Subgroups — #11716906

Order 800.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊(C₅⋊D₅)	C₂	1	1	yes	maximal, characteristic
2	#87867	C₂	1	2	yes	maximal, complemented
3	C₂³×C₅²	C₄	1	1	yes	characteristic
4	C₂²×C₅⋊D₅	C₄	1	1	yes	characteristic
5	C₂²⋊(C₅⋊D₅)	—	2	2	no	—
6	C₂×#1039	C₂²	1	1	yes	characteristic
7	#6034	—	2	2	no	—
8	#4013	—	5	2	no	maximal
9	C₂²×C₅²	—	2	1	no	—
10	C₂²×C₅²	—	4	1	no	—
11	C₂²×C₅²	C₂×C₄	1	1	yes	characteristic
12	C₂×C₅⋊D₅	—	2	1	no	—
13	C₂×C₅⋊D₅	—	4	1	no	—
14	#1039	D₄	1	2	yes	—
15	C₂×C₂²⋊D₅	—	5	2	no	—
16	C₂×C₂²⋊D₅	—	10	2	no	—
17	#704	—	5	4	no	—
18	C₂×C₅²	—	2	1	no	—
19	C₂×C₅²	—	4	1	no	—
20	C₂×C₅²	C₂²⋊C₄	1	1	yes	characteristic
21	C₅⋊D₅	—	4	1	no	—
22	C₂³×C₅	—	2	2	no	—
23	C₂³×C₅	AGL(1,𝔽₅)	1	2	yes	—
24	C₂²×D₅	—	5	2	no	—
25	C₂²×D₅	—	10	2	no	—
26	C₂²⋊D₅	—	10	4	no	—
27	C₂²⋊D₅	—	10	8	no	—
28	C₂×Dic₅	—	5	2	no	—
29	C₂×Dic₅	—	10	2	no	—
30	#170	—	10	4	no	—
31	#100	—	25	1	no	Sylow
32	C₅²	#100	1	1	yes	characteristic, complemented, Sylow
33	C₂²×C₅	—	2	4	no	—
34	C₂²×C₅	—	2	2	no	—
35	C₂²×C₅	—	2	2	no	—
36	C₂²×C₅	—	4	2	no	—
37	C₂²×C₅	—	4	4	no	—
38	C₂²×C₅	C₂×AGL(1,𝔽₅)	1	2	yes	—
39	C₂×D₅	—	10	4	no	—
40	C₂×D₅	—	10	2	no	—
41	C₂×D₅	—	20	4	no	—
42	C₂×D₅	—	20	2	no	—
43	Dic₅	—	5	4	no	—
44	Dic₅	—	10	4	no	—
45	C₂×D₄	—	25	1	no	—
46	M₁₆	—	25	2	no	—
47	C₂×C₅	—	2	4	no	—
48	C₂×C₅	—	2	2	no	—
49	C₂×C₅	—	2	2	no	—
50	C₂×C₅	—	4	4	no	—
51	C₂×C₅	—	4	2	no	—
52	C₂×C₅	C₂²⋊AGL(1,𝔽₅)	1	2	yes	—
53	D₅	—	20	4	no	—
54	D₅	—	20	2	no	—
55	C₂³	—	25	1	no	—
56	C₂³	#1044	1	1	yes	characteristic
57	D₄	—	50	2	no	—
58	C₂×C₄	—	25	1	no	—
59	C₈	—	50	2	no	—
60	C₅	—	2	2	no	—
61	C₅	#4013	1	2	yes	complemented
62	C₂²	—	2	1	no	—
63	C₂²	—	4	1	no	—
64	C₂²	—	50	1	no	—
65	C₂²	—	100	1	no	—
66	C₂²	C₂×#1044	1	1	yes	characteristic
67	C₄	—	25	2	no	—
68	C₂	—	2	1	no	—
69	C₂	—	4	1	no	—
70	C₂	—	100	1	no	—
71	C₂	C₂²⋊#1044	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²⋊#1044
commutator	C₂²×C₅²	C₂×C₄
Frattini	C₂²	C₂×#1044
Fitting	C₂³×C₅²	C₄
socle	C₂×C₅²	C₂²⋊C₄
prime radical	C₂×C₂²⋊(C₅⋊D₅)	C₂
cosocle	C₂×#1039	C₂²
hypercenter	C₂³	#1044
hypocenter	C₅²	#100
LCS	G ▷ C₂²×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂³

Complements

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

N	H ≅ G/N	type
2. #87867	C₂	semidirect
32. C₅²	#100	semidirect
61. C₅	#4013	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊(C₅⋊D₅)	2
#87867	2
#4013	5