Subgroups — C₂³×#6039 #13107851

Order 1600.

1936 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#1039	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₅×Dic₅	C₂	1	2	yes	maximal
3	C₂²×#6039	C₂	1	28	yes	maximal, direct product factor
4	C₂⁴×C₅²	C₂²	1	1	yes	characteristic
5	C₂²×#1039	C₂²	1	14	yes	—
6	C₂²×C₅×Dic₅	C₂²	1	28	yes	—
7	C₂×#6039	C₂²	1	112	yes	direct product factor
8	C₂³×Dic₁₀	—	5	2	no	maximal
9	C₂³×C₅²	Q₈	1	8	yes	complemented
10	C₂³×C₅²	C₂³	1	7	yes	—
11	C₂×#1039	C₂³	1	28	yes	—
12	C₂×C₅×Dic₅	C₂³	1	56	yes	—
13	#6039	C₂³	1	64	yes	direct product factor
14	C₂³×C₄×C₅	—	5	2	no	—
15	C₂³×Dic₅	—	5	2	no	—
16	C₂³×Dic₅	—	10	2	no	—
17	C₂³×Dic₅	D₅	1	2	yes	—
18	C₂²×Dic₁₀	—	5	56	no	—
19	C₂²×C₅²	C₂×Q₈	1	28	yes	complemented
20	C₂²×C₅²	C₂⁴	1	7	yes	—
21	#1039	C₂⁴	1	8	yes	—
22	C₅×Dic₅	C₂⁴	1	16	yes	—
23	C₂⁴×C₅	—	2	2	no	—
24	C₂⁴×C₅	C₂×D₅	1	2	yes	—
25	C₂²×C₄×C₅	—	5	28	no	—
26	C₂²×Dic₅	—	5	28	no	—
27	C₂²×Dic₅	—	10	28	no	—
28	C₂²×Dic₅	C₂×D₅	1	28	yes	—
29	C₂×Dic₁₀	—	5	224	no	—
30	C₂³×Q₈	—	25	1	no	Sylow
31	C₂×C₅²	C₂²×Q₈	1	14	yes	complemented
32	C₂×C₅²	C₂⁵	1	1	yes	characteristic
33	C₂³×C₅	—	2	16	no	—
34	C₂³×C₅	—	2	14	no	—
35	C₂³×C₅	Dic₁₀	1	16	yes	complemented
36	C₂³×C₅	C₂²×D₅	1	14	yes	—
37	C₂×C₄×C₅	—	5	56	no	—
38	C₂×Dic₅	—	5	56	no	—
39	C₂×Dic₅	—	10	56	no	—
40	C₂×Dic₅	C₂²×D₅	1	56	yes	—
41	Dic₁₀	—	5	128	no	—
42	C₂²×Q₈	—	25	28	no	—
43	C₂³×C₄	—	5	2	no	—
44	C₂³×C₄	—	25	1	no	—
45	C₅²	C₂³×Q₈	1	1	yes	characteristic, complemented, Sylow
46	C₂²×C₅	—	2	56	no	—
47	C₂²×C₅	—	2	14	no	—
48	C₂²×C₅	C₂×Dic₁₀	1	56	yes	complemented
49	C₂²×C₅	C₂³×D₅	1	14	yes	—
50	C₄×C₅	—	5	16	no	—
51	Dic₅	—	5	16	no	—
52	Dic₅	—	10	16	no	—
53	Dic₅	C₂³×D₅	1	16	yes	—
54	C₂⁴	D₅²	1	1	yes	characteristic
55	C₂×Q₈	—	25	112	no	—
56	C₂²×C₄	—	5	28	no	—
57	C₂²×C₄	—	25	14	no	—
58	C₂×C₅	—	2	28	no	—
59	C₂×C₅	—	2	2	no	—
60	C₂×C₅	C₂²×Dic₁₀	1	28	yes	complemented
61	C₂×C₅	C₂⁴×D₅	1	2	yes	—
62	C₂³	#6039	1	8	yes	direct product factor
63	C₂³	C₂×D₅²	1	7	yes	—
64	Q₈	—	25	64	no	—
65	C₂×C₄	—	5	56	no	—
66	C₂×C₄	—	25	28	no	—
67	C₅	—	2	2	no	—
68	C₅	C₂³×Dic₁₀	1	2	yes	complemented
69	C₂²	C₂×#6039	1	28	yes	direct product factor
70	C₂²	C₂²×D₅²	1	7	yes	—
71	C₄	—	5	16	no	—
72	C₄	—	25	8	no	—
73	C₂	C₂²×#6039	1	14	yes	direct product factor
74	C₂	C₂³×D₅²	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂⁴	D₅²
commutator, cosocle	C₂×C₅²	C₂⁵
Frattini	C₂	C₂³×D₅²
Fitting, socle, prime radical	C₂⁴×C₅²	C₂²
hypocenter	C₅²	C₂³×Q₈
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
3. C₂²×#6039	C₂	direct (N × H)
7. C₂×#6039	C₂²	direct (N × H)
9. C₂³×C₅²	Q₈	semidirect
13. #6039	C₂³	direct (N × H)
19. C₂²×C₅²	C₂×Q₈	semidirect
31. C₂×C₅²	C₂²×Q₈	semidirect
35. C₂³×C₅	Dic₁₀	semidirect
45. C₅²	C₂³×Q₈	semidirect
48. C₂²×C₅	C₂×Dic₁₀	semidirect
60. C₂×C₅	C₂²×Dic₁₀	semidirect
62. C₂³	#6039	direct (N × H)
68. C₅	C₂³×Dic₁₀	semidirect
69. C₂²	C₂×#6039	direct (N × H)
73. C₂	C₂²×#6039	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#1039	2
C₂³×C₅×Dic₅	2
C₂²×#6039	2
C₂³×Dic₁₀	5

Subgroups — C23×#6039 #13107851

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#6039 #13107851