Subgroups — C₂²⋊C₄×ΓL(1,𝔽₈) #66580

Order 336.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×ΓL(1,𝔽₈)	C₂	1	2	yes	maximal, complemented
3	C₇×C₂²⋊C₄	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
5	C₂²×ΓL(1,𝔽₈)	C₄	1	2	yes	complemented
6	C₂²×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
7	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
8	C₂³×C₇	C₂×C₃	1	1	yes	characteristic
9	C₂×C₄×C₇	C₂×C₃	1	2	yes	complemented
10	C₃×C₂²⋊C₄	—	7	1	no	maximal, Hall
11	C₂×ΓL(1,𝔽₈)	—	2	2	no	—
12	C₂×ΓL(1,𝔽₈)	C₂×C₄	1	1	yes	characteristic
13	C₂×ΓL(1,𝔽₈)	D₄	1	2	yes	—
14	C₂²×C₇	—	2	2	no	—
15	C₂²×C₇	C₃×C₄	1	2	yes	complemented
16	C₂²×C₇	C₂²×C₃	1	1	yes	characteristic
17	C₄×C₇	—	2	2	no	—
18	C₂³×C₃	—	7	1	no	—
19	C₂×C₃×C₄	—	7	2	no	—
20	ΓL(1,𝔽₈)	C₂²⋊C₄	1	1	yes	characteristic, direct product factor, Hall
21	C₂²⋊C₄	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
22	C₂×C₇	—	2	2	no	—
23	C₂×C₇	C₂×C₃×C₄	1	1	yes	characteristic
24	C₂×C₇	C₃×D₄	1	2	yes	—
25	C₂²×C₃	—	7	2	no	—
26	C₂²×C₃	—	7	1	no	—
27	C₂²×C₃	—	14	2	no	—
28	C₃×C₄	—	14	2	no	—
29	C₂³	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
30	C₂×C₄	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
31	C₇	C₃×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
32	C₂×C₃	—	7	1	no	—
33	C₂×C₃	—	7	2	no	—
34	C₂×C₃	—	14	2	no	—
35	C₂²	—	2	2	no	—
36	C₂²	C₄×ΓL(1,𝔽₈)	1	2	yes	complemented
37	C₂²	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
38	C₄	—	2	2	no	—
39	C₃	—	7	1	no	Sylow
40	C₂	—	2	2	no	—
41	C₂	C₂×C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
42	C₂	D₄×ΓL(1,𝔽₈)	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×ΓL(1,𝔽₈)	C₂	semidirect
3. C₇×C₂²⋊C₄	C₃	semidirect
5. C₂²×ΓL(1,𝔽₈)	C₄	semidirect
9. C₂×C₄×C₇	C₂×C₃	semidirect
15. C₂²×C₇	C₃×C₄	semidirect
20. ΓL(1,𝔽₈)	C₂²⋊C₄	direct (N × H)
21. C₂²⋊C₄	ΓL(1,𝔽₈)	direct (N × H)
30. C₂×C₄	C₂×ΓL(1,𝔽₈)	semidirect
31. C₇	C₃×C₂²⋊C₄	semidirect
36. C₂²	C₄×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×ΓL(1,𝔽₈)	2
C₂×C₄×ΓL(1,𝔽₈)	2
C₇×C₂²⋊C₄	3
C₃×C₂²⋊C₄	7

Subgroups — C22⋊C4×ΓL(1,𝔽8) #66580

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²⋊C₄×ΓL(1,𝔽₈) #66580