Subgroups — C₅×C₂²⋊((C₁₁²⋊C₃)⋊C₅) #49137539

Order 36300.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#10597243	C₃	1	1	yes	maximal, characteristic, complemented, Hall
2	C₅×(C₁₁²⋊C₃)⋊C₅	—	4	1	no	maximal, Hall
3	C₅×C₂²⋊(C₁₁²⋊C₃)	C₅	1	1	yes	maximal, characteristic, complemented
4	C₂²⋊((C₁₁²⋊C₃)⋊C₅)	C₅	1	5	yes	maximal, direct product factor
5	C₂×C₅×#10597243	—	3	1	no	—
6	C₅×#10597243	A₄	1	1	yes	characteristic, complemented, Hall
7	C₂²×#10597243	C₃×C₅	1	5	yes	complemented
8	C₂²×C₅×C₁₁²	C₃×C₅	1	1	yes	characteristic, complemented
9	C₅×C₁₁²⋊C₃	—	4	1	no	—
10	(C₁₁²⋊C₃)⋊C₅	—	4	5	no	—
11	C₂²⋊(C₁₁²⋊C₃)	C₅²	1	1	yes	characteristic, complemented, Hall
12	C₂×C₅×C₁₁²	—	3	1	no	—
13	C₂×#10597243	—	3	5	no	—
14	C₂²×C₅×C₁₁⋊C₅	—	33	4	no	—
15	C₅×C₁₁²	C₅×A₄	1	1	yes	characteristic, complemented
16	#10597243	C₅×A₄	1	5	yes	complemented
17	C₂×C₅×C₁₁⋊C₅	—	33	12	no	—
18	C₂²×C₁₁²	C₃×C₅²	1	1	yes	characteristic, complemented, Hall
19	C₁₁²⋊C₃	—	4	1	no	Hall
20	C₅²×A₄	—	121	1	no	maximal, Hall
21	C₅×C₁₁⋊C₅	—	33	4	no	—
22	C₂×C₁₁²	—	3	1	no	—
23	C₂²×C₅×C₁₁	—	3	4	no	—
24	C₂²×C₁₁⋊C₅	—	33	20	no	—
25	C₁₁²	C₅²×A₄	1	1	yes	characteristic, complemented, Sylow
26	C₂×C₅×C₁₁	—	3	12	no	—
27	C₂×C₁₁⋊C₅	—	33	60	no	—
28	C₂²×C₅²	—	121	1	no	Hall
29	C₃×C₅²	—	484	1	no	Hall
30	C₅×A₄	—	605	1	no	—
31	C₅×A₄	—	121	5	no	—
32	C₅×C₁₁	—	3	4	no	—
33	C₁₁⋊C₅	—	33	20	no	—
34	C₂×C₅²	—	363	1	no	—
35	C₂²×C₁₁	—	3	4	no	—
36	C₅²	—	121	1	no	Sylow
37	C₂×C₁₁	—	3	12	no	—
38	C₂²×C₅	—	121	5	no	—
39	C₂²×C₅	(C₁₁²⋊C₃)⋊C₅	1	1	yes	characteristic, complemented
40	C₃×C₅	—	2420	1	no	—
41	C₃×C₅	—	484	5	no	—
42	A₄	—	605	1	no	Hall
43	C₁₁	—	3	4	no	—
44	C₂×C₅	—	3	1	no	—
45	C₂×C₅	—	363	5	no	—
46	C₅	—	121	5	no	—
47	C₅	C₂²⋊((C₁₁²⋊C₃)⋊C₅)	1	1	yes	characteristic, direct product factor
48	C₂²	C₅×(C₁₁²⋊C₃)⋊C₅	1	1	yes	characteristic, complemented, Sylow
49	C₃	—	2420	1	no	Sylow
50	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₅	C₂²⋊((C₁₁²⋊C₃)⋊C₅)
commutator, cosocle, hypocenter	C₂²×C₁₁²	C₃×C₅²
Frattini	C₁	G
Fitting, socle	C₂²×C₅×C₁₁²	C₃×C₅
LCS	G ▷ C₂²×C₁₁²
UCS	C₁ ◁ C₅

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×#10597243	C₃	semidirect
3. C₅×C₂²⋊(C₁₁²⋊C₃)	C₅	semidirect
4. C₂²⋊((C₁₁²⋊C₃)⋊C₅)	C₅	direct (N × H)
6. C₅×#10597243	A₄	semidirect
7. C₂²×#10597243	C₃×C₅	semidirect
8. C₂²×C₅×C₁₁²	C₃×C₅	semidirect
11. C₂²⋊(C₁₁²⋊C₃)	C₅²	semidirect
15. C₅×C₁₁²	C₅×A₄	semidirect
16. #10597243	C₅×A₄	semidirect
18. C₂²×C₁₁²	C₃×C₅²	semidirect
25. C₁₁²	C₅²×A₄	semidirect
39. C₂²×C₅	(C₁₁²⋊C₃)⋊C₅	semidirect
47. C₅	C₂²⋊((C₁₁²⋊C₃)⋊C₅)	direct (N × H)
48. C₂²	C₅×(C₁₁²⋊C₃)⋊C₅	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₅×#10597243	3
C₅×(C₁₁²⋊C₃)⋊C₅	4
C₅×C₂²⋊(C₁₁²⋊C₃)	5
C₂²⋊((C₁₁²⋊C₃)⋊C₅)	5
C₅²×A₄	121

Subgroups — C5×C22⋊((C112⋊C3)⋊C5) #49137539

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×C₂²⋊((C₁₁²⋊C₃)⋊C₅) #49137539