Subgroups — C₅×#20269165 #45686366

Order 33300.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₃⋊D₅₅₅	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₅²×C₃₇	C₄	1	1	yes	characteristic, complemented, Hall
3	#20269165	C₅	1	1	yes	maximal, characteristic, direct product factor
4	C₅×#13050652	—	5	1	no	maximal
5	C₅×D₅₅₅	—	6	2	no	—
6	C₅×#10625191	—	9	1	no	maximal, Hall
7	C₃⋊D₅₅₅	C₂×C₅	1	1	yes	characteristic
8	C₅×C₃⋊D₁₁₁	—	5	1	no	—
9	C₃×C₅²×C₃₇	—	2	2	no	—
10	C₅×D₁₈₅	—	9	1	no	—
11	C₃²×C₅×C₃₇	—	4	1	no	—
12	C₃²×C₅×C₃₇	C₄×C₅	1	1	yes	characteristic, complemented
13	C₃²×C₅×C₃₇	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
14	#13050652	—	5	1	no	Hall
15	D₅₅₅	—	6	2	no	—
16	C₅×D₁₁₁	—	30	2	no	—
17	C₅²×C₃₇	#157	1	1	yes	characteristic, complemented, Hall
18	C₅×#4380	—	37	1	no	maximal, Hall
19	#10625191	—	9	1	no	—
20	C₅×#3866	—	45	1	no	—
21	C₃⋊D₁₁₁	—	5	1	no	—
22	C₃×C₅×C₃₇	—	2	2	no	—
23	C₃×C₅×C₃₇	—	2	2	no	—
24	C₃×C₅×C₃₇	—	4	4	no	—
25	C₅×C₃⋊D₁₅	—	37	1	no	—
26	D₁₈₅	—	9	1	no	—
27	C₅×D₃₇	—	45	1	no	—
28	C₃²×C₃₇	C₅×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
29	C₃²×C₅²	#3866	1	1	yes	characteristic, complemented, Hall
30	D₁₁₁	—	30	2	no	—
31	C₅×C₃₇	—	4	1	no	—
32	C₅×C₃₇	C₅×#157	1	1	yes	characteristic, complemented
33	C₅×C₃₇	#4380	1	1	yes	characteristic, complemented
34	#4380	—	37	1	no	—
35	C₅×#157	—	185	1	no	—
36	C₅×D₁₅	—	222	2	no	—
37	#3866	—	45	1	no	Hall
38	C₃×C₃₇	—	2	2	no	—
39	C₅×AGL(1,𝔽₅)	—	333	1	no	Hall
40	C₃⋊D₁₅	—	37	1	no	—
41	C₅×C₃⋊D₃	—	185	1	no	—
42	C₃×C₅²	—	2	2	no	—
43	D₃₇	—	45	1	no	—
44	C₅×D₅	—	333	1	no	—
45	C₃²×C₅	—	4	1	no	—
46	C₃²×C₅	C₅×#3866	1	1	yes	characteristic, complemented
47	C₃²×C₅	#10625191	1	1	yes	characteristic, complemented
48	C₃₇	C₅×#4380	1	1	yes	characteristic, complemented, Sylow
49	#157	—	185	1	no	Hall
50	D₁₅	—	222	2	no	—
51	C₅×D₃	—	1110	2	no	—
52	C₅²	#13050652	1	1	yes	characteristic, complemented, Sylow
53	AGL(1,𝔽₅)	—	333	1	no	—
54	C₄×C₅	—	1665	1	no	—
55	C₃⋊D₃	—	185	1	no	—
56	C₃×C₅	—	2	2	no	—
57	C₃×C₅	—	2	2	no	—
58	C₃×C₅	—	4	4	no	—
59	C₂×C₅	—	1665	1	no	—
60	D₅	—	333	1	no	—
61	C₃²	C₅×#10625191	1	1	yes	characteristic, complemented, Sylow
62	D₃	—	1110	2	no	—
63	C₅	—	4	1	no	—
64	C₅	C₅×#13050652	1	1	yes	characteristic, complemented
65	C₅	#20269165	1	1	yes	characteristic, direct product factor
66	C₄	—	1665	1	no	Sylow
67	C₃	—	2	2	no	—
68	C₂	—	1665	1	no	—

	subgroup	G/H
center, hypercenter	C₅	#20269165
commutator, hypocenter	C₃²×C₅×C₃₇	C₄×C₅
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₃₇	C₄
prime radical	C₅×C₃⋊D₅₅₅	C₂
cosocle	C₃⋊D₅₅₅	C₂×C₅
LCS	G ▷ C₃²×C₅×C₃₇
UCS	C₁ ◁ C₅

Complements

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

N	H ≅ G/N	type
2. C₃²×C₅²×C₃₇	C₄	semidirect
3. #20269165	C₅	direct (N × H)
12. C₃²×C₅×C₃₇	C₄×C₅	semidirect
13. C₃²×C₅×C₃₇	AGL(1,𝔽₅)	semidirect
17. C₅²×C₃₇	#157	semidirect
28. C₃²×C₃₇	C₅×AGL(1,𝔽₅)	semidirect
29. C₃²×C₅²	#3866	semidirect
32. C₅×C₃₇	C₅×#157	semidirect
33. C₅×C₃₇	#4380	semidirect
46. C₃²×C₅	C₅×#3866	semidirect
47. C₃²×C₅	#10625191	semidirect
48. C₃₇	C₅×#4380	semidirect
52. C₅²	#13050652	semidirect
61. C₃²	C₅×#10625191	semidirect
64. C₅	C₅×#13050652	semidirect
65. C₅	#20269165	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×C₃⋊D₅₅₅	2
#20269165	5
C₅×#13050652	5
C₅×#10625191	9
C₅×#4380	37

Subgroups — C5×#20269165 #45686366

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#20269165 #45686366