Subgroups — C₈²×Dic₅₃ #27273437

Order 13568.

750 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₈×Dic₅₃	C₂	1	3	yes	maximal
2	C₂×C₈×#88322	C₂	1	3	yes	maximal
3	C₂×C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×#88322	C₂²	1	3	yes	—
5	C₂×C₈×Dic₅₃	C₄	1	12	yes	—
6	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
7	C₈×#88322	C₄	1	12	yes	complemented
8	C₂×C₄×C₈×C₅₃	C₂²	1	3	yes	—
9	C₈²×C₅₃	C₄	1	4	yes	complemented
10	C₂²×#88322	C₂×C₄	1	6	yes	—
11	C₂×C₄×Dic₅₃	C₂×C₄	1	6	yes	—
12	C₄×#88322	C₂×C₄	1	12	yes	—
13	C₈×Dic₅₃	C₈	1	48	yes	direct product factor
14	C₂²×C₈×C₅₃	C₂×C₄	1	6	yes	—
15	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
16	C₄×C₈×C₅₃	C₂×C₄	1	6	yes	—
17	C₄×C₈×C₅₃	C₂×C₄	1	6	yes	—
18	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
19	C₂²×Dic₅₃	C₄²	1	4	yes	—
20	C₂×C₈×C₅₃	C₄²	1	12	yes	—
21	C₂×C₈×C₅₃	C₂×C₈	1	24	yes	—
22	C₄²×C₅₃	C₂²×C₄	1	3	yes	—
23	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
24	C₄×Dic₅₃	C₂×C₈	1	24	yes	—
25	C₂×#88322	C₄²	1	12	yes	—
26	C₂×#88322	C₂×C₈	1	24	yes	—
27	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
28	C₂×C₄×C₅₃	C₂×C₄²	1	3	yes	—
29	C₂×C₄×C₅₃	C₂×C₄²	1	3	yes	—
30	C₂×C₄×C₅₃	C₂²×C₈	1	12	yes	—
31	C₂×Dic₅₃	C₄×C₈	1	24	yes	—
32	C₈×C₅₃	C₄×C₈	1	24	yes	complemented
33	#88322	C₄×C₈	1	24	yes	complemented
34	C₄×C₈²	—	53	1	no	maximal, Sylow
35	C₂²×C₅₃	C₄³	1	1	yes	characteristic
36	C₂²×C₅₃	C₂×C₄×C₈	1	6	yes	—
37	C₄×C₅₃	C₂×C₄×C₈	1	6	yes	—
38	C₄×C₅₃	C₂×C₄×C₈	1	6	yes	—
39	Dic₅₃	C₈²	1	16	yes	direct product factor
40	C₄²×C₈	—	53	3	no	—
41	C₂×C₈²	—	53	3	no	—
42	C₂×C₈²	D₅₃	1	1	yes	characteristic
43	C₂×C₅₃	C₂×C₈²	1	4	yes	—
44	C₂×C₅₃	C₄²×C₈	1	3	yes	—
45	C₂×C₄×C₈	—	53	12	no	—
46	C₂×C₄×C₈	—	53	3	no	—
47	C₂×C₄×C₈	C₂×D₅₃	1	3	yes	—
48	C₄³	—	53	1	no	—
49	C₈²	—	53	12	no	—
50	C₈²	Dic₅₃	1	4	yes	direct product factor
51	C₅₃	C₄×C₈²	1	1	yes	characteristic, complemented, Sylow
52	C₂²×C₈	—	53	6	no	—
53	C₂²×C₈	C₄×D₅₃	1	6	yes	—
54	C₂×C₄²	—	53	6	no	—
55	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
56	C₄×C₈	—	53	48	no	—
57	C₄×C₈	—	53	12	no	—
58	C₄×C₈	C₄×D₅₃	1	6	yes	—
59	C₄×C₈	C₂×Dic₅₃	1	6	yes	—
60	C₂²×C₄	—	53	4	no	—
61	C₂²×C₄	C₂×C₄×D₅₃	1	3	yes	—
62	C₂×C₈	—	53	12	no	—
63	C₂×C₈	—	53	24	no	—
64	C₂×C₈	C₈×D₅₃	1	24	yes	—
65	C₂×C₈	C₄×Dic₅₃	1	12	yes	—
66	C₄²	—	53	24	no	—
67	C₄²	C₂×C₄×D₅₃	1	3	yes	—
68	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
69	C₂³	C₄²×D₅₃	1	1	yes	characteristic
70	C₂×C₄	—	53	24	no	—
71	C₂×C₄	C₄²×D₅₃	1	3	yes	—
72	C₂×C₄	C₂×C₄×Dic₅₃	1	3	yes	—
73	C₂×C₄	C₂×C₈×D₅₃	1	12	yes	—
74	C₈	—	53	24	no	—
75	C₈	C₈×Dic₅₃	1	24	yes	direct product factor
76	C₂²	C₄²×Dic₅₃	1	1	yes	characteristic
77	C₂²	C₄×C₈×D₅₃	1	6	yes	—
78	C₄	—	53	16	no	—
79	C₄	C₄×C₈×D₅₃	1	6	yes	—
80	C₄	C₂×C₈×Dic₅₃	1	6	yes	—
81	C₂	C₈²×D₅₃	1	4	yes	—
82	C₂	C₄×C₈×Dic₅₃	1	3	yes	—

	subgroup	G/H
center, hypercenter	C₂×C₈²	D₅₃
commutator, hypocenter	C₅₃	C₄×C₈²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₈²×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄²
cosocle	C₂×C₄²×C₅₃	C₂³
LCS	G ▷ C₅₃
UCS	C₁ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
7. C₈×#88322	C₄	semidirect
9. C₈²×C₅₃	C₄	semidirect
13. C₈×Dic₅₃	C₈	direct (N × H)
32. C₈×C₅₃	C₄×C₈	semidirect
33. #88322	C₄×C₈	semidirect
39. Dic₅₃	C₈²	direct (N × H)
50. C₈²	Dic₅₃	direct (N × H)
51. C₅₃	C₄×C₈²	semidirect
75. C₈	C₈×Dic₅₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₈×Dic₅₃	2
C₂×C₈×#88322	2
C₂×C₈²×C₅₃	2
C₄×C₈²	53

Subgroups — C82×Dic53 #27273437

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈²×Dic₅₃ #27273437