Subgroups — #87886

Order 400.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₅²	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×C₂²⋊D₅	C₂	1	2	yes	maximal, complemented
4	#6038	C₂	1	2	yes	maximal, complemented
5	#6036	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₅²	C₂²	1	1	yes	characteristic, complemented
7	C₂×C₅⋊D₅	—	2	2	no	—
8	C₂×C₅⋊D₅	C₂²	1	1	yes	characteristic
9	C₂×C₅⋊D₅	C₂²	1	1	yes	characteristic, complemented
10	C₂×C₅×D₅	C₂²	1	2	yes	complemented
11	D₅²	—	2	2	no	—
12	C₅×Dic₅	C₂²	1	2	yes	complemented
13	D₄×D₅	—	5	2	no	maximal
14	C₂×C₅²	—	2	1	no	—
15	C₂×C₅²	C₂³	1	1	yes	characteristic
16	C₅⋊D₅	—	2	1	no	—
17	C₅⋊D₅	D₄	1	2	yes	complemented
18	C₅×D₅	—	2	2	no	—
19	C₂²×D₅	—	5	2	no	—
20	C₂²×D₅	—	5	2	no	—
21	C₂²×D₅	—	10	2	no	—
22	C₅×D₄	—	5	2	no	—
23	C₂²⋊D₅	—	5	2	no	—
24	C₂²⋊D₅	D₅	1	2	yes	complemented
25	D₂₀	—	5	2	no	—
26	C₄×D₅	—	5	2	no	—
27	C₅²	C₂×D₄	1	1	yes	characteristic, complemented, Sylow
28	C₂²×C₅	—	2	2	no	—
29	C₂²×C₅	—	5	2	no	—
30	C₂²×C₅	C₂×D₅	1	2	yes	complemented
31	C₂×D₅	—	5	2	no	—
32	C₂×D₅	—	5	2	no	—
33	C₂×D₅	—	10	2	no	—
34	C₂×D₅	—	10	2	no	—
35	C₂×D₅	—	10	4	no	—
36	C₂×D₅	—	10	4	no	—
37	C₂×D₅	—	10	8	no	—
38	C₂×D₅	C₂×D₅	1	2	yes	complemented
39	C₄×C₅	—	5	2	no	—
40	Dic₅	C₂×D₅	1	2	yes	complemented
41	C₂×D₄	—	25	1	no	Sylow
42	C₂×C₅	—	2	4	no	—
43	C₂×C₅	—	2	2	no	—
44	C₂×C₅	—	2	2	no	—
45	C₂×C₅	—	10	2	no	—
46	C₂×C₅	C₂²×D₅	1	2	yes	—
47	D₅	—	2	2	no	—
48	D₅	—	5	4	no	—
49	D₅	—	10	4	no	—
50	D₅	—	10	4	no	—
51	D₅	—	10	2	no	—
52	C₂³	—	25	1	no	—
53	C₂³	—	25	1	no	—
54	D₄	—	5	2	no	—
55	D₄	—	25	2	no	—
56	C₂×C₄	—	25	1	no	—
57	C₅	—	2	2	no	—
58	C₅	D₄×D₅	1	2	yes	complemented
59	C₂²	—	5	2	no	—
60	C₂²	—	25	1	no	—
61	C₂²	—	25	1	no	—
62	C₂²	—	50	2	no	—
63	C₂²	—	50	2	no	—
64	C₂²	D₅²	1	1	yes	characteristic, complemented
65	C₄	—	5	2	no	—
66	C₂	—	2	1	no	—
67	C₂	—	10	2	no	—
68	C₂	—	25	2	no	—
69	C₂	—	50	1	no	—
70	C₂	C₂×D₅²	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×D₅²
commutator, socle, cosocle	C₂×C₅²	C₂³
Fitting	C₂²×C₅²	C₂²
hypercenter	C₂²	D₅²
hypocenter	C₅²	C₂×D₄
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂²

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅⋊D₅	C₂	semidirect
2. C₂×D₅²	C₂	semidirect
3. C₅×C₂²⋊D₅	C₂	semidirect
4. #6038	C₂	semidirect
5. #6036	C₂	semidirect
6. C₂²×C₅²	C₂²	semidirect
9. C₂×C₅⋊D₅	C₂²	semidirect
10. C₂×C₅×D₅	C₂²	semidirect
12. C₅×Dic₅	C₂²	semidirect
17. C₅⋊D₅	D₄	semidirect
24. C₂²⋊D₅	D₅	semidirect
27. C₅²	C₂×D₄	semidirect
30. C₂²×C₅	C₂×D₅	semidirect
38. C₂×D₅	C₂×D₅	semidirect
40. Dic₅	C₂×D₅	semidirect
58. C₅	D₄×D₅	semidirect
64. C₂²	D₅²	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₅⋊D₅	2
C₂×D₅²	2
C₅×C₂²⋊D₅	2
#6038	2
#6036	2
D₄×D₅	5