Subgroups — C₂³×ΓL(1,𝔽₈)⋊C₈ #13057514

Order 1344.

676 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic
2	C₂²×ΓL(1,𝔽₈)⋊C₈	C₂	1	14	yes	maximal, direct product factor
3	C₂³×C₇⋊C₈	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₂⁴×ΓL(1,𝔽₈)	C₄	1	1	yes	characteristic
5	C₂²×C₄×ΓL(1,𝔽₈)	C₄	1	7	yes	—
6	C₂²×C₄×ΓL(1,𝔽₈)	C₂²	1	7	yes	—
7	C₂×ΓL(1,𝔽₈)⋊C₈	C₂²	1	28	yes	direct product factor
8	C₂³×C₄×C₇	C₂×C₃	1	1	yes	characteristic
9	C₂²×C₇⋊C₈	C₂×C₃	1	14	yes	complemented
10	C₂³×C₃×C₈	—	7	1	no	maximal, Hall
11	C₂³×ΓL(1,𝔽₈)	C₈	1	8	yes	complemented
12	C₂³×ΓL(1,𝔽₈)	C₂×C₄	1	7	yes	—
13	C₂×C₄×ΓL(1,𝔽₈)	C₂×C₄	1	21	yes	—
14	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	7	yes	—
15	ΓL(1,𝔽₈)⋊C₈	C₂³	1	8	yes	direct product factor
16	C₂⁴×C₇	C₃×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₇	C₃×C₄	1	7	yes	—
18	C₂²×C₄×C₇	C₂²×C₃	1	7	yes	—
19	C₂×C₇⋊C₈	C₂²×C₃	1	28	yes	complemented
20	C₂³×C₃×C₄	—	7	1	no	—
21	C₂²×C₃×C₈	—	7	14	no	—
22	C₂²×ΓL(1,𝔽₈)	C₂×C₈	1	28	yes	complemented
23	C₂²×ΓL(1,𝔽₈)	C₂²×C₄	1	7	yes	—
24	C₄×ΓL(1,𝔽₈)	C₂²×C₄	1	7	yes	—
25	C₄×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
26	C₂³×C₈	—	7	1	no	Sylow
27	C₂³×C₇	C₃×C₈	1	8	yes	complemented
28	C₂³×C₇	C₂×C₃×C₄	1	7	yes	—
29	C₂×C₄×C₇	C₂×C₃×C₄	1	21	yes	—
30	C₂×C₄×C₇	C₂³×C₃	1	7	yes	—
31	C₇⋊C₈	C₂³×C₃	1	8	yes	complemented
32	C₂⁴×C₃	—	7	1	no	—
33	C₂²×C₃×C₄	—	7	7	no	—
34	C₂²×C₃×C₄	—	7	7	no	—
35	C₂×C₃×C₈	—	7	28	no	—
36	C₂×ΓL(1,𝔽₈)	C₂²×C₈	1	14	yes	complemented
37	C₂×ΓL(1,𝔽₈)	C₂³×C₄	1	1	yes	characteristic
38	C₂³×C₄	AGL(1,𝔽₇)	1	1	yes	characteristic
39	C₂²×C₈	—	7	14	no	—
40	C₂²×C₇	C₂×C₃×C₈	1	28	yes	complemented
41	C₂²×C₇	C₂²×C₃×C₄	1	7	yes	—
42	C₄×C₇	C₂²×C₃×C₄	1	7	yes	—
43	C₄×C₇	C₂⁴×C₃	1	1	yes	characteristic
44	C₂³×C₃	—	7	8	no	—
45	C₂³×C₃	—	7	7	no	—
46	C₂×C₃×C₄	—	7	21	no	—
47	C₂×C₃×C₄	—	7	7	no	—
48	C₃×C₈	—	7	8	no	—
49	ΓL(1,𝔽₈)	C₂³×C₈	1	1	yes	characteristic, complemented, Hall
50	C₂⁴	ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic
51	C₂²×C₄	ΓL(1,𝔽₈)⋊C₄	1	7	yes	—
52	C₂²×C₄	C₂×AGL(1,𝔽₇)	1	7	yes	—
53	C₂×C₈	—	7	28	no	—
54	C₂×C₇	C₂²×C₃×C₈	1	14	yes	complemented
55	C₂×C₇	C₂³×C₃×C₄	1	1	yes	characteristic
56	C₂²×C₃	—	7	7	no	—
57	C₂²×C₃	—	7	28	no	—
58	C₃×C₄	—	7	7	no	—
59	C₃×C₄	—	7	1	no	—
60	C₂³	ΓL(1,𝔽₈)⋊C₈	1	8	yes	direct product factor
61	C₂³	C₂×ΓL(1,𝔽₈)⋊C₄	1	7	yes	—
62	C₂×C₄	C₂×ΓL(1,𝔽₈)⋊C₄	1	21	yes	—
63	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	7	yes	—
64	C₈	—	7	8	no	—
65	C₇	C₂³×C₃×C₈	1	1	yes	characteristic, complemented, Sylow
66	C₂×C₃	—	7	1	no	—
67	C₂×C₃	—	7	14	no	—
68	C₂²	C₂×ΓL(1,𝔽₈)⋊C₈	1	28	yes	direct product factor
69	C₂²	C₂²×ΓL(1,𝔽₈)⋊C₄	1	7	yes	—
70	C₄	C₂²×ΓL(1,𝔽₈)⋊C₄	1	7	yes	—
71	C₄	C₂³×AGL(1,𝔽₇)	1	1	yes	characteristic
72	C₃	—	7	1	no	Sylow
73	C₂	C₂²×ΓL(1,𝔽₈)⋊C₈	1	14	yes	direct product factor
74	C₂	C₂³×ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³×C₄	AGL(1,𝔽₇)
commutator, hypocenter	C₇	C₂³×C₃×C₈
Frattini	C₄	C₂³×AGL(1,𝔽₇)
Fitting	C₂³×C₄×C₇	C₂×C₃
socle	C₂⁴×C₇	C₃×C₄
prime radical	C₂⁴×ΓL(1,𝔽₈)	C₄
cosocle	C₄×C₇	C₂⁴×C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
2. C₂²×ΓL(1,𝔽₈)⋊C₈	C₂	direct (N × H)
3. C₂³×C₇⋊C₈	C₃	semidirect
7. C₂×ΓL(1,𝔽₈)⋊C₈	C₂²	direct (N × H)
9. C₂²×C₇⋊C₈	C₂×C₃	semidirect
11. C₂³×ΓL(1,𝔽₈)	C₈	semidirect
15. ΓL(1,𝔽₈)⋊C₈	C₂³	direct (N × H)
19. C₂×C₇⋊C₈	C₂²×C₃	semidirect
22. C₂²×ΓL(1,𝔽₈)	C₂×C₈	semidirect
27. C₂³×C₇	C₃×C₈	semidirect
31. C₇⋊C₈	C₂³×C₃	semidirect
36. C₂×ΓL(1,𝔽₈)	C₂²×C₈	semidirect
40. C₂²×C₇	C₂×C₃×C₈	semidirect
49. ΓL(1,𝔽₈)	C₂³×C₈	semidirect
54. C₂×C₇	C₂²×C₃×C₈	semidirect
60. C₂³	ΓL(1,𝔽₈)⋊C₈	direct (N × H)
65. C₇	C₂³×C₃×C₈	semidirect
68. C₂²	C₂×ΓL(1,𝔽₈)⋊C₈	direct (N × H)
73. C₂	C₂²×ΓL(1,𝔽₈)⋊C₈	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×C₄×ΓL(1,𝔽₈)	2
C₂²×ΓL(1,𝔽₈)⋊C₈	2
C₂³×C₇⋊C₈	3
C₂³×C₃×C₈	7

Subgroups — C23×ΓL(1,𝔽8)⋊C8 #13057514

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×ΓL(1,𝔽₈)⋊C₈ #13057514