Subgroups — C₂²×C₂²⋊ΓL(1,𝔽₈) #66753

Order 336.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊ΓL(1,𝔽₈)	C₂	1	3	yes	maximal, direct product factor
2	C₂⁴×C₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₂²⋊ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, direct product factor
4	C₂²×ΓL(1,𝔽₈)	—	4	1	no	maximal
5	C₂³×C₇	—	3	3	no	—
6	C₂³×C₇	—	3	1	no	—
7	C₂³×C₇	C₂×C₃	1	3	yes	complemented
8	C₂²×A₄	—	7	1	no	maximal, Hall
9	C₂×ΓL(1,𝔽₈)	—	4	3	no	—
10	C₂²×C₇	—	3	3	no	—
11	C₂²×C₇	—	3	2	no	—
12	C₂²×C₇	—	3	3	no	—
13	C₂²×C₇	—	3	3	no	—
14	C₂²×C₇	A₄	1	1	yes	characteristic, complemented
15	C₂²×C₇	C₂²×C₃	1	1	yes	characteristic, complemented
16	C₂×A₄	—	7	3	no	—
17	ΓL(1,𝔽₈)	—	4	1	no	Hall
18	C₂⁴	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
19	C₂×C₇	—	3	3	no	—
20	C₂×C₇	—	3	1	no	—
21	C₂×C₇	C₂×A₄	1	3	yes	complemented
22	C₂²×C₃	—	28	1	no	—
23	A₄	—	7	1	no	—
24	C₂³	—	3	3	no	—
25	C₂³	—	3	1	no	—
26	C₂³	C₂×ΓL(1,𝔽₈)	1	3	yes	complemented
27	C₇	C₂²×A₄	1	1	yes	characteristic, complemented, Sylow
28	C₂×C₃	—	28	3	no	—
29	C₂²	—	3	3	no	—
30	C₂²	—	3	2	no	—
31	C₂²	—	3	3	no	—
32	C₂²	—	3	3	no	—
33	C₂²	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
34	C₂²	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
35	C₃	—	28	1	no	Sylow
36	C₂	—	3	1	no	—
37	C₂	—	3	3	no	—
38	C₂	C₂×C₂²⋊ΓL(1,𝔽₈)	1	3	yes	direct product factor

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

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

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