Subgroups — C₂×C₅²×C₁₆×C₂₃ #72196124

Order 18400.

224 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₅²×C₁₆×C₂₃	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₄×C₅²×C₂₃	C₄	1	1	yes	characteristic
4	C₅²×C₈×C₂₃	C₄	1	1	yes	characteristic
5	C₅²×C₈×C₂₃	C₂²	1	1	yes	characteristic
6	C₂×C₅×C₁₆×C₂₃	C₅	1	6	yes	maximal, direct product factor
7	C₂²×C₅²×C₂₃	C₈	1	1	yes	characteristic
8	C₄×C₅²×C₂₃	C₈	1	1	yes	characteristic
9	C₄×C₅²×C₂₃	C₂×C₄	1	1	yes	characteristic
10	C₂×C₅×C₈×C₂₃	C₂×C₅	1	6	yes	—
11	C₅×C₁₆×C₂₃	C₂×C₅	1	12	yes	direct product factor
12	C₂×C₅²×C₂₃	C₁₆	1	2	yes	direct product factor
13	C₂×C₅²×C₂₃	C₂×C₈	1	1	yes	characteristic
14	C₂×C₄×C₅×C₂₃	C₄×C₅	1	6	yes	—
15	C₅×C₈×C₂₃	C₄×C₅	1	6	yes	—
16	C₅×C₈×C₂₃	C₂²×C₅	1	6	yes	—
17	C₂×C₅²×C₁₆	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂×C₁₆×C₂₃	C₅²	1	1	yes	characteristic, direct product factor, Hall
19	C₅²×C₂₃	C₂×C₁₆	1	1	yes	characteristic, direct product factor, Hall
20	C₂²×C₅×C₂₃	C₅×C₈	1	6	yes	—
21	C₄×C₅×C₂₃	C₅×C₈	1	6	yes	—
22	C₄×C₅×C₂₃	C₂×C₄×C₅	1	6	yes	—
23	C₂×C₅²×C₈	C₂×C₂₃	1	1	yes	characteristic
24	C₅²×C₁₆	C₂×C₂₃	1	2	yes	direct product factor
25	C₂×C₈×C₂₃	C₂×C₅²	1	1	yes	characteristic
26	C₁₆×C₂₃	C₂×C₅²	1	2	yes	direct product factor
27	C₂×C₅×C₂₃	C₅×C₁₆	1	12	yes	direct product factor
28	C₂×C₅×C₂₃	C₂×C₅×C₈	1	6	yes	—
29	C₂×C₄×C₅²	C₄×C₂₃	1	1	yes	characteristic
30	C₅²×C₈	C₄×C₂₃	1	1	yes	characteristic
31	C₅²×C₈	C₂²×C₂₃	1	1	yes	characteristic
32	C₂×C₄×C₂₃	C₄×C₅²	1	1	yes	characteristic
33	C₈×C₂₃	C₄×C₅²	1	1	yes	characteristic
34	C₈×C₂₃	C₂²×C₅²	1	1	yes	characteristic
35	C₂×C₅×C₁₆	C₅×C₂₃	1	6	yes	direct product factor
36	C₅×C₂₃	C₂×C₅×C₁₆	1	6	yes	direct product factor
37	C₂²×C₅²	C₈×C₂₃	1	1	yes	characteristic
38	C₄×C₅²	C₈×C₂₃	1	1	yes	characteristic
39	C₄×C₅²	C₂×C₄×C₂₃	1	1	yes	characteristic
40	C₂²×C₂₃	C₅²×C₈	1	1	yes	characteristic
41	C₄×C₂₃	C₅²×C₈	1	1	yes	characteristic
42	C₄×C₂₃	C₂×C₄×C₅²	1	1	yes	characteristic
43	C₂×C₅×C₈	C₂×C₅×C₂₃	1	6	yes	—
44	C₅×C₁₆	C₂×C₅×C₂₃	1	12	yes	direct product factor
45	C₂×C₅²	C₁₆×C₂₃	1	2	yes	direct product factor
46	C₂×C₅²	C₂×C₈×C₂₃	1	1	yes	characteristic
47	C₂×C₂₃	C₅²×C₁₆	1	2	yes	direct product factor
48	C₂×C₂₃	C₂×C₅²×C₈	1	1	yes	characteristic
49	C₂×C₄×C₅	C₄×C₅×C₂₃	1	6	yes	—
50	C₅×C₈	C₄×C₅×C₂₃	1	6	yes	—
51	C₅×C₈	C₂²×C₅×C₂₃	1	6	yes	—
52	C₂×C₁₆	C₅²×C₂₃	1	1	yes	characteristic, direct product factor, Sylow
53	C₅²	C₂×C₁₆×C₂₃	1	1	yes	characteristic, direct product factor, Sylow
54	C₂₃	C₂×C₅²×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
55	C₂²×C₅	C₅×C₈×C₂₃	1	6	yes	—
56	C₄×C₅	C₅×C₈×C₂₃	1	6	yes	—
57	C₄×C₅	C₂×C₄×C₅×C₂₃	1	6	yes	—
58	C₂×C₈	C₂×C₅²×C₂₃	1	1	yes	characteristic
59	C₁₆	C₂×C₅²×C₂₃	1	2	yes	direct product factor
60	C₂×C₅	C₅×C₁₆×C₂₃	1	12	yes	direct product factor
61	C₂×C₅	C₂×C₅×C₈×C₂₃	1	6	yes	—
62	C₂×C₄	C₄×C₅²×C₂₃	1	1	yes	characteristic
63	C₈	C₄×C₅²×C₂₃	1	1	yes	characteristic
64	C₈	C₂²×C₅²×C₂₃	1	1	yes	characteristic
65	C₅	C₂×C₅×C₁₆×C₂₃	1	6	yes	direct product factor
66	C₂²	C₅²×C₈×C₂₃	1	1	yes	characteristic
67	C₄	C₅²×C₈×C₂₃	1	1	yes	characteristic
68	C₄	C₂×C₄×C₅²×C₂₃	1	1	yes	characteristic
69	C₂	C₅²×C₁₆×C₂₃	1	2	yes	direct product factor
70	C₂	C₂×C₅²×C₈×C₂₃	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₈	C₂²×C₅²×C₂₃
socle, prime radical	C₂²×C₅²×C₂₃	C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅²×C₁₆×C₂₃	C₂	direct (N × H)
6. C₂×C₅×C₁₆×C₂₃	C₅	direct (N × H)
11. C₅×C₁₆×C₂₃	C₂×C₅	direct (N × H)
12. C₂×C₅²×C₂₃	C₁₆	direct (N × H)
17. C₂×C₅²×C₁₆	C₂₃	direct (N × H)
18. C₂×C₁₆×C₂₃	C₅²	direct (N × H)
19. C₅²×C₂₃	C₂×C₁₆	direct (N × H)
24. C₅²×C₁₆	C₂×C₂₃	direct (N × H)
26. C₁₆×C₂₃	C₂×C₅²	direct (N × H)
27. C₂×C₅×C₂₃	C₅×C₁₆	direct (N × H)
35. C₂×C₅×C₁₆	C₅×C₂₃	direct (N × H)
36. C₅×C₂₃	C₂×C₅×C₁₆	direct (N × H)
44. C₅×C₁₆	C₂×C₅×C₂₃	direct (N × H)
45. C₂×C₅²	C₁₆×C₂₃	direct (N × H)
47. C₂×C₂₃	C₅²×C₁₆	direct (N × H)
52. C₂×C₁₆	C₅²×C₂₃	direct (N × H)
53. C₅²	C₂×C₁₆×C₂₃	direct (N × H)
54. C₂₃	C₂×C₅²×C₁₆	direct (N × H)
59. C₁₆	C₂×C₅²×C₂₃	direct (N × H)
60. C₂×C₅	C₅×C₁₆×C₂₃	direct (N × H)
65. C₅	C₂×C₅×C₁₆×C₂₃	direct (N × H)
69. C₂	C₅²×C₁₆×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×C₅²×C₈×C₂₃	2
C₅²×C₁₆×C₂₃	2
C₂×C₅×C₁₆×C₂₃	5
C₂×C₅²×C₁₆	23

Subgroups — C2×C52×C16×C23 #72196124

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅²×C₁₆×C₂₃ #72196124