Subgroups — #49142017

Order 36544.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₅₇₁⋊C₈)	C₂	1	2	yes	maximal
2	C₄×C₈×C₅₇₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₇₁⋊C₈	—	2	2	no	—
4	C₂×C₈×C₅₇₁	C₄	1	1	yes	characteristic
5	C₂×C₈×C₅₇₁	C₄	1	1	yes	characteristic
6	C₄²×C₅₇₁	C₂²	1	1	yes	characteristic
7	C₅₇₁⋊C₈	—	4	2	no	—
8	C₂×C₄×C₅₇₁	C₂×C₄	1	1	yes	characteristic
9	C₂×C₄×C₅₇₁	D₄	1	1	yes	characteristic
10	C₂×C₄×C₅₇₁	Q₈	1	1	yes	characteristic
11	C₈×C₅₇₁	—	2	1	no	—
12	C₈×C₅₇₁	C₈	1	2	yes	complemented
13	C₂²×C₅₇₁	C₄⋊C₄	1	1	yes	characteristic
14	C₄×C₅₇₁	—	2	1	no	—
15	C₄×C₅₇₁	C₂×C₈	1	1	yes	characteristic
16	C₄×C₅₇₁	M₁₆	1	1	yes	characteristic
17	C₄×C₅₇₁	SD₁₆	1	2	yes	—
18	C₂×C₅₇₁	C₄⋊C₈	1	1	yes	characteristic
19	C₂×C₅₇₁	#106	1	1	yes	characteristic
20	C₂×C₅₇₁	#108	1	1	yes	characteristic
21	C₅₇₁	#334	1	1	yes	characteristic, complemented, Sylow
22	#334	—	571	1	no	maximal, Sylow
23	C₄⋊C₈	—	571	2	no	—
24	C₄×C₈	D₅₇₁	1	1	yes	characteristic
25	C₂×C₈	—	1142	2	no	—
26	C₂×C₈	Dic₅₇₁	1	1	yes	characteristic
27	C₂×C₈	Dic₅₇₁	1	1	yes	characteristic
28	C₄²	C₂×D₅₇₁	1	1	yes	characteristic
29	C₂×C₄	C₂×Dic₅₇₁	1	1	yes	characteristic
30	C₂×C₄	D₂₂₈₄	1	1	yes	characteristic
31	C₂×C₄	Dic₁₁₄₂	1	1	yes	characteristic
32	C₈	—	2	1	no	—
33	C₈	—	2284	2	no	—
34	C₈	C₅₇₁⋊C₈	1	2	yes	complemented
35	C₂²	C₄⋊Dic₅₇₁	1	1	yes	characteristic
36	C₄	—	2	1	no	—
37	C₄	C₂×C₅₇₁⋊C₈	1	1	yes	characteristic
38	C₄	#22603717	1	1	yes	characteristic
39	C₄	#22603722	1	2	yes	—
40	C₂	C₄⋊(C₅₇₁⋊C₈)	1	1	yes	characteristic
41	C₂	#32912626	1	1	yes	characteristic
42	C₂	#32912630	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₂₂₈₄
commutator	C₄×C₅₇₁	C₂×C₈
Frattini	C₄²	C₂×D₅₇₁
Fitting	C₄×C₈×C₅₇₁	C₂
socle, prime radical	C₂²×C₅₇₁	C₄⋊C₄
cosocle	C₄²×C₅₇₁	C₂²
hypercenter	C₄×C₈	D₅₇₁
hypocenter	C₅₇₁	#334
LCS	G ▷ C₄×C₅₇₁ ▷ C₂×C₅₇₁ ▷ C₅₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
12. C₈×C₅₇₁	C₈	semidirect
21. C₅₇₁	#334	semidirect
34. C₈	C₅₇₁⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊(C₅₇₁⋊C₈)	2
C₄×C₈×C₅₇₁	2
#334	571