Subgroups — C₂⁴×C₂²⋊ΓL(1,𝔽₈) #13062423

Order 1344.

2330 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₂²⋊ΓL(1,𝔽₈)	C₂	1	15	yes	maximal, direct product factor
2	C₂⁶×C₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₂²×C₂²⋊ΓL(1,𝔽₈)	C₂²	1	35	yes	direct product factor
4	C₂⁴×ΓL(1,𝔽₈)	—	4	1	no	maximal
5	C₂⁵×C₇	—	3	15	no	—
6	C₂⁵×C₇	—	3	1	no	—
7	C₂⁵×C₇	C₂×C₃	1	15	yes	complemented
8	C₂⁴×A₄	—	7	1	no	maximal, Hall
9	C₂×C₂²⋊ΓL(1,𝔽₈)	C₂³	1	15	yes	direct product factor
10	C₂³×ΓL(1,𝔽₈)	—	4	15	no	—
11	C₂⁴×C₇	—	3	70	no	—
12	C₂⁴×C₇	—	3	15	no	—
13	C₂⁴×C₇	—	3	15	no	—
14	C₂⁴×C₇	—	3	105	no	—
15	C₂⁴×C₇	A₄	1	1	yes	characteristic, complemented
16	C₂⁴×C₇	C₂²×C₃	1	35	yes	complemented
17	C₂³×A₄	—	7	15	no	—
18	C₂²⋊ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic, direct product factor
19	C₂²×ΓL(1,𝔽₈)	—	4	35	no	—
20	C₂⁶	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
21	C₂³×C₇	—	3	105	no	—
22	C₂³×C₇	—	3	210	no	—
23	C₂³×C₇	—	3	105	no	—
24	C₂³×C₇	—	3	35	no	—
25	C₂³×C₇	C₂×A₄	1	15	yes	complemented
26	C₂³×C₇	C₂³×C₃	1	15	yes	complemented
27	C₂⁴×C₃	—	28	1	no	—
28	C₂²×A₄	—	7	35	no	—
29	C₂×ΓL(1,𝔽₈)	—	4	15	no	—
30	C₂⁵	—	3	15	no	—
31	C₂⁵	—	3	1	no	—
32	C₂⁵	C₂×ΓL(1,𝔽₈)	1	15	yes	complemented
33	C₂²×C₇	—	3	15	no	—
34	C₂²×C₇	—	3	70	no	—
35	C₂²×C₇	—	3	105	no	—
36	C₂²×C₇	—	3	15	no	—
37	C₂²×C₇	C₂²×A₄	1	35	yes	complemented
38	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic, complemented
39	C₂³×C₃	—	28	15	no	—
40	C₂×A₄	—	7	15	no	—
41	ΓL(1,𝔽₈)	—	4	1	no	Hall
42	C₂⁴	—	3	105	no	—
43	C₂⁴	—	3	70	no	—
44	C₂⁴	—	3	15	no	—
45	C₂⁴	—	3	15	no	—
46	C₂⁴	C₂²×ΓL(1,𝔽₈)	1	35	yes	complemented
47	C₂⁴	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
48	C₂×C₇	—	3	15	no	—
49	C₂×C₇	—	3	1	no	—
50	C₂×C₇	C₂³×A₄	1	15	yes	complemented
51	C₂²×C₃	—	28	35	no	—
52	A₄	—	7	1	no	—
53	C₂³	—	3	105	no	—
54	C₂³	—	3	210	no	—
55	C₂³	—	3	105	no	—
56	C₂³	—	3	35	no	—
57	C₂³	C₂³×ΓL(1,𝔽₈)	1	15	yes	complemented
58	C₂³	C₂×C₂²⋊ΓL(1,𝔽₈)	1	15	yes	direct product factor
59	C₇	C₂⁴×A₄	1	1	yes	characteristic, complemented, Sylow
60	C₂×C₃	—	28	15	no	—
61	C₂²	—	3	15	no	—
62	C₂²	—	3	70	no	—
63	C₂²	—	3	15	no	—
64	C₂²	—	3	105	no	—
65	C₂²	C₂⁴×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
66	C₂²	C₂²×C₂²⋊ΓL(1,𝔽₈)	1	35	yes	direct product factor
67	C₃	—	28	1	no	Sylow
68	C₂	—	3	1	no	—
69	C₂	—	3	15	no	—
70	C₂	C₂³×C₂²⋊ΓL(1,𝔽₈)	1	15	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂⁴	C₂²⋊ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₂²×C₇	C₂⁴×C₃
Frattini	C₁	G
Fitting, socle	C₂⁶×C₇	C₃
LCS	G ▷ C₂²×C₇
UCS	C₁ ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
1. C₂³×C₂²⋊ΓL(1,𝔽₈)	C₂	direct (N × H)
2. C₂⁶×C₇	C₃	semidirect
3. C₂²×C₂²⋊ΓL(1,𝔽₈)	C₂²	direct (N × H)
7. C₂⁵×C₇	C₂×C₃	semidirect
9. C₂×C₂²⋊ΓL(1,𝔽₈)	C₂³	direct (N × H)
15. C₂⁴×C₇	A₄	semidirect
16. C₂⁴×C₇	C₂²×C₃	semidirect
18. C₂²⋊ΓL(1,𝔽₈)	C₂⁴	direct (N × H)
20. C₂⁶	ΓL(1,𝔽₈)	semidirect
25. C₂³×C₇	C₂×A₄	semidirect
26. C₂³×C₇	C₂³×C₃	semidirect
32. C₂⁵	C₂×ΓL(1,𝔽₈)	semidirect
37. C₂²×C₇	C₂²×A₄	semidirect
38. C₂²×C₇	C₂⁴×C₃	semidirect
46. C₂⁴	C₂²×ΓL(1,𝔽₈)	semidirect
47. C₂⁴	C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
50. C₂×C₇	C₂³×A₄	semidirect
57. C₂³	C₂³×ΓL(1,𝔽₈)	semidirect
58. C₂³	C₂×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
59. C₇	C₂⁴×A₄	semidirect
65. C₂²	C₂⁴×ΓL(1,𝔽₈)	semidirect
66. C₂²	C₂²×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
70. C₂	C₂³×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
C₂³×C₂²⋊ΓL(1,𝔽₈)	2
C₂⁶×C₇	3
C₂⁴×ΓL(1,𝔽₈)	4
C₂⁴×A₄	7

Subgroups — C24×C22⋊ΓL(1,𝔽8) #13062423

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×C₂²⋊ΓL(1,𝔽₈) #13062423