Subgroups — C₂×C₃×Dic₅₀ #11924939

Order 1200.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₂₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×Dic₂₅	C₂	1	2	yes	maximal
3	C₃×Dic₅₀	C₂	1	4	yes	maximal, direct product factor
4	C₂×Dic₅₀	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂²×C₃×C₂₅	C₂²	1	1	yes	characteristic
6	C₃×C₄×C₂₅	C₂²	1	2	yes	—
7	C₃×Dic₂₅	C₂²	1	4	yes	—
8	C₂×C₃×Dic₁₀	—	5	1	no	maximal
9	C₂×C₄×C₂₅	C₂×C₃	1	1	yes	characteristic
10	C₂×Dic₂₅	C₂×C₃	1	2	yes	—
11	Dic₅₀	C₂×C₃	1	4	yes	direct product factor
12	C₂×C₃×C₂₅	Q₈	1	2	yes	complemented
13	C₂×C₃×C₂₅	C₂³	1	1	yes	characteristic
14	C₂×C₃×C₄×C₅	D₅	1	1	yes	characteristic
15	C₂×C₃×Dic₅	—	5	2	no	—
16	C₃×Dic₁₀	—	5	4	no	—
17	C₂²×C₂₅	C₂²×C₃	1	1	yes	characteristic
18	C₄×C₂₅	C₂²×C₃	1	2	yes	—
19	Dic₂₅	C₂²×C₃	1	4	yes	—
20	C₂×Dic₁₀	—	5	1	no	—
21	C₃×C₂₅	C₂×Q₈	1	1	yes	characteristic, complemented, Hall
22	C₂²×C₃×C₅	C₂×D₅	1	1	yes	characteristic
23	C₃×C₄×C₅	C₂×D₅	1	2	yes	—
24	C₃×Dic₅	—	5	4	no	—
25	C₂×C₂₅	C₃×Q₈	1	2	yes	complemented
26	C₂×C₂₅	C₂³×C₃	1	1	yes	characteristic
27	C₂×C₃×Q₈	—	25	1	no	Hall
28	C₂×C₄×C₅	C₃×D₅	1	1	yes	characteristic
29	C₂×Dic₅	—	5	2	no	—
30	Dic₁₀	—	5	4	no	—
31	C₂×C₃×C₅	Dic₁₀	1	2	yes	—
32	C₂×C₃×C₅	C₂²×D₅	1	1	yes	characteristic
33	C₂₅	C₂×C₃×Q₈	1	1	yes	characteristic, complemented, Sylow
34	C₃×Q₈	—	25	4	no	—
35	C₂×C₃×C₄	—	25	2	no	—
36	C₂×C₃×C₄	D₂₅	1	1	yes	characteristic
37	C₂²×C₅	C₂×C₃×D₅	1	1	yes	characteristic
38	C₄×C₅	C₂×C₃×D₅	1	2	yes	—
39	Dic₅	—	5	4	no	—
40	C₂×Q₈	—	25	1	no	Sylow
41	C₃×C₅	C₂×Dic₁₀	1	1	yes	characteristic
42	C₂²×C₃	C₂×D₂₅	1	1	yes	characteristic
43	C₃×C₄	—	25	4	no	—
44	C₃×C₄	C₂×D₂₅	1	2	yes	—
45	C₂×C₅	C₃×Dic₁₀	1	2	yes	—
46	C₂×C₅	C₂²×C₃×D₅	1	1	yes	characteristic
47	Q₈	—	25	4	no	—
48	C₂×C₄	—	25	2	no	—
49	C₂×C₄	C₃×D₂₅	1	1	yes	characteristic
50	C₂×C₃	Dic₅₀	1	2	yes	direct product factor
51	C₂×C₃	C₂²×D₂₅	1	1	yes	characteristic
52	C₅	C₂×C₃×Dic₁₀	1	1	yes	characteristic
53	C₂²	C₂×C₃×D₂₅	1	1	yes	characteristic
54	C₄	—	25	4	no	—
55	C₄	C₂×C₃×D₂₅	1	2	yes	—
56	C₃	C₂×Dic₅₀	1	1	yes	characteristic, direct product factor, Sylow
57	C₂	C₃×Dic₅₀	1	2	yes	direct product factor
58	C₂	C₂²×C₃×D₂₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×D₂₅
commutator, cosocle	C₂×C₂₅	C₂³×C₃
Frattini	C₂×C₅	C₂²×C₃×D₅
Fitting	C₂×C₃×C₄×C₂₅	C₂
socle, prime radical	C₂²×C₃×C₅	C₂×D₅
hypercenter	C₂×C₃×C₄	D₂₅
hypocenter	C₂₅	C₂×C₃×Q₈
LCS	G ▷ C₂×C₂₅ ▷ C₂₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄

Complements

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

N	H ≅ G/N	type
3. C₃×Dic₅₀	C₂	direct (N × H)
4. C₂×Dic₅₀	C₃	direct (N × H)
11. Dic₅₀	C₂×C₃	direct (N × H)
12. C₂×C₃×C₂₅	Q₈	semidirect
21. C₃×C₂₅	C₂×Q₈	semidirect
25. C₂×C₂₅	C₃×Q₈	semidirect
33. C₂₅	C₂×C₃×Q₈	semidirect
50. C₂×C₃	Dic₅₀	direct (N × H)
56. C₃	C₂×Dic₅₀	direct (N × H)
57. C₂	C₃×Dic₅₀	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×C₂₅	2
C₂×C₃×Dic₂₅	2
C₃×Dic₅₀	2
C₂×Dic₅₀	3
C₂×C₃×Dic₁₀	5

Subgroups — C2×C3×Dic50 #11924939

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×Dic₅₀ #11924939