Subgroups — C₂×C₃²×C₄×C₁₃×C₂₃ #72199048

Order 21528.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₁₃×C₂₃	C₃	1	4	yes	maximal, direct product factor
2	C₂×C₄×C₁₃×C₂₃	C₃²	1	1	yes	characteristic, direct product factor, Hall
3	C₂²×C₃²×C₁₃×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₃²×C₄×C₁₃×C₂₃	C₂	1	2	yes	maximal, direct product factor
5	C₂×C₃²×C₁₃×C₂₃	C₄	1	2	yes	direct product factor
6	C₂×C₃²×C₁₃×C₂₃	C₂²	1	1	yes	characteristic
7	C₂²×C₃×C₁₃×C₂₃	C₂×C₃	1	4	yes	—
8	C₃×C₄×C₁₃×C₂₃	C₂×C₃	1	8	yes	direct product factor
9	C₃²×C₁₃×C₂₃	C₂×C₄	1	1	yes	characteristic, direct product factor, Hall
10	C₂×C₃×C₁₃×C₂₃	C₃×C₄	1	8	yes	direct product factor
11	C₂×C₃×C₁₃×C₂₃	C₂²×C₃	1	4	yes	—
12	C₂×C₃²×C₄×C₂₃	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂²×C₁₃×C₂₃	C₂×C₃²	1	1	yes	characteristic
14	C₄×C₁₃×C₂₃	C₂×C₃²	1	2	yes	direct product factor
15	C₂×C₃²×C₄×C₁₃	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₃×C₁₃×C₂₃	C₂×C₃×C₄	1	4	yes	direct product factor
17	C₂²×C₃²×C₂₃	C₂×C₁₃	1	1	yes	characteristic
18	C₃²×C₄×C₂₃	C₂×C₁₃	1	2	yes	direct product factor
19	C₂×C₁₃×C₂₃	C₃²×C₄	1	2	yes	direct product factor
20	C₂×C₁₃×C₂₃	C₂²×C₃²	1	1	yes	characteristic
21	C₂×C₃×C₄×C₂₃	C₃×C₁₃	1	4	yes	direct product factor
22	C₂²×C₃²×C₁₃	C₂×C₂₃	1	1	yes	characteristic
23	C₃²×C₄×C₁₃	C₂×C₂₃	1	2	yes	direct product factor
24	C₂×C₃²×C₂₃	C₄×C₁₃	1	2	yes	direct product factor
25	C₂×C₃²×C₂₃	C₂²×C₁₃	1	1	yes	characteristic
26	C₂×C₃×C₄×C₁₃	C₃×C₂₃	1	4	yes	direct product factor
27	C₁₃×C₂₃	C₂×C₃²×C₄	1	1	yes	characteristic, direct product factor, Hall
28	C₂²×C₃×C₂₃	C₂×C₃×C₁₃	1	4	yes	—
29	C₃×C₄×C₂₃	C₂×C₃×C₁₃	1	8	yes	direct product factor
30	C₂×C₃²×C₁₃	C₄×C₂₃	1	2	yes	direct product factor
31	C₂×C₃²×C₁₃	C₂²×C₂₃	1	1	yes	characteristic
32	C₃²×C₂₃	C₂×C₄×C₁₃	1	1	yes	characteristic, direct product factor, Hall
33	C₂×C₄×C₂₃	C₃²×C₁₃	1	1	yes	characteristic, direct product factor, Hall
34	C₂²×C₃×C₁₃	C₂×C₃×C₂₃	1	4	yes	—
35	C₃×C₄×C₁₃	C₂×C₃×C₂₃	1	8	yes	direct product factor
36	C₂×C₃×C₂₃	C₃×C₄×C₁₃	1	8	yes	direct product factor
37	C₂×C₃×C₂₃	C₂²×C₃×C₁₃	1	4	yes	—
38	C₃²×C₁₃	C₂×C₄×C₂₃	1	1	yes	characteristic, direct product factor, Hall
39	C₂×C₄×C₁₃	C₃²×C₂₃	1	1	yes	characteristic, direct product factor, Hall
40	C₂²×C₂₃	C₂×C₃²×C₁₃	1	1	yes	characteristic
41	C₄×C₂₃	C₂×C₃²×C₁₃	1	2	yes	direct product factor
42	C₂×C₃×C₁₃	C₃×C₄×C₂₃	1	8	yes	direct product factor
43	C₂×C₃×C₁₃	C₂²×C₃×C₂₃	1	4	yes	—
44	C₂×C₃²×C₄	C₁₃×C₂₃	1	1	yes	characteristic, direct product factor, Hall
45	C₃×C₂₃	C₂×C₃×C₄×C₁₃	1	4	yes	direct product factor
46	C₂²×C₁₃	C₂×C₃²×C₂₃	1	1	yes	characteristic
47	C₄×C₁₃	C₂×C₃²×C₂₃	1	2	yes	direct product factor
48	C₂×C₂₃	C₃²×C₄×C₁₃	1	2	yes	direct product factor
49	C₂×C₂₃	C₂²×C₃²×C₁₃	1	1	yes	characteristic
50	C₃×C₁₃	C₂×C₃×C₄×C₂₃	1	4	yes	direct product factor
51	C₂²×C₃²	C₂×C₁₃×C₂₃	1	1	yes	characteristic
52	C₃²×C₄	C₂×C₁₃×C₂₃	1	2	yes	direct product factor
53	C₂×C₁₃	C₃²×C₄×C₂₃	1	2	yes	direct product factor
54	C₂×C₁₃	C₂²×C₃²×C₂₃	1	1	yes	characteristic
55	C₂×C₃×C₄	C₃×C₁₃×C₂₃	1	4	yes	direct product factor
56	C₂₃	C₂×C₃²×C₄×C₁₃	1	1	yes	characteristic, direct product factor, Sylow
57	C₂×C₃²	C₄×C₁₃×C₂₃	1	2	yes	direct product factor
58	C₂×C₃²	C₂²×C₁₃×C₂₃	1	1	yes	characteristic
59	C₁₃	C₂×C₃²×C₄×C₂₃	1	1	yes	characteristic, direct product factor, Sylow
60	C₂²×C₃	C₂×C₃×C₁₃×C₂₃	1	4	yes	—
61	C₃×C₄	C₂×C₃×C₁₃×C₂₃	1	8	yes	direct product factor
62	C₃²	C₂×C₄×C₁₃×C₂₃	1	1	yes	characteristic, direct product factor, Sylow
63	C₂×C₄	C₃²×C₁₃×C₂₃	1	1	yes	characteristic, direct product factor, Sylow
64	C₂×C₃	C₃×C₄×C₁₃×C₂₃	1	8	yes	direct product factor
65	C₂×C₃	C₂²×C₃×C₁₃×C₂₃	1	4	yes	—
66	C₂²	C₂×C₃²×C₁₃×C₂₃	1	1	yes	characteristic
67	C₄	C₂×C₃²×C₁₃×C₂₃	1	2	yes	direct product factor
68	C₃	C₂×C₃×C₄×C₁₃×C₂₃	1	4	yes	direct product factor
69	C₂	C₃²×C₄×C₁₃×C₂₃	1	2	yes	direct product factor
70	C₂	C₂²×C₃²×C₁₃×C₂₃	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂	C₂²×C₃²×C₁₃×C₂₃
socle, prime radical	C₂²×C₃²×C₁₃×C₂₃	C₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₄×C₁₃×C₂₃	C₃	direct (N × H)
2. C₂×C₄×C₁₃×C₂₃	C₃²	direct (N × H)
4. C₃²×C₄×C₁₃×C₂₃	C₂	direct (N × H)
5. C₂×C₃²×C₁₃×C₂₃	C₄	direct (N × H)
8. C₃×C₄×C₁₃×C₂₃	C₂×C₃	direct (N × H)
9. C₃²×C₁₃×C₂₃	C₂×C₄	direct (N × H)
10. C₂×C₃×C₁₃×C₂₃	C₃×C₄	direct (N × H)
12. C₂×C₃²×C₄×C₂₃	C₁₃	direct (N × H)
14. C₄×C₁₃×C₂₃	C₂×C₃²	direct (N × H)
15. C₂×C₃²×C₄×C₁₃	C₂₃	direct (N × H)
16. C₃×C₁₃×C₂₃	C₂×C₃×C₄	direct (N × H)
18. C₃²×C₄×C₂₃	C₂×C₁₃	direct (N × H)
19. C₂×C₁₃×C₂₃	C₃²×C₄	direct (N × H)
21. C₂×C₃×C₄×C₂₃	C₃×C₁₃	direct (N × H)
23. C₃²×C₄×C₁₃	C₂×C₂₃	direct (N × H)
24. C₂×C₃²×C₂₃	C₄×C₁₃	direct (N × H)
26. C₂×C₃×C₄×C₁₃	C₃×C₂₃	direct (N × H)
27. C₁₃×C₂₃	C₂×C₃²×C₄	direct (N × H)
29. C₃×C₄×C₂₃	C₂×C₃×C₁₃	direct (N × H)
30. C₂×C₃²×C₁₃	C₄×C₂₃	direct (N × H)
32. C₃²×C₂₃	C₂×C₄×C₁₃	direct (N × H)
33. C₂×C₄×C₂₃	C₃²×C₁₃	direct (N × H)
35. C₃×C₄×C₁₃	C₂×C₃×C₂₃	direct (N × H)
36. C₂×C₃×C₂₃	C₃×C₄×C₁₃	direct (N × H)
38. C₃²×C₁₃	C₂×C₄×C₂₃	direct (N × H)
39. C₂×C₄×C₁₃	C₃²×C₂₃	direct (N × H)
41. C₄×C₂₃	C₂×C₃²×C₁₃	direct (N × H)
42. C₂×C₃×C₁₃	C₃×C₄×C₂₃	direct (N × H)
44. C₂×C₃²×C₄	C₁₃×C₂₃	direct (N × H)
45. C₃×C₂₃	C₂×C₃×C₄×C₁₃	direct (N × H)
47. C₄×C₁₃	C₂×C₃²×C₂₃	direct (N × H)
48. C₂×C₂₃	C₃²×C₄×C₁₃	direct (N × H)
50. C₃×C₁₃	C₂×C₃×C₄×C₂₃	direct (N × H)
52. C₃²×C₄	C₂×C₁₃×C₂₃	direct (N × H)
53. C₂×C₁₃	C₃²×C₄×C₂₃	direct (N × H)
55. C₂×C₃×C₄	C₃×C₁₃×C₂₃	direct (N × H)
56. C₂₃	C₂×C₃²×C₄×C₁₃	direct (N × H)
57. C₂×C₃²	C₄×C₁₃×C₂₃	direct (N × H)
59. C₁₃	C₂×C₃²×C₄×C₂₃	direct (N × H)
61. C₃×C₄	C₂×C₃×C₁₃×C₂₃	direct (N × H)
62. C₃²	C₂×C₄×C₁₃×C₂₃	direct (N × H)
63. C₂×C₄	C₃²×C₁₃×C₂₃	direct (N × H)
64. C₂×C₃	C₃×C₄×C₁₃×C₂₃	direct (N × H)
67. C₄	C₂×C₃²×C₁₃×C₂₃	direct (N × H)
68. C₃	C₂×C₃×C₄×C₁₃×C₂₃	direct (N × H)
69. C₂	C₃²×C₄×C₁₃×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×C₃×C₄×C₁₃×C₂₃	3
C₂²×C₃²×C₁₃×C₂₃	2
C₃²×C₄×C₁₃×C₂₃	2
C₂×C₃²×C₄×C₂₃	13
C₂×C₃²×C₄×C₁₃	23

Subgroups — C2×C32×C4×C13×C23 #72199048

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×C₄×C₁₃×C₂₃ #72199048