Subgroups — #27271903

Order 13568.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20272926	C₂	1	2	yes	maximal, complemented
2	C₂×C₅₃×M₆₄	C₂	1	1	yes	maximal, characteristic
3	#16810342	—	2	2	no	—
4	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic
5	C₅₃×M₆₄	—	2	2	no	—
6	C₂×C₃₂×C₅₃	C₄	1	1	yes	characteristic
7	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
8	C₂²×C₈×C₅₃	C₈	1	1	yes	characteristic
9	C₂×C₁₆×C₅₃	—	2	1	no	—
10	C₂×C₁₆×C₅₃	—	2	1	no	—
11	C₂×C₁₆×C₅₃	C₈	1	1	yes	characteristic
12	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
13	C₃₂×C₅₃	—	2	1	no	—
14	C₃₂×C₅₃	D₄	1	2	yes	—
15	C₂²×C₄×C₅₃	C₁₆	1	1	yes	characteristic
16	C₂×C₈×C₅₃	—	2	1	no	—
17	C₂×C₈×C₅₃	—	2	1	no	—
18	C₂×C₈×C₅₃	C₁₆	1	1	yes	characteristic
19	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
20	C₁₆×C₅₃	—	2	1	no	—
21	C₁₆×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₁₆×C₅₃	M₁₆	1	1	yes	characteristic
23	C₂³×C₅₃	C₃₂	1	1	yes	characteristic
24	C₂×C₄×C₅₃	—	2	1	no	—
25	C₂×C₄×C₅₃	—	2	1	no	—
26	C₂×C₄×C₅₃	C₃₂	1	1	yes	characteristic
27	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
28	C₈×C₅₃	—	2	1	no	—
29	C₈×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
30	C₈×C₅₃	M₃₂	1	1	yes	characteristic
31	#7513	—	53	1	no	maximal, Sylow
32	C₂²×C₅₃	—	2	1	no	—
33	C₂²×C₅₃	—	4	1	no	—
34	C₂²×C₅₃	C₂×C₃₂	1	1	yes	characteristic
35	C₄×C₅₃	—	2	1	no	—
36	C₄×C₅₃	C₂²⋊C₁₆	1	1	yes	characteristic
37	C₄×C₅₃	M₆₄	1	1	yes	characteristic
38	C₂×M₆₄	D₅₃	1	1	yes	characteristic
39	M₁₂₈	—	53	2	no	—
40	C₂×C₅₃	—	2	1	no	—
41	C₂×C₅₃	—	4	1	no	—
42	C₂×C₅₃	C₂²⋊C₃₂	1	1	yes	characteristic
43	C₂²×C₁₆	Dic₅₃	1	1	yes	characteristic
44	M₆₄	—	2	2	no	—
45	C₂×C₃₂	Dic₅₃	1	1	yes	characteristic
46	C₂×C₃₂	C₂×D₅₃	1	1	yes	characteristic
47	C₆₄	—	106	2	no	—
48	C₅₃	#7513	1	1	yes	characteristic, complemented, Sylow
49	C₂²×C₈	#88322	1	1	yes	characteristic
50	C₂×C₁₆	—	2	1	no	—
51	C₂×C₁₆	—	2	1	no	—
52	C₂×C₁₆	#88322	1	1	yes	characteristic
53	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
54	C₃₂	—	2	1	no	—
55	C₃₂	C₂²⋊D₅₃	1	2	yes	—
56	C₂²×C₄	#11719908	1	1	yes	characteristic
57	C₂×C₈	—	2	1	no	—
58	C₂×C₈	—	2	1	no	—
59	C₂×C₈	#11719908	1	1	yes	characteristic
60	C₂×C₈	C₂×#88322	1	1	yes	characteristic
61	C₁₆	—	2	1	no	—
62	C₁₆	#11719917	1	1	yes	characteristic
63	C₁₆	C₂²⋊Dic₅₃	1	1	yes	characteristic
64	C₂³	#13133123	1	1	yes	characteristic
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	#13133123	1	1	yes	characteristic
68	C₂×C₄	C₂×#11719908	1	1	yes	characteristic
69	C₈	—	2	1	no	—
70	C₈	C₂²⋊#88322	1	1	yes	characteristic
71	C₈	#13133157	1	1	yes	characteristic
72	C₂²	—	2	1	no	—
73	C₂²	—	4	1	no	—
74	C₂²	C₂×#13133123	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	C₂²⋊#11719908	1	1	yes	characteristic
77	C₄	#16810457	1	1	yes	characteristic
78	C₂	—	2	1	no	—
79	C₂	—	4	1	no	—
80	C₂	C₂²⋊#13133123	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₂²⋊Dic₅₃
commutator	C₂²×C₅₃	C₂×C₃₂
Frattini	C₂×C₃₂	C₂×D₅₃
Fitting	C₂×C₅₃×M₆₄	C₂
socle	C₂×C₅₃	C₂²⋊C₃₂
prime radical	C₂³×C₅₃	C₃₂
cosocle	C₂×C₃₂×C₅₃	C₂²
hypercenter	C₂×M₆₄	D₅₃
hypocenter	C₅₃	#7513
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₁₆ ◁ C₂×C₃₂ ◁ C₂×M₆₄

Complements

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

N	H ≅ G/N	type
1. #20272926	C₂	semidirect
48. C₅₃	#7513	semidirect

Maximal subgroups

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

subgroup	index
#20272926	2
C₂×C₅₃×M₆₄	2
#7513	53