Subgroups — #11717389

Order 800.

177 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₅²⋊C₂	C₂	1	4	yes	maximal, complemented
2	#87877	C₂	1	2	yes	maximal, complemented
3	C₄×#1044	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₅²	C₂²	1	4	yes	—
5	C₂×#1044	C₂²	1	2	yes	—
6	D₅²⋊C₂	—	2	8	no	—
7	C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
8	C₅×D₂₀	—	2	2	no	—
9	#6037	—	2	2	no	—
10	C₂×C₅⋊D₅	C₂³	1	1	yes	characteristic
11	C₂×C₅×D₅	—	2	4	no	—
12	D₅²	—	2	8	no	—
13	#1044	D₄	1	4	yes	complemented
14	C₄×C₅²	D₄	1	1	yes	characteristic, complemented
15	#1039	D₄	1	1	yes	characteristic, complemented
16	D₄×D₅	—	10	2	no	—
17	C₄×AGL(1,𝔽₅)	—	10	1	no	—
18	C₂×C₅²	C₂×D₄	1	1	yes	characteristic
19	C₅⋊D₅	C₂×D₄	1	2	yes	—
20	C₅×D₅	—	4	4	no	—
21	C₂²×D₅	—	10	4	no	—
22	C₂×AGL(1,𝔽₅)	—	10	2	no	—
23	C₅×D₄	—	10	2	no	—
24	C₂²⋊D₅	—	10	4	no	—
25	D₂₀	—	2	2	no	—
26	C₄×D₅	—	10	1	no	—
27	C₄×D₅	—	10	2	no	—
28	#127	—	25	1	no	maximal, Sylow
29	C₅²	#127	1	1	yes	characteristic, complemented, Sylow
30	C₂²×C₅	—	10	4	no	—
31	C₂×D₅	—	2	4	no	—
32	C₂×D₅	—	10	1	no	—
33	C₂×D₅	—	10	2	no	—
34	C₂×D₅	—	20	8	no	—
35	AGL(1,𝔽₅)	—	10	4	no	—
36	C₄×C₅	—	2	1	no	—
37	C₄×C₅	—	2	2	no	—
38	Dic₅	—	10	1	no	—
39	Dic₅	—	10	2	no	—
40	C₂×D₄	—	25	2	no	—
41	C₂×D₄	—	25	4	no	—
42	C₄²	—	25	1	no	—
43	C₂×C₅	—	2	2	no	—
44	C₂×C₅	—	2	1	no	—
45	C₂×C₅	—	20	4	no	—
46	D₅	—	4	4	no	—
47	D₅	—	10	2	no	—
48	D₅	—	10	4	no	—
49	C₂³	—	25	4	no	—
50	D₄	—	10	2	no	—
51	D₄	—	50	2	no	—
52	D₄	—	50	8	no	—
53	C₂×C₄	—	25	2	no	—
54	C₂×C₄	—	25	1	no	—
55	C₅	—	2	2	no	—
56	C₅	—	2	1	no	—
57	C₂²	—	10	4	no	—
58	C₂²	—	25	1	no	—
59	C₂²	—	50	8	no	—
60	C₄	—	25	4	no	—
61	C₄	—	25	1	no	—
62	C₄	D₅²⋊C₂	1	1	yes	characteristic, complemented
63	C₂	—	20	4	no	—
64	C₂	—	25	2	no	—
65	C₂	C₂×D₅²⋊C₂	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×D₅²⋊C₂
commutator, cosocle	C₂×C₅⋊D₅	C₂³
Fitting	C₄×C₅²	D₄
socle	C₂×C₅²	C₂×D₄
hypercenter	C₄	D₅²⋊C₂
hypocenter	C₅²	#127
LCS	G ▷ C₂×C₅⋊D₅ ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
1. C₂×D₅²⋊C₂	C₂	semidirect
2. #87877	C₂	semidirect
3. C₄×#1044	C₂	semidirect
13. #1044	D₄	semidirect
14. C₄×C₅²	D₄	semidirect
15. #1039	D₄	semidirect
29. C₅²	#127	semidirect
62. C₄	D₅²⋊C₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₅²⋊C₂	2
#87877	2
C₄×#1044	2
#127	25