yes	direct product factor
62	C₂	C₃³×C₄×C₅×C₂₃	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄	C₂×C₃³×C₅×C₂₃
socle, prime radical	C₂×C₃³×C₅×C₂₃	C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃²×C₅×C₈×C₂₃	C₃	direct (N × H)
4. C₃³×C₈×C₂₃	C₅	direct (N × H)
5. C₃³×C₅×C₂₃	C₈	direct (N × H)
6. C₃×C₅×C₈×C₂₃	C₃²	direct (N × H)
10. C₃²×C₈×C₂₃	C₃×C₅	direct (N × H)
13. C₃³×C₅×C₈	C₂₃	direct (N × H)
14. C₃²×C₅×C₂₃	C₃×C₈	direct (N × H)
15. C₅×C₈×C₂₃	C₃³	direct (N × H)
18. C₃³×C₂₃	C₅×C₈	direct (N × H)
19. C₃×C₈×C₂₃	C₃²×C₅	direct (N × H)
23. C₃²×C₅×C₈	C₃×C₂₃	direct (N × H)
24. C₃×C₅×C₂₃	C₃²×C₈	direct (N × H)
28. C₃³×C₈	C₅×C₂₃	direct (N × H)
29. C₃²×C₂₃	C₃×C₅×C₈	direct (N × H)
30. C₈×C₂₃	C₃³×C₅	direct (N × H)
33. C₃³×C₅	C₈×C₂₃	direct (N × H)
34. C₃×C₅×C₈	C₃²×C₂₃	direct (N × H)
35. C₅×C₂₃	C₃³×C₈	direct (N × H)
39. C₃²×C₈	C₃×C₅×C₂₃	direct (N × H)
40. C₃×C₂₃	C₃²×C₅×C₈	direct (N × H)
44. C₃²×C₅	C₃×C₈×C₂₃	direct (N × H)
45. C₅×C₈	C₃³×C₂₃	direct (N × H)
48. C₃³	C₅×C₈×C₂₃	direct (N × H)
49. C₃×C₈	C₃²×C₅×C₂₃	direct (N × H)
50. C₂₃	C₃³×C₅×C₈	direct (N × H)
53. C₃×C₅	C₃²×C₈×C₂₃	direct (N × H)
56. C₃²	C₃×C₅×C₈×C₂₃	direct (N × H)
57. C₈	C₃³×C₅×C₂₃	direct (N × H)
59. C₅	C₃³×C₈×C₂₃	direct (N × H)
61. C₃	C₃²×C₅×C₈×C₂₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃³×C₄×C₅×C₂₃	2
C₃²×C₅×C₈×C₂₃	3
C₃³×C₈×C₂₃	5
C₃³×C₅×C₈	23

Subgroups — C33×C5×C8×C23 #72202113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃³×C₅×C₈×C₂₃ #72202113