Subgroups — C₇²×C₁₃×ΓL(1,𝔽₁₆) #50304415

Order 38220.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇²×D₅×C₁₃	C₂	1	1	yes	maximal, characteristic
2	C₇²×C₁₃×AGL(1,𝔽₅)	—	3	1	no	maximal, Hall
3	C₃×C₅×C₇²×C₁₃	C₄	1	1	yes	characteristic, complemented, Hall
4	C₇²×Dic₃×C₁₃	—	5	1	no	maximal, Hall
5	C₇²×D₅×C₁₃	D₃	1	1	yes	characteristic
6	C₇×C₁₃×ΓL(1,𝔽₁₆)	C₇	1	8	yes	maximal, direct product factor
7	C₂×C₃×C₇²×C₁₃	—	5	1	no	—
8	C₅×C₇²×C₁₃	Dic₃	1	1	yes	characteristic, complemented, Hall
9	C₇²×ΓL(1,𝔽₁₆)	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₃×C₇×D₅×C₁₃	C₂×C₇	1	8	yes	—
11	C₄×C₇²×C₁₃	—	15	1	no	Hall
12	C₃×C₇²×C₁₃	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
13	C₇×C₁₃×AGL(1,𝔽₅)	—	3	8	no	—
14	C₃×C₇²×D₅	C₂×C₁₃	1	1	yes	characteristic
15	C₃×C₅×C₇×C₁₃	C₄×C₇	1	8	yes	complemented
16	C₂×C₇²×C₁₃	—	5	1	no	—
17	C₇×Dic₃×C₁₃	—	5	8	no	—
18	C₇²×AGL(1,𝔽₅)	—	3	1	no	Hall
19	C₇×D₅×C₁₃	D₃×C₇	1	8	yes	—
20	C₁₃×ΓL(1,𝔽₁₆)	C₇²	1	1	yes	characteristic, direct product factor, Hall
21	C₃×C₅×C₇²	C₄×C₁₃	1	1	yes	characteristic, complemented, Hall
22	C₇²×C₁₃	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, direct product factor, Hall
23	C₇²×Dic₃	—	5	1	no	Hall
24	C₂×C₃×C₇×C₁₃	—	5	8	no	—
25	C₇²×D₅	D₃×C₁₃	1	1	yes	characteristic
26	C₅×C₇×C₁₃	C₇×Dic₃	1	8	yes	complemented
27	C₇×ΓL(1,𝔽₁₆)	C₇×C₁₃	1	8	yes	direct product factor
28	C₃×D₅×C₁₃	C₂×C₇²	1	1	yes	characteristic
29	C₄×C₇×C₁₃	—	15	8	no	—
30	C₂×C₃×C₇²	—	5	1	no	—
31	C₃×C₇×C₁₃	C₇×AGL(1,𝔽₅)	1	8	yes	complemented
32	C₁₃×AGL(1,𝔽₅)	—	3	1	no	Hall
33	C₅×C₇²	Dic₃×C₁₃	1	1	yes	characteristic, complemented, Hall
34	C₃×C₇×D₅	C₂×C₇×C₁₃	1	8	yes	—
35	C₄×C₇²	—	15	1	no	Hall
36	C₃×C₅×C₁₃	C₄×C₇²	1	1	yes	characteristic, complemented, Hall
37	C₂×C₇×C₁₃	—	5	8	no	—
38	Dic₃×C₁₃	—	5	1	no	Hall
39	C₃×C₇²	C₁₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
40	C₇×AGL(1,𝔽₅)	—	3	8	no	—
41	D₅×C₁₃	D₃×C₇²	1	1	yes	characteristic
42	C₃×C₅×C₇	C₄×C₇×C₁₃	1	8	yes	complemented
43	C₂×C₇²	—	5	1	no	—
44	C₇×C₁₃	C₇×ΓL(1,𝔽₁₆)	1	8	yes	direct product factor
45	C₇×Dic₃	—	5	8	no	—
46	C₂×C₃×C₁₃	—	5	1	no	—
47	C₇×D₅	D₃×C₇×C₁₃	1	8	yes	—
48	C₅×C₁₃	C₇²×Dic₃	1	1	yes	characteristic, complemented, Hall
49	ΓL(1,𝔽₁₆)	C₇²×C₁₃	1	1	yes	characteristic, direct product factor, Hall
50	C₄×C₁₃	—	15	1	no	Hall
51	C₇²	C₁₃×ΓL(1,𝔽₁₆)	1	1	yes	characteristic, direct product factor, Sylow
52	C₂×C₃×C₇	—	5	8	no	—
53	C₃×C₁₃	C₇²×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
54	C₅×C₇	C₇×Dic₃×C₁₃	1	8	yes	complemented
55	C₃×D₅	C₂×C₇²×C₁₃	1	1	yes	characteristic
56	C₄×C₇	—	15	8	no	—
57	C₂×C₁₃	—	5	1	no	—
58	C₃×C₇	C₇×C₁₃×AGL(1,𝔽₅)	1	8	yes	complemented
59	AGL(1,𝔽₅)	—	3	1	no	Hall
60	C₃×C₅	C₄×C₇²×C₁₃	1	1	yes	characteristic, complemented, Hall
61	C₂×C₇	—	5	8	no	—
62	C₁₃	C₇²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic, direct product factor, Sylow
63	Dic₃	—	5	1	no	Hall
64	D₅	D₃×C₇²×C₁₃	1	1	yes	characteristic
65	C₇	C₇×C₁₃×ΓL(1,𝔽₁₆)	1	8	yes	direct product factor
66	C₂×C₃	—	5	1	no	—
67	C₅	C₇²×Dic₃×C₁₃	1	1	yes	characteristic, complemented, Sylow
68	C₄	—	15	1	no	Sylow
69	C₃	C₇²×C₁₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Sylow
70	C₂	—	5	1	no	—

	subgroup	G/H
center, hypercenter	C₇²×C₁₃	ΓL(1,𝔽₁₆)
commutator, hypocenter	C₃×C₅	C₄×C₇²×C₁₃
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₇²×C₁₃	C₄
prime radical	C₃×C₇²×D₅×C₁₃	C₂
cosocle	C₃×D₅	C₂×C₇²×C₁₃
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₇²×C₁₃

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×C₇²×C₁₃	C₄	semidirect
6. C₇×C₁₃×ΓL(1,𝔽₁₆)	C₇	direct (N × H)
8. C₅×C₇²×C₁₃	Dic₃	semidirect
9. C₇²×ΓL(1,𝔽₁₆)	C₁₃	direct (N × H)
12. C₃×C₇²×C₁₃	AGL(1,𝔽₅)	semidirect
15. C₃×C₅×C₇×C₁₃	C₄×C₇	semidirect
20. C₁₃×ΓL(1,𝔽₁₆)	C₇²	direct (N × H)
21. C₃×C₅×C₇²	C₄×C₁₃	semidirect
22. C₇²×C₁₃	ΓL(1,𝔽₁₆)	direct (N × H)
26. C₅×C₇×C₁₃	C₇×Dic₃	semidirect
27. C₇×ΓL(1,𝔽₁₆)	C₇×C₁₃	direct (N × H)
31. C₃×C₇×C₁₃	C₇×AGL(1,𝔽₅)	semidirect
33. C₅×C₇²	Dic₃×C₁₃	semidirect
36. C₃×C₅×C₁₃	C₄×C₇²	semidirect
39. C₃×C₇²	C₁₃×AGL(1,𝔽₅)	semidirect
42. C₃×C₅×C₇	C₄×C₇×C₁₃	semidirect
44. C₇×C₁₃	C₇×ΓL(1,𝔽₁₆)	direct (N × H)
48. C₅×C₁₃	C₇²×Dic₃	semidirect
49. ΓL(1,𝔽₁₆)	C₇²×C₁₃	direct (N × H)
51. C₇²	C₁₃×ΓL(1,𝔽₁₆)	direct (N × H)
53. C₃×C₁₃	C₇²×AGL(1,𝔽₅)	semidirect
54. C₅×C₇	C₇×Dic₃×C₁₃	semidirect
58. C₃×C₇	C₇×C₁₃×AGL(1,𝔽₅)	semidirect
60. C₃×C₅	C₄×C₇²×C₁₃	semidirect
62. C₁₃	C₇²×ΓL(1,𝔽₁₆)	direct (N × H)
65. C₇	C₇×C₁₃×ΓL(1,𝔽₁₆)	direct (N × H)
67. C₅	C₇²×Dic₃×C₁₃	semidirect
69. C₃	C₇²×C₁₃×AGL(1,𝔽₅)	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₇²×D₅×C₁₃	2
C₇²×C₁₃×AGL(1,𝔽₅)	3
C₇²×Dic₃×C₁₃	5
C₇×C₁₃×ΓL(1,𝔽₁₆)	7
C₇²×ΓL(1,𝔽₁₆)	13

Subgroups — C72×C13×ΓL(1,𝔽16) #50304415

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇²×C₁₃×ΓL(1,𝔽₁₆) #50304415