Subgroups — Dic₅₈₆₅ #37553289

Order 23460.

48 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×C₁₇×C₂₃	C₂	1	1	yes	maximal, characteristic
2	Dic₁₉₅₅	—	3	1	no	maximal, Hall
3	C₃×C₅×C₁₇×C₂₃	C₄	1	1	yes	characteristic, complemented, Hall
4	Dic₁₁₇₃	—	5	1	no	maximal, Hall
5	C₂×C₅×C₁₇×C₂₃	D₃	1	1	yes	characteristic
6	C₂×C₃×C₁₇×C₂₃	D₅	1	1	yes	characteristic
7	C₅×C₁₇×C₂₃	Dic₃	1	1	yes	characteristic, complemented, Hall
8	Dic₃₉₁	—	15	1	no	Hall
9	Dic₃₄₅	—	17	1	no	maximal, Hall
10	C₃×C₁₇×C₂₃	Dic₅	1	1	yes	characteristic, complemented, Hall
11	Dic₂₅₅	—	23	1	no	maximal, Hall
12	C₂×C₁₇×C₂₃	D₁₅	1	1	yes	characteristic
13	C₂×C₃×C₅×C₂₃	D₁₇	1	1	yes	characteristic
14	C₂×C₃×C₅×C₁₇	D₂₃	1	1	yes	characteristic
15	Dic₁₁₅	—	51	1	no	Hall
16	C₁₇×C₂₃	Dic₁₅	1	1	yes	characteristic, complemented, Hall
17	C₃×C₅×C₂₃	Dic₁₇	1	1	yes	characteristic, complemented, Hall
18	Dic₈₅	—	69	1	no	Hall
19	Dic₆₉	—	85	1	no	Hall
20	C₃×C₅×C₁₇	Dic₂₃	1	1	yes	characteristic, complemented, Hall
21	C₂×C₅×C₂₃	D₅₁	1	1	yes	characteristic
22	Dic₅₁	—	115	1	no	Hall
23	C₂×C₅×C₁₇	D₆₉	1	1	yes	characteristic
24	C₂×C₃×C₂₃	D₈₅	1	1	yes	characteristic
25	C₅×C₂₃	Dic₅₁	1	1	yes	characteristic, complemented, Hall
26	C₂×C₃×C₁₇	D₁₁₅	1	1	yes	characteristic
27	Dic₂₃	—	255	1	no	Hall
28	C₅×C₁₇	Dic₆₉	1	1	yes	characteristic, complemented, Hall
29	C₃×C₂₃	Dic₈₅	1	1	yes	characteristic, complemented, Hall
30	Dic₁₇	—	345	1	no	Hall
31	Dic₁₅	—	391	1	no	Hall
32	C₃×C₁₇	Dic₁₁₅	1	1	yes	characteristic, complemented, Hall
33	C₂×C₂₃	D₂₅₅	1	1	yes	characteristic
34	C₂×C₁₇	D₃₄₅	1	1	yes	characteristic
35	C₂×C₃×C₅	D₃₉₁	1	1	yes	characteristic
36	C₂₃	Dic₂₅₅	1	1	yes	characteristic, complemented, Sylow
37	Dic₅	—	1173	1	no	Hall
38	C₁₇	Dic₃₄₅	1	1	yes	characteristic, complemented, Sylow
39	C₃×C₅	Dic₃₉₁	1	1	yes	characteristic, complemented, Hall
40	Dic₃	—	1955	1	no	Hall
41	C₂×C₅	D₁₁₇₃	1	1	yes	characteristic
42	C₂×C₃	D₁₉₅₅	1	1	yes	characteristic
43	C₅	Dic₁₁₇₃	1	1	yes	characteristic, complemented, Sylow
44	C₄	—	5865	1	no	Sylow
45	C₃	Dic₁₉₅₅	1	1	yes	characteristic, complemented, Sylow
46	C₂	D₅₈₆₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅₈₆₅
commutator, hypocenter	C₃×C₅×C₁₇×C₂₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₅×C₁₇×C₂₃	C₂
LCS	G ▷ C₃×C₅×C₁₇×C₂₃
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×C₁₇×C₂₃	C₄	semidirect
7. C₅×C₁₇×C₂₃	Dic₃	semidirect
10. C₃×C₁₇×C₂₃	Dic₅	semidirect
16. C₁₇×C₂₃	Dic₁₅	semidirect
17. C₃×C₅×C₂₃	Dic₁₇	semidirect
20. C₃×C₅×C₁₇	Dic₂₃	semidirect
25. C₅×C₂₃	Dic₅₁	semidirect
28. C₅×C₁₇	Dic₆₉	semidirect
29. C₃×C₂₃	Dic₈₅	semidirect
32. C₃×C₁₇	Dic₁₁₅	semidirect
36. C₂₃	Dic₂₅₅	semidirect
38. C₁₇	Dic₃₄₅	semidirect
39. C₃×C₅	Dic₃₉₁	semidirect
43. C₅	Dic₁₁₇₃	semidirect
45. C₃	Dic₁₉₅₅	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×C₁₇×C₂₃	2
Dic₁₉₅₅	3
Dic₁₁₇₃	5
Dic₃₄₅	17
Dic₂₅₅	23

Subgroups — Dic5865 #37553289

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₅₈₆₅ #37553289