Subgroups — C₈²×#6157 #28300028

Order 13568.

750 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₈×#6157	C₂	1	3	yes	maximal
2	C₈×D₅₃⋊C₈	C₂	1	3	yes	maximal
3	C₈²×D₅₃	C₂	1	1	yes	maximal, characteristic
4	C₄×D₅₃⋊C₈	C₂²	1	3	yes	—
5	C₂×C₈×#6157	C₄	1	12	yes	—
6	C₄²×#6157	C₂²	1	1	yes	characteristic
7	C₈×#88324	C₄	1	12	yes	complemented
8	C₄×C₈×D₅₃	C₂²	1	3	yes	—
9	C₈×#88322	C₄	1	3	yes	complemented
10	C₈²×C₅₃	C₄	1	1	yes	characteristic, complemented
11	C₂×D₅₃⋊C₈	C₂×C₄	1	6	yes	—
12	C₂×C₄×#6157	C₂×C₄	1	6	yes	—
13	C₄×#88324	C₂×C₄	1	12	yes	—
14	C₈×#6157	C₈	1	48	yes	direct product factor
15	C₂×C₈×D₅₃	C₂×C₄	1	6	yes	—
16	C₄²×D₅₃	C₂³	1	1	yes	characteristic
17	C₄×#88322	C₂×C₄	1	3	yes	—
18	C₈×Dic₅₃	C₂×C₄	1	6	yes	—
19	C₄×C₈×C₅₃	C₂×C₄	1	3	yes	—
20	C₂²×#6157	C₄²	1	4	yes	—
21	C₂×C₄×D₅₃	C₂²×C₄	1	3	yes	—
22	C₂×#88324	C₄²	1	12	yes	—
23	C₄×#6157	C₂×C₈	1	24	yes	—
24	D₅₃⋊C₈	C₂×C₈	1	24	yes	—
25	C₂×C₈×C₅₃	C₄²	1	6	yes	—
26	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
27	C₄×Dic₅₃	C₂²×C₄	1	3	yes	—
28	C₂×#88322	C₄²	1	6	yes	—
29	C₈×D₅₃	C₂×C₈	1	24	yes	—
30	C₂²×D₅₃	C₂×C₄²	1	1	yes	characteristic
31	C₂×#6157	C₄×C₈	1	24	yes	—
32	C₂×C₄×C₅₃	C₂×C₄²	1	3	yes	—
33	C₂×Dic₅₃	C₂×C₄²	1	3	yes	—
34	C₄×D₅₃	C₂²×C₈	1	12	yes	—
35	#88324	C₄×C₈	1	24	yes	complemented
36	C₈×C₅₃	C₄×C₈	1	12	yes	complemented
37	#88322	C₄×C₈	1	12	yes	complemented
38	C₄×C₈²	—	53	1	no	maximal, Sylow
39	C₂²×C₅₃	C₄³	1	1	yes	characteristic
40	C₂×D₅₃	C₂×C₄×C₈	1	6	yes	—
41	#6157	C₈²	1	16	yes	direct product factor
42	C₄×C₅₃	C₂×C₄×C₈	1	6	yes	—
43	Dic₅₃	C₂×C₄×C₈	1	6	yes	—
44	C₄²×C₈	—	53	3	no	—
45	C₂×C₈²	—	53	3	no	—
46	C₂×C₈²	—	53	1	no	—
47	C₂×C₅₃	C₄²×C₈	1	3	yes	—
48	D₅₃	C₂×C₈²	1	4	yes	—
49	C₂×C₄×C₈	—	53	12	no	—
50	C₂×C₄×C₈	—	53	3	no	—
51	C₂×C₄×C₈	—	53	3	no	—
52	C₄³	—	53	1	no	—
53	C₈²	—	53	12	no	—
54	C₈²	—	53	3	no	—
55	C₈²	#6157	1	1	yes	characteristic, direct product factor
56	C₅₃	C₄×C₈²	1	1	yes	characteristic, complemented, Sylow
57	C₂²×C₈	—	53	6	no	—
58	C₂²×C₈	—	53	6	no	—
59	C₂×C₄²	—	53	6	no	—
60	C₂×C₄²	—	53	1	no	—
61	C₄×C₈	—	53	12	no	—
62	C₄×C₈	—	53	48	no	—
63	C₄×C₈	—	53	6	no	—
64	C₄×C₈	—	53	3	no	—
65	C₄×C₈	C₂×#6157	1	3	yes	—
66	C₂²×C₄	—	53	4	no	—
67	C₂²×C₄	—	53	3	no	—
68	C₂×C₈	—	53	12	no	—
69	C₂×C₈	—	53	24	no	—
70	C₂×C₈	—	53	6	no	—
71	C₂×C₈	—	53	24	no	—
72	C₂×C₈	C₄×#6157	1	6	yes	—
73	C₄²	—	53	24	no	—
74	C₄²	—	53	3	no	—
75	C₄²	C₂²×#6157	1	1	yes	characteristic
76	C₂³	—	53	1	no	—
77	C₂×C₄	—	53	24	no	—
78	C₂×C₄	—	53	3	no	—
79	C₂×C₄	—	53	12	no	—
80	C₂×C₄	C₂×C₄×#6157	1	3	yes	—
81	C₈	—	53	24	no	—
82	C₈	—	53	12	no	—
83	C₈	C₈×#6157	1	12	yes	direct product factor
84	C₂²	—	53	6	no	—
85	C₂²	C₄²×#6157	1	1	yes	characteristic
86	C₄	—	53	16	no	—
87	C₄	—	53	6	no	—
88	C₄	C₂×C₈×#6157	1	6	yes	—
89	C₂	—	53	4	no	—
90	C₂	C₄×C₈×#6157	1	3	yes	—

	subgroup	G/H
center, hypercenter	C₈²	#6157
commutator, hypocenter	C₅₃	C₄×C₈²
Frattini	C₄²	C₂²×#6157
Fitting	C₈²×C₅₃	C₄
socle	C₂²×C₅₃	C₄³
prime radical	C₂²×D₅₃	C₂×C₄²
cosocle	C₄²×D₅₃	C₂³
LCS	G ▷ C₅₃
UCS	C₁ ◁ C₈²

Complements

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

N	H ≅ G/N	type
7. C₈×#88324	C₄	semidirect
9. C₈×#88322	C₄	semidirect
10. C₈²×C₅₃	C₄	semidirect
14. C₈×#6157	C₈	direct (N × H)
35. #88324	C₄×C₈	semidirect
36. C₈×C₅₃	C₄×C₈	semidirect
37. #88322	C₄×C₈	semidirect
41. #6157	C₈²	direct (N × H)
55. C₈²	#6157	direct (N × H)
56. C₅₃	C₄×C₈²	semidirect
83. C₈	C₈×#6157	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×#6157	2
C₈×D₅₃⋊C₈	2
C₈²×D₅₃	2
C₄×C₈²	53

Subgroups — C82×#6157 #28300028

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈²×#6157 #28300028