Subgroups — C₂²×C₅²×C₈×C₂₃ #72196128

Order 18400.

608 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₅²×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅²×C₈×C₂₃	C₂	1	6	yes	maximal, direct product factor
3	C₂³×C₅²×C₂₃	C₄	1	1	yes	characteristic
4	C₂×C₄×C₅²×C₂₃	C₄	1	3	yes	—
5	C₂×C₄×C₅²×C₂₃	C₂²	1	3	yes	—
6	C₅²×C₈×C₂₃	C₂²	1	4	yes	direct product factor
7	C₂²×C₅×C₈×C₂₃	C₅	1	6	yes	maximal, direct product factor
8	C₂²×C₅²×C₂₃	C₈	1	4	yes	direct product factor
9	C₂²×C₅²×C₂₃	C₂×C₄	1	3	yes	—
10	C₄×C₅²×C₂₃	C₂×C₄	1	3	yes	—
11	C₄×C₅²×C₂₃	C₂³	1	1	yes	characteristic
12	C₂²×C₄×C₅×C₂₃	C₂×C₅	1	6	yes	—
13	C₂×C₅×C₈×C₂₃	C₂×C₅	1	36	yes	direct product factor
14	C₂×C₅²×C₂₃	C₂×C₈	1	6	yes	direct product factor
15	C₂×C₅²×C₂₃	C₂²×C₄	1	1	yes	characteristic
16	C₂³×C₅×C₂₃	C₄×C₅	1	6	yes	—
17	C₂×C₄×C₅×C₂₃	C₄×C₅	1	18	yes	—
18	C₂×C₄×C₅×C₂₃	C₂²×C₅	1	18	yes	—
19	C₅×C₈×C₂₃	C₂²×C₅	1	24	yes	direct product factor
20	C₂²×C₅²×C₈	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₂²×C₈×C₂₃	C₅²	1	1	yes	characteristic, direct product factor, Hall
22	C₅²×C₂₃	C₂²×C₈	1	1	yes	characteristic, direct product factor, Hall
23	C₂²×C₅×C₂₃	C₅×C₈	1	24	yes	direct product factor
24	C₂²×C₅×C₂₃	C₂×C₄×C₅	1	18	yes	—
25	C₄×C₅×C₂₃	C₂×C₄×C₅	1	18	yes	—
26	C₄×C₅×C₂₃	C₂³×C₅	1	6	yes	—
27	C₂²×C₄×C₅²	C₂×C₂₃	1	1	yes	characteristic
28	C₂×C₅²×C₈	C₂×C₂₃	1	6	yes	direct product factor
29	C₂²×C₄×C₂₃	C₂×C₅²	1	1	yes	characteristic
30	C₂×C₈×C₂₃	C₂×C₅²	1	6	yes	direct product factor
31	C₂×C₅×C₂₃	C₂×C₅×C₈	1	36	yes	direct product factor
32	C₂×C₅×C₂₃	C₂²×C₄×C₅	1	6	yes	—
33	C₂³×C₅²	C₄×C₂₃	1	1	yes	characteristic
34	C₂×C₄×C₅²	C₄×C₂₃	1	3	yes	—
35	C₂×C₄×C₅²	C₂²×C₂₃	1	3	yes	—
36	C₅²×C₈	C₂²×C₂₃	1	4	yes	direct product factor
37	C₂³×C₂₃	C₄×C₅²	1	1	yes	characteristic
38	C₂×C₄×C₂₃	C₄×C₅²	1	3	yes	—
39	C₂×C₄×C₂₃	C₂²×C₅²	1	3	yes	—
40	C₈×C₂₃	C₂²×C₅²	1	4	yes	direct product factor
41	C₂²×C₅×C₈	C₅×C₂₃	1	6	yes	direct product factor
42	C₅×C₂₃	C₂²×C₅×C₈	1	6	yes	direct product factor
43	C₂²×C₅²	C₈×C₂₃	1	4	yes	direct product factor
44	C₂²×C₅²	C₂×C₄×C₂₃	1	3	yes	—
45	C₄×C₅²	C₂×C₄×C₂₃	1	3	yes	—
46	C₄×C₅²	C₂³×C₂₃	1	1	yes	characteristic
47	C₂²×C₂₃	C₅²×C₈	1	4	yes	direct product factor
48	C₂²×C₂₃	C₂×C₄×C₅²	1	3	yes	—
49	C₄×C₂₃	C₂×C₄×C₅²	1	3	yes	—
50	C₄×C₂₃	C₂³×C₅²	1	1	yes	characteristic
51	C₂²×C₄×C₅	C₂×C₅×C₂₃	1	6	yes	—
52	C₂×C₅×C₈	C₂×C₅×C₂₃	1	36	yes	direct product factor
53	C₂×C₅²	C₂×C₈×C₂₃	1	6	yes	direct product factor
54	C₂×C₅²	C₂²×C₄×C₂₃	1	1	yes	characteristic
55	C₂×C₂₃	C₂×C₅²×C₈	1	6	yes	direct product factor
56	C₂×C₂₃	C₂²×C₄×C₅²	1	1	yes	characteristic
57	C₂³×C₅	C₄×C₅×C₂₃	1	6	yes	—
58	C₂×C₄×C₅	C₄×C₅×C₂₃	1	18	yes	—
59	C₂×C₄×C₅	C₂²×C₅×C₂₃	1	18	yes	—
60	C₅×C₈	C₂²×C₅×C₂₃	1	24	yes	direct product factor
61	C₂²×C₈	C₅²×C₂₃	1	1	yes	characteristic, direct product factor, Sylow
62	C₅²	C₂²×C₈×C₂₃	1	1	yes	characteristic, direct product factor, Sylow
63	C₂₃	C₂²×C₅²×C₈	1	1	yes	characteristic, direct product factor, Sylow
64	C₂²×C₅	C₅×C₈×C₂₃	1	24	yes	direct product factor
65	C₂²×C₅	C₂×C₄×C₅×C₂₃	1	18	yes	—
66	C₄×C₅	C₂×C₄×C₅×C₂₃	1	18	yes	—
67	C₄×C₅	C₂³×C₅×C₂₃	1	6	yes	—
68	C₂²×C₄	C₂×C₅²×C₂₃	1	1	yes	characteristic
69	C₂×C₈	C₂×C₅²×C₂₃	1	6	yes	direct product factor
70	C₂×C₅	C₂×C₅×C₈×C₂₃	1	36	yes	direct product factor
71	C₂×C₅	C₂²×C₄×C₅×C₂₃	1	6	yes	—
72	C₂³	C₄×C₅²×C₂₃	1	1	yes	characteristic
73	C₂×C₄	C₄×C₅²×C₂₃	1	3	yes	—
74	C₂×C₄	C₂²×C₅²×C₂₃	1	3	yes	—
75	C₈	C₂²×C₅²×C₂₃	1	4	yes	direct product factor
76	C₅	C₂²×C₅×C₈×C₂₃	1	6	yes	direct product factor
77	C₂²	C₅²×C₈×C₂₃	1	4	yes	direct product factor
78	C₂²	C₂×C₄×C₅²×C₂₃	1	3	yes	—
79	C₄	C₂×C₄×C₅²×C₂₃	1	3	yes	—
80	C₄	C₂³×C₅²×C₂₃	1	1	yes	characteristic
81	C₂	C₂×C₅²×C₈×C₂₃	1	6	yes	direct product factor
82	C₂	C₂²×C₄×C₅²×C₂₃	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄	C₂³×C₅²×C₂₃
socle, prime radical	C₂³×C₅²×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. C₂×C₅²×C₈×C₂₃	C₂	direct (N × H)
6. C₅²×C₈×C₂₃	C₂²	direct (N × H)
7. C₂²×C₅×C₈×C₂₃	C₅	direct (N × H)
8. C₂²×C₅²×C₂₃	C₈	direct (N × H)
13. C₂×C₅×C₈×C₂₃	C₂×C₅	direct (N × H)
14. C₂×C₅²×C₂₃	C₂×C₈	direct (N × H)
19. C₅×C₈×C₂₃	C₂²×C₅	direct (N × H)
20. C₂²×C₅²×C₈	C₂₃	direct (N × H)
21. C₂²×C₈×C₂₃	C₅²	direct (N × H)
22. C₅²×C₂₃	C₂²×C₈	direct (N × H)
23. C₂²×C₅×C₂₃	C₅×C₈	direct (N × H)
28. C₂×C₅²×C₈	C₂×C₂₃	direct (N × H)
30. C₂×C₈×C₂₃	C₂×C₅²	direct (N × H)
31. C₂×C₅×C₂₃	C₂×C₅×C₈	direct (N × H)
36. C₅²×C₈	C₂²×C₂₃	direct (N × H)
40. C₈×C₂₃	C₂²×C₅²	direct (N × H)
41. C₂²×C₅×C₈	C₅×C₂₃	direct (N × H)
42. C₅×C₂₃	C₂²×C₅×C₈	direct (N × H)
43. C₂²×C₅²	C₈×C₂₃	direct (N × H)
47. C₂²×C₂₃	C₅²×C₈	direct (N × H)
52. C₂×C₅×C₈	C₂×C₅×C₂₃	direct (N × H)
53. C₂×C₅²	C₂×C₈×C₂₃	direct (N × H)
55. C₂×C₂₃	C₂×C₅²×C₈	direct (N × H)
60. C₅×C₈	C₂²×C₅×C₂₃	direct (N × H)
61. C₂²×C₈	C₅²×C₂₃	direct (N × H)
62. C₅²	C₂²×C₈×C₂₃	direct (N × H)
63. C₂₃	C₂²×C₅²×C₈	direct (N × H)
64. C₂²×C₅	C₅×C₈×C₂₃	direct (N × H)
69. C₂×C₈	C₂×C₅²×C₂₃	direct (N × H)
70. C₂×C₅	C₂×C₅×C₈×C₂₃	direct (N × H)
75. C₈	C₂²×C₅²×C₂₃	direct (N × H)
76. C₅	C₂²×C₅×C₈×C₂₃	direct (N × H)
77. C₂²	C₅²×C₈×C₂₃	direct (N × H)
81. C₂	C₂×C₅²×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
C₂²×C₄×C₅²×C₂₃	2
C₂×C₅²×C₈×C₂₃	2
C₂²×C₅×C₈×C₂₃	5
C₂²×C₅²×C₈	23

Subgroups — C22×C52×C8×C23 #72196128

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅²×C₈×C₂₃ #72196128