Subgroups — C₂×C₅₃×D₆₄ #27221250

Order 13568.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×D₃₂	C₂	1	2	yes	maximal, complemented
2	C₅₃×D₆₄	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₅₃×C₆₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₆×C₅₃	—	2	2	no	—
5	C₅₃×D₃₂	—	2	2	no	—
6	C₅₃×D₃₂	C₂²	1	4	yes	complemented
7	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
8	C₅₃×C₆₄	C₂²	1	2	yes	complemented
9	C₂×D₈×C₅₃	—	4	2	no	—
10	D₁₆×C₅₃	—	2	4	no	—
11	D₁₆×C₅₃	—	4	2	no	—
12	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
13	C₃₂×C₅₃	D₄	1	1	yes	characteristic
14	C₃₂×C₅₃	C₂³	1	1	yes	characteristic
15	C₂×D₄×C₅₃	—	8	2	no	—
16	D₈×C₅₃	—	4	4	no	—
17	D₈×C₅₃	—	8	2	no	—
18	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
19	C₁₆×C₅₃	D₈	1	1	yes	characteristic
20	C₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
21	C₂³×C₅₃	—	16	2	no	—
22	D₄×C₅₃	—	8	4	no	—
23	D₄×C₅₃	—	16	2	no	—
24	C₂×C₄×C₅₃	D₁₆	1	1	yes	characteristic
25	C₈×C₅₃	D₁₆	1	1	yes	characteristic
26	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
27	C₂×D₆₄	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂²×C₅₃	—	16	4	no	—
29	C₂²×C₅₃	—	32	4	no	—
30	C₂²×C₅₃	D₃₂	1	1	yes	characteristic
31	C₄×C₅₃	D₃₂	1	1	yes	characteristic
32	C₄×C₅₃	C₂×D₁₆	1	1	yes	characteristic
33	C₂×D₃₂	C₂×C₅₃	1	2	yes	complemented
34	D₆₄	C₂×C₅₃	1	4	yes	direct product factor
35	C₂×C₆₄	C₂×C₅₃	1	1	yes	characteristic, complemented
36	C₂×C₅₃	—	32	4	no	—
37	C₂×C₅₃	D₆₄	1	2	yes	direct product factor
38	C₂×C₅₃	C₂×D₃₂	1	1	yes	characteristic
39	C₂×D₁₆	—	2	2	no	—
40	D₃₂	—	2	2	no	—
41	D₃₂	C₂²×C₅₃	1	4	yes	complemented
42	C₂×C₃₂	C₂²×C₅₃	1	1	yes	characteristic
43	C₆₄	C₂²×C₅₃	1	2	yes	complemented
44	C₅₃	C₂×D₆₄	1	1	yes	characteristic, direct product factor, Sylow
45	C₂×D₈	—	4	2	no	—
46	D₁₆	—	2	4	no	—
47	D₁₆	—	4	2	no	—
48	C₂×C₁₆	D₄×C₅₃	1	1	yes	characteristic
49	C₃₂	D₄×C₅₃	1	1	yes	characteristic
50	C₃₂	C₂³×C₅₃	1	1	yes	characteristic
51	C₂×D₄	—	8	2	no	—
52	D₈	—	4	4	no	—
53	D₈	—	8	2	no	—
54	C₂×C₈	D₈×C₅₃	1	1	yes	characteristic
55	C₁₆	D₈×C₅₃	1	1	yes	characteristic
56	C₁₆	C₂×D₄×C₅₃	1	1	yes	characteristic
57	C₂³	—	16	2	no	—
58	D₄	—	8	4	no	—
59	D₄	—	16	2	no	—
60	C₂×C₄	D₁₆×C₅₃	1	1	yes	characteristic
61	C₈	D₁₆×C₅₃	1	1	yes	characteristic
62	C₈	C₂×D₈×C₅₃	1	1	yes	characteristic
63	C₂²	—	16	4	no	—
64	C₂²	—	32	4	no	—
65	C₂²	C₅₃×D₃₂	1	1	yes	characteristic
66	C₄	C₅₃×D₃₂	1	1	yes	characteristic
67	C₄	C₂×D₁₆×C₅₃	1	1	yes	characteristic
68	C₂	—	32	4	no	—
69	C₂	C₅₃×D₆₄	1	2	yes	direct product factor
70	C₂	C₂×C₅₃×D₃₂	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	D₃₂
commutator, Frattini, cosocle	C₃₂	C₂³×C₅₃
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄×C₅₃ ◁ C₂×C₈×C₅₃ ◁ C₂×C₁₆×C₅₃ ◁ C₂×C₃₂×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₃×D₃₂	C₂	semidirect
2. C₅₃×D₆₄	C₂	direct (N × H)
3. C₂×C₅₃×C₆₄	C₂	semidirect
6. C₅₃×D₃₂	C₂²	semidirect
8. C₅₃×C₆₄	C₂²	semidirect
27. C₂×D₆₄	C₅₃	direct (N × H)
33. C₂×D₃₂	C₂×C₅₃	semidirect
34. D₆₄	C₂×C₅₃	direct (N × H)
35. C₂×C₆₄	C₂×C₅₃	semidirect
37. C₂×C₅₃	D₆₄	direct (N × H)
41. D₃₂	C₂²×C₅₃	semidirect
43. C₆₄	C₂²×C₅₃	semidirect
44. C₅₃	C₂×D₆₄	direct (N × H)
69. C₂	C₅₃×D₆₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×D₃₂	2
C₅₃×D₆₄	2
C₂×C₅₃×C₆₄	2
C₂×D₆₄	53

Subgroups — C2×C53×D64 #27221250

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×D₆₄ #27221250