Subgroups — #87780

Order 400.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#1039	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×Dic₅	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×Dic₅	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₅²	C₂²	1	1	yes	characteristic
5	#1039	—	2	1	no	—
6	C₅×Dic₅	—	2	1	no	—
7	C₅×Dic₅	C₄	1	2	yes	complemented
8	C₄⋊Dic₅	—	5	1	no	maximal
9	#683	—	5	1	no	maximal
10	C₂×C₅²	C₂×C₄	1	1	yes	characteristic
11	C₂×C₅²	D₄	1	1	yes	characteristic
12	C₂×C₅²	Q₈	1	1	yes	characteristic
13	C₂×C₄×C₅	—	5	1	no	—
14	C₂×C₄×C₅	—	5	1	no	—
15	C₂×Dic₅	—	5	1	no	—
16	C₂×Dic₅	—	5	1	no	—
17	C₂×Dic₅	—	10	2	no	—
18	C₂×Dic₅	D₅	1	1	yes	characteristic
19	C₂×Dic₅	D₅	1	1	yes	characteristic
20	C₅²	C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
21	C₂²×C₅	—	2	2	no	—
22	C₂²×C₅	C₂×D₅	1	1	yes	characteristic
23	C₂²×C₅	C₂×D₅	1	1	yes	characteristic
24	C₄×C₅	—	5	2	no	—
25	C₄×C₅	—	10	1	no	—
26	Dic₅	—	2	1	no	—
27	Dic₅	—	10	4	no	—
28	Dic₅	—	10	1	no	—
29	Dic₅	—	10	1	no	—
30	Dic₅	Dic₅	1	2	yes	complemented
31	C₄⋊C₄	—	25	1	no	Sylow
32	C₂×C₅	—	2	2	no	—
33	C₂×C₅	—	2	2	no	—
34	C₂×C₅	—	2	2	no	—
35	C₂×C₅	Dic₁₀	1	1	yes	characteristic
36	C₂×C₅	Dic₁₀	1	1	yes	characteristic
37	C₂×C₅	C₄×D₅	1	1	yes	characteristic
38	C₂×C₅	D₂₀	1	1	yes	characteristic
39	C₂×C₅	C₂×Dic₅	1	1	yes	characteristic
40	C₂×C₅	C₂²⋊D₅	1	1	yes	characteristic
41	C₂×C₄	—	5	1	no	—
42	C₂×C₄	—	5	1	no	—
43	C₂×C₄	—	25	1	no	—
44	C₅	—	2	2	no	—
45	C₅	#683	1	1	yes	characteristic, complemented
46	C₅	C₄⋊Dic₅	1	1	yes	characteristic, complemented
47	C₂²	D₅²	1	1	yes	characteristic
48	C₄	—	5	2	no	—
49	C₄	—	10	1	no	—
50	C₄	—	50	1	no	—
51	C₂	D₅×Dic₅	1	1	yes	characteristic
52	C₂	#6038	1	1	yes	characteristic
53	C₂	#6039	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂²	D₅²
commutator	C₂×C₅²	C₂×C₄
Fitting, socle, prime radical, cosocle	C₂²×C₅²	C₂²
hypocenter	C₅²	C₄⋊C₄
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂²

Complements

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

N	H ≅ G/N	type
7. C₅×Dic₅	C₄	semidirect
20. C₅²	C₄⋊C₄	semidirect
30. Dic₅	Dic₅	semidirect
45. C₅	#683	semidirect
46. C₅	C₄⋊Dic₅	semidirect

Maximal subgroups

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

subgroup	index
C₂×#1039	2
C₂×C₅×Dic₅	2
C₄⋊Dic₅	5
#683	5