Subgroups — #13102543

Order 1600.

392 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11717189	C₂	1	2	yes	maximal
2	C₄×C₅²×C₈	C₂	1	1	yes	maximal, characteristic
3	#11716684	C₂	1	4	yes	maximal
4	C₂×#6045	—	2	2	no	—
5	C₂×C₅²×C₈	C₂²	1	2	yes	—
6	C₄²×C₅²	C₂²	1	1	yes	characteristic
7	C₄⋊#1039	—	2	2	no	—
8	C₄⋊#1039	C₂²	1	4	yes	—
9	#64969	—	5	6	no	maximal
10	C₂×C₄×C₅²	D₄	1	2	yes	—
11	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
12	C₂×#1039	—	2	4	no	—
13	#6045	—	4	4	no	—
14	C₅²×C₈	Q₈	1	4	yes	complemented
15	#4015	—	5	12	no	—
16	C₄×C₅×C₈	D₅	1	6	yes	—
17	#3949	—	5	24	no	—
18	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
19	C₄×C₅²	SD₁₆	1	4	yes	—
20	C₄×C₅²	C₂×Q₈	1	2	yes	—
21	#1039	—	4	4	no	—
22	C₂×Dic₁₀	—	10	12	no	—
23	C₂×C₅×C₈	C₂×D₅	1	12	yes	—
24	C₄²×C₅	C₂×D₅	1	6	yes	—
25	C₄⋊Dic₅	—	5	24	no	—
26	C₄⋊Dic₅	—	10	12	no	—
27	#498	—	25	1	no	Sylow
28	C₂×C₅²	C₄⋊Q₈	1	1	yes	characteristic
29	C₂×C₅²	C₂×SD₁₆	1	2	yes	—
30	C₂×C₄×C₅	D₂₀	1	12	yes	—
31	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
32	C₂×Dic₅	—	10	24	no	—
33	Dic₁₀	—	20	24	no	—
34	C₅×C₈	Dic₁₀	1	24	yes	complemented
35	C₄⋊Q₈	—	25	2	no	—
36	#106	—	25	4	no	—
37	C₄×C₈	C₅⋊D₅	1	1	yes	characteristic
38	C₅²	#498	1	1	yes	characteristic, complemented, Sylow
39	C₂²×C₅	C₂×D₂₀	1	6	yes	—
40	C₄×C₅	#677	1	24	yes	—
41	C₄×C₅	C₂×Dic₁₀	1	12	yes	—
42	Dic₅	—	20	24	no	—
43	C₂×Q₈	—	50	2	no	—
44	C₂×C₈	C₂×C₅⋊D₅	1	2	yes	—
45	C₄⋊C₄	—	25	4	no	—
46	C₄⋊C₄	—	50	2	no	—
47	C₄²	C₂×C₅⋊D₅	1	1	yes	characteristic
48	C₂×C₅	#4015	1	6	yes	—
49	C₂×C₅	C₂×#677	1	12	yes	—
50	Q₈	—	100	4	no	—
51	C₂×C₄	—	50	4	no	—
52	C₂×C₄	C₄⋊(C₅⋊D₅)	1	2	yes	—
53	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
54	C₈	#6045	1	4	yes	complemented
55	C₅	#64969	1	6	yes	complemented
56	C₂²	C₂×C₄⋊(C₅⋊D₅)	1	1	yes	characteristic
57	C₄	—	100	4	no	—
58	C₄	#87800	1	4	yes	—
59	C₄	C₂×#6045	1	2	yes	—
60	C₂	#11717189	1	1	yes	characteristic
61	C₂	C₂×#87800	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₄⋊(C₅⋊D₅)
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂²×C₅⋊D₅
Fitting	C₄×C₅²×C₈	C₂
socle, prime radical	C₂²×C₅²	C₂×D₄
hypercenter	C₄×C₈	C₅⋊D₅
hypocenter	C₅²	#498
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
14. C₅²×C₈	Q₈	semidirect
34. C₅×C₈	Dic₁₀	semidirect
38. C₅²	#498	semidirect
54. C₈	#6045	semidirect
55. C₅	#64969	semidirect

Maximal subgroups

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

subgroup	index
#11717189	2
C₄×C₅²×C₈	2
#11716684	2
#64969	5