Subgroups — C₂×C₄×C₅³×C₂₃ #72200331

Order 23000.

1024 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅³×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₄×C₅³×C₂₃	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₅³×C₂₃	C₄	1	2	yes	direct product factor
4	C₂×C₅³×C₂₃	C₂²	1	1	yes	characteristic
5	C₂×C₄×C₅²×C₂₃	C₅	1	31	yes	maximal, direct product factor
6	C₅³×C₂₃	C₂×C₄	1	1	yes	characteristic, direct product factor, Hall
7	C₂²×C₅²×C₂₃	C₂×C₅	1	31	yes	—
8	C₄×C₅²×C₂₃	C₂×C₅	1	62	yes	direct product factor
9	C₂×C₅²×C₂₃	C₄×C₅	1	62	yes	direct product factor
10	C₂×C₅²×C₂₃	C₂²×C₅	1	31	yes	—
11	C₂×C₄×C₅³	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₂×C₄×C₅×C₂₃	C₅²	1	31	yes	direct product factor
13	C₅²×C₂₃	C₂×C₄×C₅	1	31	yes	direct product factor
14	C₂²×C₅³	C₂×C₂₃	1	1	yes	characteristic
15	C₄×C₅³	C₂×C₂₃	1	2	yes	direct product factor
16	C₂²×C₅×C₂₃	C₂×C₅²	1	31	yes	—
17	C₄×C₅×C₂₃	C₂×C₅²	1	62	yes	direct product factor
18	C₂×C₅³	C₄×C₂₃	1	2	yes	direct product factor
19	C₂×C₅³	C₂²×C₂₃	1	1	yes	characteristic
20	C₂×C₅×C₂₃	C₄×C₅²	1	62	yes	direct product factor
21	C₂×C₅×C₂₃	C₂²×C₅²	1	31	yes	—
22	C₂×C₄×C₅²	C₅×C₂₃	1	31	yes	direct product factor
23	C₂×C₄×C₂₃	C₅³	1	1	yes	characteristic, direct product factor, Hall
24	C₅³	C₂×C₄×C₂₃	1	1	yes	characteristic, direct product factor, Sylow
25	C₅×C₂₃	C₂×C₄×C₅²	1	31	yes	direct product factor
26	C₂²×C₅²	C₂×C₅×C₂₃	1	31	yes	—
27	C₄×C₅²	C₂×C₅×C₂₃	1	62	yes	direct product factor
28	C₂²×C₂₃	C₂×C₅³	1	1	yes	characteristic
29	C₄×C₂₃	C₂×C₅³	1	2	yes	direct product factor
30	C₂×C₅²	C₄×C₅×C₂₃	1	62	yes	direct product factor
31	C₂×C₅²	C₂²×C₅×C₂₃	1	31	yes	—
32	C₂×C₂₃	C₄×C₅³	1	2	yes	direct product factor
33	C₂×C₂₃	C₂²×C₅³	1	1	yes	characteristic
34	C₂×C₄×C₅	C₅²×C₂₃	1	31	yes	direct product factor
35	C₅²	C₂×C₄×C₅×C₂₃	1	31	yes	direct product factor
36	C₂₃	C₂×C₄×C₅³	1	1	yes	characteristic, direct product factor, Sylow
37	C₂²×C₅	C₂×C₅²×C₂₃	1	31	yes	—
38	C₄×C₅	C₂×C₅²×C₂₃	1	62	yes	direct product factor
39	C₂×C₅	C₄×C₅²×C₂₃	1	62	yes	direct product factor
40	C₂×C₅	C₂²×C₅²×C₂₃	1	31	yes	—
41	C₂×C₄	C₅³×C₂₃	1	1	yes	characteristic, direct product factor, Sylow
42	C₅	C₂×C₄×C₅²×C₂₃	1	31	yes	direct product factor
43	C₂²	C₂×C₅³×C₂₃	1	1	yes	characteristic
44	C₄	C₂×C₅³×C₂₃	1	2	yes	direct product factor
45	C₂	C₄×C₅³×C₂₃	1	2	yes	direct product factor
46	C₂	C₂²×C₅³×C₂₃	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂	C₂²×C₅³×C₂₃
socle, prime radical	C₂²×C₅³×C₂₃	C₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₅³×C₂₃	C₂	direct (N × H)
3. C₂×C₅³×C₂₃	C₄	direct (N × H)
5. C₂×C₄×C₅²×C₂₃	C₅	direct (N × H)
6. C₅³×C₂₃	C₂×C₄	direct (N × H)
8. C₄×C₅²×C₂₃	C₂×C₅	direct (N × H)
9. C₂×C₅²×C₂₃	C₄×C₅	direct (N × H)
11. C₂×C₄×C₅³	C₂₃	direct (N × H)
12. C₂×C₄×C₅×C₂₃	C₅²	direct (N × H)
13. C₅²×C₂₃	C₂×C₄×C₅	direct (N × H)
15. C₄×C₅³	C₂×C₂₃	direct (N × H)
17. C₄×C₅×C₂₃	C₂×C₅²	direct (N × H)
18. C₂×C₅³	C₄×C₂₃	direct (N × H)
20. C₂×C₅×C₂₃	C₄×C₅²	direct (N × H)
22. C₂×C₄×C₅²	C₅×C₂₃	direct (N × H)
23. C₂×C₄×C₂₃	C₅³	direct (N × H)
24. C₅³	C₂×C₄×C₂₃	direct (N × H)
25. C₅×C₂₃	C₂×C₄×C₅²	direct (N × H)
27. C₄×C₅²	C₂×C₅×C₂₃	direct (N × H)
29. C₄×C₂₃	C₂×C₅³	direct (N × H)
30. C₂×C₅²	C₄×C₅×C₂₃	direct (N × H)
32. C₂×C₂₃	C₄×C₅³	direct (N × H)
34. C₂×C₄×C₅	C₅²×C₂₃	direct (N × H)
35. C₅²	C₂×C₄×C₅×C₂₃	direct (N × H)
36. C₂₃	C₂×C₄×C₅³	direct (N × H)
38. C₄×C₅	C₂×C₅²×C₂₃	direct (N × H)
39. C₂×C₅	C₄×C₅²×C₂₃	direct (N × H)
41. C₂×C₄	C₅³×C₂₃	direct (N × H)
42. C₅	C₂×C₄×C₅²×C₂₃	direct (N × H)
44. C₄	C₂×C₅³×C₂₃	direct (N × H)
45. C₂	C₄×C₅³×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
C₂²×C₅³×C₂₃	2
C₄×C₅³×C₂₃	2
C₂×C₄×C₅²×C₂₃	5
C₂×C₄×C₅³	23

Subgroups — C2×C4×C53×C23 #72200331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₅³×C₂₃ #72200331