Subgroups — C₇⋊(C₂₂₃⋊AGL(1,𝔽₅)) #44530239

Order 31220.

24 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×D₅×C₂₂₃	C₂	1	1	yes	maximal, characteristic
2	C₅×C₇×C₂₂₃	C₄	1	1	yes	characteristic, complemented, Hall
3	Dic₁₅₆₁	—	5	1	no	maximal, Hall
4	C₂₂₃⋊AGL(1,𝔽₅)	—	7	1	no	maximal, Hall
5	C₂×C₇×C₂₂₃	—	5	1	no	—
6	D₅×C₂₂₃	D₇	1	1	yes	characteristic
7	C₇×C₂₂₃	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
8	C₅×C₂₂₃	Dic₇	1	1	yes	characteristic, complemented, Hall
9	Dic₂₂₃	—	35	1	no	Hall
10	C₂×C₂₂₃	—	5	1	no	—
11	C₂₂₃	C₇⋊AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Sylow
12	C₇⋊AGL(1,𝔽₅)	—	223	1	no	maximal, Hall
13	C₇×D₅	D₂₂₃	1	1	yes	characteristic
14	C₅×C₇	Dic₂₂₃	1	1	yes	characteristic, complemented, Hall
15	Dic₇	—	1115	1	no	Hall
16	AGL(1,𝔽₅)	—	1561	1	no	Hall
17	C₂×C₇	—	5	1	no	—
18	D₅	D₁₅₆₁	1	1	yes	characteristic
19	C₇	C₂₂₃⋊AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Sylow
20	C₅	Dic₁₅₆₁	1	1	yes	characteristic, complemented, Sylow
21	C₄	—	7805	1	no	Sylow
22	C₂	—	5	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅×C₇×C₂₂₃	C₄
prime radical, cosocle	C₇×D₅×C₂₂₃	C₂
LCS	G ▷ C₅×C₇×C₂₂₃
UCS	C₁

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.