Subgroups — #27271766

Order 13568.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20272913	C₂	1	1	yes	maximal, characteristic
2	C₁₆×#88322	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1422	C₂	1	1	yes	maximal, characteristic
4	#16810457	—	2	1	no	—
5	C₄×#11719908	C₄	1	1	yes	characteristic
6	C₈×#88322	C₄	1	1	yes	characteristic
7	C₅₃×M₆₄	—	2	1	no	—
8	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×#11719908	C₈	1	1	yes	characteristic
10	C₂×#11719908	C₈	1	1	yes	characteristic
11	C₄×#88322	—	2	1	no	—
12	#13133123	—	4	1	no	—
13	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
14	C₂×C₁₆×C₅₃	Q₈	1	1	yes	characteristic
15	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₃₂×C₅₃	—	4	1	no	—
17	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
18	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
19	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
20	C₂×#88322	—	2	2	no	—
21	C₁₆×C₅₃	—	2	1	no	—
22	C₁₆×C₅₃	D₈	1	1	yes	characteristic
23	C₁₆×C₅₃	Q₁₆	1	1	yes	characteristic
24	#11719908	—	2	1	no	—
25	#11719908	C₁₆	1	2	yes	—
26	C₂×C₄×C₅₃	—	2	1	no	—
27	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
28	C₈×C₅₃	#107	1	1	yes	characteristic
29	C₈×C₅₃	#108	1	1	yes	characteristic
30	C₈×C₅₃	C₂×C₁₆	1	1	yes	characteristic
31	C₈×C₅₃	M₃₂	1	1	yes	characteristic
32	#88322	—	2	4	no	—
33	#7459	—	53	1	no	maximal, Sylow
34	C₂²×C₅₃	#364	1	1	yes	characteristic
35	C₄×C₅₃	—	2	2	no	—
36	C₄×C₅₃	#335	1	1	yes	characteristic
37	C₄×C₅₃	C₄⋊C₁₆	1	1	yes	characteristic
38	#1422	—	53	1	no	—
39	#1422	D₅₃	1	1	yes	characteristic
40	C₈×C₁₆	—	53	1	no	—
41	C₂×C₅₃	—	2	1	no	—
42	C₂×C₅₃	#1370	1	1	yes	characteristic
43	M₆₄	—	2	1	no	—
44	M₆₄	—	106	1	no	—
45	C₄×C₁₆	—	53	1	no	—
46	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
47	C₈²	—	53	1	no	—
48	C₅₃	#7459	1	1	yes	characteristic, complemented, Sylow
49	C₂×C₁₆	—	53	1	no	—
50	C₂×C₁₆	—	53	1	no	—
51	C₂×C₁₆	Dic₁₀₆	1	1	yes	characteristic
52	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
53	C₄×C₈	—	106	1	no	—
54	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
55	C₃₂	—	4	1	no	—
56	C₃₂	—	212	1	no	—
57	C₂×C₈	—	106	2	no	—
58	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
59	C₂×C₈	#11719919	1	1	yes	characteristic
60	C₄²	#11719912	1	1	yes	characteristic
61	C₁₆	—	2	1	no	—
62	C₁₆	—	53	2	no	—
63	C₁₆	—	106	1	no	—
64	C₁₆	#11719922	1	1	yes	characteristic
65	C₁₆	#11719925	1	1	yes	characteristic
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	#13133146	1	1	yes	characteristic
68	C₈	—	106	4	no	—
69	C₈	#13133150	1	1	yes	characteristic
70	C₈	#13133152	1	1	yes	characteristic
71	C₈	C₁₆×D₅₃	1	1	yes	characteristic
72	C₈	#13133156	1	1	yes	characteristic
73	C₂²	#16810443	1	1	yes	characteristic
74	C₄	—	2	2	no	—
75	C₄	#16810375	1	1	yes	characteristic
76	C₄	#16810441	1	1	yes	characteristic
77	C₂	—	2	1	no	—
78	C₂	#20272784	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	#11719922
commutator	C₈×C₅₃	C₂×C₁₆
Frattini	C₄×C₁₆	C₂×D₅₃
Fitting	C₅₃×#1422	C₂
socle	C₂×C₅₃	#1370
prime radical	C₂²×C₅₃	#364
cosocle	C₄×C₁₆×C₅₃	C₂²
hypercenter	#1422	D₅₃
hypocenter	C₅₃	#7459
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ C₄×C₁₆ ◁ #1422

Complements

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

N	H ≅ G/N	type
48. C₅₃	#7459	semidirect

Maximal subgroups

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

subgroup	index
#20272913	2
C₁₆×#88322	2
C₅₃×#1422	2
#7459	53