Subgroups — C₅×C₂²⋊(C₁₁²⋊C₉) #36344949

Order 21780.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×C₁₁²	C₃	1	1	yes	maximal, characteristic
2	C₅×C₁₁²⋊C₉	—	4	1	no	maximal, Hall
3	C₂²⋊(C₁₁²⋊C₉)	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂×C₃×C₅×C₁₁²	—	3	1	no	—
5	C₂²×C₅×C₁₁²	C₉	1	1	yes	characteristic, complemented, Hall
6	C₃×C₅×C₁₁²	A₄	1	1	yes	characteristic
7	C₂²×C₃×C₁₁²	C₃×C₅	1	1	yes	characteristic
8	C₂×C₅×C₁₁²	—	3	1	no	—
9	C₁₁²⋊C₉	—	4	1	no	Hall
10	C₂×C₃×C₁₁²	—	3	1	no	—
11	C₂²×C₃×C₅×C₁₁	—	3	4	no	—
12	C₅×C₁₁²	C₂²⋊C₉	1	1	yes	characteristic, complemented, Hall
13	C₂²×C₁₁²	C₅×C₉	1	1	yes	characteristic, complemented, Hall
14	C₃×C₁₁²	C₅×A₄	1	1	yes	characteristic
15	C₂×C₃×C₅×C₁₁	—	3	12	no	—
16	C₂×C₁₁²	—	3	1	no	—
17	C₂²×C₅×C₁₁	—	3	4	no	—
18	C₅×C₂²⋊C₉	—	121	1	no	maximal, Hall
19	C₃×C₅×C₁₁	—	3	4	no	—
20	C₂²×C₃×C₁₁	—	3	4	no	—
21	C₁₁²	C₅×C₂²⋊C₉	1	1	yes	characteristic, complemented, Sylow
22	C₂×C₅×C₁₁	—	3	12	no	—
23	C₂×C₃×C₁₁	—	3	12	no	—
24	C₂²×C₃×C₅	C₁₁²⋊C₃	1	1	yes	characteristic
25	C₅×C₁₁	—	3	4	no	—
26	C₅×C₉	—	484	1	no	Hall
27	C₂²×C₁₁	—	3	4	no	—
28	C₂²⋊C₉	—	121	1	no	Hall
29	C₃×C₁₁	—	3	4	no	—
30	C₂×C₃×C₅	—	3	1	no	—
31	C₂×C₁₁	—	3	12	no	—
32	C₂²×C₅	C₁₁²⋊C₉	1	1	yes	characteristic, complemented, Hall
33	C₃×C₅	C₂²⋊(C₁₁²⋊C₃)	1	1	yes	characteristic
34	C₂²×C₃	C₅×C₁₁²⋊C₃	1	1	yes	characteristic
35	C₁₁	—	3	4	no	—
36	C₂×C₅	—	3	1	no	—
37	C₉	—	484	1	no	Sylow
38	C₂×C₃	—	3	1	no	—
39	C₅	C₂²⋊(C₁₁²⋊C₉)	1	1	yes	characteristic, direct product factor, Sylow
40	C₂²	C₅×C₁₁²⋊C₉	1	1	yes	characteristic, complemented, Sylow
41	C₃	C₅×C₂²⋊(C₁₁²⋊C₃)	1	1	yes	characteristic
42	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₃×C₅	C₂²⋊(C₁₁²⋊C₃)
commutator, hypocenter	C₂²×C₁₁²	C₅×C₉
Frattini	C₃	C₅×C₂²⋊(C₁₁²⋊C₃)
Fitting, socle, prime radical	C₂²×C₃×C₅×C₁₁²	C₃
cosocle	C₂²×C₃×C₁₁²	C₃×C₅
LCS	G ▷ C₂²×C₁₁²
UCS	C₁ ◁ C₃×C₅

Complements

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

N	H ≅ G/N	type
3. C₂²⋊(C₁₁²⋊C₉)	C₅	direct (N × H)
5. C₂²×C₅×C₁₁²	C₉	semidirect
12. C₅×C₁₁²	C₂²⋊C₉	semidirect
13. C₂²×C₁₁²	C₅×C₉	semidirect
21. C₁₁²	C₅×C₂²⋊C₉	semidirect
32. C₂²×C₅	C₁₁²⋊C₉	semidirect
39. C₅	C₂²⋊(C₁₁²⋊C₉)	direct (N × H)
40. C₂²	C₅×C₁₁²⋊C₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₅×C₁₁²	3
C₅×C₁₁²⋊C₉	4
C₂²⋊(C₁₁²⋊C₉)	5
C₅×C₂²⋊C₉	121

Subgroups — C5×C22⋊(C112⋊C9) #36344949

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×C₂²⋊(C₁₁²⋊C₉) #36344949