Subgroups — C₁₂₁×Dic₅₇ #42119918

Order 27588.

36 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₁₉×C₁₂₁	C₂	1	1	yes	maximal, characteristic
2	Dic₁₉×C₁₂₁	—	3	1	no	maximal, Hall
3	C₃×C₁₉×C₁₂₁	C₄	1	1	yes	characteristic, complemented, Hall
4	C₂×C₁₉×C₁₂₁	D₃	1	1	yes	characteristic
5	C₁₁×Dic₅₇	C₁₁	1	1	yes	maximal, characteristic
6	C₁₉×C₁₂₁	Dic₃	1	1	yes	characteristic, complemented, Hall
7	Dic₃×C₁₂₁	—	19	1	no	maximal, Hall
8	C₂×C₃×C₁₁×C₁₉	C₂×C₁₁	1	1	yes	characteristic
9	C₁₁×Dic₁₉	—	3	1	no	—
10	C₂×C₃×C₁₂₁	D₁₉	1	1	yes	characteristic
11	C₃×C₁₁×C₁₉	C₄×C₁₁	1	1	yes	characteristic
12	C₄×C₁₂₁	—	57	1	no	Hall
13	C₂×C₁₁×C₁₉	D₃×C₁₁	1	1	yes	characteristic
14	C₃×C₁₂₁	Dic₁₉	1	1	yes	characteristic, complemented, Hall
15	C₂×C₁₂₁	D₅₇	1	1	yes	characteristic
16	Dic₅₇	C₁₂₁	1	1	yes	characteristic, direct product factor, Hall
17	C₁₁×C₁₉	C₁₁×Dic₃	1	1	yes	characteristic
18	C₁₁×Dic₃	—	19	1	no	—
19	C₁₂₁	Dic₅₇	1	1	yes	characteristic, direct product factor, Sylow
20	C₂×C₃×C₁₉	C₂×C₁₂₁	1	1	yes	characteristic
21	Dic₁₉	—	3	1	no	Hall
22	C₂×C₃×C₁₁	C₁₁×D₁₉	1	1	yes	characteristic
23	C₃×C₁₉	C₄×C₁₂₁	1	1	yes	characteristic, complemented, Hall
24	C₄×C₁₁	—	57	1	no	—
25	C₂×C₁₉	D₃×C₁₂₁	1	1	yes	characteristic
26	C₃×C₁₁	C₁₁×Dic₁₉	1	1	yes	characteristic
27	C₂×C₁₁	C₁₁×D₅₇	1	1	yes	characteristic
28	C₁₉	Dic₃×C₁₂₁	1	1	yes	characteristic, complemented, Sylow
29	Dic₃	—	19	1	no	Hall
30	C₁₁	C₁₁×Dic₅₇	1	1	yes	characteristic
31	C₂×C₃	D₁₉×C₁₂₁	1	1	yes	characteristic
32	C₄	—	57	1	no	Sylow
33	C₃	Dic₁₉×C₁₂₁	1	1	yes	characteristic, complemented, Sylow
34	C₂	D₅₇×C₁₂₁	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₁₂₁	D₅₇
commutator, hypocenter	C₃×C₁₉	C₄×C₁₂₁
Frattini	C₂×C₁₁	C₁₁×D₅₇
Fitting	C₂×C₃×C₁₉×C₁₂₁	C₂
socle, prime radical, cosocle	C₂×C₃×C₁₁×C₁₉	C₂×C₁₁
LCS	G ▷ C₃×C₁₉
UCS	C₁ ◁ C₂×C₁₂₁

Complements

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

N	H ≅ G/N	type
3. C₃×C₁₉×C₁₂₁	C₄	semidirect
6. C₁₉×C₁₂₁	Dic₃	semidirect
14. C₃×C₁₂₁	Dic₁₉	semidirect
16. Dic₅₇	C₁₂₁	direct (N × H)
19. C₁₂₁	Dic₅₇	direct (N × H)
23. C₃×C₁₉	C₄×C₁₂₁	semidirect
28. C₁₉	Dic₃×C₁₂₁	semidirect
33. C₃	Dic₁₉×C₁₂₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₁₉×C₁₂₁	2
Dic₁₉×C₁₂₁	3
C₁₁×Dic₅₇	11
Dic₃×C₁₂₁	19

Subgroups — C121×Dic57 #42119918

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₁×Dic₅₇ #42119918