Subgroups — ΓL(1,𝔽₁₆)×#10597244 #49137449

Order 36300.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×D₅×#10597244	C₂	1	1	yes	maximal, characteristic
2	AGL(1,𝔽₅)×#10597244	—	3	1	no	maximal, Hall
3	C₃×C₅×#10597244	C₄	1	1	yes	characteristic, complemented, Hall
4	C₁₁²×ΓL(1,𝔽₁₆)	C₅	1	1	yes	maximal, characteristic, complemented
5	Dic₃×#10597244	—	5	1	no	maximal
6	D₅×#10597244	D₃	1	1	yes	characteristic
7	C₃×D₅×C₁₁²	C₂×C₅	1	1	yes	characteristic
8	C₂×C₃×#10597244	—	5	1	no	—
9	C₁₁⋊C₅×ΓL(1,𝔽₁₆)	—	11	2	no	maximal
10	C₅×#10597244	Dic₃	1	1	yes	characteristic, complemented, Hall
11	C₁₁²×AGL(1,𝔽₅)	—	3	1	no	—
12	C₄×#10597244	—	15	1	no	—
13	C₃×C₅×C₁₁²	C₄×C₅	1	1	yes	characteristic, complemented
14	C₃×#10597244	—	4	1	no	—
15	C₃×#10597244	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
16	C₃×D₅×C₁₁⋊C₅	—	11	2	no	—
17	C₁₁²×Dic₃	—	5	1	no	Hall
18	D₅×C₁₁²	C₅×D₃	1	1	yes	characteristic
19	C₂×#10597244	—	5	1	no	—
20	AGL(1,𝔽₅)×C₁₁⋊C₅	—	33	2	no	—
21	C₃×C₅×C₁₁⋊C₅	—	11	2	no	—
22	C₂×C₃×C₁₁²	—	5	1	no	—
23	C₁₁×ΓL(1,𝔽₁₆)	—	5	2	no	—
24	C₁₁×ΓL(1,𝔽₁₆)	C₁₁⋊C₅	1	2	yes	complemented
25	Dic₃×C₁₁⋊C₅	—	55	2	no	—
26	C₅×C₁₁²	C₅×Dic₃	1	1	yes	characteristic, complemented
27	#10597244	—	4	1	no	—
28	#10597244	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, direct product factor
29	D₅×C₁₁⋊C₅	—	11	2	no	—
30	C₄×C₁₁²	—	15	1	no	Hall
31	C₃×C₁₁²	C₅×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
32	C₃×D₅×C₁₁	—	5	2	no	—
33	C₃×D₅×C₁₁	C₂×C₁₁⋊C₅	1	2	yes	—
34	C₂×C₃×C₁₁⋊C₅	—	55	2	no	—
35	C₅×ΓL(1,𝔽₁₆)	—	121	1	no	Hall
36	C₅×C₁₁⋊C₅	—	11	2	no	—
37	C₂×C₁₁²	—	5	1	no	—
38	C₁₁×AGL(1,𝔽₅)	—	3	2	no	—
39	C₁₁×AGL(1,𝔽₅)	—	15	2	no	—
40	C₄×C₁₁⋊C₅	—	165	2	no	—
41	C₃×C₅×C₁₁	—	5	2	no	—
42	C₃×C₅×C₁₁	C₄×C₁₁⋊C₅	1	2	yes	complemented
43	C₃×C₁₁⋊C₅	—	11	2	no	—
44	C₃×C₁₁⋊C₅	—	44	2	no	—
45	C₃×C₅×D₅	—	121	1	no	—
46	C₁₁×Dic₃	—	5	2	no	—
47	C₁₁×Dic₃	—	25	2	no	—
48	C₁₁²	C₅×ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented, Sylow
49	D₅×C₁₁	—	5	2	no	—
50	D₅×C₁₁	D₃×C₁₁⋊C₅	1	2	yes	—
51	C₂×C₁₁⋊C₅	—	55	2	no	—
52	C₅×AGL(1,𝔽₅)	—	363	1	no	Hall
53	C₃×C₅²	—	121	1	no	Hall
54	C₂×C₃×C₁₁	—	5	2	no	—
55	C₂×C₃×C₁₁	—	25	2	no	—
56	ΓL(1,𝔽₁₆)	#10597244	1	1	yes	characteristic, direct product factor
57	C₅×Dic₃	—	605	1	no	—
58	C₅×C₁₁	—	5	2	no	—
59	C₅×C₁₁	Dic₃×C₁₁⋊C₅	1	2	yes	complemented
60	C₁₁⋊C₅	—	11	2	no	—
61	C₁₁⋊C₅	—	44	2	no	—
62	C₅×D₅	—	121	1	no	—
63	C₄×C₁₁	—	15	2	no	—
64	C₄×C₁₁	—	75	2	no	—
65	C₃×C₁₁	—	5	2	no	—
66	C₃×C₁₁	AGL(1,𝔽₅)×C₁₁⋊C₅	1	2	yes	complemented
67	C₂×C₃×C₅	—	605	1	no	—
68	C₃×D₅	C₂×#10597244	1	1	yes	characteristic
69	C₅²	—	121	1	no	Sylow
70	C₂×C₁₁	—	5	2	no	—
71	C₂×C₁₁	—	25	2	no	—
72	AGL(1,𝔽₅)	—	3	1	no	—
73	C₄×C₅	—	1815	1	no	—
74	C₃×C₅	—	121	1	no	—
75	C₃×C₅	—	484	1	no	—
76	C₃×C₅	C₄×#10597244	1	1	yes	characteristic, complemented
77	Dic₃	—	5	1	no	Hall
78	C₁₁	—	5	2	no	—
79	C₁₁	C₁₁⋊C₅×ΓL(1,𝔽₁₆)	1	2	yes	complemented
80	C₂×C₅	—	605	1	no	—
81	D₅	D₃×#10597244	1	1	yes	characteristic
82	C₂×C₃	—	5	1	no	—
83	C₅	—	121	1	no	—
84	C₅	—	484	1	no	—
85	C₅	Dic₃×#10597244	1	1	yes	characteristic, complemented
86	C₄	—	15	1	no	Sylow
87	C₃	AGL(1,𝔽₅)×#10597244	1	1	yes	characteristic, complemented, Sylow
88	C₂	—	5	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃×C₅×C₁₁²	C₄×C₅
prime radical	C₃×D₅×#10597244	C₂
cosocle	C₃×D₅×C₁₁²	C₂×C₅
LCS	G ▷ C₃×C₅×C₁₁²
UCS	C₁

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×#10597244	C₄	semidirect
4. C₁₁²×ΓL(1,𝔽₁₆)	C₅	semidirect
10. C₅×#10597244	Dic₃	semidirect
13. C₃×C₅×C₁₁²	C₄×C₅	semidirect
15. C₃×#10597244	AGL(1,𝔽₅)	semidirect
24. C₁₁×ΓL(1,𝔽₁₆)	C₁₁⋊C₅	semidirect
26. C₅×C₁₁²	C₅×Dic₃	semidirect
28. #10597244	ΓL(1,𝔽₁₆)	direct (N × H)
31. C₃×C₁₁²	C₅×AGL(1,𝔽₅)	semidirect
42. C₃×C₅×C₁₁	C₄×C₁₁⋊C₅	semidirect
48. C₁₁²	C₅×ΓL(1,𝔽₁₆)	semidirect
56. ΓL(1,𝔽₁₆)	#10597244	direct (N × H)
59. C₅×C₁₁	Dic₃×C₁₁⋊C₅	semidirect
66. C₃×C₁₁	AGL(1,𝔽₅)×C₁₁⋊C₅	semidirect
76. C₃×C₅	C₄×#10597244	semidirect
79. C₁₁	C₁₁⋊C₅×ΓL(1,𝔽₁₆)	semidirect
85. C₅	Dic₃×#10597244	semidirect
87. C₃	AGL(1,𝔽₅)×#10597244	semidirect

Maximal subgroups

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

subgroup	index
C₃×D₅×#10597244	2
AGL(1,𝔽₅)×#10597244	3
C₁₁²×ΓL(1,𝔽₁₆)	5
Dic₃×#10597244	5
C₁₁⋊C₅×ΓL(1,𝔽₁₆)	11

Subgroups — ΓL(1,𝔽16)×#10597244 #49137449

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₁₆)×#10597244 #49137449