Subgroups — #13099547

Order 1600.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×M₃₂	C₂	1	1	yes	maximal, characteristic
2	#11716686	C₂	1	1	yes	maximal, characteristic
3	#11716684	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
5	C₄⋊#1039	—	2	1	no	—
6	#87804	—	2	1	no	—
7	C₅²×C₁₆	C₄	1	2	yes	complemented
8	#64733	—	5	6	no	maximal
9	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
10	C₂×#1039	—	4	1	no	—
11	C₅²×C₈	C₂×C₄	1	1	yes	characteristic
12	C₅²×C₈	Q₈	1	1	yes	characteristic
13	#6029	—	4	1	no	—
14	C₅×M₃₂	D₅	1	6	yes	—
15	#3951	—	5	6	no	—
16	#3949	—	5	6	no	—
17	C₂²×C₅²	SD₁₆	1	1	yes	characteristic
18	C₄×C₅²	C₄⋊C₄	1	1	yes	characteristic
19	C₄×C₅²	SD₁₆	1	1	yes	characteristic
20	#1039	—	8	1	no	—
21	C₂×C₅×C₈	C₂×D₅	1	6	yes	—
22	C₄⋊Dic₅	—	10	6	no	—
23	#681	—	10	6	no	—
24	C₅×C₁₆	Dic₅	1	12	yes	complemented
25	#365	—	25	1	no	Sylow
26	C₂×C₅²	—	2	1	no	—
27	C₂×C₅²	#106	1	1	yes	characteristic
28	C₂×C₄×C₅	D₂₀	1	6	yes	—
29	C₂×Dic₅	—	20	6	no	—
30	C₅×C₈	Dic₁₀	1	6	yes	—
31	C₅×C₈	C₂×Dic₅	1	6	yes	—
32	#168	—	20	6	no	—
33	M₃₂	C₅⋊D₅	1	1	yes	characteristic
34	#108	—	25	1	no	—
35	#106	—	25	1	no	—
36	C₅²	#365	1	1	yes	characteristic, complemented, Sylow
37	C₂²×C₅	#677	1	6	yes	—
38	C₄×C₅	#677	1	6	yes	—
39	C₄×C₅	C₄⋊Dic₅	1	6	yes	—
40	Dic₅	—	40	6	no	—
41	M₁₆	—	50	1	no	—
42	C₂×C₈	C₂×C₅⋊D₅	1	1	yes	characteristic
43	C₄⋊C₄	—	50	1	no	—
44	C₁₆	#1039	1	2	yes	complemented
45	C₂×C₅	—	2	6	no	—
46	C₂×C₅	#3949	1	6	yes	—
47	C₂×C₄	—	100	1	no	—
48	C₂×C₄	C₄⋊(C₅⋊D₅)	1	1	yes	characteristic
49	C₈	—	100	1	no	—
50	C₈	#6045	1	1	yes	characteristic
51	C₈	C₂×#1039	1	1	yes	characteristic
52	C₅	#64733	1	6	yes	complemented
53	C₂²	#87800	1	1	yes	characteristic
54	C₄	—	200	1	no	—
55	C₄	#87800	1	1	yes	characteristic
56	C₄	C₄⋊#1039	1	1	yes	characteristic
57	C₂	—	2	1	no	—
58	C₂	#11716684	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#11716684
commutator	C₅²×C₈	C₂×C₄
Frattini	C₂×C₈	C₂×C₅⋊D₅
Fitting	C₅²×M₃₂	C₂
socle	C₂×C₅²	#106
prime radical	C₂²×C₅²	SD₁₆
cosocle	C₂×C₅²×C₈	C₂²
hypercenter	M₃₂	C₅⋊D₅
hypocenter	C₅²	#365
LCS	G ▷ C₅²×C₈ ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
7. C₅²×C₁₆	C₄	semidirect
24. C₅×C₁₆	Dic₅	semidirect
36. C₅²	#365	semidirect
44. C₁₆	#1039	semidirect
52. C₅	#64733	semidirect

Maximal subgroups

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

subgroup	index
C₅²×M₃₂	2
#11716686	2
#11716684	2
#64733	5