Subgroups — C₅²×C₁₆×C₇₁ #72205346

Order 28400.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
2	C₅×C₁₆×C₇₁	C₅	1	6	yes	maximal, direct product factor
3	C₅×C₈×C₇₁	C₂×C₅	1	6	yes	—
4	C₄×C₅²×C₇₁	C₄	1	1	yes	characteristic
5	C₂×C₅²×C₇₁	C₈	1	1	yes	characteristic
6	C₅²×C₇₁	C₁₆	1	1	yes	characteristic, direct product factor, Hall
7	C₄×C₅×C₇₁	C₄×C₅	1	6	yes	—
8	C₁₆×C₇₁	C₅²	1	1	yes	characteristic, direct product factor, Hall
9	C₂×C₅×C₇₁	C₅×C₈	1	6	yes	—
10	C₈×C₇₁	C₂×C₅²	1	1	yes	characteristic
11	C₅²×C₁₆	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₅×C₇₁	C₅×C₁₆	1	6	yes	direct product factor
13	C₄×C₇₁	C₄×C₅²	1	1	yes	characteristic
14	C₅²×C₈	C₂×C₇₁	1	1	yes	characteristic
15	C₂×C₇₁	C₅²×C₈	1	1	yes	characteristic
16	C₄×C₅²	C₄×C₇₁	1	1	yes	characteristic
17	C₅×C₁₆	C₅×C₇₁	1	6	yes	direct product factor
18	C₇₁	C₅²×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
19	C₂×C₅²	C₈×C₇₁	1	1	yes	characteristic
20	C₅×C₈	C₂×C₅×C₇₁	1	6	yes	—
21	C₅²	C₁₆×C₇₁	1	1	yes	characteristic, direct product factor, Sylow
22	C₄×C₅	C₄×C₅×C₇₁	1	6	yes	—
23	C₁₆	C₅²×C₇₁	1	1	yes	characteristic, direct product factor, Sylow
24	C₂×C₅	C₅×C₈×C₇₁	1	6	yes	—
25	C₈	C₂×C₅²×C₇₁	1	1	yes	characteristic
26	C₅	C₅×C₁₆×C₇₁	1	6	yes	direct product factor
27	C₄	C₄×C₅²×C₇₁	1	1	yes	characteristic
28	C₂	C₅²×C₈×C₇₁	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₈	C₂×C₅²×C₇₁
socle, prime radical	C₂×C₅²×C₇₁	C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×C₁₆×C₇₁	C₅	direct (N × H)
6. C₅²×C₇₁	C₁₆	direct (N × H)
8. C₁₆×C₇₁	C₅²	direct (N × H)
11. C₅²×C₁₆	C₇₁	direct (N × H)
12. C₅×C₇₁	C₅×C₁₆	direct (N × H)
17. C₅×C₁₆	C₅×C₇₁	direct (N × H)
18. C₇₁	C₅²×C₁₆	direct (N × H)
21. C₅²	C₁₆×C₇₁	direct (N × H)
23. C₁₆	C₅²×C₇₁	direct (N × H)
26. C₅	C₅×C₁₆×C₇₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅²×C₈×C₇₁	2
C₅×C₁₆×C₇₁	5
C₅²×C₁₆	71

Subgroups — C52×C16×C71 #72205346

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×C₁₆×C₇₁ #72205346