Subgroups — C₅×C₂²⋊C₄×ΓL(1,𝔽₈) #13132248

Order 1680.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₅×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₅×ΓL(1,𝔽₈)	C₂	1	2	yes	maximal, complemented
3	C₅×C₇×C₂²⋊C₄	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₂²×C₅×ΓL(1,𝔽₈)	—	2	2	no	—
5	C₂²×C₅×ΓL(1,𝔽₈)	C₄	1	2	yes	complemented
6	C₂²×C₅×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
7	C₄×C₅×ΓL(1,𝔽₈)	—	2	2	no	—
8	C₂²⋊C₄×ΓL(1,𝔽₈)	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂³×C₅×C₇	C₂×C₃	1	1	yes	characteristic
10	C₂×C₄×C₅×C₇	C₂×C₃	1	2	yes	complemented
11	C₃×C₅×C₂²⋊C₄	—	7	1	no	maximal, Hall
12	C₂×C₅×ΓL(1,𝔽₈)	—	2	2	no	—
13	C₂×C₅×ΓL(1,𝔽₈)	C₂×C₄	1	1	yes	characteristic
14	C₂×C₅×ΓL(1,𝔽₈)	D₄	1	2	yes	—
15	C₂³×ΓL(1,𝔽₈)	C₂×C₅	1	1	yes	characteristic
16	C₂×C₄×ΓL(1,𝔽₈)	C₂×C₅	1	2	yes	complemented
17	C₂²×C₅×C₇	—	2	2	no	—
18	C₂²×C₅×C₇	C₃×C₄	1	2	yes	complemented
19	C₂²×C₅×C₇	C₂²×C₃	1	1	yes	characteristic
20	C₄×C₅×C₇	—	2	2	no	—
21	C₂³×C₃×C₅	—	7	1	no	—
22	C₂×C₃×C₄×C₅	—	7	2	no	—
23	C₇×C₂²⋊C₄	C₃×C₅	1	1	yes	characteristic, complemented, Hall
24	C₅×ΓL(1,𝔽₈)	C₂²⋊C₄	1	1	yes	characteristic, direct product factor, Hall
25	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
26	C₂²×ΓL(1,𝔽₈)	C₄×C₅	1	2	yes	complemented
27	C₂²×ΓL(1,𝔽₈)	C₂²×C₅	1	1	yes	characteristic
28	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
29	C₅×C₂²⋊C₄	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
30	C₂×C₅×C₇	—	2	2	no	—
31	C₂×C₅×C₇	C₂×C₃×C₄	1	1	yes	characteristic
32	C₂×C₅×C₇	C₃×D₄	1	2	yes	—
33	C₂²×C₃×C₅	—	7	2	no	—
34	C₂²×C₃×C₅	—	7	1	no	—
35	C₂²×C₃×C₅	—	14	2	no	—
36	C₃×C₄×C₅	—	14	2	no	—
37	C₂³×C₇	C₂×C₃×C₅	1	1	yes	characteristic
38	C₂×C₄×C₇	C₂×C₃×C₅	1	2	yes	complemented
39	C₃×C₂²⋊C₄	—	7	1	no	Hall
40	C₂×ΓL(1,𝔽₈)	—	2	2	no	—
41	C₂×ΓL(1,𝔽₈)	C₂×C₄×C₅	1	1	yes	characteristic
42	C₂×ΓL(1,𝔽₈)	C₅×D₄	1	2	yes	—
43	C₂³×C₅	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
44	C₂×C₄×C₅	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
45	C₅×C₇	C₃×C₂²⋊C₄	1	1	yes	characteristic, complemented, Hall
46	C₂×C₃×C₅	—	7	1	no	—
47	C₂×C₃×C₅	—	7	2	no	—
48	C₂×C₃×C₅	—	14	2	no	—
49	C₂²×C₇	—	2	2	no	—
50	C₂²×C₇	C₃×C₄×C₅	1	2	yes	complemented
51	C₂²×C₇	C₂²×C₃×C₅	1	1	yes	characteristic
52	C₄×C₇	—	2	2	no	—
53	C₂³×C₃	—	7	1	no	—
54	C₂×C₃×C₄	—	7	2	no	—
55	ΓL(1,𝔽₈)	C₅×C₂²⋊C₄	1	1	yes	characteristic, direct product factor, Hall
56	C₂²×C₅	—	2	2	no	—
57	C₂²×C₅	C₄×ΓL(1,𝔽₈)	1	2	yes	complemented
58	C₂²×C₅	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
59	C₄×C₅	—	2	2	no	—
60	C₂²⋊C₄	C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
61	C₃×C₅	—	7	1	no	Hall
62	C₂×C₇	—	2	2	no	—
63	C₂×C₇	C₂×C₃×C₄×C₅	1	1	yes	characteristic
64	C₂×C₇	C₃×C₅×D₄	1	2	yes	—
65	C₂²×C₃	—	7	2	no	—
66	C₂²×C₃	—	7	1	no	—
67	C₂²×C₃	—	14	2	no	—
68	C₃×C₄	—	14	2	no	—
69	C₂×C₅	—	2	2	no	—
70	C₂×C₅	C₂×C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
71	C₂×C₅	D₄×ΓL(1,𝔽₈)	1	2	yes	—
72	C₂³	C₂×C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic
73	C₂×C₄	C₂×C₅×ΓL(1,𝔽₈)	1	2	yes	complemented
74	C₇	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
75	C₂×C₃	—	7	1	no	—
76	C₂×C₃	—	7	2	no	—
77	C₂×C₃	—	14	2	no	—
78	C₅	C₂²⋊C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
79	C₂²	—	2	2	no	—
80	C₂²	C₄×C₅×ΓL(1,𝔽₈)	1	2	yes	complemented
81	C₂²	C₂²×C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic
82	C₄	—	2	2	no	—
83	C₃	—	7	1	no	Sylow
84	C₂	—	2	2	no	—
85	C₂	C₂×C₄×C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic
86	C₂	C₅×D₄×ΓL(1,𝔽₈)	1	2	yes	—

	subgroup	G/H
center	C₂²×C₅	C₂²×ΓL(1,𝔽₈)
commutator	C₂×C₇	C₂×C₃×C₄×C₅
Frattini	C₂²	C₂²×C₅×ΓL(1,𝔽₈)
Fitting	C₅×C₇×C₂²⋊C₄	C₃
socle	C₂²×C₅×C₇	C₂²×C₃
prime radical	C₂³×C₅×ΓL(1,𝔽₈)	C₂
cosocle	C₂²×C₇	C₂²×C₃×C₅
hypercenter	C₅×C₂²⋊C₄	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×C₅×C₂²⋊C₄
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×C₅×ΓL(1,𝔽₈)	C₂	semidirect
3. C₅×C₇×C₂²⋊C₄	C₃	semidirect
5. C₂²×C₅×ΓL(1,𝔽₈)	C₄	semidirect
8. C₂²⋊C₄×ΓL(1,𝔽₈)	C₅	direct (N × H)
10. C₂×C₄×C₅×C₇	C₂×C₃	semidirect
16. C₂×C₄×ΓL(1,𝔽₈)	C₂×C₅	semidirect
18. C₂²×C₅×C₇	C₃×C₄	semidirect
23. C₇×C₂²⋊C₄	C₃×C₅	semidirect
24. C₅×ΓL(1,𝔽₈)	C₂²⋊C₄	direct (N × H)
26. C₂²×ΓL(1,𝔽₈)	C₄×C₅	semidirect
29. C₅×C₂²⋊C₄	ΓL(1,𝔽₈)	direct (N × H)
38. C₂×C₄×C₇	C₂×C₃×C₅	semidirect
44. C₂×C₄×C₅	C₂×ΓL(1,𝔽₈)	semidirect
45. C₅×C₇	C₃×C₂²⋊C₄	semidirect
50. C₂²×C₇	C₃×C₄×C₅	semidirect
55. ΓL(1,𝔽₈)	C₅×C₂²⋊C₄	direct (N × H)
57. C₂²×C₅	C₄×ΓL(1,𝔽₈)	semidirect
60. C₂²⋊C₄	C₅×ΓL(1,𝔽₈)	direct (N × H)
73. C₂×C₄	C₂×C₅×ΓL(1,𝔽₈)	semidirect
74. C₇	C₃×C₅×C₂²⋊C₄	semidirect
78. C₅	C₂²⋊C₄×ΓL(1,𝔽₈)	direct (N × H)
80. C₂²	C₄×C₅×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
C₂³×C₅×ΓL(1,𝔽₈)	2
C₂×C₄×C₅×ΓL(1,𝔽₈)	2
C₅×C₇×C₂²⋊C₄	3
C₂²⋊C₄×ΓL(1,𝔽₈)	5
C₃×C₅×C₂²⋊C₄	7

Subgroups — C5×C22⋊C4×ΓL(1,𝔽8) #13132248

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×C₂²⋊C₄×ΓL(1,𝔽₈) #13132248