Subgroups — C₂×C₅₃×SD₆₄ #20271197

Order 6784.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×SD₆₄	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Q₁₆×C₅₃	—	2	1	no	—
6	C₂×D₈×C₅₃	—	2	1	no	—
7	Q₃₂×C₅₃	—	2	1	no	—
8	Q₃₂×C₅₃	C₂²	1	2	yes	complemented
9	D₁₆×C₅₃	—	2	1	no	—
10	D₁₆×C₅₃	C₂²	1	2	yes	—
11	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
12	C₃₂×C₅₃	C₂²	1	2	yes	complemented
13	C₂×Q₈×C₅₃	—	4	1	no	—
14	C₂×D₄×C₅₃	—	4	1	no	—
15	Q₁₆×C₅₃	—	2	2	no	—
16	Q₁₆×C₅₃	—	4	1	no	—
17	D₈×C₅₃	—	2	2	no	—
18	D₈×C₅₃	—	4	1	no	—
19	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
20	C₁₆×C₅₃	D₄	1	1	yes	characteristic
21	C₁₆×C₅₃	C₂³	1	1	yes	characteristic
22	C₂³×C₅₃	—	8	1	no	—
23	Q₈×C₅₃	—	4	2	no	—
24	Q₈×C₅₃	—	8	1	no	—
25	D₄×C₅₃	—	4	2	no	—
26	D₄×C₅₃	—	8	1	no	—
27	C₂×C₄×C₅₃	—	8	1	no	—
28	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
29	C₈×C₅₃	D₈	1	1	yes	characteristic
30	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
31	C₂²×C₅₃	—	8	2	no	—
32	C₂²×C₅₃	—	16	2	no	—
33	C₂²×C₅₃	D₁₆	1	1	yes	characteristic
34	C₄×C₅₃	—	8	2	no	—
35	C₄×C₅₃	D₁₆	1	1	yes	characteristic
36	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
37	C₂×SD₆₄	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
38	C₂×C₅₃	—	16	2	no	—
39	C₂×C₅₃	SD₆₄	1	2	yes	direct product factor
40	C₂×C₅₃	C₂×D₁₆	1	1	yes	characteristic
41	C₂×Q₃₂	C₂×C₅₃	1	1	yes	characteristic, complemented
42	C₂×D₁₆	C₂×C₅₃	1	1	yes	characteristic
43	SD₆₄	C₂×C₅₃	1	4	yes	direct product factor
44	C₂×C₃₂	C₂×C₅₃	1	1	yes	characteristic, complemented
45	C₅₃	C₂×SD₆₄	1	1	yes	characteristic, direct product factor, Sylow
46	C₂×Q₁₆	—	2	1	no	—
47	C₂×D₈	—	2	1	no	—
48	Q₃₂	—	2	1	no	—
49	Q₃₂	C₂²×C₅₃	1	2	yes	complemented
50	D₁₆	—	2	1	no	—
51	D₁₆	C₂²×C₅₃	1	2	yes	—
52	C₂×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
53	C₃₂	C₂²×C₅₃	1	2	yes	complemented
54	C₂×Q₈	—	4	1	no	—
55	C₂×D₄	—	4	1	no	—
56	Q₁₆	—	2	2	no	—
57	Q₁₆	—	4	1	no	—
58	D₈	—	2	2	no	—
59	D₈	—	4	1	no	—
60	C₂×C₈	D₄×C₅₃	1	1	yes	characteristic
61	C₁₆	D₄×C₅₃	1	1	yes	characteristic
62	C₁₆	C₂³×C₅₃	1	1	yes	characteristic
63	C₂³	—	8	1	no	—
64	Q₈	—	4	2	no	—
65	Q₈	—	8	1	no	—
66	D₄	—	4	2	no	—
67	D₄	—	8	1	no	—
68	C₂×C₄	—	8	1	no	—
69	C₂×C₄	D₈×C₅₃	1	1	yes	characteristic
70	C₈	D₈×C₅₃	1	1	yes	characteristic
71	C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
72	C₂²	—	8	2	no	—
73	C₂²	—	16	2	no	—
74	C₂²	D₁₆×C₅₃	1	1	yes	characteristic
75	C₄	—	8	2	no	—
76	C₄	D₁₆×C₅₃	1	1	yes	characteristic
77	C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
78	C₂	—	16	2	no	—
79	C₂	C₅₃×SD₆₄	1	2	yes	direct product factor
80	C₂	C₂×D₁₆×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	D₁₆
commutator, Frattini, cosocle	C₁₆	C₂³×C₅₃
prime radical	C₂×D₁₆×C₅₃	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄×C₅₃ ◁ C₂×C₈×C₅₃ ◁ C₂×C₁₆×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×Q₃₂×C₅₃	C₂	semidirect
3. C₅₃×SD₆₄	C₂	direct (N × H)
4. C₂×C₃₂×C₅₃	C₂	semidirect
8. Q₃₂×C₅₃	C₂²	semidirect
12. C₃₂×C₅₃	C₂²	semidirect
37. C₂×SD₆₄	C₅₃	direct (N × H)
39. C₂×C₅₃	SD₆₄	direct (N × H)
41. C₂×Q₃₂	C₂×C₅₃	semidirect
43. SD₆₄	C₂×C₅₃	direct (N × H)
44. C₂×C₃₂	C₂×C₅₃	semidirect
45. C₅₃	C₂×SD₆₄	direct (N × H)
49. Q₃₂	C₂²×C₅₃	semidirect
53. C₃₂	C₂²×C₅₃	semidirect
79. C₂	C₅₃×SD₆₄	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×Q₃₂×C₅₃	2
C₂×D₁₆×C₅₃	2
C₅₃×SD₆₄	2
C₂×C₃₂×C₅₃	2
C₂×SD₆₄	53

Subgroups — C2×C53×SD64 #20271197

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×SD₆₄ #20271197