Subgroups — #49137618

Order 36300.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅⋊D₅×C₁₁²⋊C₃	C₂	1	1	yes	maximal, characteristic
2	C₁₁⋊(C₁₁⋊#1043)	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₅²×C₁₁²⋊C₃	C₄	1	1	yes	characteristic, complemented, Hall
4	(C₁₁⋊(C₁₁⋊AGL(1,𝔽₅)))⋊C₃	—	5	6	no	maximal
5	C₁₁²×C₅⋊D₅	C₂×C₃	1	1	yes	characteristic
6	D₅×C₁₁²⋊C₃	—	5	6	no	—
7	C₅²×C₁₁²	C₃×C₄	1	1	yes	characteristic, complemented, Hall
8	C₁₁⋊(C₁₁⋊AGL(1,𝔽₅))	—	5	6	no	—
9	C₅×C₁₁²⋊C₃	AGL(1,𝔽₅)	1	6	yes	complemented
10	(C₁₁⋊Dic₁₁)⋊C₃	—	25	1	no	Hall
11	D₅×C₁₁²	—	5	6	no	—
12	C₁₁⋊#1043	—	33	4	no	—
13	C₂×C₁₁²⋊C₃	—	25	1	no	—
14	C₅×C₁₁²	C₃×AGL(1,𝔽₅)	1	6	yes	complemented
15	C₁₁×C₅⋊D₅	—	3	4	no	—
16	C₁₁⋊Dic₁₁	—	25	1	no	Hall
17	C₁₁²⋊C₃	#1043	1	1	yes	characteristic, complemented, Hall
18	C₃×#1043	—	121	1	no	maximal, Hall
19	C₅²×C₁₁	—	3	4	no	—
20	C₂×C₁₁²	—	25	1	no	—
21	C₁₁⋊AGL(1,𝔽₅)	—	165	24	no	—
22	C₃×C₅⋊D₅	—	242	1	no	—
23	C₁₁²	C₃×#1043	1	1	yes	characteristic, complemented, Sylow
24	D₅×C₁₁	—	15	24	no	—
25	#1043	—	121	1	no	Hall
26	C₃×C₅²	—	242	1	no	Hall
27	C₃×AGL(1,𝔽₅)	—	605	6	no	—
28	C₅×C₁₁	—	3	24	no	—
29	C₅⋊D₅	(C₁₁⋊D₁₁)⋊C₃	1	1	yes	characteristic
30	Dic₁₁	—	825	4	no	—
31	C₃×D₅	—	1210	6	no	—
32	C₅²	(C₁₁⋊Dic₁₁)⋊C₃	1	1	yes	characteristic, complemented, Sylow
33	C₂×C₁₁	—	75	4	no	—
34	AGL(1,𝔽₅)	—	605	6	no	—
35	C₃×C₅	—	242	6	no	—
36	C₃×C₄	—	3025	1	no	Hall
37	C₁₁	—	3	4	no	—
38	D₅	—	5	6	no	—
39	C₂×C₃	—	6050	1	no	—
40	C₅	(C₁₁⋊(C₁₁⋊AGL(1,𝔽₅)))⋊C₃	1	6	yes	complemented
41	C₄	—	3025	1	no	Sylow
42	C₃	—	242	1	no	Sylow
43	C₂	—	25	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅²×C₁₁²	C₃×C₄
prime radical	C₅⋊D₅×C₁₁²⋊C₃	C₂
cosocle	C₁₁²×C₅⋊D₅	C₂×C₃
LCS	G ▷ C₅²×C₁₁²
UCS	C₁

Complements

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

N	H ≅ G/N	type
2. C₁₁⋊(C₁₁⋊#1043)	C₃	semidirect
3. C₅²×C₁₁²⋊C₃	C₄	semidirect
7. C₅²×C₁₁²	C₃×C₄	semidirect
9. C₅×C₁₁²⋊C₃	AGL(1,𝔽₅)	semidirect
14. C₅×C₁₁²	C₃×AGL(1,𝔽₅)	semidirect
17. C₁₁²⋊C₃	#1043	semidirect
23. C₁₁²	C₃×#1043	semidirect
32. C₅²	(C₁₁⋊Dic₁₁)⋊C₃	semidirect
40. C₅	(C₁₁⋊(C₁₁⋊AGL(1,𝔽₅)))⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₅⋊D₅×C₁₁²⋊C₃	2
C₁₁⋊(C₁₁⋊#1043)	3
(C₁₁⋊(C₁₁⋊AGL(1,𝔽₅)))⋊C₃	5
C₃×#1043	121