Subgroups — C₇²×C₁₆×C₃₁ #72201593

Order 24304.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇²×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₇×C₁₆×C₃₁	C₇	1	8	yes	maximal, direct product factor
3	C₄×C₇²×C₃₁	C₄	1	1	yes	characteristic
4	C₂×C₇²×C₃₁	C₈	1	1	yes	characteristic
5	C₇×C₈×C₃₁	C₂×C₇	1	8	yes	—
6	C₇²×C₃₁	C₁₆	1	1	yes	characteristic, direct product factor, Hall
7	C₄×C₇×C₃₁	C₄×C₇	1	8	yes	—
8	C₇²×C₁₆	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₁₆×C₃₁	C₇²	1	1	yes	characteristic, direct product factor, Hall
10	C₂×C₇×C₃₁	C₇×C₈	1	8	yes	—
11	C₇²×C₈	C₂×C₃₁	1	1	yes	characteristic
12	C₈×C₃₁	C₂×C₇²	1	1	yes	characteristic
13	C₇×C₃₁	C₇×C₁₆	1	8	yes	direct product factor
14	C₄×C₇²	C₄×C₃₁	1	1	yes	characteristic
15	C₄×C₃₁	C₄×C₇²	1	1	yes	characteristic
16	C₇×C₁₆	C₇×C₃₁	1	8	yes	direct product factor
17	C₂×C₇²	C₈×C₃₁	1	1	yes	characteristic
18	C₂×C₃₁	C₇²×C₈	1	1	yes	characteristic
19	C₇×C₈	C₂×C₇×C₃₁	1	8	yes	—
20	C₇²	C₁₆×C₃₁	1	1	yes	characteristic, direct product factor, Sylow
21	C₃₁	C₇²×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
22	C₄×C₇	C₄×C₇×C₃₁	1	8	yes	—
23	C₁₆	C₇²×C₃₁	1	1	yes	characteristic, direct product factor, Sylow
24	C₂×C₇	C₇×C₈×C₃₁	1	8	yes	—
25	C₈	C₂×C₇²×C₃₁	1	1	yes	characteristic
26	C₇	C₇×C₁₆×C₃₁	1	8	yes	direct product factor
27	C₄	C₄×C₇²×C₃₁	1	1	yes	characteristic
28	C₂	C₇²×C₈×C₃₁	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₈	C₂×C₇²×C₃₁
socle, prime radical	C₂×C₇²×C₃₁	C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×C₁₆×C₃₁	C₇	direct (N × H)
6. C₇²×C₃₁	C₁₆	direct (N × H)
8. C₇²×C₁₆	C₃₁	direct (N × H)
9. C₁₆×C₃₁	C₇²	direct (N × H)
13. C₇×C₃₁	C₇×C₁₆	direct (N × H)
16. C₇×C₁₆	C₇×C₃₁	direct (N × H)
20. C₇²	C₁₆×C₃₁	direct (N × H)
21. C₃₁	C₇²×C₁₆	direct (N × H)
23. C₁₆	C₇²×C₃₁	direct (N × H)
26. C₇	C₇×C₁₆×C₃₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇²×C₈×C₃₁	2
C₇×C₁₆×C₃₁	7
C₇²×C₁₆	31

Subgroups — C72×C16×C31 #72201593

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇²×C₁₆×C₃₁ #72201593