Subgroups — C₅×SL(2,𝔽₇) #13132377

Order 1680.

38 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	SL(2,𝔽₇)	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
2	C₅×#226	—	7	2	no	maximal, Hall
3	C₂×C₅×ΓL(1,𝔽₈)	—	8	1	no	maximal
4	C₅×SL(2,𝔽₃)	—	7	2	no	—
5	C₅×ΓL(1,𝔽₈)	—	8	1	no	Hall
6	C₅×Q₁₆	—	21	1	no	Hall
7	C₂×C₅×C₇	—	8	1	no	—
8	C₅×Dic₃	—	28	1	no	—
9	#226	—	7	2	no	Hall
10	C₂×ΓL(1,𝔽₈)	—	8	1	no	—
11	C₅×Q₈	—	7	2	no	—
12	C₅×C₈	—	21	1	no	—
13	C₅×C₇	—	8	1	no	Hall
14	C₂×C₃×C₅	—	28	1	no	—
15	SL(2,𝔽₃)	—	7	2	no	—
16	ΓL(1,𝔽₈)	—	8	1	no	Hall
17	C₄×C₅	—	21	1	no	—
18	Q₁₆	—	21	1	no	Sylow
19	C₃×C₅	—	28	1	no	Hall
20	C₂×C₇	—	8	1	no	—
21	Dic₃	—	28	1	no	—
22	C₂×C₅	GL(3,𝔽₂)	1	1	yes	maximal normal, characteristic
23	Q₈	—	7	2	no	—
24	C₈	—	21	1	no	—
25	C₇	—	8	1	no	Sylow
26	C₂×C₃	—	28	1	no	—
27	C₅	SL(2,𝔽₇)	1	1	yes	characteristic, direct product factor, Sylow
28	C₄	—	21	1	no	—
29	C₃	—	28	1	no	Sylow
30	C₂	C₅×GL(3,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, Fitting, socle, hypercenter	C₂×C₅	GL(3,𝔽₂)
commutator, hypocenter	SL(2,𝔽₇)	C₅
Frattini, cosocle	C₂	C₅×GL(3,𝔽₂)
LCS	G ▷ SL(2,𝔽₇)
UCS	C₁ ◁ C₂×C₅

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

N	H ≅ G/N	type
1. SL(2,𝔽₇)	C₅	direct (N × H)
27. C₅	SL(2,𝔽₇)	direct (N × H)

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