Subgroups — #11717195

Order 800.

304 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₅⋊D₅)	C₂	1	1	yes	maximal, characteristic
2	C₂×#6045	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄²×C₅²	C₂	1	1	yes	maximal, characteristic, complemented
4	#87808	C₂	1	4	yes	maximal, complemented
5	C₂²×C₅⋊D₅	C₂²	1	2	yes	—
6	C₂×C₄×C₅²	C₂²	1	2	yes	—
7	C₂×C₄×C₅²	C₂²	1	1	yes	characteristic
8	C₂×#1039	C₂²	1	2	yes	—
9	C₄⋊(C₅⋊D₅)	—	2	2	no	—
10	#6045	—	2	2	no	—
11	#4021	—	5	6	no	maximal
12	C₂²×C₅²	C₂³	1	1	yes	characteristic
13	C₂×C₅⋊D₅	—	2	2	no	—
14	C₂×C₅⋊D₅	—	2	4	no	—
15	C₄×C₅²	—	2	2	no	—
16	C₄×C₅²	D₄	1	2	yes	—
17	#1039	—	2	2	no	—
18	C₂×D₂₀	—	5	6	no	—
19	C₂×Dic₁₀	—	5	6	no	—
20	C₄²×C₅	D₅	1	6	yes	complemented
21	#685	—	5	24	no	—
22	C₂×C₅²	C₂×D₄	1	1	yes	characteristic
23	C₂×C₅²	#41	1	2	yes	—
24	C₅⋊D₅	—	4	2	no	—
25	C₂²×D₅	—	5	12	no	—
26	C₂×C₄×C₅	C₂×D₅	1	12	yes	—
27	C₂×C₄×C₅	C₂×D₅	1	6	yes	—
28	C₂×Dic₅	—	5	12	no	—
29	D₂₀	—	10	12	no	—
30	Dic₁₀	—	10	12	no	—
31	#124	—	25	1	no	Sylow
32	C₅²	#124	1	1	yes	characteristic, complemented, Sylow
33	C₂²×C₅	C₂²×D₅	1	6	yes	—
34	C₂×D₅	—	10	12	no	—
35	C₂×D₅	—	10	24	no	—
36	C₄×C₅	—	2	12	no	—
37	C₄×C₅	D₂₀	1	12	yes	—
38	Dic₅	—	10	12	no	—
39	C₂×Q₈	—	25	1	no	—
40	C₂×D₄	—	25	1	no	—
41	C₂²⋊C₄	—	25	4	no	—
42	C₄²	C₅⋊D₅	1	1	yes	characteristic, complemented
43	C₂×C₅	C₂×D₂₀	1	6	yes	—
44	C₂×C₅	#709	1	12	yes	—
45	D₅	—	20	12	no	—
46	C₂³	—	25	2	no	—
47	Q₈	—	50	2	no	—
48	D₄	—	50	2	no	—
49	C₂×C₄	—	25	2	no	—
50	C₂×C₄	C₂×C₅⋊D₅	1	2	yes	—
51	C₂×C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
52	C₅	#4021	1	6	yes	complemented
53	C₂²	—	50	2	no	—
54	C₂²	—	50	4	no	—
55	C₂²	C₂²×C₅⋊D₅	1	1	yes	characteristic
56	C₄	—	2	2	no	—
57	C₄	—	50	2	no	—
58	C₄	C₄⋊(C₅⋊D₅)	1	2	yes	—
59	C₂	—	100	2	no	—
60	C₂	C₂×C₄⋊(C₅⋊D₅)	1	1	yes	characteristic
61	C₂	#87900	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×C₅⋊D₅
commutator, socle, cosocle	C₂²×C₅²	C₂³
Fitting	C₄²×C₅²	C₂
prime radical	C₂×C₄⋊(C₅⋊D₅)	C₂
hypercenter	C₄²	C₅⋊D₅
hypocenter	C₅²	#124
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×#6045	C₂	semidirect
3. C₄²×C₅²	C₂	semidirect
4. #87808	C₂	semidirect
20. C₄²×C₅	D₅	semidirect
32. C₅²	#124	semidirect
42. C₄²	C₅⋊D₅	semidirect
52. C₅	#4021	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₅⋊D₅)	2
C₂×#6045	2
C₄²×C₅²	2
#87808	2
#4021	5