Subgroups — C₂²×C₅×#11924780 #37561434

Order 23940.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#11924780	C₂	1	3	yes	maximal, direct product factor
2	C₂²×C₅×C₇×C₁₉⋊C₃	C₃	1	1	yes	maximal, characteristic
3	C₅×#11924780	C₂²	1	1	yes	characteristic, direct product factor, Hall
4	C₂²×#11924780	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂×C₅×C₇×C₁₉⋊C₃	C₂×C₃	1	3	yes	—
6	C₂²×C₅×#4296	—	7	1	no	maximal, Hall
7	C₂²×C₅×C₇×C₁₉	C₉	1	1	yes	characteristic, complemented, Hall
8	C₂×#11924780	C₂×C₅	1	3	yes	direct product factor
9	C₅×C₇×C₁₉⋊C₃	C₂²×C₃	1	1	yes	characteristic
10	C₂×C₅×#4296	—	7	3	no	—
11	C₂²×C₇×C₁₉⋊C₃	C₃×C₅	1	1	yes	characteristic
12	C₂×C₅×C₇×C₁₉	C₂×C₉	1	3	yes	complemented
13	C₂²×C₅×C₇⋊C₉	—	19	1	no	maximal, Hall
14	#11924780	C₂²×C₅	1	1	yes	characteristic, direct product factor, Hall
15	C₂²×C₅×C₁₉⋊C₃	ΓL(1,𝔽₈)	1	1	yes	characteristic
16	C₅×#4296	—	7	1	no	Hall
17	C₂×C₇×C₁₉⋊C₃	C₂×C₃×C₅	1	3	yes	—
18	C₂²×#4296	—	7	1	no	Hall
19	C₅×C₇×C₁₉	C₂²×C₉	1	1	yes	characteristic, complemented, Hall
20	C₂×C₅×C₇⋊C₉	—	19	3	no	—
21	C₂×C₅×C₁₉⋊C₃	C₂×ΓL(1,𝔽₈)	1	3	yes	—
22	C₂²×C₇×C₁₉	C₅×C₉	1	1	yes	characteristic, complemented, Hall
23	C₂²×C₃×C₅×C₇	—	19	1	no	—
24	C₇×C₁₉⋊C₃	C₂²×C₃×C₅	1	1	yes	characteristic
25	C₂²×C₅×C₁₉	C₇⋊C₉	1	1	yes	characteristic, complemented, Hall
26	C₂×#4296	—	7	3	no	—
27	C₅×C₇⋊C₉	—	19	1	no	Hall
28	C₅×C₁₉⋊C₃	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
29	C₂×C₇×C₁₉	C₂×C₅×C₉	1	3	yes	complemented
30	C₂²×C₇⋊C₉	—	19	1	no	Hall
31	C₂²×C₁₉⋊C₃	C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic
32	C₂×C₃×C₅×C₇	—	19	3	no	—
33	C₂×C₅×C₁₉	C₂×C₇⋊C₉	1	3	yes	complemented
34	C₂²×C₅×C₉	—	133	1	no	Hall
35	#4296	—	7	1	no	Hall
36	C₂²×C₅×C₇	#4296	1	1	yes	characteristic, complemented, Hall
37	C₇×C₁₉	C₂²×C₅×C₉	1	1	yes	characteristic, complemented, Hall
38	C₂×C₇⋊C₉	—	19	3	no	—
39	C₂×C₁₉⋊C₃	C₂×C₅×ΓL(1,𝔽₈)	1	3	yes	—
40	C₃×C₅×C₇	—	19	1	no	—
41	C₅×C₁₉	C₂²×C₇⋊C₉	1	1	yes	characteristic, complemented, Hall
42	C₂×C₅×C₉	—	133	3	no	—
43	C₂²×C₃×C₇	—	19	1	no	—
44	C₂²×C₁₉	C₅×C₇⋊C₉	1	1	yes	characteristic, complemented, Hall
45	C₂×C₅×C₇	C₂×#4296	1	3	yes	complemented
46	C₇⋊C₉	—	19	1	no	Hall
47	C₂²×C₃×C₅	—	19	1	no	—
48	C₁₉⋊C₃	C₂²×C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic
49	C₅×C₉	—	133	1	no	Hall
50	C₂×C₃×C₇	—	19	3	no	—
51	C₂×C₁₉	C₂×C₅×C₇⋊C₉	1	3	yes	complemented
52	C₂²×C₉	—	133	1	no	Hall
53	C₅×C₇	C₂²×#4296	1	1	yes	characteristic, complemented, Hall
54	C₂×C₃×C₅	—	19	3	no	—
55	C₂²×C₇	C₅×#4296	1	1	yes	characteristic, complemented, Hall
56	C₃×C₇	—	19	1	no	—
57	C₂²×C₅	#11924780	1	1	yes	characteristic, direct product factor, Hall
58	C₁₉	C₂²×C₅×C₇⋊C₉	1	1	yes	characteristic, complemented, Sylow
59	C₂×C₉	—	133	3	no	—
60	C₃×C₅	—	19	1	no	—
61	C₂×C₇	C₂×C₅×#4296	1	3	yes	complemented
62	C₂²×C₃	—	19	1	no	—
63	C₂×C₅	C₂×#11924780	1	3	yes	direct product factor
64	C₉	—	133	1	no	Sylow
65	C₇	C₂²×C₅×#4296	1	1	yes	characteristic, complemented, Sylow
66	C₂×C₃	—	19	3	no	—
67	C₅	C₂²×#11924780	1	1	yes	characteristic, direct product factor, Sylow
68	C₂²	C₅×#11924780	1	1	yes	characteristic, direct product factor, Sylow
69	C₃	—	19	1	no	—
70	C₂	C₂×C₅×#11924780	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₅	#11924780
commutator, hypocenter	C₇×C₁₉	C₂²×C₅×C₉
Frattini	C₁	G
Fitting, socle	C₂²×C₅×C₇×C₁₉	C₉
prime radical	C₂²×C₅×C₇×C₁₉⋊C₃	C₃
cosocle	C₇×C₁₉⋊C₃	C₂²×C₃×C₅
LCS	G ▷ C₇×C₁₉
UCS	C₁ ◁ C₂²×C₅

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#11924780	C₂	direct (N × H)
3. C₅×#11924780	C₂²	direct (N × H)
4. C₂²×#11924780	C₅	direct (N × H)
7. C₂²×C₅×C₇×C₁₉	C₉	semidirect
8. C₂×#11924780	C₂×C₅	direct (N × H)
12. C₂×C₅×C₇×C₁₉	C₂×C₉	semidirect
14. #11924780	C₂²×C₅	direct (N × H)
19. C₅×C₇×C₁₉	C₂²×C₉	semidirect
22. C₂²×C₇×C₁₉	C₅×C₉	semidirect
25. C₂²×C₅×C₁₉	C₇⋊C₉	semidirect
29. C₂×C₇×C₁₉	C₂×C₅×C₉	semidirect
33. C₂×C₅×C₁₉	C₂×C₇⋊C₉	semidirect
36. C₂²×C₅×C₇	#4296	semidirect
37. C₇×C₁₉	C₂²×C₅×C₉	semidirect
41. C₅×C₁₉	C₂²×C₇⋊C₉	semidirect
44. C₂²×C₁₉	C₅×C₇⋊C₉	semidirect
45. C₂×C₅×C₇	C₂×#4296	semidirect
51. C₂×C₁₉	C₂×C₅×C₇⋊C₉	semidirect
53. C₅×C₇	C₂²×#4296	semidirect
55. C₂²×C₇	C₅×#4296	semidirect
57. C₂²×C₅	#11924780	direct (N × H)
58. C₁₉	C₂²×C₅×C₇⋊C₉	semidirect
61. C₂×C₇	C₂×C₅×#4296	semidirect
63. C₂×C₅	C₂×#11924780	direct (N × H)
65. C₇	C₂²×C₅×#4296	semidirect
67. C₅	C₂²×#11924780	direct (N × H)
68. C₂²	C₅×#11924780	direct (N × H)
70. C₂	C₂×C₅×#11924780	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×#11924780	2
C₂²×C₅×C₇×C₁₉⋊C₃	3
C₂²×#11924780	5
C₂²×C₅×#4296	7
C₂²×C₅×C₇⋊C₉	19

Subgroups — C22×C5×#11924780 #37561434

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×#11924780 #37561434