Subgroups — Dic₄₁₂₅ #72215474

Order 16500.

48 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Dic₁₃₇₅	—	3	1	no	maximal, Hall
2	C₂×C₃×C₁₁×C₁₂₅	C₂	1	1	yes	maximal, characteristic
3	C₃×C₁₁×C₁₂₅	C₄	1	1	yes	characteristic, complemented, Hall
4	C₂×C₁₁×C₁₂₅	D₃	1	1	yes	characteristic
5	Dic₈₂₅	—	5	1	no	maximal
6	C₂×C₃×C₁₁×C₂₅	D₅	1	1	yes	characteristic
7	Dic₃₇₅	—	11	1	no	maximal, Hall
8	C₁₁×C₁₂₅	Dic₃	1	1	yes	characteristic, complemented, Hall
9	Dic₂₇₅	—	15	1	no	—
10	C₃×C₁₁×C₂₅	Dic₅	1	1	yes	characteristic
11	C₂×C₃×C₁₂₅	D₁₁	1	1	yes	characteristic
12	Dic₁₆₅	—	25	1	no	—
13	C₂×C₁₁×C₂₅	D₁₅	1	1	yes	characteristic
14	Dic₁₂₅	—	33	1	no	Hall
15	C₃×C₁₂₅	Dic₁₁	1	1	yes	characteristic, complemented, Hall
16	C₂×C₃×C₅×C₁₁	D₂₅	1	1	yes	characteristic
17	Dic₇₅	—	55	1	no	—
18	C₁₁×C₂₅	Dic₁₅	1	1	yes	characteristic
19	C₂×C₁₂₅	D₃₃	1	1	yes	characteristic
20	Dic₅₅	—	75	1	no	—
21	C₃×C₅×C₁₁	Dic₂₅	1	1	yes	characteristic
22	C₂×C₃×C₂₅	D₅₅	1	1	yes	characteristic
23	Dic₃₃	—	125	1	no	Hall
24	C₁₂₅	Dic₃₃	1	1	yes	characteristic, complemented, Sylow
25	C₂×C₅×C₁₁	D₇₅	1	1	yes	characteristic
26	Dic₂₅	—	165	1	no	—
27	C₃×C₂₅	Dic₅₅	1	1	yes	characteristic
28	C₂×C₃×C₁₁	D₁₂₅	1	1	yes	characteristic
29	Dic₁₅	—	275	1	no	—
30	C₅×C₁₁	Dic₇₅	1	1	yes	characteristic
31	C₂×C₂₅	D₁₆₅	1	1	yes	characteristic
32	Dic₁₁	—	375	1	no	Hall
33	C₃×C₁₁	Dic₁₂₅	1	1	yes	characteristic, complemented, Hall
34	C₂×C₃×C₅	D₂₇₅	1	1	yes	characteristic
35	C₂₅	Dic₁₆₅	1	1	yes	characteristic
36	C₂×C₁₁	D₃₇₅	1	1	yes	characteristic
37	Dic₅	—	825	1	no	—
38	C₃×C₅	Dic₂₇₅	1	1	yes	characteristic
39	Dic₃	—	1375	1	no	Hall
40	C₁₁	Dic₃₇₅	1	1	yes	characteristic, complemented, Sylow
41	C₂×C₅	D₈₂₅	1	1	yes	characteristic
42	C₂×C₃	D₁₃₇₅	1	1	yes	characteristic
43	C₅	Dic₈₂₅	1	1	yes	characteristic
44	C₄	—	4125	1	no	Sylow
45	C₃	Dic₁₃₇₅	1	1	yes	characteristic, complemented, Sylow
46	C₂	D₄₁₂₅	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂	D₄₁₂₅
commutator, hypocenter	C₃×C₁₁×C₁₂₅	C₄
Frattini	C₂×C₂₅	D₁₆₅
Fitting, cosocle	C₂×C₃×C₁₁×C₁₂₅	C₂
socle, prime radical	C₂×C₃×C₅×C₁₁	D₂₅
LCS	G ▷ C₃×C₁₁×C₁₂₅
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
3. C₃×C₁₁×C₁₂₅	C₄	semidirect
8. C₁₁×C₁₂₅	Dic₃	semidirect
15. C₃×C₁₂₅	Dic₁₁	semidirect
24. C₁₂₅	Dic₃₃	semidirect
33. C₃×C₁₁	Dic₁₂₅	semidirect
40. C₁₁	Dic₃₇₅	semidirect
45. C₃	Dic₁₃₇₅	semidirect

Maximal subgroups

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

subgroup	index
Dic₁₃₇₅	3
C₂×C₃×C₁₁×C₁₂₅	2
Dic₈₂₅	5
Dic₃₇₅	11

Subgroups — Dic4125 #72215474

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₄₁₂₅ #72215474