Subgroups — #49137624

Order 36300.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅⋊(C₅⋊((C₁₁⋊D₁₁)⋊C₃))	C₂	1	1	yes	maximal, characteristic
2	#27130981	—	3	1	no	maximal, Hall
3	C₅²×C₁₁²⋊C₃	C₄	1	1	yes	characteristic, complemented, Hall
4	#20300038	—	5	2	no	maximal
5	C₅⋊(C₅⋊(C₁₁⋊D₁₁))	D₃	1	1	yes	characteristic
6	C₅⋊((C₁₁⋊D₁₁)⋊C₃)	—	5	2	no	—
7	C₅⋊((C₁₁⋊D₁₁)⋊C₃)	—	10	2	no	—
8	C₅²×C₁₁²	Dic₃	1	1	yes	characteristic, complemented, Hall
9	#14559097	—	15	2	no	—
10	C₅×C₁₁²⋊C₃	—	2	2	no	—
11	C₅×C₁₁²⋊C₃	AGL(1,𝔽₅)	1	2	yes	complemented
12	#13089721	—	25	1	no	Hall
13	C₅⋊(C₁₁⋊D₁₁)	—	5	2	no	—
14	C₅⋊(C₁₁⋊D₁₁)	—	10	2	no	—
15	(C₁₁⋊D₁₁)⋊C₃	—	25	1	no	—
16	C₅×C₁₁²	—	2	2	no	—
17	C₅×C₁₁²	ΓL(1,𝔽₁₆)	1	2	yes	complemented
18	C₅⋊D₅₅	—	66	2	no	—
19	#92150	—	75	1	no	Hall
20	C₁₁²⋊C₃	#1044	1	1	yes	characteristic, complemented, Hall
21	C₃⋊#1044	—	121	1	no	maximal, Hall
22	C₅²×C₁₁	—	6	2	no	—
23	C₁₁⋊D₁₁	—	25	1	no	—
24	C₃×C₅⋊D₅	—	121	1	no	—
25	C₁₁²	C₃⋊#1044	1	1	yes	characteristic, complemented, Sylow
26	D₅₅	—	330	4	no	—
27	D₅₅	—	330	8	no	—
28	#1044	—	363	1	no	Hall
29	C₃×C₅²	—	484	1	no	Hall
30	ΓL(1,𝔽₁₆)	—	605	2	no	—
31	C₅×C₁₁	—	6	8	no	—
32	C₅×C₁₁	—	6	4	no	—
33	C₅⋊D₅	—	121	1	no	—
34	C₃×D₅	—	605	2	no	—
35	C₃×D₅	—	1210	2	no	—
36	C₅²	#13089721	1	1	yes	characteristic, complemented, Sylow
37	D₁₁	—	1650	2	no	—
38	AGL(1,𝔽₅)	—	1815	2	no	—
39	C₃×C₅	—	484	2	no	—
40	C₃×C₅	—	484	2	no	—
41	Dic₃	—	3025	1	no	Hall
42	C₁₁	—	6	2	no	—
43	D₅	—	605	2	no	—
44	D₅	—	1210	2	no	—
45	C₂×C₃	—	3025	1	no	—
46	C₅	—	2	2	no	—
47	C₅	#20300038	1	2	yes	complemented
48	C₄	—	9075	1	no	Sylow
49	C₃	—	484	1	no	Sylow
50	C₂	—	3025	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, hypocenter	C₅²×C₁₁²⋊C₃	C₄
Fitting, socle	C₅²×C₁₁²	Dic₃
prime radical, cosocle	C₅⋊(C₅⋊((C₁₁⋊D₁₁)⋊C₃))	C₂
LCS	G ▷ C₅²×C₁₁²⋊C₃
UCS	C₁

Complements

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

N	H ≅ G/N	type
3. C₅²×C₁₁²⋊C₃	C₄	semidirect
8. C₅²×C₁₁²	Dic₃	semidirect
11. C₅×C₁₁²⋊C₃	AGL(1,𝔽₅)	semidirect
17. C₅×C₁₁²	ΓL(1,𝔽₁₆)	semidirect
20. C₁₁²⋊C₃	#1044	semidirect
25. C₁₁²	C₃⋊#1044	semidirect
36. C₅²	#13089721	semidirect
47. C₅	#20300038	semidirect

Maximal subgroups

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

subgroup	index
C₅⋊(C₅⋊((C₁₁⋊D₁₁)⋊C₃))	2
#27130981	3
#20300038	5
C₃⋊#1044	121