Subgroups — D₃×AGL(1,𝔽₅)×#5935 #72260191

Order 23040.

At least 9046 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

The subgroup lattice has 115 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	AGL(1,𝔽₅)×C₃⋊#5935	C₂	1	1	yes	maximal, characteristic, complemented
2	#72231175	C₂	1	1	yes	maximal, characteristic, complemented
3	D₃×#14514395	C₂	1	1	yes	maximal, characteristic, complemented
4	ΓL(1,𝔽₁₆)×#5935	C₂	1	1	yes	maximal, characteristic, complemented
5	D₃×D₅×#5935	C₂	1	1	yes	maximal, characteristic
6	#72231064	C₄	1	1	yes	characteristic, complemented
7	D₃×C₅⋊#5935	C₄	1	1	yes	characteristic, complemented
8	D₁₅×#5935	C₄	1	1	yes	characteristic, complemented
9	C₅×D₃×#5935	C₄	1	1	yes	characteristic, complemented
10	C₃×AGL(1,𝔽₅)×#5935	C₂	1	1	yes	maximal, characteristic, complemented
11	D₃×AGL(1,𝔽₅)×#997	C₂	1	1	yes	maximal, characteristic, complemented
12	#72230966	C₂²	1	1	yes	characteristic, complemented
13	D₅×C₃⋊#5935	C₂²	1	1	yes	characteristic
14	#72230913	C₂×C₄	1	1	yes	characteristic, complemented
15	C₅×C₃⋊#5935	C₂×C₄	1	1	yes	characteristic, complemented
16	C₃×#14514395	C₂²	1	1	yes	characteristic, complemented
17	C₃×D₅×#5935	C₂²	1	1	yes	characteristic
18	C₃×C₅⋊#5935	C₂×C₄	1	1	yes	characteristic, complemented
19	C₃×C₅×#5935	C₂×C₄	1	1	yes	characteristic, complemented
20	ΓL(1,𝔽₁₆)×#997	C₂²	1	1	yes	characteristic, complemented
21	D₃×D₅×#997	C₂²	1	1	yes	characteristic
22	D₁₅×#997	C₂×C₄	1	1	yes	characteristic, complemented
23	C₅×D₃×#997	C₂×C₄	1	1	yes	characteristic, complemented
24	C₃×AGL(1,𝔽₅)×#997	C₂²	1	1	yes	characteristic, complemented
25	C₃×D₅×#997	C₂³	1	1	yes	characteristic
26	D₃×AGL(1,𝔽₅)×H(5,𝔽₂)	D₃	1	1	yes	characteristic, complemented
27	AGL(1,𝔽₅)×#5935	D₃	1	1	yes	characteristic, direct product factor
28	D₅×#5935	C₂×D₃	1	1	yes	characteristic
29	AGL(1,𝔽₅)×#997	C₂×D₃	1	1	yes	characteristic, complemented
30	#14514395	C₂×D₃	1	1	yes	characteristic, complemented
31	D₃×D₅×H(5,𝔽₂)	C₂×D₃	1	1	yes	characteristic
32	H(5,𝔽₂)×ΓL(1,𝔽₁₆)	C₂×D₃	1	1	yes	characteristic, complemented
33	C₃×AGL(1,𝔽₅)×H(5,𝔽₂)	C₂×D₃	1	1	yes	characteristic, complemented
34	C₃×C₅×#997	C₂²×C₄	1	1	yes	characteristic, complemented
35	D₃×#5935	AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
36	C₂³×D₃×AGL(1,𝔽₅)	S₄	1	1	yes	characteristic
37	D₁₅×H(5,𝔽₂)	C₄×D₃	1	1	yes	characteristic, complemented
38	C₅×D₃×H(5,𝔽₂)	C₄×D₃	1	1	yes	characteristic, complemented
39	C₃×D₅×H(5,𝔽₂)	C₂²×D₃	1	1	yes	characteristic
40	D₅×#997	C₂²×D₃	1	1	yes	characteristic
41	C₅⋊#5935	C₄×D₃	1	1	yes	characteristic, complemented
42	C₅×#5935	C₄×D₃	1	1	yes	characteristic, complemented
43	D₃×Q₈×AGL(1,𝔽₅)	S₄	1	2	yes	complemented
44	AGL(1,𝔽₅)×H(5,𝔽₂)	D₃²	1	1	yes	characteristic, complemented
45	D₃×#997	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
46	C₃⋊#5935	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
47	C₃×#5935	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
48	C₂³×D₃×D₅	C₂×S₄	1	1	yes	characteristic
49	C₂³×ΓL(1,𝔽₁₆)	C₂×S₄	1	1	yes	characteristic
50	C₂³×C₃×AGL(1,𝔽₅)	C₂×S₄	1	1	yes	characteristic
51	C₃×C₅×H(5,𝔽₂)	C₂×C₄×D₃	1	1	yes	characteristic, complemented
52	C₅×#997	C₂×C₄×D₃	1	1	yes	characteristic, complemented
53	D₃×Q₈×D₅	C₂×S₄	1	2	yes	—
54	Q₈×ΓL(1,𝔽₁₆)	C₂×S₄	1	2	yes	complemented
55	C₃×Q₈×AGL(1,𝔽₅)	C₂×S₄	1	2	yes	complemented
56	D₅×H(5,𝔽₂)	C₂×D₃²	1	1	yes	characteristic
57	C₃×#997	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
58	C₂³×D₁₅	C₄×S₄	1	1	yes	characteristic
59	C₂³×C₅×D₃	C₄×S₄	1	1	yes	characteristic
60	C₂³×C₃×D₅	C₂²×S₄	1	1	yes	characteristic
61	C₂×D₃×AGL(1,𝔽₅)	#1020	1	1	yes	characteristic
62	Q₈×D₁₅	C₄×S₄	1	2	yes	complemented
63	C₅×D₃×Q₈	C₄×S₄	1	2	yes	complemented
64	C₃×Q₈×D₅	C₂²×S₄	1	2	yes	—
65	D₃×H(5,𝔽₂)	D₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
66	#5935	D₃×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
67	C₂³×AGL(1,𝔽₅)	D₃×S₄	1	1	yes	characteristic
68	C₅×H(5,𝔽₂)	C₄×D₃²	1	1	yes	characteristic, complemented
69	Q₈×AGL(1,𝔽₅)	D₃×S₄	1	2	yes	complemented
70	C₂³×C₃×C₅	C₂×C₄×S₄	1	1	yes	characteristic
71	C₂×D₃×D₅	C₂×#1020	1	1	yes	characteristic
72	C₂×ΓL(1,𝔽₁₆)	C₂×#1020	1	1	yes	characteristic
73	C₂×C₃×AGL(1,𝔽₅)	C₂×#1020	1	1	yes	characteristic
74	D₃×AGL(1,𝔽₅)	#5935	1	4	yes	direct product factor
75	C₃×C₅×Q₈	C₂×C₄×S₄	1	2	yes	complemented
76	C₃×H(5,𝔽₂)	C₂×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
77	#997	C₂×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
78	C₂³×D₅	C₂×D₃×S₄	1	1	yes	characteristic
79	Q₈×D₅	C₂×D₃×S₄	1	2	yes	—
80	C₂×D₁₅	C₄×#1020	1	1	yes	characteristic
81	C₂×C₅×D₃	C₄×#1020	1	1	yes	characteristic
82	C₂×C₃×D₅	C₂²×#1020	1	1	yes	characteristic
83	D₃×D₅	C₂×#5935	1	2	yes	—
84	D₃×D₅	#87520	1	2	yes	—
85	ΓL(1,𝔽₁₆)	C₂×#5935	1	2	yes	complemented
86	C₃×AGL(1,𝔽₅)	C₂×#5935	1	2	yes	complemented
87	C₂³×D₃	AGL(1,𝔽₅)×S₄	1	1	yes	characteristic
88	D₃×Q₈	AGL(1,𝔽₅)×S₄	1	2	yes	complemented
89	C₂³×C₅	C₄×D₃×S₄	1	1	yes	characteristic
90	C₂×AGL(1,𝔽₅)	D₃×#1020	1	1	yes	characteristic
91	C₅×Q₈	C₄×D₃×S₄	1	2	yes	complemented
92	H(5,𝔽₂)	D₃²×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
93	C₂×C₃×C₅	C₂×C₄×#1020	1	1	yes	characteristic
94	D₁₅	C₄×#5935	1	2	yes	complemented
95	C₃×D₅	C₂²×#5935	1	1	yes	characteristic
96	C₃×D₅	C₂×#87520	1	1	yes	characteristic
97	C₅×D₃	C₄×#5935	1	2	yes	complemented
98	C₂³×C₃	C₂×AGL(1,𝔽₅)×S₄	1	1	yes	characteristic
99	C₃×Q₈	C₂×AGL(1,𝔽₅)×S₄	1	2	yes	complemented
100	C₂×D₅	C₂×D₃×#1020	1	1	yes	characteristic
101	AGL(1,𝔽₅)	D₃×#5935	1	2	yes	direct product factor
102	C₃×C₅	C₂×C₄×#5935	1	1	yes	characteristic, complemented
103	C₂×D₃	AGL(1,𝔽₅)×#1020	1	1	yes	characteristic
104	C₂×C₅	C₄×D₃×#1020	1	1	yes	characteristic
105	D₅	C₂×D₃×#5935	1	1	yes	characteristic
106	D₅	D₃×#87520	1	1	yes	characteristic
107	C₂³	D₃×AGL(1,𝔽₅)×S₄	1	1	yes	characteristic
108	Q₈	D₃×AGL(1,𝔽₅)×S₄	1	2	yes	complemented
109	C₂×C₃	C₂×AGL(1,𝔽₅)×#1020	1	1	yes	characteristic
110	D₃	AGL(1,𝔽₅)×#5935	1	2	yes	direct product factor
111	C₅	C₄×D₃×#5935	1	1	yes	characteristic, complemented, Sylow
112	C₃	C₂×AGL(1,𝔽₅)×#5935	1	1	yes	characteristic, complemented
113	C₂	D₃×AGL(1,𝔽₅)×#1020	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃×AGL(1,𝔽₅)×#1020
commutator, hypocenter	C₃×C₅×#997	C₂²×C₄
Fitting	C₃×C₅×H(5,𝔽₂)	C₂×C₄×D₃
socle	C₂×C₃×C₅	C₂×C₄×#1020
prime radical	D₃×D₅×#5935	C₂
cosocle	C₃×D₅×#997	C₂³
LCS	G ▷ C₃×C₅×#997
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. AGL(1,𝔽₅)×C₃⋊#5935	C₂	semidirect
2. #72231175	C₂	semidirect
3. D₃×#14514395	C₂	semidirect
4. ΓL(1,𝔽₁₆)×#5935	C₂	semidirect
6. #72231064	C₄	semidirect
7. D₃×C₅⋊#5935	C₄	semidirect
8. D₁₅×#5935	C₄	semidirect
9. C₅×D₃×#5935	C₄	semidirect
10. C₃×AGL(1,𝔽₅)×#5935	C₂	semidirect
11. D₃×AGL(1,𝔽₅)×#997	C₂	semidirect
12. #72230966	C₂²	semidirect
14. #72230913	C₂×C₄	semidirect
15. C₅×C₃⋊#5935	C₂×C₄	semidirect
16. C₃×#14514395	C₂²	semidirect
18. C₃×C₅⋊#5935	C₂×C₄	semidirect
19. C₃×C₅×#5935	C₂×C₄	semidirect
20. ΓL(1,𝔽₁₆)×#997	C₂²	semidirect
22. D₁₅×#997	C₂×C₄	semidirect
23. C₅×D₃×#997	C₂×C₄	semidirect
24. C₃×AGL(1,𝔽₅)×#997	C₂²	semidirect
26. D₃×AGL(1,𝔽₅)×H(5,𝔽₂)	D₃	semidirect
27. AGL(1,𝔽₅)×#5935	D₃	direct (N × H)
29. AGL(1,𝔽₅)×#997	C₂×D₃	semidirect
30. #14514395	C₂×D₃	semidirect
32. H(5,𝔽₂)×ΓL(1,𝔽₁₆)	C₂×D₃	semidirect
33. C₃×AGL(1,𝔽₅)×H(5,𝔽₂)	C₂×D₃	semidirect
34. C₃×C₅×#997	C₂²×C₄	semidirect
35. D₃×#5935	AGL(1,𝔽₅)	direct (N × H)
37. D₁₅×H(5,𝔽₂)	C₄×D₃	semidirect
38. C₅×D₃×H(5,𝔽₂)	C₄×D₃	semidirect
41. C₅⋊#5935	C₄×D₃	semidirect
42. C₅×#5935	C₄×D₃	semidirect
43. D₃×Q₈×AGL(1,𝔽₅)	S₄	semidirect
44. AGL(1,𝔽₅)×H(5,𝔽₂)	D₃²	semidirect
45. D₃×#997	C₂×AGL(1,𝔽₅)	semidirect
46. C₃⋊#5935	C₂×AGL(1,𝔽₅)	semidirect
47. C₃×#5935	C₂×AGL(1,𝔽₅)	semidirect
51. C₃×C₅×H(5,𝔽₂)	C₂×C₄×D₃	semidirect
52. C₅×#997	C₂×C₄×D₃	semidirect
54. Q₈×ΓL(1,𝔽₁₆)	C₂×S₄	semidirect
55. C₃×Q₈×AGL(1,𝔽₅)	C₂×S₄	semidirect
57. C₃×#997	C₂²×AGL(1,𝔽₅)	semidirect
62. Q₈×D₁₅	C₄×S₄	semidirect
63. C₅×D₃×Q₈	C₄×S₄	semidirect
65. D₃×H(5,𝔽₂)	D₃×AGL(1,𝔽₅)	semidirect
66. #5935	D₃×AGL(1,𝔽₅)	direct (N × H)
68. C₅×H(5,𝔽₂)	C₄×D₃²	semidirect
69. Q₈×AGL(1,𝔽₅)	D₃×S₄	semidirect
74. D₃×AGL(1,𝔽₅)	#5935	direct (N × H)
75. C₃×C₅×Q₈	C₂×C₄×S₄	semidirect
76. C₃×H(5,𝔽₂)	C₂×D₃×AGL(1,𝔽₅)	semidirect
77. #997	C₂×D₃×AGL(1,𝔽₅)	semidirect
85. ΓL(1,𝔽₁₆)	C₂×#5935	semidirect
86. C₃×AGL(1,𝔽₅)	C₂×#5935	semidirect
88. D₃×Q₈	AGL(1,𝔽₅)×S₄	semidirect
91. C₅×Q₈	C₄×D₃×S₄	semidirect
92. H(5,𝔽₂)	D₃²×AGL(1,𝔽₅)	semidirect
94. D₁₅	C₄×#5935	semidirect
97. C₅×D₃	C₄×#5935	semidirect
99. C₃×Q₈	C₂×AGL(1,𝔽₅)×S₄	semidirect
101. AGL(1,𝔽₅)	D₃×#5935	direct (N × H)
102. C₃×C₅	C₂×C₄×#5935	semidirect
108. Q₈	D₃×AGL(1,𝔽₅)×S₄	semidirect
110. D₃	AGL(1,𝔽₅)×#5935	direct (N × H)
111. C₅	C₄×D₃×#5935	semidirect
112. C₃	C₂×AGL(1,𝔽₅)×#5935	semidirect

Maximal subgroups

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

subgroup	index
D₃×AGL(1,𝔽₅)×GL(2,𝔽₃)	4
C₂×D₃×AGL(1,𝔽₅)×S₄	4
C₄×D₃×#5935	5
D₃×AGL(1,𝔽₅)×#453	3
AGL(1,𝔽₅)×C₃⋊#5935	2
#72231175	2
D₃×#14514395	2
ΓL(1,𝔽₁₆)×#5935	2
D₃×D₅×#5935	2
C₃×AGL(1,𝔽₅)×#5935	2
D₃×AGL(1,𝔽₅)×#997	2
C₂×AGL(1,𝔽₅)×#5935	3

Subgroups — D3×AGL(1,𝔽5)×#5935 #72260191

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃×AGL(1,𝔽₅)×#5935 #72260191