Subgroups — C₃²×C₄²×C₅×C₂₃ #72194574

Order 16560.

360 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₄×C₅×C₂₃	C₂	1	3	yes	maximal
2	C₃×C₄²×C₅×C₂₃	C₃	1	4	yes	maximal, direct product factor
3	C₃²×C₄²×C₂₃	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂×C₃×C₄×C₅×C₂₃	C₂×C₃	1	12	yes	—
5	C₂²×C₃²×C₅×C₂₃	C₂²	1	1	yes	characteristic
6	C₃²×C₄×C₅×C₂₃	C₄	1	6	yes	direct product factor
7	C₂×C₃²×C₅×C₂₃	C₂×C₄	1	3	yes	—
8	C₄²×C₅×C₂₃	C₃²	1	1	yes	characteristic, direct product factor, Hall
9	C₂×C₃²×C₄×C₂₃	C₂×C₅	1	3	yes	—
10	C₂²×C₃×C₅×C₂₃	C₂²×C₃	1	4	yes	—
11	C₃×C₄×C₅×C₂₃	C₃×C₄	1	24	yes	direct product factor
12	C₃×C₄²×C₂₃	C₃×C₅	1	4	yes	direct product factor
13	C₃²×C₅×C₂₃	C₄²	1	1	yes	characteristic, direct product factor, Hall
14	C₂×C₄×C₅×C₂₃	C₂×C₃²	1	3	yes	—
15	C₂²×C₃²×C₂₃	C₂²×C₅	1	1	yes	characteristic
16	C₃²×C₄×C₂₃	C₄×C₅	1	6	yes	direct product factor
17	C₃²×C₄²×C₅	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂×C₃×C₅×C₂₃	C₂×C₃×C₄	1	12	yes	—
19	C₂×C₃×C₄×C₂₃	C₂×C₃×C₅	1	12	yes	—
20	C₂²×C₅×C₂₃	C₂²×C₃²	1	1	yes	characteristic
21	C₄×C₅×C₂₃	C₃²×C₄	1	6	yes	direct product factor
22	C₂×C₃²×C₂₃	C₂×C₄×C₅	1	3	yes	—
23	C₄²×C₂₃	C₃²×C₅	1	1	yes	characteristic, direct product factor, Hall
24	C₂×C₃²×C₄×C₅	C₂×C₂₃	1	3	yes	—
25	C₃×C₅×C₂₃	C₃×C₄²	1	4	yes	direct product factor
26	C₂²×C₃×C₂₃	C₂²×C₃×C₅	1	4	yes	—
27	C₃×C₄×C₂₃	C₃×C₄×C₅	1	24	yes	direct product factor
28	C₃×C₄²×C₅	C₃×C₂₃	1	4	yes	direct product factor
29	C₂×C₅×C₂₃	C₂×C₃²×C₄	1	3	yes	—
30	C₃²×C₂₃	C₄²×C₅	1	1	yes	characteristic, direct product factor, Hall
31	C₂×C₄×C₂₃	C₂×C₃²×C₅	1	3	yes	—
32	C₂²×C₃²×C₅	C₂²×C₂₃	1	1	yes	characteristic
33	C₃²×C₄×C₅	C₄×C₂₃	1	6	yes	direct product factor
34	C₃²×C₄²	C₅×C₂₃	1	1	yes	characteristic, direct product factor, Hall
35	C₂×C₃×C₂₃	C₂×C₃×C₄×C₅	1	12	yes	—
36	C₂×C₃×C₄×C₅	C₂×C₃×C₂₃	1	12	yes	—
37	C₅×C₂₃	C₃²×C₄²	1	1	yes	characteristic, direct product factor, Hall
38	C₂²×C₂₃	C₂²×C₃²×C₅	1	1	yes	characteristic
39	C₄×C₂₃	C₃²×C₄×C₅	1	6	yes	direct product factor
40	C₂×C₃²×C₅	C₂×C₄×C₂₃	1	3	yes	—
41	C₄²×C₅	C₃²×C₂₃	1	1	yes	characteristic, direct product factor, Hall
42	C₂×C₃²×C₄	C₂×C₅×C₂₃	1	3	yes	—
43	C₃×C₂₃	C₃×C₄²×C₅	1	4	yes	direct product factor
44	C₂²×C₃×C₅	C₂²×C₃×C₂₃	1	4	yes	—
45	C₃×C₄×C₅	C₃×C₄×C₂₃	1	24	yes	direct product factor
46	C₃×C₄²	C₃×C₅×C₂₃	1	4	yes	direct product factor
47	C₂×C₂₃	C₂×C₃²×C₄×C₅	1	3	yes	—
48	C₃²×C₅	C₄²×C₂₃	1	1	yes	characteristic, direct product factor, Hall
49	C₂×C₄×C₅	C₂×C₃²×C₂₃	1	3	yes	—
50	C₂²×C₃²	C₂²×C₅×C₂₃	1	1	yes	characteristic
51	C₃²×C₄	C₄×C₅×C₂₃	1	6	yes	direct product factor
52	C₂×C₃×C₅	C₂×C₃×C₄×C₂₃	1	12	yes	—
53	C₂×C₃×C₄	C₂×C₃×C₅×C₂₃	1	12	yes	—
54	C₂₃	C₃²×C₄²×C₅	1	1	yes	characteristic, direct product factor, Sylow
55	C₂²×C₅	C₂²×C₃²×C₂₃	1	1	yes	characteristic
56	C₄×C₅	C₃²×C₄×C₂₃	1	6	yes	direct product factor
57	C₂×C₃²	C₂×C₄×C₅×C₂₃	1	3	yes	—
58	C₄²	C₃²×C₅×C₂₃	1	1	yes	characteristic, direct product factor, Sylow
59	C₃×C₅	C₃×C₄²×C₂₃	1	4	yes	direct product factor
60	C₂²×C₃	C₂²×C₃×C₅×C₂₃	1	4	yes	—
61	C₃×C₄	C₃×C₄×C₅×C₂₃	1	24	yes	direct product factor
62	C₂×C₅	C₂×C₃²×C₄×C₂₃	1	3	yes	—
63	C₃²	C₄²×C₅×C₂₃	1	1	yes	characteristic, direct product factor, Sylow
64	C₂×C₄	C₂×C₃²×C₅×C₂₃	1	3	yes	—
65	C₂×C₃	C₂×C₃×C₄×C₅×C₂₃	1	12	yes	—
66	C₅	C₃²×C₄²×C₂₃	1	1	yes	characteristic, direct product factor, Sylow
67	C₂²	C₂²×C₃²×C₅×C₂₃	1	1	yes	characteristic
68	C₄	C₃²×C₄×C₅×C₂₃	1	6	yes	direct product factor
69	C₃	C₃×C₄²×C₅×C₂₃	1	4	yes	direct product factor
70	C₂	C₂×C₃²×C₄×C₅×C₂₃	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²	C₂²×C₃²×C₅×C₂₃
socle, prime radical	C₂²×C₃²×C₅×C₂₃	C₂²
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₄²×C₅×C₂₃	C₃	direct (N × H)
3. C₃²×C₄²×C₂₃	C₅	direct (N × H)
6. C₃²×C₄×C₅×C₂₃	C₄	direct (N × H)
8. C₄²×C₅×C₂₃	C₃²	direct (N × H)
11. C₃×C₄×C₅×C₂₃	C₃×C₄	direct (N × H)
12. C₃×C₄²×C₂₃	C₃×C₅	direct (N × H)
13. C₃²×C₅×C₂₃	C₄²	direct (N × H)
16. C₃²×C₄×C₂₃	C₄×C₅	direct (N × H)
17. C₃²×C₄²×C₅	C₂₃	direct (N × H)
21. C₄×C₅×C₂₃	C₃²×C₄	direct (N × H)
23. C₄²×C₂₃	C₃²×C₅	direct (N × H)
25. C₃×C₅×C₂₃	C₃×C₄²	direct (N × H)
27. C₃×C₄×C₂₃	C₃×C₄×C₅	direct (N × H)
28. C₃×C₄²×C₅	C₃×C₂₃	direct (N × H)
30. C₃²×C₂₃	C₄²×C₅	direct (N × H)
33. C₃²×C₄×C₅	C₄×C₂₃	direct (N × H)
34. C₃²×C₄²	C₅×C₂₃	direct (N × H)
37. C₅×C₂₃	C₃²×C₄²	direct (N × H)
39. C₄×C₂₃	C₃²×C₄×C₅	direct (N × H)
41. C₄²×C₅	C₃²×C₂₃	direct (N × H)
43. C₃×C₂₃	C₃×C₄²×C₅	direct (N × H)
45. C₃×C₄×C₅	C₃×C₄×C₂₃	direct (N × H)
46. C₃×C₄²	C₃×C₅×C₂₃	direct (N × H)
48. C₃²×C₅	C₄²×C₂₃	direct (N × H)
51. C₃²×C₄	C₄×C₅×C₂₃	direct (N × H)
54. C₂₃	C₃²×C₄²×C₅	direct (N × H)
56. C₄×C₅	C₃²×C₄×C₂₃	direct (N × H)
58. C₄²	C₃²×C₅×C₂₃	direct (N × H)
59. C₃×C₅	C₃×C₄²×C₂₃	direct (N × H)
61. C₃×C₄	C₃×C₄×C₅×C₂₃	direct (N × H)
63. C₃²	C₄²×C₅×C₂₃	direct (N × H)
66. C₅	C₃²×C₄²×C₂₃	direct (N × H)
68. C₄	C₃²×C₄×C₅×C₂₃	direct (N × H)
69. C₃	C₃×C₄²×C₅×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
C₂×C₃²×C₄×C₅×C₂₃	2
C₃×C₄²×C₅×C₂₃	3
C₃²×C₄²×C₂₃	5
C₃²×C₄²×C₅	23

Subgroups — C32×C42×C5×C23 #72194574

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₄²×C₅×C₂₃ #72194574