Subgroups — C₂⁴×ΓL(1,𝔽₈)⋊C₄ #13062159

Order 1344.

2724 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic
2	C₂³×ΓL(1,𝔽₈)⋊C₄	C₂	1	30	yes	maximal, direct product factor
3	C₂⁴×Dic₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₂⁴×ΓL(1,𝔽₈)	C₄	1	16	yes	complemented
5	C₂⁴×ΓL(1,𝔽₈)	C₂²	1	15	yes	—
6	C₂²×ΓL(1,𝔽₈)⋊C₄	C₂²	1	140	yes	direct product factor
7	C₂⁵×C₇	C₂×C₃	1	1	yes	characteristic
8	C₂³×Dic₇	C₂×C₃	1	30	yes	complemented
9	C₂⁴×C₃×C₄	—	7	1	no	maximal, Hall
10	C₂³×ΓL(1,𝔽₈)	C₂×C₄	1	120	yes	complemented
11	C₂³×ΓL(1,𝔽₈)	C₂³	1	35	yes	—
12	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	120	yes	direct product factor
13	C₂⁴×C₇	C₃×C₄	1	16	yes	complemented
14	C₂⁴×C₇	C₂²×C₃	1	15	yes	—
15	C₂²×Dic₇	C₂²×C₃	1	140	yes	complemented
16	C₂⁵×C₃	—	7	1	no	—
17	C₂³×C₃×C₄	—	7	30	no	—
18	C₂²×ΓL(1,𝔽₈)	C₂²×C₄	1	140	yes	complemented
19	C₂²×ΓL(1,𝔽₈)	C₂⁴	1	15	yes	—
20	ΓL(1,𝔽₈)⋊C₄	C₂⁴	1	16	yes	direct product factor
21	C₂⁴×C₄	—	7	1	no	Sylow
22	C₂³×C₇	C₂×C₃×C₄	1	120	yes	complemented
23	C₂³×C₇	C₂³×C₃	1	35	yes	—
24	C₂×Dic₇	C₂³×C₃	1	120	yes	complemented
25	C₂⁴×C₃	—	7	16	no	—
26	C₂⁴×C₃	—	7	15	no	—
27	C₂²×C₃×C₄	—	7	140	no	—
28	C₂×ΓL(1,𝔽₈)	C₂³×C₄	1	30	yes	complemented
29	C₂×ΓL(1,𝔽₈)	C₂⁵	1	1	yes	characteristic
30	C₂⁵	AGL(1,𝔽₇)	1	1	yes	characteristic
31	C₂³×C₄	—	7	30	no	—
32	C₂²×C₇	C₂²×C₃×C₄	1	140	yes	complemented
33	C₂²×C₇	C₂⁴×C₃	1	15	yes	—
34	Dic₇	C₂⁴×C₃	1	16	yes	complemented
35	C₂³×C₃	—	7	120	no	—
36	C₂³×C₃	—	7	35	no	—
37	C₂×C₃×C₄	—	7	120	no	—
38	ΓL(1,𝔽₈)	C₂⁴×C₄	1	1	yes	characteristic, complemented, Hall
39	C₂⁴	ΓL(1,𝔽₈)⋊C₄	1	16	yes	direct product factor
40	C₂⁴	C₂×AGL(1,𝔽₇)	1	15	yes	—
41	C₂²×C₄	—	7	140	no	—
42	C₂×C₇	C₂³×C₃×C₄	1	30	yes	complemented
43	C₂×C₇	C₂⁵×C₃	1	1	yes	characteristic
44	C₂²×C₃	—	7	140	no	—
45	C₂²×C₃	—	7	15	no	—
46	C₃×C₄	—	7	16	no	—
47	C₂³	C₂×ΓL(1,𝔽₈)⋊C₄	1	120	yes	direct product factor
48	C₂³	C₂²×AGL(1,𝔽₇)	1	35	yes	—
49	C₂×C₄	—	7	120	no	—
50	C₇	C₂⁴×C₃×C₄	1	1	yes	characteristic, complemented, Sylow
51	C₂×C₃	—	7	30	no	—
52	C₂×C₃	—	7	1	no	—
53	C₂²	C₂²×ΓL(1,𝔽₈)⋊C₄	1	140	yes	direct product factor
54	C₂²	C₂³×AGL(1,𝔽₇)	1	15	yes	—
55	C₄	—	7	16	no	—
56	C₃	—	7	1	no	Sylow
57	C₂	C₂³×ΓL(1,𝔽₈)⋊C₄	1	30	yes	direct product factor
58	C₂	C₂⁴×AGL(1,𝔽₇)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂⁵	AGL(1,𝔽₇)
commutator, hypocenter	C₇	C₂⁴×C₃×C₄
Frattini	C₂	C₂⁴×AGL(1,𝔽₇)
Fitting, socle	C₂⁵×C₇	C₂×C₃
prime radical	C₂⁵×ΓL(1,𝔽₈)	C₂
cosocle	C₂×C₇	C₂⁵×C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₂⁵

Complements

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

N	H ≅ G/N	type
2. C₂³×ΓL(1,𝔽₈)⋊C₄	C₂	direct (N × H)
3. C₂⁴×Dic₇	C₃	semidirect
4. C₂⁴×ΓL(1,𝔽₈)	C₄	semidirect
6. C₂²×ΓL(1,𝔽₈)⋊C₄	C₂²	direct (N × H)
8. C₂³×Dic₇	C₂×C₃	semidirect
10. C₂³×ΓL(1,𝔽₈)	C₂×C₄	semidirect
12. C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	direct (N × H)
13. C₂⁴×C₇	C₃×C₄	semidirect
15. C₂²×Dic₇	C₂²×C₃	semidirect
18. C₂²×ΓL(1,𝔽₈)	C₂²×C₄	semidirect
20. ΓL(1,𝔽₈)⋊C₄	C₂⁴	direct (N × H)
22. C₂³×C₇	C₂×C₃×C₄	semidirect
24. C₂×Dic₇	C₂³×C₃	semidirect
28. C₂×ΓL(1,𝔽₈)	C₂³×C₄	semidirect
32. C₂²×C₇	C₂²×C₃×C₄	semidirect
34. Dic₇	C₂⁴×C₃	semidirect
38. ΓL(1,𝔽₈)	C₂⁴×C₄	semidirect
39. C₂⁴	ΓL(1,𝔽₈)⋊C₄	direct (N × H)
42. C₂×C₇	C₂³×C₃×C₄	semidirect
47. C₂³	C₂×ΓL(1,𝔽₈)⋊C₄	direct (N × H)
50. C₇	C₂⁴×C₃×C₄	semidirect
53. C₂²	C₂²×ΓL(1,𝔽₈)⋊C₄	direct (N × H)
57. C₂	C₂³×ΓL(1,𝔽₈)⋊C₄	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂⁵×ΓL(1,𝔽₈)	2
C₂³×ΓL(1,𝔽₈)⋊C₄	2
C₂⁴×Dic₇	3
C₂⁴×C₃×C₄	7

Subgroups — C24×ΓL(1,𝔽8)⋊C4 #13062159

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×ΓL(1,𝔽₈)⋊C₄ #13062159