Subgroups — C₅²×ΓL(1,𝔽₈)×C₃₇ #34085447

Order 19425.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×C₇×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
2	C₅×ΓL(1,𝔽₈)×C₃₇	C₅	1	6	yes	maximal, direct product factor
3	C₃×C₅²×C₃₇	—	7	1	no	maximal, Hall
4	C₅×C₇×C₃₇	C₃×C₅	1	6	yes	complemented
5	C₅²×C₃₇	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
6	ΓL(1,𝔽₈)×C₃₇	C₅²	1	1	yes	characteristic, direct product factor, Hall
7	C₃×C₅×C₃₇	—	7	6	no	—
8	C₅²×ΓL(1,𝔽₈)	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₇×C₃₇	C₃×C₅²	1	1	yes	characteristic, complemented, Hall
10	C₅×C₃₇	C₅×ΓL(1,𝔽₈)	1	6	yes	direct product factor
11	C₅²×C₇	C₃×C₃₇	1	1	yes	characteristic, complemented, Hall
12	C₃×C₃₇	—	7	1	no	Hall
13	C₅×ΓL(1,𝔽₈)	C₅×C₃₇	1	6	yes	direct product factor
14	C₃×C₅²	—	7	1	no	Hall
15	C₃₇	C₅²×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
16	C₅×C₇	C₃×C₅×C₃₇	1	6	yes	complemented
17	C₅²	ΓL(1,𝔽₈)×C₃₇	1	1	yes	characteristic, direct product factor, Sylow
18	ΓL(1,𝔽₈)	C₅²×C₃₇	1	1	yes	characteristic, direct product factor, Hall
19	C₃×C₅	—	7	6	no	—
20	C₇	C₃×C₅²×C₃₇	1	1	yes	characteristic, complemented, Sylow
21	C₅	C₅×ΓL(1,𝔽₈)×C₃₇	1	6	yes	direct product factor
22	C₃	—	7	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₅²×C₃₇	ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇	C₃×C₅²×C₃₇
Frattini	C₁	G
Fitting, socle	C₅²×C₇×C₃₇	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₅²×C₃₇

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.