Subgroups — C₂³×#6036 #13107848

Order 1600.

3472 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₅×Dic₅	C₂	1	2	yes	maximal, complemented
3	C₂²×#6036	C₂	1	28	yes	maximal, direct product factor
4	C₂⁴×C₅²	C₂²	1	1	yes	characteristic
5	C₂³×C₅⋊D₅	C₄	1	16	yes	complemented
6	C₂³×C₅⋊D₅	C₂²	1	14	yes	—
7	C₂²×C₅×Dic₅	C₂²	1	28	yes	complemented
8	C₂×#6036	C₂²	1	112	yes	direct product factor
9	C₂³×C₄×D₅	—	5	2	no	maximal
10	C₂³×C₅²	C₂×C₄	1	8	yes	complemented
11	C₂³×C₅²	C₂³	1	7	yes	—
12	C₂²×C₅⋊D₅	C₂×C₄	1	112	yes	complemented
13	C₂²×C₅⋊D₅	C₂³	1	28	yes	—
14	C₂×C₅×Dic₅	C₂³	1	56	yes	complemented
15	#6036	C₂³	1	64	yes	direct product factor
16	C₂⁴×D₅	—	5	2	no	—
17	C₂⁴×D₅	—	10	2	no	—
18	C₂³×C₄×C₅	—	5	2	no	—
19	C₂³×Dic₅	D₅	1	2	yes	complemented
20	C₂²×C₄×D₅	—	5	56	no	—
21	C₂²×C₅²	C₂²×C₄	1	28	yes	complemented
22	C₂²×C₅²	C₂⁴	1	7	yes	—
23	C₂×C₅⋊D₅	C₂²×C₄	1	112	yes	complemented
24	C₂×C₅⋊D₅	C₂⁴	1	8	yes	—
25	C₅×Dic₅	C₂⁴	1	16	yes	complemented
26	C₂⁴×C₅	—	2	2	no	—
27	C₂⁴×C₅	C₂×D₅	1	2	yes	—
28	C₂³×D₅	—	5	32	no	—
29	C₂³×D₅	—	5	28	no	—
30	C₂³×D₅	—	10	32	no	—
31	C₂³×D₅	—	10	28	no	—
32	C₂²×C₄×C₅	—	5	28	no	—
33	C₂²×Dic₅	C₂×D₅	1	28	yes	complemented
34	C₂×C₄×D₅	—	5	224	no	—
35	C₂⁴×C₄	—	25	1	no	Sylow
36	C₂×C₅²	C₂³×C₄	1	14	yes	complemented
37	C₂×C₅²	C₂⁵	1	1	yes	characteristic
38	C₅⋊D₅	C₂³×C₄	1	16	yes	complemented
39	C₂³×C₅	—	2	16	no	—
40	C₂³×C₅	—	2	14	no	—
41	C₂³×C₅	C₄×D₅	1	16	yes	complemented
42	C₂³×C₅	C₂²×D₅	1	14	yes	—
43	C₂²×D₅	—	5	224	no	—
44	C₂²×D₅	—	5	56	no	—
45	C₂²×D₅	—	10	224	no	—
46	C₂²×D₅	—	10	56	no	—
47	C₂×C₄×C₅	—	5	56	no	—
48	C₂×Dic₅	C₂²×D₅	1	56	yes	complemented
49	C₄×D₅	—	5	128	no	—
50	C₂⁵	—	25	1	no	—
51	C₂³×C₄	—	5	2	no	—
52	C₂³×C₄	—	25	28	no	—
53	C₅²	C₂⁴×C₄	1	1	yes	characteristic, complemented, Sylow
54	C₂²×C₅	—	2	56	no	—
55	C₂²×C₅	—	2	14	no	—
56	C₂²×C₅	C₂×C₄×D₅	1	56	yes	complemented
57	C₂²×C₅	C₂³×D₅	1	14	yes	—
58	C₂×D₅	—	5	224	no	—
59	C₂×D₅	—	5	16	no	—
60	C₂×D₅	—	10	224	no	—
61	C₂×D₅	—	10	16	no	—
62	C₄×C₅	—	5	16	no	—
63	Dic₅	C₂³×D₅	1	16	yes	complemented
64	C₂⁴	—	25	16	no	—
65	C₂⁴	—	25	14	no	—
66	C₂⁴	D₅²	1	1	yes	characteristic
67	C₂²×C₄	—	5	28	no	—
68	C₂²×C₄	—	25	112	no	—
69	C₂×C₅	—	2	28	no	—
70	C₂×C₅	—	2	2	no	—
71	C₂×C₅	C₂²×C₄×D₅	1	28	yes	complemented
72	C₂×C₅	C₂⁴×D₅	1	2	yes	—
73	D₅	—	5	32	no	—
74	D₅	—	10	32	no	—
75	C₂³	—	25	112	no	—
76	C₂³	—	25	28	no	—
77	C₂³	#6036	1	8	yes	direct product factor
78	C₂³	C₂×D₅²	1	7	yes	—
79	C₂×C₄	—	5	56	no	—
80	C₂×C₄	—	25	64	no	—
81	C₅	—	2	2	no	—
82	C₅	C₂³×C₄×D₅	1	2	yes	complemented
83	C₂²	—	25	112	no	—
84	C₂²	—	25	8	no	—
85	C₂²	C₂×#6036	1	28	yes	direct product factor
86	C₂²	C₂²×D₅²	1	7	yes	—
87	C₄	—	5	16	no	—
88	C₂	—	25	16	no	—
89	C₂	C₂²×#6036	1	14	yes	direct product factor
90	C₂	C₂³×D₅²	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂⁴	D₅²
commutator, hypocenter	C₅²	C₂⁴×C₄
Frattini	C₂	C₂³×D₅²
Fitting, socle	C₂⁴×C₅²	C₂²
prime radical	C₂⁴×C₅⋊D₅	C₂
cosocle	C₂×C₅²	C₂⁵
LCS	G ▷ C₅²
UCS	C₁ ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
2. C₂³×C₅×Dic₅	C₂	semidirect
3. C₂²×#6036	C₂	direct (N × H)
5. C₂³×C₅⋊D₅	C₄	semidirect
7. C₂²×C₅×Dic₅	C₂²	semidirect
8. C₂×#6036	C₂²	direct (N × H)
10. C₂³×C₅²	C₂×C₄	semidirect
12. C₂²×C₅⋊D₅	C₂×C₄	semidirect
14. C₂×C₅×Dic₅	C₂³	semidirect
15. #6036	C₂³	direct (N × H)
19. C₂³×Dic₅	D₅	semidirect
21. C₂²×C₅²	C₂²×C₄	semidirect
23. C₂×C₅⋊D₅	C₂²×C₄	semidirect
25. C₅×Dic₅	C₂⁴	semidirect
33. C₂²×Dic₅	C₂×D₅	semidirect
36. C₂×C₅²	C₂³×C₄	semidirect
38. C₅⋊D₅	C₂³×C₄	semidirect
41. C₂³×C₅	C₄×D₅	semidirect
48. C₂×Dic₅	C₂²×D₅	semidirect
53. C₅²	C₂⁴×C₄	semidirect
56. C₂²×C₅	C₂×C₄×D₅	semidirect
63. Dic₅	C₂³×D₅	semidirect
71. C₂×C₅	C₂²×C₄×D₅	semidirect
77. C₂³	#6036	direct (N × H)
82. C₅	C₂³×C₄×D₅	semidirect
85. C₂²	C₂×#6036	direct (N × H)
89. C₂	C₂²×#6036	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂⁴×C₅⋊D₅	2
C₂³×C₅×Dic₅	2
C₂²×#6036	2
C₂³×C₄×D₅	5

Subgroups — C23×#6036 #13107848

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#6036 #13107848