Subgroups — #1042×#66526 #45686172

Order 33300.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×D₅×#66526	C₂	1	1	yes	maximal, characteristic
2	#1042×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
3	C₅²×#66526	C₄	1	1	yes	characteristic, complemented, Hall
4	AGL(1,𝔽₅)×#66526	—	5	1	no	maximal
5	Dic₅×#66526	—	5	1	no	maximal
6	C₅×D₅×C₃₇⋊C₃	C₂×C₃	1	1	yes	characteristic
7	C₃₇×#1042	C₉	1	1	yes	characteristic, complemented, Hall
8	C₂×C₅×#66526	—	5	1	no	—
9	D₅×#66526	D₅	1	1	yes	characteristic
10	C₅²×C₃₇⋊C₃	C₃×C₄	1	1	yes	characteristic
11	AGL(1,𝔽₅)×C₃₇⋊C₃	—	5	1	no	—
12	Dic₅×C₃₇⋊C₃	—	5	1	no	—
13	C₅×D₅×C₃₇	C₂×C₉	1	1	yes	characteristic
14	C₅×#66526	—	4	1	no	—
15	C₅×#66526	Dic₅	1	1	yes	characteristic, complemented
16	C₅×#66526	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
17	C₄×#66526	—	25	1	no	Hall
18	C₂×C₅×C₃₇⋊C₃	—	5	1	no	—
19	D₅×C₃₇⋊C₃	C₃×D₅	1	1	yes	characteristic
20	C₅²×C₃₇	C₄×C₉	1	1	yes	characteristic, complemented, Hall
21	C₉×#1042	—	37	1	no	maximal, Hall
22	AGL(1,𝔽₅)×C₃₇	—	5	1	no	—
23	Dic₅×C₃₇	—	5	1	no	—
24	C₂×#66526	—	5	1	no	—
25	C₅×C₃₇⋊C₃	—	4	1	no	—
26	C₅×C₃₇⋊C₃	C₃×Dic₅	1	1	yes	characteristic
27	C₅×C₃₇⋊C₃	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
28	C₅×C₉×D₅	—	37	1	no	—
29	C₄×C₃₇⋊C₃	—	25	1	no	—
30	C₂×C₅×C₃₇	—	5	1	no	—
31	D₅×C₃₇	C₉×D₅	1	1	yes	characteristic
32	#66526	#1042	1	1	yes	characteristic, direct product factor, Hall
33	C₃×#1042	—	37	1	no	—
34	C₅²×C₉	—	37	1	no	Hall
35	C₂×C₃₇⋊C₃	—	5	1	no	—
36	C₅×C₃₇	—	4	1	no	—
37	C₅×C₃₇	C₉×Dic₅	1	1	yes	characteristic, complemented
38	C₅×C₃₇	C₉×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
39	C₉×AGL(1,𝔽₅)	—	185	1	no	—
40	C₉×Dic₅	—	185	1	no	—
41	C₃×C₅×D₅	—	37	1	no	—
42	C₄×C₃₇	—	25	1	no	Hall
43	C₃₇⋊C₃	C₃×#1042	1	1	yes	characteristic
44	#1042	#66526	1	1	yes	characteristic, direct product factor, Hall
45	C₂×C₅×C₉	—	185	1	no	—
46	C₉×D₅	—	37	1	no	—
47	C₃×C₅²	—	37	1	no	—
48	C₂×C₃₇	—	5	1	no	—
49	C₃×AGL(1,𝔽₅)	—	185	1	no	—
50	C₃×Dic₅	—	185	1	no	—
51	C₅×D₅	C₂×#66526	1	1	yes	characteristic
52	C₅×C₉	—	37	1	no	—
53	C₅×C₉	—	37	1	no	—
54	C₅×C₉	—	148	1	no	—
55	C₃₇	C₉×#1042	1	1	yes	characteristic, complemented, Sylow
56	C₄×C₉	—	925	1	no	Hall
57	C₂×C₃×C₅	—	185	1	no	—
58	C₃×D₅	—	37	1	no	—
59	C₅²	C₄×#66526	1	1	yes	characteristic, complemented, Sylow
60	AGL(1,𝔽₅)	—	5	1	no	—
61	Dic₅	—	5	1	no	—
62	C₂×C₉	—	185	1	no	—
63	C₃×C₅	—	37	1	no	—
64	C₃×C₅	—	37	1	no	—
65	C₃×C₅	—	148	1	no	—
66	C₃×C₄	—	925	1	no	—
67	C₂×C₅	—	5	1	no	—
68	D₅	D₅×#66526	1	1	yes	characteristic
69	C₉	—	37	1	no	Sylow
70	C₂×C₃	—	185	1	no	—
71	C₅	—	4	1	no	—
72	C₅	Dic₅×#66526	1	1	yes	characteristic, complemented
73	C₅	AGL(1,𝔽₅)×#66526	1	1	yes	characteristic, complemented
74	C₄	—	25	1	no	Sylow
75	C₃	—	37	1	no	—
76	C₂	—	5	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅²×C₃₇	C₄×C₉
prime radical, cosocle	C₅×D₅×C₃₇⋊C₃	C₂×C₃
LCS	G ▷ C₅²×C₃₇
UCS	C₁

Complements

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

N	H ≅ G/N	type
3. C₅²×#66526	C₄	semidirect
7. C₃₇×#1042	C₉	semidirect
15. C₅×#66526	Dic₅	semidirect
16. C₅×#66526	AGL(1,𝔽₅)	semidirect
20. C₅²×C₃₇	C₄×C₉	semidirect
32. #66526	#1042	direct (N × H)
37. C₅×C₃₇	C₉×Dic₅	semidirect
38. C₅×C₃₇	C₉×AGL(1,𝔽₅)	semidirect
44. #1042	#66526	direct (N × H)
55. C₃₇	C₉×#1042	semidirect
59. C₅²	C₄×#66526	semidirect
72. C₅	Dic₅×#66526	semidirect
73. C₅	AGL(1,𝔽₅)×#66526	semidirect

Maximal subgroups

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

subgroup	index
C₅×D₅×#66526	2
#1042×C₃₇⋊C₃	3
AGL(1,𝔽₅)×#66526	5
Dic₅×#66526	5
C₉×#1042	37