Subgroups — #27271980

Order 13568.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20277024	C₂	1	1	yes	maximal, characteristic
2	#20272917	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×M₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#16811055	—	2	1	no	—
5	Dic₈₄₈	C₄	1	1	yes	characteristic, complemented
6	#16810462	—	2	1	no	—
7	D₁₆₉₆	C₄	1	1	yes	characteristic, complemented
8	#16810449	—	2	1	no	—
9	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
10	C₅₃×C₆₄	—	2	1	no	—
11	#13133239	—	4	1	no	—
12	Dic₄₂₄	—	2	1	no	—
13	#13133162	—	4	1	no	—
14	D₈₄₈	—	2	1	no	—
15	#13133151	—	4	1	no	—
16	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
17	C₃₂×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₃₂×C₅₃	D₄	1	1	yes	characteristic
19	#11719945	—	8	1	no	—
20	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
21	C₄⋊Dic₅₃	—	8	1	no	—
22	Dic₂₁₂	—	4	1	no	—
23	D₄₂₄	—	4	1	no	—
24	#11719913	—	8	1	no	—
25	C₁₆×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₁₆×C₅₃	SD₁₆	1	1	yes	characteristic
27	C₂×C₄×C₅₃	D₁₆	1	1	yes	characteristic
28	C₂²⋊D₅₃	—	16	1	no	—
29	C₂×Dic₅₃	—	16	1	no	—
30	D₂₁₂	—	8	1	no	—
31	C₄×D₅₃	—	16	1	no	—
32	Dic₁₀₆	—	8	1	no	—
33	C₈×C₅₃	#102	1	1	yes	characteristic
34	C₈×C₅₃	SD₃₂	1	1	yes	characteristic
35	#7540	—	53	1	no	maximal, Sylow
36	C₂²×C₅₃	D₃₂	1	1	yes	characteristic
37	C₂×D₅₃	—	16	1	no	—
38	C₄×C₅₃	#357	1	1	yes	characteristic
39	C₄×C₅₃	SD₆₄	1	1	yes	characteristic
40	Dic₅₃	—	16	1	no	—
41	Dic₅₃	—	32	1	no	—
42	#2262	—	53	1	no	—
43	M₁₂₈	D₅₃	1	1	yes	characteristic, complemented
44	#1424	—	53	1	no	—
45	C₂×C₅₃	—	2	1	no	—
46	C₂×C₅₃	#1415	1	1	yes	characteristic
47	D₅₃	—	32	1	no	—
48	#508	—	106	1	no	—
49	Q₆₄	—	53	1	no	—
50	SD₆₄	—	106	1	no	—
51	D₃₂	—	53	1	no	—
52	C₂×C₃₂	C₂×D₅₃	1	1	yes	characteristic
53	#366	—	106	1	no	—
54	C₆₄	—	2	1	no	—
55	C₅₃	#7540	1	1	yes	characteristic, complemented, Sylow
56	#135	—	212	1	no	—
57	Q₃₂	—	106	1	no	—
58	SD₃₂	—	212	1	no	—
59	D₁₆	—	106	1	no	—
60	C₂×C₁₆	D₂₁₂	1	1	yes	characteristic
61	#107	—	212	1	no	—
62	C₃₂	C₄×D₅₃	1	1	yes	characteristic
63	C₃₂	C₂²⋊D₅₃	1	1	yes	characteristic
64	#41	—	424	1	no	—
65	Q₁₆	—	212	1	no	—
66	SD₁₆	—	424	1	no	—
67	D₈	—	212	1	no	—
68	C₂×C₈	D₄₂₄	1	1	yes	characteristic
69	C₄⋊C₄	—	424	1	no	—
70	C₁₆	#11719913	1	1	yes	characteristic
71	C₁₆	#11719921	1	1	yes	characteristic
72	Q₈	—	424	1	no	—
73	D₄	—	848	1	no	—
74	D₄	—	424	1	no	—
75	C₂×C₄	—	848	1	no	—
76	C₂×C₄	—	848	1	no	—
77	C₂×C₄	D₈₄₈	1	1	yes	characteristic
78	C₈	#13133139	1	1	yes	characteristic
79	C₈	#13133162	1	1	yes	characteristic
80	C₂²	—	848	1	no	—
81	C₂²	D₁₆₉₆	1	1	yes	characteristic
82	C₄	—	848	1	no	—
83	C₄	—	1696	1	no	—
84	C₄	#16810425	1	1	yes	characteristic
85	C₄	#16810462	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	1696	1	no	—
88	C₂	#20272894	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20272894
commutator	C₃₂×C₅₃	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₅₃
Fitting	C₅₃×M₁₂₈	C₂
socle	C₂×C₅₃	#1415
prime radical	#20277024	C₂
cosocle	C₂×C₃₂×C₅₃	C₂²
hypercenter	M₁₂₈	D₅₃
hypocenter	C₅₃	#7540
LCS	G ▷ C₃₂×C₅₃ ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ M₁₂₈

Complements

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

N	H ≅ G/N	type
2. #20272917	C₂	semidirect
3. C₅₃×M₁₂₈	C₂	semidirect
5. Dic₈₄₈	C₄	semidirect
7. D₁₆₉₆	C₄	semidirect
43. M₁₂₈	D₅₃	semidirect
55. C₅₃	#7540	semidirect

Maximal subgroups

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

subgroup	index
#20277024	2
#20272917	2
C₅₃×M₁₂₈	2
#7540	53