Subgroups — D₅×ΓL(1,𝔽₈)×D₇₃ #44501356

Order 30660.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)×D₃₆₅	C₂	1	1	yes	maximal, characteristic, complemented
2	D₅×ΓL(1,𝔽₈)×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×ΓL(1,𝔽₈)×D₇₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×D₅×D₇₃	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	C₅×ΓL(1,𝔽₈)×C₇₃	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₂×ΓL(1,𝔽₈)×D₇₃	—	5	1	no	maximal, Hall
7	C₇×D₃₆₅	C₂×C₃	1	1	yes	characteristic, complemented
8	C₇×D₅×C₇₃	C₂×C₃	1	1	yes	characteristic, complemented
9	C₅×C₇×D₇₃	C₂×C₃	1	1	yes	characteristic, complemented
10	C₃×D₅×D₇₃	—	7	1	no	maximal, Hall
11	ΓL(1,𝔽₈)×D₇₃	—	5	1	no	—
12	ΓL(1,𝔽₈)×D₇₃	D₅	1	1	yes	characteristic, direct product factor
13	C₂×ΓL(1,𝔽₈)×C₇₃	—	5	1	no	—
14	C₅×C₇×C₇₃	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
15	C₃×D₃₆₅	—	7	1	no	—
16	C₃×D₅×C₇₃	—	7	1	no	—
17	C₃×C₅×D₇₃	—	7	1	no	—
18	C₂×C₇×D₇₃	—	5	1	no	Hall
19	ΓL(1,𝔽₈)×C₇₃	C₂×D₅	1	1	yes	characteristic, complemented, Hall
20	D₅×D₇₃	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
21	C₃×C₅×C₇₃	—	7	1	no	Hall
22	C₂×C₇×C₇₃	—	5	1	no	—
23	C₇×D₇₃	—	5	1	no	—
24	C₇×D₇₃	C₃×D₅	1	1	yes	characteristic, complemented
25	C₂×C₃×D₇₃	—	35	1	no	Hall
26	D₃₆₅	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
27	C₅×D₇₃	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
28	D₅×C₇₃	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
29	C₇×C₇₃	C₂×C₃×D₅	1	1	yes	characteristic, complemented, Hall
30	C₂×C₃×C₇₃	—	35	1	no	—
31	C₃×D₇₃	—	7	1	no	—
32	C₃×D₇₃	—	35	1	no	—
33	C₂×D₅×ΓL(1,𝔽₈)	—	73	1	no	maximal, Hall
34	C₅×C₇₃	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
35	C₂×D₇₃	—	5	1	no	Hall
36	C₃×C₇₃	—	7	1	no	Hall
37	C₂×C₅×ΓL(1,𝔽₈)	—	73	1	no	—
38	D₅×ΓL(1,𝔽₈)	—	73	1	no	—
39	D₅×ΓL(1,𝔽₈)	D₇₃	1	1	yes	characteristic, direct product factor
40	C₂×C₇₃	—	5	1	no	—
41	D₇₃	—	5	1	no	—
42	D₇₃	D₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
43	C₂×C₇×D₅	—	73	1	no	Hall
44	C₅×ΓL(1,𝔽₈)	C₂×D₇₃	1	1	yes	characteristic, complemented, Hall
45	C₂²×ΓL(1,𝔽₈)	—	365	1	no	Hall
46	C₇₃	C₂×D₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
47	C₂×C₅×C₇	—	73	1	no	—
48	C₇×D₅	—	73	1	no	—
49	C₇×D₅	C₃×D₇₃	1	1	yes	characteristic, complemented
50	C₂×C₃×D₅	—	511	1	no	Hall
51	C₂×ΓL(1,𝔽₈)	—	5	1	no	—
52	C₂×ΓL(1,𝔽₈)	—	73	1	no	—
53	C₂×ΓL(1,𝔽₈)	—	365	1	no	—
54	C₅×C₇	C₂×C₃×D₇₃	1	1	yes	characteristic, complemented, Hall
55	C₂×C₃×C₅	—	511	1	no	—
56	C₃×D₅	—	7	1	no	—
57	C₃×D₅	—	511	1	no	—
58	C₂²×C₇	—	365	1	no	Hall
59	ΓL(1,𝔽₈)	D₅×D₇₃	1	1	yes	characteristic, direct product factor, Hall
60	C₂×D₅	—	73	1	no	Hall
61	C₃×C₅	—	7	1	no	Hall
62	C₂×C₇	—	5	1	no	—
63	C₂×C₇	—	73	1	no	—
64	C₂×C₇	—	365	1	no	—
65	C₂²×C₃	—	2555	1	no	Hall
66	C₂×C₅	—	73	1	no	—
67	D₅	—	73	1	no	—
68	D₅	ΓL(1,𝔽₈)×D₇₃	1	1	yes	characteristic, direct product factor
69	C₇	C₃×D₅×D₇₃	1	1	yes	characteristic, complemented, Sylow
70	C₂×C₃	—	35	1	no	—
71	C₂×C₃	—	2555	1	no	—
72	C₂×C₃	—	511	1	no	—
73	C₅	C₂×ΓL(1,𝔽₈)×D₇₃	1	1	yes	characteristic, complemented, Sylow
74	C₂²	—	365	1	no	Sylow
75	C₃	—	7	1	no	Sylow
76	C₂	—	5	1	no	—
77	C₂	—	73	1	no	—
78	C₂	—	365	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₅×C₇×C₇₃	C₂²×C₃
LCS	G ▷ C₅×C₇×C₇₃
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. ΓL(1,𝔽₈)×D₃₆₅	C₂	semidirect
2. D₅×ΓL(1,𝔽₈)×C₇₃	C₂	semidirect
3. C₅×ΓL(1,𝔽₈)×D₇₃	C₂	semidirect
4. C₇×D₅×D₇₃	C₃	semidirect
5. C₅×ΓL(1,𝔽₈)×C₇₃	C₂²	semidirect
7. C₇×D₃₆₅	C₂×C₃	semidirect
8. C₇×D₅×C₇₃	C₂×C₃	semidirect
9. C₅×C₇×D₇₃	C₂×C₃	semidirect
12. ΓL(1,𝔽₈)×D₇₃	D₅	direct (N × H)
14. C₅×C₇×C₇₃	C₂²×C₃	semidirect
19. ΓL(1,𝔽₈)×C₇₃	C₂×D₅	semidirect
20. D₅×D₇₃	ΓL(1,𝔽₈)	direct (N × H)
24. C₇×D₇₃	C₃×D₅	semidirect
26. D₃₆₅	C₂×ΓL(1,𝔽₈)	semidirect
27. C₅×D₇₃	C₂×ΓL(1,𝔽₈)	semidirect
28. D₅×C₇₃	C₂×ΓL(1,𝔽₈)	semidirect
29. C₇×C₇₃	C₂×C₃×D₅	semidirect
34. C₅×C₇₃	C₂²×ΓL(1,𝔽₈)	semidirect
39. D₅×ΓL(1,𝔽₈)	D₇₃	direct (N × H)
42. D₇₃	D₅×ΓL(1,𝔽₈)	direct (N × H)
44. C₅×ΓL(1,𝔽₈)	C₂×D₇₃	semidirect
46. C₇₃	C₂×D₅×ΓL(1,𝔽₈)	semidirect
49. C₇×D₅	C₃×D₇₃	semidirect
54. C₅×C₇	C₂×C₃×D₇₃	semidirect
59. ΓL(1,𝔽₈)	D₅×D₇₃	direct (N × H)
68. D₅	ΓL(1,𝔽₈)×D₇₃	direct (N × H)
69. C₇	C₃×D₅×D₇₃	semidirect
73. C₅	C₂×ΓL(1,𝔽₈)×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
ΓL(1,𝔽₈)×D₃₆₅	2
D₅×ΓL(1,𝔽₈)×C₇₃	2
C₅×ΓL(1,𝔽₈)×D₇₃	2
C₇×D₅×D₇₃	3
C₂×ΓL(1,𝔽₈)×D₇₃	5
C₃×D₅×D₇₃	7
C₂×D₅×ΓL(1,𝔽₈)	73

Subgroups — D5×ΓL(1,𝔽8)×D73 #44501356

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅×ΓL(1,𝔽₈)×D₇₃ #44501356