Finite Group Databasebeta

← C₇×C₁₆×C₂₂₃ #72202201

Subgroups — C₇×C₁₆×C₂₂₃ #72202201

Order 24976.

20 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
2	C₄×C₇×C₂₂₃	C₄	1	1	yes	characteristic
3	C₁₆×C₂₂₃	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂×C₇×C₂₂₃	C₈	1	1	yes	characteristic
5	C₈×C₂₂₃	C₂×C₇	1	1	yes	characteristic
6	C₇×C₂₂₃	C₁₆	1	1	yes	characteristic, direct product factor, Hall
7	C₄×C₂₂₃	C₄×C₇	1	1	yes	characteristic
8	C₂×C₂₂₃	C₇×C₈	1	1	yes	characteristic
9	C₂₂₃	C₇×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
10	C₇×C₁₆	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₇×C₈	C₂×C₂₂₃	1	1	yes	characteristic
12	C₄×C₇	C₄×C₂₂₃	1	1	yes	characteristic
13	C₁₆	C₇×C₂₂₃	1	1	yes	characteristic, direct product factor, Sylow
14	C₂×C₇	C₈×C₂₂₃	1	1	yes	characteristic
15	C₈	C₂×C₇×C₂₂₃	1	1	yes	characteristic
16	C₇	C₁₆×C₂₂₃	1	1	yes	characteristic, direct product factor, Sylow
17	C₄	C₄×C₇×C₂₂₃	1	1	yes	characteristic
18	C₂	C₇×C₈×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₈	C₂×C₇×C₂₂₃
socle, prime radical	C₂×C₇×C₂₂₃	C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₆×C₂₂₃	C₇	direct (N × H)
6. C₇×C₂₂₃	C₁₆	direct (N × H)
9. C₂₂₃	C₇×C₁₆	direct (N × H)
10. C₇×C₁₆	C₂₂₃	direct (N × H)
13. C₁₆	C₇×C₂₂₃	direct (N × H)
16. C₇	C₁₆×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
C₇×C₈×C₂₂₃	2
C₁₆×C₂₂₃	7
C₇×C₁₆	223