Subgroups — C₅×C₇×C₁₁×C₂²⋊ΓL(1,𝔽₈) #44568926

Order 32340.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×C₇²×C₁₁	C₃	1	1	yes	maximal, characteristic, complemented, Hall
2	C₅×C₇×C₁₁×ΓL(1,𝔽₈)	—	4	1	no	maximal, Hall
3	C₇×C₁₁×C₂²⋊ΓL(1,𝔽₈)	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂×C₅×C₇²×C₁₁	—	3	1	no	—
5	C₅×C₁₁×C₂²⋊ΓL(1,𝔽₈)	C₇	1	1	yes	maximal, characteristic, direct product factor
6	C₅×C₇×C₁₁×A₄	—	7	1	no	maximal
7	C₅×C₇×C₂²⋊ΓL(1,𝔽₈)	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₅×C₇²×C₁₁	A₄	1	1	yes	characteristic, complemented, Hall
9	C₂²×C₇²×C₁₁	C₃×C₅	1	1	yes	characteristic, complemented, Hall
10	C₇×C₁₁×ΓL(1,𝔽₈)	—	4	1	no	Hall
11	C₂²×C₅×C₇×C₁₁	—	3	2	no	—
12	C₂²×C₅×C₇×C₁₁	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
13	C₂²×C₅×C₇×C₁₁	C₃×C₇	1	1	yes	characteristic, complemented
14	C₃×C₅×C₇×C₁₁	—	28	1	no	—
15	C₅×C₁₁×ΓL(1,𝔽₈)	—	4	1	no	—
16	C₂×C₇²×C₁₁	—	3	1	no	—
17	C₂²×C₅×C₇²	C₃×C₁₁	1	1	yes	characteristic, complemented, Hall
18	C₁₁×C₂²⋊ΓL(1,𝔽₈)	C₅×C₇	1	1	yes	characteristic, direct product factor
19	C₇×C₁₁×A₄	—	7	1	no	—
20	C₂×C₅×C₇×C₁₁	—	3	6	no	—
21	C₂×C₅×C₇×C₁₁	—	3	1	no	—
22	C₂×C₅×C₇×C₁₁	—	3	1	no	—
23	C₅×C₇×ΓL(1,𝔽₈)	—	4	1	no	Hall
24	C₅×C₁₁×A₄	—	7	1	no	Hall
25	C₇×C₂²⋊ΓL(1,𝔽₈)	C₅×C₁₁	1	1	yes	characteristic, direct product factor, Hall
26	C₇²×C₁₁	C₅×A₄	1	1	yes	characteristic, complemented, Hall
27	C₂×C₅×C₇²	—	3	1	no	—
28	C₅×C₂²⋊ΓL(1,𝔽₈)	C₇×C₁₁	1	1	yes	characteristic, direct product factor
29	C₅×C₇×A₄	—	7	1	no	—
30	C₅×C₇×C₁₁	—	3	2	no	—
31	C₅×C₇×C₁₁	C₇×A₄	1	1	yes	characteristic, complemented
32	C₅×C₇×C₁₁	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
33	C₂²×C₇×C₁₁	—	3	2	no	—
34	C₂²×C₇×C₁₁	C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
35	C₂²×C₇×C₁₁	C₃×C₅×C₇	1	1	yes	characteristic, complemented
36	C₅×C₇²	C₁₁×A₄	1	1	yes	characteristic, complemented, Hall
37	C₃×C₇×C₁₁	—	28	1	no	—
38	C₁₁×ΓL(1,𝔽₈)	—	4	1	no	—
39	C₂²×C₅×C₁₁	C₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
40	C₂²×C₇²	C₃×C₅×C₁₁	1	1	yes	characteristic, complemented, Hall
41	C₃×C₅×C₁₁	—	28	1	no	Hall
42	C₂×C₇×C₁₁	—	3	6	no	—
43	C₂×C₇×C₁₁	—	3	1	no	—
44	C₂×C₇×C₁₁	—	3	1	no	—
45	C₇×ΓL(1,𝔽₈)	—	4	1	no	Hall
46	C₂²×C₅×C₇	—	3	2	no	—
47	C₂²×C₅×C₇	C₁₁×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
48	C₂²×C₅×C₇	C₃×C₇×C₁₁	1	1	yes	characteristic, complemented
49	C₁₁×A₄	—	7	1	no	Hall
50	C₂×C₅×C₁₁	—	3	1	no	—
51	C₃×C₅×C₇	—	28	1	no	—
52	C₅×ΓL(1,𝔽₈)	—	4	1	no	—
53	C₂×C₇²	—	3	1	no	—
54	C₂²⋊ΓL(1,𝔽₈)	C₅×C₇×C₁₁	1	1	yes	characteristic, direct product factor
55	C₇×A₄	—	7	1	no	—
56	C₇×C₁₁	—	3	2	no	—
57	C₇×C₁₁	C₅×C₇×A₄	1	1	yes	characteristic, complemented
58	C₇×C₁₁	C₅×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
59	C₂×C₅×C₇	—	3	6	no	—
60	C₂×C₅×C₇	—	3	1	no	—
61	C₂×C₅×C₇	—	3	1	no	—
62	C₅×A₄	—	7	1	no	Hall
63	C₅×C₁₁	C₇×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
64	C₇²	C₅×C₁₁×A₄	1	1	yes	characteristic, complemented, Sylow
65	C₂²×C₁₁	C₅×C₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
66	C₅×C₇	—	3	2	no	—
67	C₅×C₇	C₇×C₁₁×A₄	1	1	yes	characteristic, complemented
68	C₅×C₇	C₁₁×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
69	C₃×C₁₁	—	28	1	no	Hall
70	C₂²×C₇	—	3	2	no	—
71	C₂²×C₇	C₅×C₁₁×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
72	C₂²×C₇	C₃×C₅×C₇×C₁₁	1	1	yes	characteristic, complemented
73	C₂×C₁₁	—	3	1	no	—
74	C₃×C₇	—	28	1	no	—
75	ΓL(1,𝔽₈)	—	4	1	no	—
76	C₂²×C₅	C₇×C₁₁×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
77	C₃×C₅	—	28	1	no	Hall
78	C₂×C₇	—	3	6	no	—
79	C₂×C₇	—	3	1	no	—
80	C₂×C₇	—	3	1	no	—
81	A₄	—	7	1	no	Hall
82	C₁₁	C₅×C₇×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
83	C₂×C₅	—	3	1	no	—
84	C₇	—	3	2	no	—
85	C₇	C₅×C₇×C₁₁×A₄	1	1	yes	characteristic, complemented
86	C₇	C₅×C₁₁×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
87	C₅	C₇×C₁₁×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
88	C₂²	C₅×C₇×C₁₁×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
89	C₃	—	28	1	no	Sylow
90	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₅×C₇×C₁₁	C₂²⋊ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₂²×C₇	C₃×C₅×C₇×C₁₁
Frattini	C₁	G
Fitting, socle	C₂²×C₅×C₇²×C₁₁	C₃
LCS	G ▷ C₂²×C₇
UCS	C₁ ◁ C₅×C₇×C₁₁

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×C₇²×C₁₁	C₃	semidirect
3. C₇×C₁₁×C₂²⋊ΓL(1,𝔽₈)	C₅	direct (N × H)
5. C₅×C₁₁×C₂²⋊ΓL(1,𝔽₈)	C₇	direct (N × H)
7. C₅×C₇×C₂²⋊ΓL(1,𝔽₈)	C₁₁	direct (N × H)
8. C₅×C₇²×C₁₁	A₄	semidirect
9. C₂²×C₇²×C₁₁	C₃×C₅	semidirect
12. C₂²×C₅×C₇×C₁₁	ΓL(1,𝔽₈)	semidirect
13. C₂²×C₅×C₇×C₁₁	C₃×C₇	semidirect
17. C₂²×C₅×C₇²	C₃×C₁₁	semidirect
18. C₁₁×C₂²⋊ΓL(1,𝔽₈)	C₅×C₇	direct (N × H)
25. C₇×C₂²⋊ΓL(1,𝔽₈)	C₅×C₁₁	direct (N × H)
26. C₇²×C₁₁	C₅×A₄	semidirect
28. C₅×C₂²⋊ΓL(1,𝔽₈)	C₇×C₁₁	direct (N × H)
31. C₅×C₇×C₁₁	C₇×A₄	semidirect
32. C₅×C₇×C₁₁	C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
34. C₂²×C₇×C₁₁	C₅×ΓL(1,𝔽₈)	semidirect
35. C₂²×C₇×C₁₁	C₃×C₅×C₇	semidirect
36. C₅×C₇²	C₁₁×A₄	semidirect
39. C₂²×C₅×C₁₁	C₇×ΓL(1,𝔽₈)	semidirect
40. C₂²×C₇²	C₃×C₅×C₁₁	semidirect
47. C₂²×C₅×C₇	C₁₁×ΓL(1,𝔽₈)	semidirect
48. C₂²×C₅×C₇	C₃×C₇×C₁₁	semidirect
54. C₂²⋊ΓL(1,𝔽₈)	C₅×C₇×C₁₁	direct (N × H)
57. C₇×C₁₁	C₅×C₇×A₄	semidirect
58. C₇×C₁₁	C₅×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
63. C₅×C₁₁	C₇×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
64. C₇²	C₅×C₁₁×A₄	semidirect
65. C₂²×C₁₁	C₅×C₇×ΓL(1,𝔽₈)	semidirect
67. C₅×C₇	C₇×C₁₁×A₄	semidirect
68. C₅×C₇	C₁₁×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
71. C₂²×C₇	C₅×C₁₁×ΓL(1,𝔽₈)	semidirect
72. C₂²×C₇	C₃×C₅×C₇×C₁₁	semidirect
76. C₂²×C₅	C₇×C₁₁×ΓL(1,𝔽₈)	semidirect
82. C₁₁	C₅×C₇×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
85. C₇	C₅×C₇×C₁₁×A₄	semidirect
86. C₇	C₅×C₁₁×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
87. C₅	C₇×C₁₁×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
88. C₂²	C₅×C₇×C₁₁×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
C₂²×C₅×C₇²×C₁₁	3
C₅×C₇×C₁₁×ΓL(1,𝔽₈)	4
C₇×C₁₁×C₂²⋊ΓL(1,𝔽₈)	5
C₅×C₁₁×C₂²⋊ΓL(1,𝔽₈)	7
C₅×C₇×C₁₁×A₄	7
C₅×C₇×C₂²⋊ΓL(1,𝔽₈)	11

Subgroups — C5×C7×C11×C22⋊ΓL(1,𝔽8) #44568926

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×C₇×C₁₁×C₂²⋊ΓL(1,𝔽₈) #44568926