Subgroups — C₅×A₄×C₁₃×ΓL(1,𝔽₈) #30655549

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	2	yes	maximal, complemented
2	C₅×C₇×A₄×C₁₃	C₃	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₅×C₁₃×ΓL(1,𝔽₈)	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×C₁₃×ΓL(1,𝔽₈)	—	4	1	no	maximal, Hall
5	A₄×C₁₃×ΓL(1,𝔽₈)	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₅×C₁₃×ΓL(1,𝔽₈)	—	3	1	no	—
7	C₃×C₅×A₄×C₁₃	—	7	1	no	maximal, Hall
8	C₂²×C₅×C₇×C₁₃	C₃²	1	1	yes	characteristic, complemented, Hall
9	C₃×C₅×C₇×C₁₃	—	4	1	no	—
10	C₅×C₁₃×ΓL(1,𝔽₈)	—	4	2	no	—
11	C₅×C₁₃×ΓL(1,𝔽₈)	A₄	1	1	yes	characteristic, direct product factor
12	C₅×A₄×ΓL(1,𝔽₈)	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₁₃×C₂²⋊ΓL(1,𝔽₈)	C₃×C₅	1	2	yes	complemented
14	C₇×A₄×C₁₃	C₃×C₅	1	1	yes	characteristic, complemented
15	C₂²×C₁₃×ΓL(1,𝔽₈)	C₃×C₅	1	1	yes	characteristic, complemented
16	C₂×C₅×C₇×C₁₃	—	3	1	no	—
17	C₃×C₁₃×ΓL(1,𝔽₈)	—	4	1	no	Hall
18	C₂²×C₃×C₅×C₁₃	—	7	1	no	—
19	C₅×A₄×C₁₃	—	7	2	no	—
20	C₅×A₄×C₁₃	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
21	C₃²×C₅×C₁₃	—	28	1	no	Hall
22	C₂×C₁₃×ΓL(1,𝔽₈)	—	3	1	no	—
23	C₃×A₄×C₁₃	—	7	1	no	Hall
24	C₅×C₇×C₁₃	C₃×A₄	1	1	yes	characteristic, complemented, Hall
25	C₅×C₂²⋊ΓL(1,𝔽₈)	C₃×C₁₃	1	2	yes	complemented
26	C₅×C₇×A₄	C₃×C₁₃	1	1	yes	characteristic, complemented
27	C₂²×C₅×ΓL(1,𝔽₈)	C₃×C₁₃	1	1	yes	characteristic, complemented
28	C₂×C₃×C₅×C₁₃	—	21	1	no	—
29	C₂²×C₇×C₁₃	C₃²×C₅	1	1	yes	characteristic, complemented, Hall
30	C₃×C₅×ΓL(1,𝔽₈)	—	4	1	no	Hall
31	C₃×C₇×C₁₃	—	4	1	no	—
32	C₁₃×ΓL(1,𝔽₈)	—	4	2	no	—
33	C₁₃×ΓL(1,𝔽₈)	C₅×A₄	1	1	yes	characteristic, direct product factor
34	C₂²×C₅×C₁₃	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
35	A₄×ΓL(1,𝔽₈)	C₅×C₁₃	1	1	yes	characteristic, direct product factor, Hall
36	C₂×C₅×ΓL(1,𝔽₈)	—	3	1	no	—
37	C₃×C₅×C₁₃	—	4	1	no	—
38	C₃×C₅×C₁₃	—	7	1	no	—
39	C₃×C₅×C₁₃	—	28	2	no	—
40	C₂×C₇×C₁₃	—	3	1	no	—
41	C₃×C₅×A₄	—	7	1	no	Hall
42	C₂²×C₃×C₁₃	—	7	1	no	—
43	A₄×C₁₃	—	7	2	no	—
44	A₄×C₁₃	C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
45	C₂²×C₅×C₇	C₃²×C₁₃	1	1	yes	characteristic, complemented, Hall
46	C₂×C₅×C₁₃	—	3	1	no	—
47	C₃²×C₁₃	—	28	1	no	Hall
48	C₃×C₅×C₇	—	4	1	no	—
49	C₅×ΓL(1,𝔽₈)	—	4	2	no	—
50	C₅×ΓL(1,𝔽₈)	A₄×C₁₃	1	1	yes	characteristic, direct product factor
51	C₇×C₁₃	C₃×C₅×A₄	1	1	yes	characteristic, complemented, Hall
52	C₂²⋊ΓL(1,𝔽₈)	C₃×C₅×C₁₃	1	2	yes	complemented
53	C₇×A₄	C₃×C₅×C₁₃	1	1	yes	characteristic, complemented
54	C₂²×ΓL(1,𝔽₈)	C₃×C₅×C₁₃	1	1	yes	characteristic, complemented
55	C₂×C₃×C₁₃	—	21	1	no	—
56	C₂×C₅×C₇	—	3	1	no	—
57	C₅×C₁₃	A₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
58	C₃×ΓL(1,𝔽₈)	—	4	1	no	Hall
59	C₂²×C₃×C₅	—	7	1	no	—
60	C₅×A₄	—	7	2	no	—
61	C₅×A₄	C₁₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
62	C₂²×C₁₃	C₃×C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
63	C₃²×C₅	—	28	1	no	Hall
64	C₂×ΓL(1,𝔽₈)	—	3	1	no	—
65	C₃×C₁₃	—	4	1	no	—
66	C₃×C₁₃	—	7	1	no	—
67	C₃×C₁₃	—	28	2	no	—
68	C₃×A₄	—	7	1	no	Hall
69	C₅×C₇	C₃×A₄×C₁₃	1	1	yes	characteristic, complemented, Hall
70	C₂×C₃×C₅	—	21	1	no	—
71	C₂²×C₇	C₃²×C₅×C₁₃	1	1	yes	characteristic, complemented, Hall
72	C₂×C₁₃	—	3	1	no	—
73	C₃×C₇	—	4	1	no	—
74	ΓL(1,𝔽₈)	—	4	2	no	—
75	ΓL(1,𝔽₈)	C₅×A₄×C₁₃	1	1	yes	characteristic, direct product factor
76	C₂²×C₅	C₃×C₁₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
77	C₃×C₅	—	4	1	no	—
78	C₃×C₅	—	7	1	no	—
79	C₃×C₅	—	28	2	no	—
80	C₂×C₇	—	3	1	no	—
81	C₁₃	C₅×A₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
82	C₂²×C₃	—	7	1	no	—
83	A₄	—	7	2	no	—
84	A₄	C₅×C₁₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
85	C₂×C₅	—	3	1	no	—
86	C₃²	—	28	1	no	Sylow
87	C₇	C₃×C₅×A₄×C₁₃	1	1	yes	characteristic, complemented, Sylow
88	C₂×C₃	—	21	1	no	—
89	C₅	A₄×C₁₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
90	C₂²	C₃×C₅×C₁₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
91	C₃	—	4	1	no	—
92	C₃	—	7	1	no	—
93	C₃	—	28	2	no	—
94	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₅×C₁₃	A₄×ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₂²×C₇	C₃²×C₅×C₁₃
Frattini	C₁	G
Fitting, socle	C₂²×C₅×C₇×C₁₃	C₃²
LCS	G ▷ C₂²×C₇
UCS	C₁ ◁ 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	semidirect
2. C₅×C₇×A₄×C₁₃	C₃	semidirect
3. C₂²×C₅×C₁₃×ΓL(1,𝔽₈)	C₃	semidirect
5. A₄×C₁₃×ΓL(1,𝔽₈)	C₅	direct (N × H)
8. C₂²×C₅×C₇×C₁₃	C₃²	semidirect
11. C₅×C₁₃×ΓL(1,𝔽₈)	A₄	direct (N × H)
12. C₅×A₄×ΓL(1,𝔽₈)	C₁₃	direct (N × H)
13. C₁₃×C₂²⋊ΓL(1,𝔽₈)	C₃×C₅	semidirect
14. C₇×A₄×C₁₃	C₃×C₅	semidirect
15. C₂²×C₁₃×ΓL(1,𝔽₈)	C₃×C₅	semidirect
20. C₅×A₄×C₁₃	ΓL(1,𝔽₈)	direct (N × H)
24. C₅×C₇×C₁₃	C₃×A₄	semidirect
25. C₅×C₂²⋊ΓL(1,𝔽₈)	C₃×C₁₃	semidirect
26. C₅×C₇×A₄	C₃×C₁₃	semidirect
27. C₂²×C₅×ΓL(1,𝔽₈)	C₃×C₁₃	semidirect
29. C₂²×C₇×C₁₃	C₃²×C₅	semidirect
33. C₁₃×ΓL(1,𝔽₈)	C₅×A₄	direct (N × H)
34. C₂²×C₅×C₁₃	C₃×ΓL(1,𝔽₈)	semidirect
35. A₄×ΓL(1,𝔽₈)	C₅×C₁₃	direct (N × H)
44. A₄×C₁₃	C₅×ΓL(1,𝔽₈)	direct (N × H)
45. C₂²×C₅×C₇	C₃²×C₁₃	semidirect
50. C₅×ΓL(1,𝔽₈)	A₄×C₁₃	direct (N × H)
51. C₇×C₁₃	C₃×C₅×A₄	semidirect
52. C₂²⋊ΓL(1,𝔽₈)	C₃×C₅×C₁₃	semidirect
53. C₇×A₄	C₃×C₅×C₁₃	semidirect
54. C₂²×ΓL(1,𝔽₈)	C₃×C₅×C₁₃	semidirect
57. C₅×C₁₃	A₄×ΓL(1,𝔽₈)	direct (N × H)
61. C₅×A₄	C₁₃×ΓL(1,𝔽₈)	direct (N × H)
62. C₂²×C₁₃	C₃×C₅×ΓL(1,𝔽₈)	semidirect
69. C₅×C₇	C₃×A₄×C₁₃	semidirect
71. C₂²×C₇	C₃²×C₅×C₁₃	semidirect
75. ΓL(1,𝔽₈)	C₅×A₄×C₁₃	direct (N × H)
76. C₂²×C₅	C₃×C₁₃×ΓL(1,𝔽₈)	semidirect
81. C₁₃	C₅×A₄×ΓL(1,𝔽₈)	direct (N × H)
84. A₄	C₅×C₁₃×ΓL(1,𝔽₈)	direct (N × H)
87. C₇	C₃×C₅×A₄×C₁₃	semidirect
89. C₅	A₄×C₁₃×ΓL(1,𝔽₈)	direct (N × H)
90. C₂²	C₃×C₅×C₁₃×ΓL(1,𝔽₈)	semidirect

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

Subgroups — C5×A4×C13×ΓL(1,𝔽8) #30655549

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×A₄×C₁₃×ΓL(1,𝔽₈) #30655549