Subgroups — #87778

Order 400.

84 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
2	C₂×#1039	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₅²	C₂²	1	1	yes	characteristic
5	C₂×C₅×D₅	—	2	1	no	—
6	C₂×C₅×D₅	—	2	1	no	—
7	C₂×C₅×D₅	C₄	1	2	yes	complemented
8	#1039	—	2	1	no	—
9	C₅×Dic₅	—	2	1	no	—
10	C₂²⋊Dic₅	—	5	1	no	maximal
11	#685	—	5	1	no	maximal
12	C₂×C₅²	C₂×C₄	1	1	yes	characteristic
13	C₂×C₅²	D₄	1	1	yes	characteristic
14	C₂×C₅²	D₄	1	1	yes	characteristic
15	C₅×D₅	—	2	2	no	—
16	C₂³×C₅	—	5	1	no	—
17	C₂²×D₅	D₅	1	1	yes	characteristic
18	C₂×C₄×C₅	—	5	1	no	—
19	C₂×Dic₅	—	5	1	no	—
20	C₂×Dic₅	—	5	1	no	—
21	C₂×Dic₅	—	10	2	no	—
22	C₂×Dic₅	D₅	1	1	yes	characteristic, complemented
23	C₅²	C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
24	C₂²×C₅	—	2	2	no	—
25	C₂²×C₅	—	5	2	no	—
26	C₂²×C₅	—	10	1	no	—
27	C₂²×C₅	—	10	1	no	—
28	C₂²×C₅	C₂×D₅	1	1	yes	characteristic
29	C₂²×C₅	C₂×D₅	1	1	yes	characteristic
30	C₂×D₅	—	2	1	no	—
31	C₂×D₅	—	2	1	no	—
32	C₂×D₅	Dic₅	1	2	yes	complemented
33	C₄×C₅	—	10	1	no	—
34	Dic₅	—	2	1	no	—
35	Dic₅	—	10	4	no	—
36	Dic₅	—	10	1	no	—
37	Dic₅	—	10	1	no	—
38	C₂²⋊C₄	—	25	1	no	Sylow
39	C₂×C₅	—	2	2	no	—
40	C₂×C₅	—	2	2	no	—
41	C₂×C₅	—	2	2	no	—
42	C₂×C₅	—	10	2	no	—
43	C₂×C₅	C₄×D₅	1	1	yes	characteristic
44	C₂×C₅	D₂₀	1	1	yes	characteristic
45	C₂×C₅	C₂×Dic₅	1	1	yes	characteristic
46	C₂×C₅	C₂²⋊D₅	1	1	yes	characteristic
47	C₂×C₅	C₂²⋊D₅	1	1	yes	characteristic
48	C₂×C₅	C₂²⋊D₅	1	1	yes	characteristic
49	D₅	—	2	2	no	—
50	C₂³	—	5	1	no	—
51	C₂×C₄	—	5	1	no	—
52	C₂×C₄	—	25	1	no	—
53	C₅	—	2	2	no	—
54	C₅	#685	1	1	yes	characteristic, complemented
55	C₅	C₂²⋊Dic₅	1	1	yes	characteristic, complemented
56	C₂²	—	5	2	no	—
57	C₂²	—	10	1	no	—
58	C₂²	—	10	1	no	—
59	C₂²	D₅²	1	1	yes	characteristic
60	C₄	—	10	1	no	—
61	C₄	—	50	1	no	—
62	C₂	—	10	2	no	—
63	C₂	D₅×Dic₅	1	1	yes	characteristic
64	C₂	#6037	1	1	yes	characteristic
65	C₂	#6038	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂²	D₅²
commutator	C₂×C₅²	C₂×C₄
Fitting, socle, cosocle	C₂²×C₅²	C₂²
prime radical	C₂²×C₅×D₅	C₂
hypocenter	C₅²	C₂²⋊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
2. C₂×#1039	C₂	semidirect
3. C₂×C₅×Dic₅	C₂	semidirect
7. C₂×C₅×D₅	C₄	semidirect
22. C₂×Dic₅	D₅	semidirect
23. C₅²	C₂²⋊C₄	semidirect
32. C₂×D₅	Dic₅	semidirect
54. C₅	#685	semidirect
55. C₅	C₂²⋊Dic₅	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×#1039	2
C₂×C₅×Dic₅	2
C₂²⋊Dic₅	5
#685	5