Subgroups — (ΓL(1,𝔽₈)⋊C₄)⋊C₄ #66546

Order 336.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×ΓL(1,𝔽₈)⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂×ΓL(1,𝔽₈)⋊C₄	C₂	1	1	yes	maximal, characteristic
4	Dic₇⋊C₄	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	C₂²×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
6	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
7	ΓL(1,𝔽₈)⋊C₄	—	2	1	no	—
8	ΓL(1,𝔽₈)⋊C₄	C₄	1	2	yes	complemented
9	C₂×C₄×C₇	C₂×C₃	1	1	yes	characteristic
10	C₂×Dic₇	C₂×C₃	1	1	yes	characteristic
11	C₂×Dic₇	C₂×C₃	1	1	yes	characteristic
12	C₃×C₄⋊C₄	—	7	1	no	maximal, Hall
13	C₂×ΓL(1,𝔽₈)	C₂×C₄	1	1	yes	characteristic
14	C₂×ΓL(1,𝔽₈)	D₄	1	1	yes	characteristic
15	C₂×ΓL(1,𝔽₈)	Q₈	1	1	yes	characteristic
16	C₂²×C₇	C₂²×C₃	1	1	yes	characteristic
17	C₄×C₇	—	2	1	no	—
18	Dic₇	—	2	1	no	—
19	Dic₇	C₃×C₄	1	2	yes	complemented
20	C₂×C₃×C₄	—	7	1	no	—
21	C₂×C₃×C₄	—	7	1	no	—
22	C₂×C₃×C₄	—	7	1	no	—
23	ΓL(1,𝔽₈)	C₄⋊C₄	1	1	yes	characteristic, complemented, Hall
24	C₄⋊C₄	—	7	1	no	Sylow
25	C₂×C₇	C₂×C₃×C₄	1	1	yes	characteristic
26	C₂×C₇	C₃×D₄	1	1	yes	characteristic
27	C₂×C₇	C₃×Q₈	1	1	yes	characteristic
28	C₂²×C₃	—	7	1	no	—
29	C₃×C₄	—	7	2	no	—
30	C₃×C₄	—	14	1	no	—
31	C₃×C₄	—	14	1	no	—
32	C₂×C₄	—	7	1	no	—
33	C₂×C₄	—	7	1	no	—
34	C₂×C₄	AGL(1,𝔽₇)	1	1	yes	characteristic
35	C₇	C₃×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
36	C₂×C₃	—	7	1	no	—
37	C₂×C₃	—	7	1	no	—
38	C₂×C₃	—	7	1	no	—
39	C₂²	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
40	C₄	—	2	1	no	—
41	C₄	—	7	2	no	—
42	C₄	—	14	1	no	—
43	C₃	—	7	1	no	Sylow
44	C₂	ΓL(1,𝔽₈)⋊Q₈	1	1	yes	characteristic
45	C₂	C₄×AGL(1,𝔽₇)	1	1	yes	characteristic
46	C₂	(C₂²⋊D₇)⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂×AGL(1,𝔽₇)
commutator	C₂×C₇	C₂×C₃×C₄
Fitting	C₂×C₄×C₇	C₂×C₃
socle, cosocle	C₂²×C₇	C₂²×C₃
prime radical	C₂²×ΓL(1,𝔽₈)	C₂²
hypercenter	C₂×C₄	AGL(1,𝔽₇)
hypocenter	C₇	C₃×C₄⋊C₄
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
4. Dic₇⋊C₄	C₃	semidirect
8. ΓL(1,𝔽₈)⋊C₄	C₄	semidirect
19. Dic₇	C₃×C₄	semidirect
23. ΓL(1,𝔽₈)	C₄⋊C₄	semidirect
35. C₇	C₃×C₄⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×ΓL(1,𝔽₈)	2
C₂×ΓL(1,𝔽₈)⋊C₄	2
Dic₇⋊C₄	3
C₃×C₄⋊C₄	7

Subgroups — (ΓL(1,𝔽8)⋊C4)⋊C4 #66546

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — (ΓL(1,𝔽₈)⋊C₄)⋊C₄ #66546