Subgroups — C₆₄×Dic₅₃ #27271894

Order 13568.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810342	C₂	1	1	yes	maximal, characteristic
2	C₃₂×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×C₆₄	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133123	C₄	1	1	yes	characteristic
5	C₁₆×Dic₅₃	C₄	1	1	yes	characteristic
6	#16810342	C₄	1	2	yes	complemented
7	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
8	C₅₃×C₆₄	C₄	1	2	yes	complemented
9	C₂×#11719908	C₈	1	1	yes	characteristic
10	C₈×Dic₅₃	C₈	1	1	yes	characteristic
11	#13133123	C₈	1	2	yes	—
12	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
13	C₃₂×C₅₃	C₂×C₄	1	1	yes	characteristic
14	C₃₂×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
16	C₄×Dic₅₃	C₁₆	1	1	yes	characteristic
17	C₂×#88322	C₁₆	1	1	yes	characteristic
18	C₁₆×C₅₃	C₄²	1	1	yes	characteristic
19	C₁₆×C₅₃	C₂×C₈	1	1	yes	characteristic
20	#11719908	C₁₆	1	2	yes	—
21	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
22	C₂×Dic₅₃	C₃₂	1	2	yes	—
23	C₈×C₅₃	C₄×C₈	1	1	yes	characteristic
24	C₈×C₅₃	C₂×C₁₆	1	1	yes	characteristic
25	#88322	C₃₂	1	2	yes	—
26	C₄×C₆₄	—	53	1	no	maximal, Sylow
27	C₂²×C₅₃	C₂×C₃₂	1	1	yes	characteristic
28	C₄×C₅₃	C₄×C₁₆	1	1	yes	characteristic
29	C₄×C₅₃	C₂×C₃₂	1	1	yes	characteristic
30	Dic₅₃	C₆₄	1	4	yes	direct product factor
31	C₂×C₆₄	—	53	1	no	—
32	C₂×C₆₄	D₅₃	1	1	yes	characteristic
33	C₄×C₃₂	—	53	1	no	—
34	C₂×C₅₃	C₄×C₃₂	1	1	yes	characteristic
35	C₂×C₅₃	C₂×C₆₄	1	2	yes	—
36	C₂×C₃₂	—	53	1	no	—
37	C₂×C₃₂	C₂×D₅₃	1	1	yes	characteristic
38	C₄×C₁₆	—	53	1	no	—
39	C₆₄	—	53	2	no	—
40	C₆₄	Dic₅₃	1	2	yes	direct product factor
41	C₅₃	C₄×C₆₄	1	1	yes	characteristic, complemented, Sylow
42	C₂×C₁₆	—	53	1	no	—
43	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
44	C₄×C₈	—	53	1	no	—
45	C₃₂	—	53	2	no	—
46	C₃₂	C₄×D₅₃	1	1	yes	characteristic
47	C₃₂	C₂×Dic₅₃	1	1	yes	characteristic
48	C₂×C₈	—	53	1	no	—
49	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
50	C₄²	—	53	1	no	—
51	C₁₆	—	53	2	no	—
52	C₁₆	C₈×D₅₃	1	1	yes	characteristic
53	C₁₆	C₄×Dic₅₃	1	1	yes	characteristic
54	C₂×C₄	—	53	2	no	—
55	C₂×C₄	C₁₆×D₅₃	1	1	yes	characteristic
56	C₈	—	53	2	no	—
57	C₈	C₈×Dic₅₃	1	1	yes	characteristic
58	C₈	C₁₆×D₅₃	1	1	yes	characteristic
59	C₂²	C₃₂×D₅₃	1	1	yes	characteristic
60	C₄	—	53	4	no	—
61	C₄	C₁₆×Dic₅₃	1	1	yes	characteristic
62	C₄	C₃₂×D₅₃	1	1	yes	characteristic
63	C₂	C₃₂×Dic₅₃	1	1	yes	characteristic
64	C₂	C₆₄×D₅₃	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₂×C₆₄	D₅₃
commutator, hypocenter	C₅₃	C₄×C₆₄
Frattini	C₂×C₃₂	C₂×D₅₃
Fitting	C₂×C₅₃×C₆₄	C₂
socle, prime radical	C₂²×C₅₃	C₂×C₃₂
cosocle	C₂×C₃₂×C₅₃	C₂²
LCS	G ▷ C₅₃
UCS	C₁ ◁ C₂×C₆₄

Complements

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

N	H ≅ G/N	type
6. #16810342	C₄	semidirect
8. C₅₃×C₆₄	C₄	semidirect
30. Dic₅₃	C₆₄	direct (N × H)
40. C₆₄	Dic₅₃	direct (N × H)
41. C₅₃	C₄×C₆₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810342	2
C₃₂×Dic₅₃	2
C₂×C₅₃×C₆₄	2
C₄×C₆₄	53

Subgroups — C64×Dic53 #27271894

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₄×Dic₅₃ #27271894