Subgroups — C₂²×#1042 #87921

Order 400.

113 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×#1042	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₅²	C₄	1	1	yes	characteristic, complemented
4	C₂×C₅×D₅	C₄	1	3	yes	complemented
5	C₂×C₅×D₅	C₂²	1	3	yes	—
6	#1042	C₂²	1	4	yes	direct product factor
7	C₂²×AGL(1,𝔽₅)	—	5	1	no	maximal
8	C₂²×Dic₅	—	5	1	no	maximal
9	C₂×C₅²	C₂×C₄	1	3	yes	complemented
10	C₅×D₅	C₂×C₄	1	3	yes	complemented
11	C₅×D₅	C₂³	1	1	yes	characteristic
12	C₂³×C₅	—	5	1	no	—
13	C₂²×D₅	D₅	1	1	yes	characteristic
14	C₂×AGL(1,𝔽₅)	—	5	6	no	—
15	C₂×Dic₅	—	5	6	no	—
16	C₅²	C₂²×C₄	1	1	yes	characteristic, complemented, Sylow
17	C₂²×C₅	—	4	1	no	—
18	C₂²×C₅	—	5	3	no	—
19	C₂²×C₅	—	5	3	no	—
20	C₂²×C₅	Dic₅	1	1	yes	characteristic, complemented
21	C₂²×C₅	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
22	C₂×D₅	Dic₅	1	3	yes	complemented
23	C₂×D₅	C₂×D₅	1	3	yes	—
24	AGL(1,𝔽₅)	—	5	4	no	—
25	Dic₅	—	5	4	no	—
26	C₂²×C₄	—	25	1	no	Sylow
27	C₂×C₅	—	4	3	no	—
28	C₂×C₅	—	5	1	no	—
29	C₂×C₅	—	5	3	no	—
30	C₂×C₅	C₂×Dic₅	1	3	yes	complemented
31	C₂×C₅	C₂×AGL(1,𝔽₅)	1	3	yes	complemented
32	D₅	C₂×Dic₅	1	3	yes	complemented
33	D₅	C₂²×D₅	1	1	yes	characteristic
34	C₂³	—	5	1	no	—
35	C₂×C₄	—	25	6	no	—
36	C₅	—	4	1	no	—
37	C₅	C₂²×Dic₅	1	1	yes	characteristic, complemented
38	C₅	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
39	C₂²	—	5	3	no	—
40	C₂²	—	5	3	no	—
41	C₂²	#1042	1	1	yes	characteristic, direct product factor
42	C₄	—	25	4	no	—
43	C₂	—	5	1	no	—
44	C₂	—	5	3	no	—
45	C₂	C₂×#1042	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²	#1042
commutator, hypocenter	C₅²	C₂²×C₄
Frattini	C₁	G
Fitting, socle	C₂²×C₅²	C₄
prime radical	C₂²×C₅×D₅	C₂
cosocle	C₅×D₅	C₂³
LCS	G ▷ C₅²
UCS	C₁ ◁ C₂²

Complements

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

N	H ≅ G/N	type
2. C₂×#1042	C₂	direct (N × H)
3. C₂²×C₅²	C₄	semidirect
4. C₂×C₅×D₅	C₄	semidirect
6. #1042	C₂²	direct (N × H)
9. C₂×C₅²	C₂×C₄	semidirect
10. C₅×D₅	C₂×C₄	semidirect
16. C₅²	C₂²×C₄	semidirect
20. C₂²×C₅	Dic₅	semidirect
21. C₂²×C₅	AGL(1,𝔽₅)	semidirect
22. C₂×D₅	Dic₅	semidirect
30. C₂×C₅	C₂×Dic₅	semidirect
31. C₂×C₅	C₂×AGL(1,𝔽₅)	semidirect
32. D₅	C₂×Dic₅	semidirect
37. C₅	C₂²×Dic₅	semidirect
38. C₅	C₂²×AGL(1,𝔽₅)	semidirect
41. C₂²	#1042	direct (N × H)
45. C₂	C₂×#1042	direct (N × H)

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×#1042	2
C₂²×AGL(1,𝔽₅)	5
C₂²×Dic₅	5

Subgroups — C22×#1042 #87921

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#1042 #87921