Finite Group Databasebeta

← C₂³×C₁₇×C₂₂₃ #72207113

Subgroups — C₂³×C₁₇×C₂₂₃ #72207113

Order 30328.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₇×C₂₂₃	C₂	1	7	yes	maximal, direct product factor
2	C₂×C₁₇×C₂₂₃	C₂²	1	7	yes	direct product factor
3	C₁₇×C₂₂₃	C₂³	1	1	yes	characteristic, direct product factor, Hall
4	C₂³×C₂₂₃	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂²×C₂₂₃	C₂×C₁₇	1	7	yes	direct product factor
6	C₂×C₂₂₃	C₂²×C₁₇	1	7	yes	direct product factor
7	C₂₂₃	C₂³×C₁₇	1	1	yes	characteristic, direct product factor, Sylow
8	C₂³×C₁₇	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂²×C₁₇	C₂×C₂₂₃	1	7	yes	direct product factor
10	C₂×C₁₇	C₂²×C₂₂₃	1	7	yes	direct product factor
11	C₁₇	C₂³×C₂₂₃	1	1	yes	characteristic, direct product factor, Sylow
12	C₂³	C₁₇×C₂₂₃	1	1	yes	characteristic, direct product factor, Sylow
13	C₂²	C₂×C₁₇×C₂₂₃	1	7	yes	direct product factor
14	C₂	C₂²×C₁₇×C₂₂₃	1	7	yes	direct product factor

	subgroup	G/H
center	G	C₁
commutator, Frattini	C₁	G
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₁₇×C₂₂₃	C₂	direct (N × H)
2. C₂×C₁₇×C₂₂₃	C₂²	direct (N × H)
3. C₁₇×C₂₂₃	C₂³	direct (N × H)
4. C₂³×C₂₂₃	C₁₇	direct (N × H)
5. C₂²×C₂₂₃	C₂×C₁₇	direct (N × H)
6. C₂×C₂₂₃	C₂²×C₁₇	direct (N × H)
7. C₂₂₃	C₂³×C₁₇	direct (N × H)
8. C₂³×C₁₇	C₂₂₃	direct (N × H)
9. C₂²×C₁₇	C₂×C₂₂₃	direct (N × H)
10. C₂×C₁₇	C₂²×C₂₂₃	direct (N × H)
11. C₁₇	C₂³×C₂₂₃	direct (N × H)
12. C₂³	C₁₇×C₂₂₃	direct (N × H)
13. C₂²	C₂×C₁₇×C₂₂₃	direct (N × H)
14. C₂	C₂²×C₁₇×C₂₂₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₇×C₂₂₃	2
C₂³×C₂₂₃	17
C₂³×C₁₇	223