Subgroups — C₂²×C₅×C₂²⋊ΓL(1,𝔽₈) #13132975

Order 1680.

156 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₂²⋊ΓL(1,𝔽₈)	C₂	1	3	yes	maximal, direct product factor
2	C₂⁴×C₅×C₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₅×C₂²⋊ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, direct product factor
4	C₂²×C₅×ΓL(1,𝔽₈)	—	4	1	no	maximal
5	C₂²×C₂²⋊ΓL(1,𝔽₈)	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂³×C₅×C₇	—	3	1	no	—
7	C₂³×C₅×C₇	—	3	3	no	—
8	C₂³×C₅×C₇	C₂×C₃	1	3	yes	complemented
9	C₂²×C₅×A₄	—	7	1	no	maximal, Hall
10	C₂×C₅×ΓL(1,𝔽₈)	—	4	3	no	—
11	C₂×C₂²⋊ΓL(1,𝔽₈)	C₂×C₅	1	3	yes	direct product factor
12	C₂²×C₅×C₇	—	3	3	no	—
13	C₂²×C₅×C₇	—	3	2	no	—
14	C₂²×C₅×C₇	—	3	3	no	—
15	C₂²×C₅×C₇	—	3	3	no	—
16	C₂²×C₅×C₇	A₄	1	1	yes	characteristic, complemented
17	C₂²×C₅×C₇	C₂²×C₃	1	1	yes	characteristic, complemented
18	C₂×C₅×A₄	—	7	3	no	—
19	C₂⁴×C₇	C₃×C₅	1	1	yes	characteristic, complemented, Hall
20	C₅×ΓL(1,𝔽₈)	—	4	1	no	Hall
21	C₂²⋊ΓL(1,𝔽₈)	C₂²×C₅	1	1	yes	characteristic, direct product factor
22	C₂²×ΓL(1,𝔽₈)	—	4	1	no	—
23	C₂⁴×C₅	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
24	C₂×C₅×C₇	—	3	3	no	—
25	C₂×C₅×C₇	—	3	1	no	—
26	C₂×C₅×C₇	C₂×A₄	1	3	yes	complemented
27	C₂²×C₃×C₅	—	28	1	no	—
28	C₅×A₄	—	7	1	no	—
29	C₂³×C₇	—	3	3	no	—
30	C₂³×C₇	—	3	1	no	—
31	C₂³×C₇	C₂×C₃×C₅	1	3	yes	complemented
32	C₂²×A₄	—	7	1	no	Hall
33	C₂×ΓL(1,𝔽₈)	—	4	3	no	—
34	C₂³×C₅	—	3	3	no	—
35	C₂³×C₅	—	3	1	no	—
36	C₂³×C₅	C₂×ΓL(1,𝔽₈)	1	3	yes	complemented
37	C₅×C₇	C₂²×A₄	1	1	yes	characteristic, complemented, Hall
38	C₂×C₃×C₅	—	28	3	no	—
39	C₂²×C₇	—	3	3	no	—
40	C₂²×C₇	—	3	2	no	—
41	C₂²×C₇	—	3	3	no	—
42	C₂²×C₇	—	3	3	no	—
43	C₂²×C₇	C₅×A₄	1	1	yes	characteristic, complemented
44	C₂²×C₇	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
45	C₂×A₄	—	7	3	no	—
46	ΓL(1,𝔽₈)	—	4	1	no	Hall
47	C₂²×C₅	—	3	3	no	—
48	C₂²×C₅	—	3	2	no	—
49	C₂²×C₅	—	3	3	no	—
50	C₂²×C₅	—	3	3	no	—
51	C₂²×C₅	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
52	C₂²×C₅	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
53	C₂⁴	C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
54	C₃×C₅	—	28	1	no	Hall
55	C₂×C₇	—	3	1	no	—
56	C₂×C₇	—	3	3	no	—
57	C₂×C₇	C₂×C₅×A₄	1	3	yes	complemented
58	C₂²×C₃	—	28	1	no	—
59	A₄	—	7	1	no	—
60	C₂×C₅	—	3	1	no	—
61	C₂×C₅	—	3	3	no	—
62	C₂×C₅	C₂×C₂²⋊ΓL(1,𝔽₈)	1	3	yes	direct product factor
63	C₂³	—	3	3	no	—
64	C₂³	—	3	1	no	—
65	C₂³	C₂×C₅×ΓL(1,𝔽₈)	1	3	yes	complemented
66	C₇	C₂²×C₅×A₄	1	1	yes	characteristic, complemented, Sylow
67	C₂×C₃	—	28	3	no	—
68	C₅	C₂²×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
69	C₂²	—	3	3	no	—
70	C₂²	—	3	2	no	—
71	C₂²	—	3	3	no	—
72	C₂²	—	3	3	no	—
73	C₂²	C₂²×C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
74	C₂²	C₅×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
75	C₃	—	28	1	no	Sylow
76	C₂	—	3	1	no	—
77	C₂	—	3	3	no	—
78	C₂	C₂×C₅×C₂²⋊ΓL(1,𝔽₈)	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₅	C₂²⋊ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₂²×C₇	C₂²×C₃×C₅
Frattini	C₁	G
Fitting, socle	C₂⁴×C₅×C₇	C₃
LCS	G ▷ C₂²×C₇
UCS	C₁ ◁ C₂²×C₅

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×C₂²⋊ΓL(1,𝔽₈)	C₂	direct (N × H)
2. C₂⁴×C₅×C₇	C₃	semidirect
3. C₅×C₂²⋊ΓL(1,𝔽₈)	C₂²	direct (N × H)
5. C₂²×C₂²⋊ΓL(1,𝔽₈)	C₅	direct (N × H)
8. C₂³×C₅×C₇	C₂×C₃	semidirect
11. C₂×C₂²⋊ΓL(1,𝔽₈)	C₂×C₅	direct (N × H)
16. C₂²×C₅×C₇	A₄	semidirect
17. C₂²×C₅×C₇	C₂²×C₃	semidirect
19. C₂⁴×C₇	C₃×C₅	semidirect
21. C₂²⋊ΓL(1,𝔽₈)	C₂²×C₅	direct (N × H)
23. C₂⁴×C₅	ΓL(1,𝔽₈)	semidirect
26. C₂×C₅×C₇	C₂×A₄	semidirect
31. C₂³×C₇	C₂×C₃×C₅	semidirect
36. C₂³×C₅	C₂×ΓL(1,𝔽₈)	semidirect
37. C₅×C₇	C₂²×A₄	semidirect
43. C₂²×C₇	C₅×A₄	semidirect
44. C₂²×C₇	C₂²×C₃×C₅	semidirect
51. C₂²×C₅	C₂²×ΓL(1,𝔽₈)	semidirect
52. C₂²×C₅	C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
53. C₂⁴	C₅×ΓL(1,𝔽₈)	semidirect
57. C₂×C₇	C₂×C₅×A₄	semidirect
62. C₂×C₅	C₂×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
65. C₂³	C₂×C₅×ΓL(1,𝔽₈)	semidirect
66. C₇	C₂²×C₅×A₄	semidirect
68. C₅	C₂²×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
73. C₂²	C₂²×C₅×ΓL(1,𝔽₈)	semidirect
74. C₂²	C₅×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
78. C₂	C₂×C₅×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
C₂×C₅×C₂²⋊ΓL(1,𝔽₈)	2
C₂⁴×C₅×C₇	3
C₂²×C₅×ΓL(1,𝔽₈)	4
C₂²×C₂²⋊ΓL(1,𝔽₈)	5
C₂²×C₅×A₄	7

Subgroups — C22×C5×C22⋊ΓL(1,𝔽8) #13132975

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×C₂²⋊ΓL(1,𝔽₈) #13132975