LMFDB - Subgroup search results

Refine search

Subgroup label	Subgroup order	Cyclic	Abelian	Solvable

Normal	Characteristic	Perfect	Maximal	Central

Ambient label	Ambient order	Direct product	Semidirect product

Quotient label	Subgroup index	Cyclic quotient	Abelian quotient	Solvable quotient

Maximal quotient	Non-trivial proper	Hall subgroup	Sylow subgroup

		Sort order	Select

Results (1-50 of 832 matches)

Next Download displayed columns for results to

Label	Subgroup								Ambient			Quotient
Label	Name	Order	Sylow	norm	char	max	cent	cyc	ab	nilp		sup solv	solv	perf	simp	Agp	Zgp	metab	metacyc	Name	Order	Name	Size	max	cyc	ab	nilp	sup solv	solv	simp	Agp	metab
63281250.b.1.a1.a1	$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$		✓	✓						✓								$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$		$C_1$	$1$		✓	✓	✓	✓	✓		✓	✓
63281250.b.2.a1.a1	$C_5^8.C_3.\He_3$	$3^{4} \cdot 5^{8}$		✓	✓	✓					✓								$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$		$C_2$	$2$		✓	✓	✓	✓	✓	✓	✓	✓
63281250.b.3._.A	$C_5^6.C_{15}^2:S_3$	$2 \cdot 3^{3} \cdot 5^{8}$		✓	✓	✓					✓								$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$		$C_3$	$3$		✓	✓	✓	✓	✓	✓	✓	✓
63281250.b.3._.B	$C_5^7.C_{15}.C_3.C_6$	$2 \cdot 3^{3} \cdot 5^{8}$				✓					✓			✓					$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$			$3$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
63281250.b.6._.A	$C_5^8.C_3^3$	$3^{3} \cdot 5^{8}$		✓	✓						✓			✓		✓			$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$		$S_3$	$2 \cdot 3$					✓	✓		✓	✓
63281250.b.6._.B	$C_5^8.\He_3$	$3^{3} \cdot 5^{8}$		✓	✓						✓								$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$		$C_6$	$2 \cdot 3$		✓	✓	✓	✓	✓		✓	✓
63281250.b.6._.C	$C_5^8.C_9:C_3$	$3^{3} \cdot 5^{8}$									✓								$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$			$2 \cdot 3$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
63281250.b.9._.A	$C_5^7.C_{15}.C_6$	$2 \cdot 3^{2} \cdot 5^{8}$									✓			✓					$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$			$3^{2}$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
63281250.b.9._.B	$C_5^7.C_{15}.C_6$	$2 \cdot 3^{2} \cdot 5^{8}$									✓			✓					$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$			$3^{2}$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
63281250.b.9._.C	$C_5^5.C_3.C_5^3.C_6$	$2 \cdot 3^{2} \cdot 5^{8}$									✓			✓					$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$			$3^{2}$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
63281250.b.9._.D	$C_5^7.C_{15}.C_6$	$2 \cdot 3^{2} \cdot 5^{8}$									✓			✓					$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$			$3^{2}$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
63281250.b.9._.E	$C_5^8.C_3.C_6$	$2 \cdot 3^{2} \cdot 5^{8}$									✓			✓		✓			$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$			$3^{2}$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
63281250.b.9._.F	$C_5^8.C_3.C_6$	$2 \cdot 3^{2} \cdot 5^{8}$									✓			✓					$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$			$3^{2}$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
63281250.b.15._.A	$C_5^7.C_3^3.C_2$	$2 \cdot 3^{3} \cdot 5^{7}$									✓			✓					$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$			$3 \cdot 5$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
63281250.b.15._.B	$C_5^6.C_{15}.C_3.C_6$	$2 \cdot 3^{3} \cdot 5^{7}$									✓			✓					$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$			$3 \cdot 5$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
63281250.b.18._.A	$C_5^8.C_3^2$	$3^{2} \cdot 5^{8}$		✓	✓						✓			✓		✓			$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$		$C_3\times S_3$	$2 \cdot 3^{2}$					✓	✓		✓	✓
63281250.b.18._.B	$C_5^8.C_3^2$	$3^{2} \cdot 5^{8}$									✓			✓		✓			$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$			$2 \cdot 3^{2}$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
63281250.b.18._.C	$C_5^8.C_3^2$	$3^{2} \cdot 5^{8}$									✓			✓		✓			$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$			$2 \cdot 3^{2}$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
63281250.b.18._.D	$C_5^8.C_3^2$	$3^{2} \cdot 5^{8}$									✓			✓		✓			$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$			$2 \cdot 3^{2}$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
63281250.b.18._.E	$C_5^8.C_3^2$	$3^{2} \cdot 5^{8}$									✓			✓		✓			$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$			$2 \cdot 3^{2}$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
63281250.b.18._.F	$C_5^8.C_3^2$	$3^{2} \cdot 5^{8}$									✓			✓		✓			$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$			$2 \cdot 3^{2}$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
63281250.b.18._.G	$C_5^8.C_9$	$3^{2} \cdot 5^{8}$									✓			✓		✓			$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$			$2 \cdot 3^{2}$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
63281250.b.25.a1.a1	$C_5^6:C_3\wr S_3$	$2 \cdot 3^{4} \cdot 5^{6}$				✓					✓								$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$			$5^{2}$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
63281250.b.27._.A	$C_5^5.C_3.C_5^3.C_2$	$2 \cdot 3 \cdot 5^{8}$									✓			✓					$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$			$3^{3}$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
63281250.b.27._.B	$C_5^8.C_6$	$2 \cdot 3 \cdot 5^{8}$									✓			✓		✓			$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$			$3^{3}$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
63281250.b.27._.C	$C_5^8.C_6$	$2 \cdot 3 \cdot 5^{8}$									✓			✓		✓			$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$			$3^{3}$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
63281250.b.27._.D	$C_5^8.C_6$	$2 \cdot 3 \cdot 5^{8}$									✓			✓		✓			$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$			$3^{3}$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
63281250.b.27._.E	$C_5^8.C_6$	$2 \cdot 3 \cdot 5^{8}$									✓			✓		✓			$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$			$3^{3}$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
63281250.b.27._.F	$C_5^6.C_{15}.C_{10}$	$2 \cdot 3 \cdot 5^{8}$									✓			✓					$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$			$3^{3}$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
63281250.b.30._.A	$C_5^7.C_3^3$	$3^{3} \cdot 5^{7}$									✓			✓		✓			$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$			$2 \cdot 3 \cdot 5$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
63281250.b.30._.B	$C_5^7.C_3^3$	$3^{3} \cdot 5^{7}$									✓			✓		✓			$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$			$2 \cdot 3 \cdot 5$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
63281250.b.45._.A	$C_5^5.C_{15}.C_{30}$	$2 \cdot 3^{2} \cdot 5^{7}$									✓			✓					$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$			$3^{2} \cdot 5$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
63281250.b.45._.B	$C_5^7.C_3.C_6$	$2 \cdot 3^{2} \cdot 5^{7}$									✓			✓					$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$			$3^{2} \cdot 5$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
63281250.b.45._.C	$C_5^7.C_3.C_6$	$2 \cdot 3^{2} \cdot 5^{7}$									✓			✓					$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$			$3^{2} \cdot 5$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
63281250.b.45._.D	$C_5^7.C_3.C_6$	$2 \cdot 3^{2} \cdot 5^{7}$									✓			✓					$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$			$3^{2} \cdot 5$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
63281250.b.45._.E	$C_5^5.C_{15}.C_{30}$	$2 \cdot 3^{2} \cdot 5^{7}$									✓			✓					$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$			$3^{2} \cdot 5$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
63281250.b.45._.F	$C_5^5.C_{15}.C_{30}$	$2 \cdot 3^{2} \cdot 5^{7}$									✓			✓					$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$			$3^{2} \cdot 5$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
63281250.b.45._.G	$C_5^5.C_{15}.C_{30}$	$2 \cdot 3^{2} \cdot 5^{7}$									✓			✓					$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$			$3^{2} \cdot 5$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
63281250.b.45._.H	$C_5^5.C_{15}.C_{30}$	$2 \cdot 3^{2} \cdot 5^{7}$									✓			✓					$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$			$3^{2} \cdot 5$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
63281250.b.45._.I	$C_5^5.C_{15}.C_{30}$	$2 \cdot 3^{2} \cdot 5^{7}$									✓			✓					$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$			$3^{2} \cdot 5$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
63281250.b.45._.J	$C_5^5.C_{15}.C_{30}$	$2 \cdot 3^{2} \cdot 5^{7}$									✓			✓					$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$			$3^{2} \cdot 5$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
63281250.b.45._.K	$C_5^5.C_{15}.C_{30}$	$2 \cdot 3^{2} \cdot 5^{7}$									✓			✓					$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$			$3^{2} \cdot 5$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
63281250.b.45._.L	$C_5^5.C_{15}.C_{30}$	$2 \cdot 3^{2} \cdot 5^{7}$									✓			✓					$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$			$3^{2} \cdot 5$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
63281250.b.45._.M	$C_5^5.C_{15}.C_{30}$	$2 \cdot 3^{2} \cdot 5^{7}$									✓			✓					$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$			$3^{2} \cdot 5$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
63281250.b.45._.N	$C_5^5.C_{15}.C_{30}$	$2 \cdot 3^{2} \cdot 5^{7}$									✓			✓					$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$			$3^{2} \cdot 5$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
63281250.b.45._.O	$C_5^7.C_3.C_6$	$2 \cdot 3^{2} \cdot 5^{7}$									✓			✓		✓			$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$			$3^{2} \cdot 5$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
63281250.b.45._.P	$C_5^6.C_{15}.C_6$	$2 \cdot 3^{2} \cdot 5^{7}$									✓			✓					$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$			$3^{2} \cdot 5$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
63281250.b.45._.Q	$C_5^7:(C_3\times S_3)$	$2 \cdot 3^{2} \cdot 5^{7}$									✓			✓					$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$			$3^{2} \cdot 5$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
63281250.b.45._.R	$C_5^7:(C_3\times S_3)$	$2 \cdot 3^{2} \cdot 5^{7}$									✓			✓					$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$			$3^{2} \cdot 5$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$
63281250.b.45._.S	$C_5^4.C_3.C_5^3.C_6$	$2 \cdot 3^{2} \cdot 5^{7}$									✓			✓					$C_5^8.\He_3.C_6$	$2 \cdot 3^{4} \cdot 5^{8}$			$3^{2} \cdot 5$		$-$	$-$	$-$	$-$	$-$	$-$	$-$	$-$

Next Download displayed columns for results to

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Learn more

Refine search

Results (1-50 of 832 matches)

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.