Subgroups — C₂×Dic₅² #11717072

Order 800.

270 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	Dic₅²	C₂	1	4	yes	maximal, direct product factor
4	C₂³×C₅²	C₂²	1	1	yes	characteristic
5	C₂×#1039	C₄	1	4	yes	complemented
6	C₂×#1039	C₂²	1	2	yes	—
7	C₂×C₅×Dic₅	C₄	1	8	yes	complemented
8	C₂×C₅×Dic₅	C₂²	1	4	yes	—
9	C₂×C₄×Dic₅	—	5	2	no	maximal
10	C₂²×C₅²	C₂×C₄	1	2	yes	—
11	C₂²×C₅²	C₂×C₄	1	4	yes	—
12	C₂²×C₅²	C₂³	1	1	yes	characteristic
13	#1039	C₂×C₄	1	4	yes	complemented
14	C₅×Dic₅	C₂×C₄	1	8	yes	complemented
15	C₂²×C₄×C₅	—	5	2	no	—
16	C₂²×Dic₅	—	5	2	no	—
17	C₂²×Dic₅	—	10	2	no	—
18	C₂²×Dic₅	D₅	1	2	yes	—
19	C₄×Dic₅	—	5	8	no	—
20	C₂×C₅²	C₄²	1	4	yes	complemented
21	C₂×C₅²	C₂²×C₄	1	1	yes	characteristic
22	C₂×C₅²	C₂²×C₄	1	2	yes	—
23	C₂³×C₅	—	2	2	no	—
24	C₂³×C₅	C₂×D₅	1	2	yes	—
25	C₂×C₄×C₅	—	5	8	no	—
26	C₂×C₄×C₅	—	5	4	no	—
27	C₂×Dic₅	—	5	8	no	—
28	C₂×Dic₅	—	5	4	no	—
29	C₂×Dic₅	—	10	8	no	—
30	C₂×Dic₅	—	10	4	no	—
31	C₂×Dic₅	Dic₅	1	8	yes	direct product factor
32	C₂×Dic₅	C₂×D₅	1	4	yes	—
33	C₂×C₄²	—	25	1	no	Sylow
34	C₅²	C₂×C₄²	1	1	yes	characteristic, complemented, Sylow
35	C₂²×C₅	—	2	8	no	—
36	C₂²×C₅	—	2	4	no	—
37	C₂²×C₅	—	2	2	no	—
38	C₂²×C₅	C₄×D₅	1	4	yes	—
39	C₂²×C₅	C₄×D₅	1	4	yes	—
40	C₂²×C₅	C₂×Dic₅	1	4	yes	—
41	C₂²×C₅	C₂²×D₅	1	2	yes	—
42	C₄×C₅	—	5	8	no	—
43	Dic₅	—	5	8	no	—
44	Dic₅	—	10	8	no	—
45	Dic₅	C₂×Dic₅	1	8	yes	direct product factor
46	C₂²×C₄	—	5	2	no	—
47	C₂²×C₄	—	25	1	no	—
48	C₄²	—	25	4	no	—
49	C₂×C₅	—	2	4	no	—
50	C₂×C₅	—	2	2	no	—
51	C₂×C₅	—	2	8	no	—
52	C₂×C₅	C₄×Dic₅	1	8	yes	complemented
53	C₂×C₅	C₂×C₄×D₅	1	2	yes	—
54	C₂×C₅	C₂×C₄×D₅	1	2	yes	—
55	C₂×C₅	C₂²×Dic₅	1	2	yes	—
56	C₂³	D₅²	1	1	yes	characteristic
57	C₂×C₄	—	5	8	no	—
58	C₂×C₄	—	5	4	no	—
59	C₂×C₄	—	25	4	no	—
60	C₂×C₄	—	25	2	no	—
61	C₅	—	2	2	no	—
62	C₅	C₂×C₄×Dic₅	1	2	yes	complemented
63	C₂²	D₅×Dic₅	1	4	yes	—
64	C₂²	#6036	1	2	yes	—
65	C₂²	C₂×D₅²	1	1	yes	characteristic
66	C₄	—	5	8	no	—
67	C₄	—	25	4	no	—
68	C₂	Dic₅²	1	4	yes	direct product factor
69	C₂	C₂×D₅×Dic₅	1	2	yes	—
70	C₂	C₂×#6036	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³	D₅²
commutator, hypocenter	C₅²	C₂×C₄²
Frattini	C₂²	C₂×D₅²
Fitting, socle, prime radical	C₂³×C₅²	C₂²
cosocle	C₂²×C₅²	C₂³
LCS	G ▷ C₅²
UCS	C₁ ◁ C₂³

Complements

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

N	H ≅ G/N	type
3. Dic₅²	C₂	direct (N × H)
5. C₂×#1039	C₄	semidirect
7. C₂×C₅×Dic₅	C₄	semidirect
13. #1039	C₂×C₄	semidirect
14. C₅×Dic₅	C₂×C₄	semidirect
20. C₂×C₅²	C₄²	semidirect
31. C₂×Dic₅	Dic₅	direct (N × H)
34. C₅²	C₂×C₄²	semidirect
45. Dic₅	C₂×Dic₅	direct (N × H)
52. C₂×C₅	C₄×Dic₅	semidirect
62. C₅	C₂×C₄×Dic₅	semidirect
68. C₂	Dic₅²	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
Dic₅²	2
C₂×C₄×Dic₅	5

Subgroups — C2×Dic52 #11717072

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Dic₅² #11717072