Subgroups — D₅₃×Q₁₂₈ #27305140

Order 13568.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₆₄×D₅₃	C₂	1	2	yes	maximal
2	Dic₁₆₉₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#20272932	C₂	1	2	yes	maximal, complemented
4	C₆₄×D₅₃	C₂	1	1	yes	maximal, characteristic
5	C₅₃×Q₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₃₂×D₅₃	—	2	2	no	—
7	Dic₈₄₈	C₂²	1	2	yes	—
8	#16810463	—	2	2	no	—
9	C₃₂×D₅₃	C₂²	1	1	yes	characteristic
10	#16810342	C₂²	1	1	yes	characteristic
11	C₅₃×Q₆₄	C₂²	1	2	yes	—
12	C₅₃×C₆₄	C₂²	1	1	yes	characteristic
13	Q₁₆×D₅₃	—	4	2	no	—
14	Dic₄₂₄	—	2	2	no	—
15	#13133163	—	4	2	no	—
16	C₁₆×D₅₃	D₄	1	1	yes	characteristic
17	#13133123	D₄	1	1	yes	characteristic
18	Q₃₂×C₅₃	—	2	2	no	—
19	C₃₂×C₅₃	C₂³	1	1	yes	characteristic
20	Q₈×D₅₃	—	8	2	no	—
21	Q₁₆×C₅₃	—	4	2	no	—
22	#11719925	—	8	2	no	—
23	Dic₂₁₂	—	4	2	no	—
24	C₈×D₅₃	D₈	1	1	yes	characteristic
25	C₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
26	#11719908	D₈	1	1	yes	characteristic
27	Q₈×C₅₃	—	8	2	no	—
28	C₄×D₅₃	—	16	2	no	—
29	C₄×D₅₃	D₁₆	1	1	yes	characteristic
30	Dic₁₀₆	—	8	2	no	—
31	Dic₁₀₆	—	16	2	no	—
32	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
33	#88322	D₁₆	1	1	yes	characteristic
34	C₂×Q₁₂₈	—	53	1	no	maximal, Sylow
35	C₂×D₅₃	D₃₂	1	1	yes	characteristic
36	C₄×C₅₃	—	16	2	no	—
37	C₄×C₅₃	C₂×D₁₆	1	1	yes	characteristic
38	Dic₅₃	—	16	2	no	—
39	Dic₅₃	D₃₂	1	1	yes	characteristic
40	C₂×Q₆₄	—	53	2	no	—
41	Q₁₂₈	—	53	2	no	—
42	Q₁₂₈	—	53	1	no	—
43	Q₁₂₈	D₅₃	1	1	yes	characteristic, direct product factor
44	C₂×C₆₄	—	53	1	no	—
45	C₂×C₅₃	C₂×D₃₂	1	1	yes	characteristic
46	D₅₃	Q₁₂₈	1	2	yes	direct product factor
47	C₂×Q₃₂	—	106	2	no	—
48	Q₆₄	—	53	2	no	—
49	Q₆₄	—	106	2	no	—
50	Q₆₄	C₂×D₅₃	1	2	yes	—
51	C₂×C₃₂	—	53	1	no	—
52	C₆₄	—	53	1	no	—
53	C₆₄	C₂×D₅₃	1	1	yes	characteristic
54	C₅₃	C₂×Q₁₂₈	1	1	yes	characteristic, complemented, Sylow
55	C₂×Q₁₆	—	212	2	no	—
56	Q₃₂	—	2	2	no	—
57	Q₃₂	—	106	2	no	—
58	Q₃₂	—	212	2	no	—
59	C₂×C₁₆	—	53	1	no	—
60	C₃₂	—	53	1	no	—
61	C₃₂	C₂²×D₅₃	1	1	yes	characteristic
62	C₂×Q₈	—	424	2	no	—
63	Q₁₆	—	4	2	no	—
64	Q₁₆	—	424	2	no	—
65	Q₁₆	—	212	2	no	—
66	C₂×C₈	—	53	1	no	—
67	C₁₆	—	53	1	no	—
68	C₁₆	D₄×D₅₃	1	1	yes	characteristic
69	Q₈	—	8	2	no	—
70	Q₈	—	848	2	no	—
71	Q₈	—	424	2	no	—
72	C₂×C₄	—	53	1	no	—
73	C₂×C₄	—	848	2	no	—
74	C₈	—	53	1	no	—
75	C₈	D₈×D₅₃	1	1	yes	characteristic
76	C₂²	—	53	1	no	—
77	C₄	—	16	2	no	—
78	C₄	—	53	1	no	—
79	C₄	—	848	2	no	—
80	C₄	D₁₆×D₅₃	1	1	yes	characteristic
81	C₂	—	53	2	no	—
82	C₂	D₃₂×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₃₂×D₅₃
commutator, cosocle	C₃₂×C₅₃	C₂³
Frattini	C₃₂	C₂²×D₅₃
Fitting	C₅₃×Q₁₂₈	C₂
socle	C₂×C₅₃	C₂×D₃₂
prime radical	C₂×D₅₃	D₃₂
hypercenter	Q₁₂₈	D₅₃
hypocenter	C₅₃	C₂×Q₁₂₈
LCS	G ▷ C₃₂×C₅₃ ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂ ◁ Q₁₂₈

Complements

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

N	H ≅ G/N	type
2. Dic₁₆₉₆	C₂	semidirect
3. #20272932	C₂	semidirect
5. C₅₃×Q₁₂₈	C₂	semidirect
43. Q₁₂₈	D₅₃	direct (N × H)
46. D₅₃	Q₁₂₈	direct (N × H)
54. C₅₃	C₂×Q₁₂₈	semidirect

Maximal subgroups

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

subgroup	index
Q₆₄×D₅₃	2
Dic₁₆₉₆	2
#20272932	2
C₆₄×D₅₃	2
C₅₃×Q₁₂₈	2
C₂×Q₁₂₈	53

Subgroups — D53×Q128 #27305140

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃×Q₁₂₈ #27305140