Subgroups — D₁₁₆₅₅ #37552001

Order 23310.

48 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₇×C₉×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	D₃₈₈₅	—	3	1	no	maximal
3	D₂₃₃₁	—	5	1	no	maximal, Hall
4	C₃×C₅×C₇×C₃₇	D₃	1	1	yes	characteristic
5	D₁₆₆₅	—	7	1	no	maximal, Hall
6	D₁₂₉₅	—	9	1	no	Hall
7	C₇×C₉×C₃₇	D₅	1	1	yes	characteristic, complemented, Hall
8	C₅×C₉×C₃₇	D₇	1	1	yes	characteristic, complemented, Hall
9	D₇₇₇	—	15	1	no	—
10	C₅×C₇×C₃₇	D₉	1	1	yes	characteristic, complemented, Hall
11	D₅₅₅	—	21	1	no	—
12	C₃×C₇×C₃₇	D₁₅	1	1	yes	characteristic
13	D₃₃₃	—	35	1	no	Hall
14	D₃₁₅	—	37	1	no	maximal, Hall
15	C₃×C₅×C₃₇	D₂₁	1	1	yes	characteristic
16	D₂₅₉	—	45	1	no	Hall
17	D₁₈₅	—	63	1	no	Hall
18	C₉×C₃₇	D₃₅	1	1	yes	characteristic, complemented, Hall
19	C₅×C₇×C₉	D₃₇	1	1	yes	characteristic, complemented, Hall
20	C₇×C₃₇	D₄₅	1	1	yes	characteristic, complemented, Hall
21	D₁₁₁	—	105	1	no	—
22	D₁₀₅	—	111	1	no	—
23	C₅×C₃₇	D₆₃	1	1	yes	characteristic, complemented, Hall
24	D₆₃	—	185	1	no	Hall
25	C₃×C₃₇	D₁₀₅	1	1	yes	characteristic
26	C₃×C₅×C₇	D₁₁₁	1	1	yes	characteristic
27	D₄₅	—	259	1	no	Hall
28	D₃₇	—	315	1	no	Hall
29	D₃₅	—	333	1	no	Hall
30	C₇×C₉	D₁₈₅	1	1	yes	characteristic, complemented, Hall
31	C₅×C₉	D₂₅₉	1	1	yes	characteristic, complemented, Hall
32	D₂₁	—	555	1	no	—
33	C₃₇	D₃₁₅	1	1	yes	characteristic, complemented, Sylow
34	C₅×C₇	D₃₃₃	1	1	yes	characteristic, complemented, Hall
35	D₁₅	—	777	1	no	—
36	C₃×C₇	D₅₅₅	1	1	yes	characteristic
37	D₉	—	1295	1	no	Hall
38	C₃×C₅	D₇₇₇	1	1	yes	characteristic
39	D₇	—	1665	1	no	Hall
40	D₅	—	2331	1	no	Hall
41	C₉	D₁₂₉₅	1	1	yes	characteristic, complemented, Sylow
42	C₇	D₁₆₆₅	1	1	yes	characteristic, complemented, Sylow
43	D₃	—	3885	1	no	—
44	C₅	D₂₃₃₁	1	1	yes	characteristic, complemented, Sylow
45	C₃	D₃₈₈₅	1	1	yes	characteristic
46	C₂	—	11655	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₅×C₇×C₉×C₃₇	C₂
Frattini	C₃	D₃₈₈₅
socle	C₃×C₅×C₇×C₃₇	D₃
LCS	G ▷ C₅×C₇×C₉×C₃₇
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₅×C₇×C₉×C₃₇	C₂	semidirect
7. C₇×C₉×C₃₇	D₅	semidirect
8. C₅×C₉×C₃₇	D₇	semidirect
10. C₅×C₇×C₃₇	D₉	semidirect
18. C₉×C₃₇	D₃₅	semidirect
19. C₅×C₇×C₉	D₃₇	semidirect
20. C₇×C₃₇	D₄₅	semidirect
23. C₅×C₃₇	D₆₃	semidirect
30. C₇×C₉	D₁₈₅	semidirect
31. C₅×C₉	D₂₅₉	semidirect
33. C₃₇	D₃₁₅	semidirect
34. C₅×C₇	D₃₃₃	semidirect
41. C₉	D₁₂₉₅	semidirect
42. C₇	D₁₆₆₅	semidirect
44. C₅	D₂₃₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₅×C₇×C₉×C₃₇	2
D₃₈₈₅	3
D₂₃₃₁	5
D₁₆₆₅	7
D₃₁₅	37

Subgroups — D11655 #37552001

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₁₁₆₅₅ #37552001