Subgroups — C₃×C₅×C₁₃×C₂²⋊ΓL(1,𝔽₈) #30655772

Order 16380.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₁₃×C₂²⋊ΓL(1,𝔽₈)	C₃	1	3	yes	maximal, direct product factor
2	C₂²×C₃×C₅×C₇×C₁₃	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×C₁₃×ΓL(1,𝔽₈)	—	4	1	no	maximal, Hall
4	C₃×C₁₃×C₂²⋊ΓL(1,𝔽₈)	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂×C₃×C₅×C₇×C₁₃	—	3	1	no	—
6	C₃×C₅×A₄×C₁₃	—	7	1	no	maximal, Hall
7	C₂²×C₅×C₇×C₁₃	C₃²	1	1	yes	characteristic, complemented, Hall
8	C₃×C₅×C₇×C₁₃	A₄	1	1	yes	characteristic, complemented
9	C₅×C₁₃×ΓL(1,𝔽₈)	—	4	3	no	—
10	C₃×C₅×C₂²⋊ΓL(1,𝔽₈)	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂²×C₃×C₇×C₁₃	C₃×C₅	1	1	yes	characteristic, complemented
12	C₁₃×C₂²⋊ΓL(1,𝔽₈)	C₃×C₅	1	3	yes	direct product factor
13	C₂×C₅×C₇×C₁₃	—	3	1	no	—
14	C₃×C₁₃×ΓL(1,𝔽₈)	—	4	1	no	Hall
15	C₂²×C₃×C₅×C₁₃	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
16	C₅×A₄×C₁₃	—	7	3	no	—
17	C₃²×C₅×C₁₃	—	28	1	no	Hall
18	C₂×C₃×C₇×C₁₃	—	3	1	no	—
19	C₃×A₄×C₁₃	—	7	1	no	Hall
20	C₅×C₇×C₁₃	C₃×A₄	1	1	yes	characteristic, complemented, Hall
21	C₂²×C₃×C₅×C₇	C₃×C₁₃	1	1	yes	characteristic, complemented
22	C₅×C₂²⋊ΓL(1,𝔽₈)	C₃×C₁₃	1	3	yes	direct product factor
23	C₂×C₃×C₅×C₁₃	—	3	1	no	—
24	C₂²×C₇×C₁₃	C₃²×C₅	1	1	yes	characteristic, complemented, Hall
25	C₃×C₅×ΓL(1,𝔽₈)	—	4	1	no	Hall
26	C₃×C₇×C₁₃	C₅×A₄	1	1	yes	characteristic, complemented
27	C₁₃×ΓL(1,𝔽₈)	—	4	3	no	—
28	C₂²×C₅×C₁₃	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
29	C₃×C₂²⋊ΓL(1,𝔽₈)	C₅×C₁₃	1	1	yes	characteristic, direct product factor, Hall
30	C₂×C₃×C₅×C₇	—	3	1	no	—
31	C₃×C₅×C₁₃	—	28	3	no	—
32	C₃×C₅×C₁₃	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
33	C₂×C₇×C₁₃	—	3	1	no	—
34	C₃×C₅×A₄	—	7	1	no	Hall
35	C₂²×C₃×C₁₃	C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
36	A₄×C₁₃	—	7	3	no	—
37	C₂²×C₅×C₇	C₃²×C₁₃	1	1	yes	characteristic, complemented, Hall
38	C₂×C₅×C₁₃	—	3	1	no	—
39	C₃²×C₁₃	—	28	1	no	Hall
40	C₃×C₅×C₇	A₄×C₁₃	1	1	yes	characteristic, complemented
41	C₅×ΓL(1,𝔽₈)	—	4	3	no	—
42	C₇×C₁₃	C₃×C₅×A₄	1	1	yes	characteristic, complemented, Hall
43	C₂²×C₃×C₇	C₃×C₅×C₁₃	1	1	yes	characteristic, complemented
44	C₂²⋊ΓL(1,𝔽₈)	C₃×C₅×C₁₃	1	3	yes	direct product factor
45	C₂×C₃×C₁₃	—	3	1	no	—
46	C₂×C₅×C₇	—	3	1	no	—
47	C₅×C₁₃	C₃×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
48	C₃×ΓL(1,𝔽₈)	—	4	1	no	Hall
49	C₂²×C₃×C₅	C₁₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
50	C₅×A₄	—	7	3	no	—
51	C₂²×C₁₃	C₃×C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
52	C₃²×C₅	—	28	1	no	Hall
53	C₂×C₃×C₇	—	3	1	no	—
54	C₃×C₁₃	—	28	3	no	—
55	C₃×C₁₃	C₅×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
56	C₃×A₄	—	7	1	no	Hall
57	C₅×C₇	C₃×A₄×C₁₃	1	1	yes	characteristic, complemented, Hall
58	C₂×C₃×C₅	—	3	1	no	—
59	C₂²×C₇	C₃²×C₅×C₁₃	1	1	yes	characteristic, complemented, Hall
60	C₂×C₁₃	—	3	1	no	—
61	C₃×C₇	C₅×A₄×C₁₃	1	1	yes	characteristic, complemented
62	ΓL(1,𝔽₈)	—	4	3	no	—
63	C₂²×C₅	C₃×C₁₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
64	C₃×C₅	—	28	3	no	—
65	C₃×C₅	C₁₃×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
66	C₂×C₇	—	3	1	no	—
67	C₁₃	C₃×C₅×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
68	C₂²×C₃	C₅×C₁₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
69	A₄	—	7	3	no	—
70	C₂×C₅	—	3	1	no	—
71	C₃²	—	28	1	no	Sylow
72	C₇	C₃×C₅×A₄×C₁₃	1	1	yes	characteristic, complemented, Sylow
73	C₂×C₃	—	3	1	no	—
74	C₅	C₃×C₁₃×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
75	C₂²	C₃×C₅×C₁₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
76	C₃	—	28	3	no	—
77	C₃	C₅×C₁₃×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
78	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₃×C₅×C₁₃	C₂²⋊ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₂²×C₇	C₃²×C₅×C₁₃
Frattini	C₁	G
Fitting, socle	C₂²×C₃×C₅×C₇×C₁₃	C₃
LCS	G ▷ C₂²×C₇
UCS	C₁ ◁ C₃×C₅×C₁₃

Complements

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

N	H ≅ G/N	type
1. C₅×C₁₃×C₂²⋊ΓL(1,𝔽₈)	C₃	direct (N × H)
2. C₂²×C₃×C₅×C₇×C₁₃	C₃	semidirect
4. C₃×C₁₃×C₂²⋊ΓL(1,𝔽₈)	C₅	direct (N × H)
7. C₂²×C₅×C₇×C₁₃	C₃²	semidirect
8. C₃×C₅×C₇×C₁₃	A₄	semidirect
10. C₃×C₅×C₂²⋊ΓL(1,𝔽₈)	C₁₃	direct (N × H)
11. C₂²×C₃×C₇×C₁₃	C₃×C₅	semidirect
12. C₁₃×C₂²⋊ΓL(1,𝔽₈)	C₃×C₅	direct (N × H)
15. C₂²×C₃×C₅×C₁₃	ΓL(1,𝔽₈)	semidirect
20. C₅×C₇×C₁₃	C₃×A₄	semidirect
21. C₂²×C₃×C₅×C₇	C₃×C₁₃	semidirect
22. C₅×C₂²⋊ΓL(1,𝔽₈)	C₃×C₁₃	direct (N × H)
24. C₂²×C₇×C₁₃	C₃²×C₅	semidirect
26. C₃×C₇×C₁₃	C₅×A₄	semidirect
28. C₂²×C₅×C₁₃	C₃×ΓL(1,𝔽₈)	semidirect
29. C₃×C₂²⋊ΓL(1,𝔽₈)	C₅×C₁₃	direct (N × H)
32. C₃×C₅×C₁₃	C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
35. C₂²×C₃×C₁₃	C₅×ΓL(1,𝔽₈)	semidirect
37. C₂²×C₅×C₇	C₃²×C₁₃	semidirect
40. C₃×C₅×C₇	A₄×C₁₃	semidirect
42. C₇×C₁₃	C₃×C₅×A₄	semidirect
43. C₂²×C₃×C₇	C₃×C₅×C₁₃	semidirect
44. C₂²⋊ΓL(1,𝔽₈)	C₃×C₅×C₁₃	direct (N × H)
47. C₅×C₁₃	C₃×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
49. C₂²×C₃×C₅	C₁₃×ΓL(1,𝔽₈)	semidirect
51. C₂²×C₁₃	C₃×C₅×ΓL(1,𝔽₈)	semidirect
55. C₃×C₁₃	C₅×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
57. C₅×C₇	C₃×A₄×C₁₃	semidirect
59. C₂²×C₇	C₃²×C₅×C₁₃	semidirect
61. C₃×C₇	C₅×A₄×C₁₃	semidirect
63. C₂²×C₅	C₃×C₁₃×ΓL(1,𝔽₈)	semidirect
65. C₃×C₅	C₁₃×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
67. C₁₃	C₃×C₅×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
68. C₂²×C₃	C₅×C₁₃×ΓL(1,𝔽₈)	semidirect
72. C₇	C₃×C₅×A₄×C₁₃	semidirect
74. C₅	C₃×C₁₃×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
75. C₂²	C₃×C₅×C₁₃×ΓL(1,𝔽₈)	semidirect
77. C₃	C₅×C₁₃×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×C₁₃×C₂²⋊ΓL(1,𝔽₈)	3
C₂²×C₃×C₅×C₇×C₁₃	3
C₃×C₅×C₁₃×ΓL(1,𝔽₈)	4
C₃×C₁₃×C₂²⋊ΓL(1,𝔽₈)	5
C₃×C₅×A₄×C₁₃	7
C₃×C₅×C₂²⋊ΓL(1,𝔽₈)	13

Subgroups — C3×C5×C13×C22⋊ΓL(1,𝔽8) #30655772

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×C₁₃×C₂²⋊ΓL(1,𝔽₈) #30655772