Subgroups — C₁₁×C₂₃×D₅₅ #42123745

Order 27830.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₁₁²×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₁₁×D₁₁×C₂₃	—	5	1	no	maximal, Hall
3	C₁₁²×C₂₃	D₅	1	1	yes	characteristic, complemented, Hall
4	C₂₃×D₅₅	C₁₁	1	1	yes	maximal, characteristic, direct product factor
5	D₅×C₁₁×C₂₃	—	11	1	no	maximal
6	C₅×C₁₁×C₂₃	—	2	5	no	—
7	C₅×C₁₁×C₂₃	D₁₁	1	1	yes	characteristic, complemented
8	C₅×C₁₁×C₂₃	C₂×C₁₁	1	1	yes	characteristic, complemented
9	C₁₁×D₅₅	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₅×C₁₁²	C₂×C₂₃	1	1	yes	characteristic, complemented, Hall
11	C₂×C₁₁×C₂₃	—	55	1	no	—
12	D₁₁×C₂₃	—	5	1	no	—
13	C₁₁×C₂₃	—	2	5	no	—
14	C₁₁×C₂₃	D₅×C₁₁	1	1	yes	characteristic, complemented
15	C₁₁×C₂₃	D₅₅	1	1	yes	characteristic, direct product factor
16	C₁₁×D₁₁	—	5	1	no	Hall
17	D₅×C₂₃	—	11	1	no	Hall
18	C₁₁²	D₅×C₂₃	1	1	yes	characteristic, complemented, Sylow
19	C₅×C₂₃	C₁₁×D₁₁	1	1	yes	characteristic, complemented, Hall
20	D₅₅	C₁₁×C₂₃	1	1	yes	characteristic, direct product factor
21	D₅×C₁₁	—	11	1	no	—
22	C₅×C₁₁	—	2	5	no	—
23	C₅×C₁₁	D₁₁×C₂₃	1	1	yes	characteristic, complemented
24	C₅×C₁₁	C₂×C₁₁×C₂₃	1	1	yes	characteristic, complemented
25	C₂×C₂₃	—	55	1	no	Hall
26	C₂₃	C₁₁×D₅₅	1	1	yes	characteristic, direct product factor, Sylow
27	C₂×C₁₁	—	55	1	no	—
28	D₁₁	—	5	1	no	—
29	C₁₁	—	2	5	no	—
30	C₁₁	D₅×C₁₁×C₂₃	1	1	yes	characteristic, complemented
31	C₁₁	C₂₃×D₅₅	1	1	yes	characteristic, direct product factor
32	D₅	—	11	1	no	Hall
33	C₅	C₁₁×D₁₁×C₂₃	1	1	yes	characteristic, complemented, Sylow
34	C₂	—	55	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₁₁×C₂₃	D₅₅
commutator, cosocle, hypocenter	C₅×C₁₁	C₂×C₁₁×C₂₃
Frattini	C₁	G
Fitting, socle	C₅×C₁₁²×C₂₃	C₂
LCS	G ▷ C₅×C₁₁
UCS	C₁ ◁ C₁₁×C₂₃

Complements

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

N	H ≅ G/N	type
1. C₅×C₁₁²×C₂₃	C₂	semidirect
3. C₁₁²×C₂₃	D₅	semidirect
4. C₂₃×D₅₅	C₁₁	direct (N × H)
7. C₅×C₁₁×C₂₃	D₁₁	semidirect
8. C₅×C₁₁×C₂₃	C₂×C₁₁	semidirect
9. C₁₁×D₅₅	C₂₃	direct (N × H)
10. C₅×C₁₁²	C₂×C₂₃	semidirect
14. C₁₁×C₂₃	D₅×C₁₁	semidirect
15. C₁₁×C₂₃	D₅₅	direct (N × H)
18. C₁₁²	D₅×C₂₃	semidirect
19. C₅×C₂₃	C₁₁×D₁₁	semidirect
20. D₅₅	C₁₁×C₂₃	direct (N × H)
23. C₅×C₁₁	D₁₁×C₂₃	semidirect
24. C₅×C₁₁	C₂×C₁₁×C₂₃	semidirect
26. C₂₃	C₁₁×D₅₅	direct (N × H)
30. C₁₁	D₅×C₁₁×C₂₃	semidirect
31. C₁₁	C₂₃×D₅₅	direct (N × H)
33. C₅	C₁₁×D₁₁×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₅×C₁₁²×C₂₃	2
C₁₁×D₁₁×C₂₃	5
C₂₃×D₅₅	11
D₅×C₁₁×C₂₃	11
C₁₁×D₅₅	23

Subgroups — C11×C23×D55 #42123745

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁×C₂₃×D₅₅ #42123745