Subgroups — #49142016

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	1	yes	maximal, characteristic
2	C₄×C₅₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₈×C₅₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₇₁⋊C₈	—	2	1	no	—
5	C₂×C₅₇₁⋊C₈	C₄	1	1	yes	characteristic
6	C₂×C₅₇₁⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₈×C₅₇₁	—	2	1	no	—
8	C₄²×C₅₇₁	C₂²	1	1	yes	characteristic
9	C₅₇₁⋊C₈	—	2	1	no	—
10	C₅₇₁⋊C₈	—	4	1	no	—
11	C₅₇₁⋊C₈	C₈	1	2	yes	complemented
12	C₂×C₄×C₅₇₁	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×C₅₇₁	D₄	1	1	yes	characteristic
14	C₂×C₄×C₅₇₁	Q₈	1	1	yes	characteristic
15	C₈×C₅₇₁	—	4	1	no	—
16	C₂²×C₅₇₁	C₄⋊C₄	1	1	yes	characteristic
17	C₄×C₅₇₁	—	2	1	no	—
18	C₄×C₅₇₁	C₂×C₈	1	1	yes	characteristic
19	C₄×C₅₇₁	M₁₆	1	1	yes	characteristic
20	C₄×C₅₇₁	SD₁₆	1	1	yes	characteristic
21	C₄×C₅₇₁	SD₁₆	1	1	yes	characteristic
22	C₂×C₅₇₁	C₄⋊C₈	1	1	yes	characteristic
23	C₂×C₅₇₁	#106	1	1	yes	characteristic
24	C₂×C₅₇₁	#108	1	1	yes	characteristic
25	C₅₇₁	#334	1	1	yes	characteristic, complemented, Sylow
26	#334	—	571	1	no	maximal, Sylow
27	C₄⋊C₈	—	571	1	no	—
28	C₄⋊C₈	D₅₇₁	1	1	yes	characteristic
29	C₄×C₈	—	571	1	no	—
30	C₂×C₈	—	2	1	no	—
31	C₂×C₈	—	571	1	no	—
32	C₂×C₈	—	571	1	no	—
33	C₂×C₈	—	1142	1	no	—
34	C₄²	C₂×D₅₇₁	1	1	yes	characteristic
35	C₂×C₄	C₄×D₅₇₁	1	1	yes	characteristic
36	C₂×C₄	C₂²⋊D₅₇₁	1	1	yes	characteristic
37	C₂×C₄	Dic₁₁₄₂	1	1	yes	characteristic
38	C₈	—	4	1	no	—
39	C₈	—	571	2	no	—
40	C₈	—	1142	1	no	—
41	C₈	—	2284	1	no	—
42	C₂²	Dic₅₇₁⋊C₄	1	1	yes	characteristic
43	C₄	—	2	1	no	—
44	C₄	C₈×D₅₇₁	1	1	yes	characteristic
45	C₄	#22603716	1	1	yes	characteristic
46	C₄	#22603720	1	1	yes	characteristic
47	C₄	#22603721	1	1	yes	characteristic
48	C₂	Dic₅₇₁⋊C₈	1	1	yes	characteristic
49	C₂	#32912625	1	1	yes	characteristic
50	C₂	#32912629	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₅₇₁
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
11. C₅₇₁⋊C₈	C₈	semidirect
25. C₅₇₁	#334	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×C₅₇₁⋊C₈	2
C₄⋊C₈×C₅₇₁	2
#334	571