Subgroups — #49147071

Order 36900.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₄₁×C₃⋊D₁₅	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₅²×C₄₁	C₄	1	1	yes	characteristic, complemented, Hall
3	#20301658	—	5	1	no	maximal
4	#20301655	—	5	1	no	maximal
5	C₅×D₁₅×C₄₁	—	6	2	no	—
6	#16861527	—	9	1	no	maximal, Hall
7	C₄₁×C₃⋊D₁₅	D₅	1	1	yes	characteristic
8	C₅×C₃⋊D₃×C₄₁	—	5	1	no	—
9	C₃×C₅²×C₄₁	—	2	2	no	—
10	C₅×D₅×C₄₁	—	9	1	no	—
11	C₃²×C₅×C₄₁	—	4	1	no	—
12	C₃²×C₅×C₄₁	Dic₅	1	1	yes	characteristic, complemented
13	C₃²×C₅×C₄₁	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
14	#13093288	—	25	1	no	Hall
15	D₁₅×C₄₁	—	6	2	no	—
16	C₅×D₃×C₄₁	—	30	2	no	—
17	C₅²×C₄₁	#157	1	1	yes	characteristic, complemented, Hall
18	#11744709	—	41	1	no	maximal, Hall
19	#11717963	—	45	1	no	—
20	Dic₂₀₅	—	45	1	no	—
21	C₃⋊D₃×C₄₁	—	5	1	no	—
22	C₃×C₅×C₄₁	—	2	2	no	—
23	C₃×C₅×C₄₁	—	2	2	no	—
24	C₃×C₅×C₄₁	—	4	4	no	—
25	C₅×C₃⋊D₁₅	D₄₁	1	1	yes	characteristic
26	C₂×C₅×C₄₁	—	45	1	no	—
27	D₅×C₄₁	—	9	1	no	—
28	C₃²×C₄₁	#1042	1	1	yes	characteristic, complemented, Hall
29	D₃×C₄₁	—	30	2	no	—
30	C₃²×C₅²	Dic₄₁	1	1	yes	characteristic, complemented, Hall
31	C₅×C₄₁	—	4	1	no	—
32	C₅×C₄₁	#4379	1	1	yes	characteristic, complemented
33	C₅×C₄₁	#4380	1	1	yes	characteristic, complemented
34	#4380	—	205	1	no	—
35	#4379	—	205	1	no	—
36	Dic₄₁	—	225	1	no	Hall
37	C₅×D₁₅	—	6	2	no	—
38	C₃×C₄₁	—	2	2	no	—
39	#1042	—	369	1	no	Hall
40	C₃⋊D₁₅	D₂₀₅	1	1	yes	characteristic
41	C₅×C₃⋊D₃	—	5	1	no	—
42	C₂×C₄₁	—	45	1	no	—
43	C₃×C₅²	—	2	2	no	—
44	C₅×D₅	—	9	1	no	—
45	C₃²×C₅	—	4	1	no	—
46	C₃²×C₅	Dic₂₀₅	1	1	yes	characteristic, complemented
47	C₃²×C₅	#11717963	1	1	yes	characteristic, complemented
48	C₄₁	#11744709	1	1	yes	characteristic, complemented, Sylow
49	#157	—	1025	1	no	Hall
50	D₁₅	—	6	2	no	—
51	C₅×D₃	—	30	2	no	—
52	C₅²	#13093288	1	1	yes	characteristic, complemented, Sylow
53	AGL(1,𝔽₅)	—	1845	1	no	—
54	Dic₅	—	1845	1	no	—
55	C₃⋊D₃	—	5	1	no	—
56	C₃×C₅	—	2	2	no	—
57	C₃×C₅	—	2	2	no	—
58	C₃×C₅	—	4	4	no	—
59	C₂×C₅	—	45	1	no	—
60	D₅	—	9	1	no	—
61	C₃²	#16861527	1	1	yes	characteristic, complemented, Sylow
62	D₃	—	30	2	no	—
63	C₅	—	4	1	no	—
64	C₅	#20301655	1	1	yes	characteristic, complemented
65	C₅	#20301658	1	1	yes	characteristic, complemented
66	C₄	—	9225	1	no	Sylow
67	C₃	—	2	2	no	—
68	C₂	—	45	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃²×C₅²×C₄₁	C₄
prime radical, cosocle	C₅×C₄₁×C₃⋊D₁₅	C₂
LCS	G ▷ C₃²×C₅²×C₄₁
UCS	C₁

Complements

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

N	H ≅ G/N	type
2. C₃²×C₅²×C₄₁	C₄	semidirect
12. C₃²×C₅×C₄₁	Dic₅	semidirect
13. C₃²×C₅×C₄₁	AGL(1,𝔽₅)	semidirect
17. C₅²×C₄₁	#157	semidirect
28. C₃²×C₄₁	#1042	semidirect
30. C₃²×C₅²	Dic₄₁	semidirect
32. C₅×C₄₁	#4379	semidirect
33. C₅×C₄₁	#4380	semidirect
46. C₃²×C₅	Dic₂₀₅	semidirect
47. C₃²×C₅	#11717963	semidirect
48. C₄₁	#11744709	semidirect
52. C₅²	#13093288	semidirect
61. C₃²	#16861527	semidirect
64. C₅	#20301655	semidirect
65. C₅	#20301658	semidirect

Maximal subgroups

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

subgroup	index
C₅×C₄₁×C₃⋊D₁₅	2
#20301658	5
#20301655	5
#16861527	9
#11744709	41