Subgroups — #87873

Order 400.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₅×D₅	C₂	1	2	yes	maximal, complemented
3	#6039	C₂	1	1	yes	maximal, characteristic, complemented
4	#6038	C₂	1	2	yes	maximal, complemented
5	#6037	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅⋊D₅	C₂²	1	1	yes	characteristic
7	C₂×C₅×D₅	C₂²	1	2	yes	—
8	C₄×C₅²	C₂²	1	1	yes	characteristic
9	#1039	C₂²	1	1	yes	characteristic
10	C₅×Dic₅	C₂²	1	2	yes	—
11	#709	—	5	2	no	maximal
12	C₂×C₅²	C₂³	1	1	yes	characteristic
13	C₅⋊D₅	—	2	1	no	—
14	C₅×D₅	—	2	2	no	—
15	C₂×C₄×C₅	—	5	2	no	—
16	C₂²⋊D₅	—	5	2	no	—
17	C₂²⋊D₅	—	5	2	no	—
18	D₂₀	—	5	2	no	—
19	C₄×D₅	—	5	2	no	—
20	C₄×D₅	—	10	2	no	—
21	C₄×D₅	D₅	1	2	yes	complemented
22	Dic₁₀	—	5	2	no	—
23	C₅²	#41	1	1	yes	characteristic, complemented, Sylow
24	C₂²×C₅	—	5	2	no	—
25	C₂×D₅	—	5	2	no	—
26	C₂×D₅	—	10	2	no	—
27	C₂×D₅	C₂×D₅	1	2	yes	—
28	C₄×C₅	—	2	2	no	—
29	C₄×C₅	—	5	2	no	—
30	C₄×C₅	C₂×D₅	1	2	yes	—
31	Dic₅	—	5	2	no	—
32	Dic₅	—	10	2	no	—
33	Dic₅	C₂×D₅	1	2	yes	—
34	#41	—	25	1	no	Sylow
35	C₂×C₅	—	2	2	no	—
36	C₂×C₅	—	10	2	no	—
37	C₂×C₅	C₂²×D₅	1	2	yes	—
38	D₅	—	2	2	no	—
39	D₅	—	10	4	no	—
40	D₅	—	10	2	no	—
41	Q₈	—	25	1	no	—
42	D₄	—	25	2	no	—
43	D₄	—	25	1	no	—
44	C₂×C₄	—	5	2	no	—
45	C₂×C₄	—	25	1	no	—
46	C₅	—	2	2	no	—
47	C₅	#709	1	2	yes	complemented
48	C₂²	—	5	2	no	—
49	C₂²	—	25	1	no	—
50	C₄	—	5	2	no	—
51	C₄	—	25	1	no	—
52	C₄	D₅²	1	1	yes	characteristic
53	C₂	—	10	2	no	—
54	C₂	—	50	1	no	—
55	C₂	C₂×D₅²	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₄	D₅²
commutator, socle, cosocle	C₂×C₅²	C₂³
Frattini	C₂	C₂×D₅²
Fitting	C₄×C₅²	C₂²
hypocenter	C₅²	#41
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₄

Complements

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

N	H ≅ G/N	type
1. C₄×C₅⋊D₅	C₂	semidirect
2. C₄×C₅×D₅	C₂	semidirect
3. #6039	C₂	semidirect
4. #6038	C₂	semidirect
5. #6037	C₂	semidirect
21. C₄×D₅	D₅	semidirect
23. C₅²	#41	semidirect
47. C₅	#709	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅⋊D₅	2
C₄×C₅×D₅	2
#6039	2
#6038	2
#6037	2
#709	5