Subgroups — C₅₃×C₂²⋊C₆₄ #27215024

Order 13568.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×C₆₄	C₂	1	2	yes	maximal, complemented
3	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic
4	C₂×C₃₂×C₅₃	—	2	2	no	—
5	C₂×C₃₂×C₅₃	C₄	1	1	yes	characteristic
6	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
7	C₅₃×C₆₄	—	2	2	no	—
8	C₂²×C₈×C₅₃	C₈	1	1	yes	characteristic
9	C₂×C₁₆×C₅₃	—	2	1	no	—
10	C₂×C₁₆×C₅₃	—	2	1	no	—
11	C₂×C₁₆×C₅₃	C₈	1	1	yes	characteristic
12	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
13	C₃₂×C₅₃	—	2	1	no	—
14	C₃₂×C₅₃	D₄	1	2	yes	—
15	C₂²×C₄×C₅₃	C₁₆	1	1	yes	characteristic
16	C₂×C₈×C₅₃	—	2	1	no	—
17	C₂×C₈×C₅₃	—	2	1	no	—
18	C₂×C₈×C₅₃	C₁₆	1	1	yes	characteristic
19	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
20	C₁₆×C₅₃	—	2	1	no	—
21	C₁₆×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₁₆×C₅₃	M₁₆	1	1	yes	characteristic
23	C₂³×C₅₃	C₃₂	1	1	yes	characteristic
24	C₂×C₄×C₅₃	—	2	1	no	—
25	C₂×C₄×C₅₃	—	2	1	no	—
26	C₂×C₄×C₅₃	C₃₂	1	1	yes	characteristic
27	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
28	C₈×C₅₃	—	2	1	no	—
29	C₈×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
30	C₈×C₅₃	M₃₂	1	1	yes	characteristic
31	C₂²⋊C₆₄	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂²×C₅₃	—	2	1	no	—
33	C₂²×C₅₃	—	2	1	no	—
34	C₂²×C₅₃	C₆₄	1	2	yes	complemented
35	C₂²×C₅₃	C₂×C₃₂	1	1	yes	characteristic
36	C₄×C₅₃	—	2	1	no	—
37	C₄×C₅₃	C₂²⋊C₁₆	1	1	yes	characteristic
38	C₄×C₅₃	M₆₄	1	1	yes	characteristic
39	C₂²×C₃₂	C₂×C₅₃	1	1	yes	characteristic
40	C₂×C₆₄	C₂×C₅₃	1	2	yes	complemented
41	C₂×C₅₃	—	2	2	no	—
42	C₂×C₅₃	C₂²⋊C₃₂	1	1	yes	characteristic
43	C₂×C₅₃	C₂×C₆₄	1	1	yes	characteristic
44	C₂×C₅₃	M₁₂₈	1	1	yes	characteristic
45	C₂²×C₁₆	C₄×C₅₃	1	1	yes	characteristic
46	C₂×C₃₂	—	2	2	no	—
47	C₂×C₃₂	C₄×C₅₃	1	1	yes	characteristic
48	C₂×C₃₂	C₂²×C₅₃	1	1	yes	characteristic
49	C₆₄	—	2	2	no	—
50	C₅₃	C₂²⋊C₆₄	1	1	yes	characteristic, direct product factor, Sylow
51	C₂²×C₈	C₈×C₅₃	1	1	yes	characteristic
52	C₂×C₁₆	—	2	1	no	—
53	C₂×C₁₆	—	2	1	no	—
54	C₂×C₁₆	C₈×C₅₃	1	1	yes	characteristic
55	C₂×C₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
56	C₃₂	—	2	1	no	—
57	C₃₂	D₄×C₅₃	1	2	yes	—
58	C₂²×C₄	C₁₆×C₅₃	1	1	yes	characteristic
59	C₂×C₈	—	2	1	no	—
60	C₂×C₈	—	2	1	no	—
61	C₂×C₈	C₁₆×C₅₃	1	1	yes	characteristic
62	C₂×C₈	C₂×C₈×C₅₃	1	1	yes	characteristic
63	C₁₆	—	2	1	no	—
64	C₁₆	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
65	C₁₆	M₁₆×C₅₃	1	1	yes	characteristic
66	C₂³	C₃₂×C₅₃	1	1	yes	characteristic
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	C₃₂×C₅₃	1	1	yes	characteristic
70	C₂×C₄	C₂×C₁₆×C₅₃	1	1	yes	characteristic
71	C₈	—	2	1	no	—
72	C₈	C₂²⋊C₈×C₅₃	1	1	yes	characteristic
73	C₈	M₃₂×C₅₃	1	1	yes	characteristic
74	C₂²	—	2	1	no	—
75	C₂²	—	2	1	no	—
76	C₂²	C₅₃×C₆₄	1	2	yes	complemented
77	C₂²	C₂×C₃₂×C₅₃	1	1	yes	characteristic
78	C₄	—	2	1	no	—
79	C₄	C₅₃×C₂²⋊C₁₆	1	1	yes	characteristic
80	C₄	C₅₃×M₆₄	1	1	yes	characteristic
81	C₂	—	2	2	no	—
82	C₂	C₅₃×C₂²⋊C₃₂	1	1	yes	characteristic
83	C₂	C₂×C₅₃×C₆₄	1	1	yes	characteristic
84	C₂	C₅₃×M₁₂₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃₂×C₅₃	C₂²
commutator	C₂	C₂×C₅₃×C₆₄
Frattini, cosocle	C₂×C₃₂	C₂²×C₅₃
socle	C₂²×C₅₃	C₂×C₃₂
prime radical	C₂³×C₅₃	C₃₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₂×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅₃×C₆₄	C₂	semidirect
31. C₂²⋊C₆₄	C₅₃	direct (N × H)
34. C₂²×C₅₃	C₆₄	semidirect
40. C₂×C₆₄	C₂×C₅₃	semidirect
50. C₅₃	C₂²⋊C₆₄	direct (N × H)
76. C₂²	C₅₃×C₆₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃₂×C₅₃	2
C₂×C₅₃×C₆₄	2
C₂²⋊C₆₄	53

Subgroups — C53×C22⋊C64 #27215024

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×C₂²⋊C₆₄ #27215024