Subgroups — C₂²×#6039 #11717492

Order 800.

366 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#1039	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×Dic₅	C₂	1	2	yes	maximal
3	C₂×#6039	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₅²	C₂²	1	1	yes	characteristic
5	C₂×#1039	C₂²	1	6	yes	—
6	C₂×C₅×Dic₅	C₂²	1	12	yes	—
7	#6039	C₂²	1	16	yes	direct product factor
8	C₂²×Dic₁₀	—	5	2	no	maximal
9	C₂²×C₅²	Q₈	1	4	yes	complemented
10	C₂²×C₅²	C₂³	1	3	yes	—
11	#1039	C₂³	1	4	yes	—
12	C₅×Dic₅	C₂³	1	8	yes	—
13	C₂²×C₄×C₅	—	5	2	no	—
14	C₂²×Dic₅	—	5	2	no	—
15	C₂²×Dic₅	—	10	2	no	—
16	C₂²×Dic₅	D₅	1	2	yes	—
17	C₂×Dic₁₀	—	5	24	no	—
18	C₂×C₅²	C₂×Q₈	1	6	yes	complemented
19	C₂×C₅²	C₂⁴	1	1	yes	characteristic
20	C₂³×C₅	—	2	2	no	—
21	C₂³×C₅	C₂×D₅	1	2	yes	—
22	C₂×C₄×C₅	—	5	12	no	—
23	C₂×Dic₅	—	5	12	no	—
24	C₂×Dic₅	—	10	12	no	—
25	C₂×Dic₅	C₂×D₅	1	12	yes	—
26	Dic₁₀	—	5	32	no	—
27	C₂²×Q₈	—	25	1	no	Sylow
28	C₅²	C₂²×Q₈	1	1	yes	characteristic, complemented, Sylow
29	C₂²×C₅	—	2	8	no	—
30	C₂²×C₅	—	2	6	no	—
31	C₂²×C₅	Dic₁₀	1	8	yes	complemented
32	C₂²×C₅	C₂²×D₅	1	6	yes	—
33	C₄×C₅	—	5	8	no	—
34	Dic₅	—	5	8	no	—
35	Dic₅	—	10	8	no	—
36	Dic₅	C₂²×D₅	1	8	yes	—
37	C₂×Q₈	—	25	12	no	—
38	C₂²×C₄	—	5	2	no	—
39	C₂²×C₄	—	25	1	no	—
40	C₂×C₅	—	2	12	no	—
41	C₂×C₅	—	2	2	no	—
42	C₂×C₅	C₂×Dic₁₀	1	12	yes	complemented
43	C₂×C₅	C₂³×D₅	1	2	yes	—
44	C₂³	D₅²	1	1	yes	characteristic
45	Q₈	—	25	16	no	—
46	C₂×C₄	—	5	12	no	—
47	C₂×C₄	—	25	6	no	—
48	C₅	—	2	2	no	—
49	C₅	C₂²×Dic₁₀	1	2	yes	complemented
50	C₂²	#6039	1	4	yes	direct product factor
51	C₂²	C₂×D₅²	1	3	yes	—
52	C₄	—	5	8	no	—
53	C₄	—	25	4	no	—
54	C₂	C₂×#6039	1	6	yes	direct product factor
55	C₂	C₂²×D₅²	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³	D₅²
commutator, cosocle	C₂×C₅²	C₂⁴
Frattini	C₂	C₂²×D₅²
Fitting, socle, prime radical	C₂³×C₅²	C₂²
hypocenter	C₅²	C₂²×Q₈
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂³

Complements

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

N	H ≅ G/N	type
3. C₂×#6039	C₂	direct (N × H)
7. #6039	C₂²	direct (N × H)
9. C₂²×C₅²	Q₈	semidirect
18. C₂×C₅²	C₂×Q₈	semidirect
28. C₅²	C₂²×Q₈	semidirect
31. C₂²×C₅	Dic₁₀	semidirect
42. C₂×C₅	C₂×Dic₁₀	semidirect
49. C₅	C₂²×Dic₁₀	semidirect
50. C₂²	#6039	direct (N × H)
54. C₂	C₂×#6039	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×#1039	2
C₂²×C₅×Dic₅	2
C₂×#6039	2
C₂²×Dic₁₀	5

Subgroups — C22×#6039 #11717492

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#6039 #11717492