Subgroups — C₂×C₄₇×D₁₂₃ #37528871

Order 23124.

40 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₇×D₁₂₃	C₂	1	2	yes	maximal, direct product factor
2	C₂×C₃×C₄₁×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄₇×D₄₁	—	3	1	no	maximal, Hall
4	C₃×C₄₁×C₄₇	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×C₄₁×C₄₇	D₃	1	1	yes	characteristic, complemented
6	C₄₇×D₄₁	—	3	2	no	—
7	C₄₁×C₄₇	C₂×D₃	1	1	yes	characteristic, complemented, Hall
8	C₂×D₃×C₄₇	—	41	1	no	maximal, Hall
9	C₂×D₁₂₃	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂×C₃×C₄₇	D₄₁	1	1	yes	characteristic, complemented
11	D₃×C₄₇	—	41	2	no	—
12	C₂×C₃×C₄₁	C₂×C₄₇	1	1	yes	characteristic, complemented
13	D₁₂₃	C₂×C₄₇	1	2	yes	direct product factor
14	C₂²×C₄₇	—	123	1	no	Hall
15	C₂×D₄₁	—	3	1	no	Hall
16	C₃×C₄₇	C₂×D₄₁	1	1	yes	characteristic, complemented, Hall
17	C₃×C₄₁	C₂²×C₄₇	1	1	yes	characteristic, complemented, Hall
18	C₂×C₄₇	—	123	2	no	—
19	C₂×C₄₇	D₁₂₃	1	1	yes	characteristic, direct product factor
20	C₂×C₄₁	D₃×C₄₇	1	1	yes	characteristic, complemented
21	D₄₁	—	3	2	no	—
22	C₄₇	C₂×D₁₂₃	1	1	yes	characteristic, direct product factor, Sylow
23	C₄₁	C₂×D₃×C₄₇	1	1	yes	characteristic, complemented, Sylow
24	C₂×D₃	—	41	1	no	Hall
25	C₂×C₃	C₄₇×D₄₁	1	1	yes	characteristic, complemented
26	D₃	—	41	2	no	—
27	C₂²	—	123	1	no	Sylow
28	C₃	C₂×C₄₇×D₄₁	1	1	yes	characteristic, complemented, Sylow
29	C₂	—	123	2	no	—
30	C₂	C₄₇×D₁₂₃	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₄₇	D₁₂₃
commutator, cosocle, hypocenter	C₃×C₄₁	C₂²×C₄₇
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₄₁×C₄₇	C₂
LCS	G ▷ C₃×C₄₁
UCS	C₁ ◁ C₂×C₄₇

Complements

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

N	H ≅ G/N	type
1. C₄₇×D₁₂₃	C₂	direct (N × H)
2. C₂×C₃×C₄₁×C₄₇	C₂	semidirect
4. C₃×C₄₁×C₄₇	C₂²	semidirect
5. C₂×C₄₁×C₄₇	D₃	semidirect
7. C₄₁×C₄₇	C₂×D₃	semidirect
9. C₂×D₁₂₃	C₄₇	direct (N × H)
10. C₂×C₃×C₄₇	D₄₁	semidirect
12. C₂×C₃×C₄₁	C₂×C₄₇	semidirect
13. D₁₂₃	C₂×C₄₇	direct (N × H)
16. C₃×C₄₇	C₂×D₄₁	semidirect
17. C₃×C₄₁	C₂²×C₄₇	semidirect
19. C₂×C₄₇	D₁₂₃	direct (N × H)
20. C₂×C₄₁	D₃×C₄₇	semidirect
22. C₄₇	C₂×D₁₂₃	direct (N × H)
23. C₄₁	C₂×D₃×C₄₇	semidirect
25. C₂×C₃	C₄₇×D₄₁	semidirect
28. C₃	C₂×C₄₇×D₄₁	semidirect
30. C₂	C₄₇×D₁₂₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄₇×D₁₂₃	2
C₂×C₃×C₄₁×C₄₇	2
C₂×C₄₇×D₄₁	3
C₂×D₃×C₄₇	41
C₂×D₁₂₃	47

Subgroups — C2×C47×D123 #37528871

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₇×D₁₂₃ #37528871