Subgroups — C₅×C₂²⋊(C₂²⋊ΓL(1,𝔽₈)) #13132977

Order 1680.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₅×C₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
2	C₅×C₂²⋊ΓL(1,𝔽₈)	—	4	5	no	maximal
3	C₂²⋊(C₂²⋊ΓL(1,𝔽₈))	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂³×C₅×C₇	—	3	5	no	—
5	C₅×C₂²⋊A₄	—	7	1	no	maximal, Hall
6	C₂²×C₅×C₇	—	3	10	no	—
7	C₂²×C₅×C₇	A₄	1	5	yes	complemented
8	C₂⁴×C₇	C₃×C₅	1	1	yes	characteristic, complemented, Hall
9	C₅×ΓL(1,𝔽₈)	—	16	1	no	Hall
10	C₂²⋊ΓL(1,𝔽₈)	—	4	5	no	—
11	C₂⁴×C₅	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
12	C₂×C₅×C₇	—	3	5	no	—
13	C₅×A₄	—	28	5	no	—
14	C₂³×C₇	—	3	5	no	—
15	C₂²⋊A₄	—	7	1	no	Hall
16	C₂³×C₅	—	3	5	no	—
17	C₅×C₇	C₂²⋊A₄	1	1	yes	characteristic, complemented, Hall
18	C₂²×C₇	—	3	10	no	—
19	C₂²×C₇	C₅×A₄	1	5	yes	complemented
20	ΓL(1,𝔽₈)	—	16	1	no	Hall
21	C₂²×C₅	—	3	10	no	—
22	C₂²×C₅	C₂²⋊ΓL(1,𝔽₈)	1	5	yes	complemented
23	C₂⁴	C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
24	C₃×C₅	—	112	1	no	Hall
25	C₂×C₇	—	3	5	no	—
26	A₄	—	28	5	no	—
27	C₂×C₅	—	3	5	no	—
28	C₂³	—	3	5	no	—
29	C₇	C₅×C₂²⋊A₄	1	1	yes	characteristic, complemented, Sylow
30	C₅	C₂²⋊(C₂²⋊ΓL(1,𝔽₈))	1	1	yes	characteristic, direct product factor, Sylow
31	C₂²	—	3	10	no	—
32	C₂²	C₅×C₂²⋊ΓL(1,𝔽₈)	1	5	yes	complemented
33	C₃	—	112	1	no	Sylow
34	C₂	—	3	5	no	—

	subgroup	G/H
center, hypercenter	C₅	C₂²⋊(C₂²⋊ΓL(1,𝔽₈))
commutator, cosocle, hypocenter	C₂⁴×C₇	C₃×C₅
Frattini	C₁	G
Fitting, socle	C₂⁴×C₅×C₇	C₃
LCS	G ▷ C₂⁴×C₇
UCS	C₁ ◁ C₅

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.