Subgroups — #27271902

Order 13568.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810456	C₂	1	1	yes	maximal, characteristic
2	#20272926	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×M₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₆×D₅₃	C₄	1	1	yes	characteristic
5	#16810456	—	2	1	no	—
6	#16810456	—	2	1	no	—
7	C₂×#13133123	C₄	1	1	yes	characteristic
8	#16810342	—	2	1	no	—
9	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
10	C₅₃×C₆₄	—	2	1	no	—
11	C₂×C₈×D₅₃	C₈	1	1	yes	characteristic
12	C₂×#11719908	C₈	1	1	yes	characteristic
13	C₁₆×D₅₃	—	2	1	no	—
14	C₁₆×D₅₃	—	2	1	no	—
15	#13133123	—	2	1	no	—
16	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₃₂×C₅₃	D₄	1	1	yes	characteristic
18	C₃₂×C₅₃	D₄	1	1	yes	characteristic
19	C₂×C₄×D₅₃	C₁₆	1	1	yes	characteristic
20	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
21	C₂×#88322	C₁₆	1	1	yes	characteristic
22	C₈×D₅₃	—	2	1	no	—
23	C₈×D₅₃	—	2	1	no	—
24	C₁₆×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₁₆×C₅₃	M₁₆	1	1	yes	characteristic
26	#11719908	—	2	1	no	—
27	C₂²×D₅₃	C₃₂	1	1	yes	characteristic
28	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
29	C₂×Dic₅₃	C₃₂	1	1	yes	characteristic
30	C₄×D₅₃	—	2	1	no	—
31	C₄×D₅₃	—	2	1	no	—
32	C₈×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
33	C₈×C₅₃	M₃₂	1	1	yes	characteristic
34	#88322	—	2	1	no	—
35	#7513	—	53	1	no	maximal, Sylow
36	C₂²×C₅₃	C₂×C₃₂	1	1	yes	characteristic
37	C₂×D₅₃	—	2	1	no	—
38	C₂×D₅₃	—	4	1	no	—
39	C₄×C₅₃	C₂²⋊C₁₆	1	1	yes	characteristic
40	C₄×C₅₃	M₆₄	1	1	yes	characteristic
41	Dic₅₃	—	2	1	no	—
42	C₂×M₆₄	—	53	1	no	—
43	M₁₂₈	—	53	1	no	—
44	M₁₂₈	D₅₃	1	1	yes	characteristic, complemented
45	C₂×C₅₃	—	2	1	no	—
46	C₂×C₅₃	C₂²⋊C₃₂	1	1	yes	characteristic
47	D₅₃	—	4	1	no	—
48	C₂²×C₁₆	—	53	1	no	—
49	M₆₄	—	106	1	no	—
50	M₆₄	—	106	1	no	—
51	C₂×C₃₂	—	53	1	no	—
52	C₂×C₃₂	C₂×D₅₃	1	1	yes	characteristic
53	C₆₄	—	2	1	no	—
54	C₆₄	—	106	1	no	—
55	C₅₃	#7513	1	1	yes	characteristic, complemented, Sylow
56	C₂²×C₈	—	53	1	no	—
57	C₂×C₁₆	—	53	1	no	—
58	C₂×C₁₆	—	106	1	no	—
59	C₂×C₁₆	—	106	1	no	—
60	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
61	C₃₂	—	106	1	no	—
62	C₃₂	D₂₁₂	1	1	yes	characteristic
63	C₃₂	C₂²⋊D₅₃	1	1	yes	characteristic
64	C₂²×C₄	—	53	1	no	—
65	C₂×C₈	—	53	1	no	—
66	C₂×C₈	—	106	1	no	—
67	C₂×C₈	—	106	1	no	—
68	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
69	C₁₆	—	106	1	no	—
70	C₁₆	#11719912	1	1	yes	characteristic
71	C₁₆	#11719921	1	1	yes	characteristic
72	C₂³	—	53	1	no	—
73	C₂×C₄	—	53	1	no	—
74	C₂×C₄	—	106	1	no	—
75	C₂×C₄	—	106	1	no	—
76	C₂×C₄	C₁₆×D₅₃	1	1	yes	characteristic
77	C₈	—	106	1	no	—
78	C₈	#13133129	1	1	yes	characteristic
79	C₈	#13133156	1	1	yes	characteristic
80	C₂²	—	106	1	no	—
81	C₂²	—	212	1	no	—
82	C₂²	C₃₂×D₅₃	1	1	yes	characteristic
83	C₄	—	106	1	no	—
84	C₄	#16810402	1	1	yes	characteristic
85	C₄	#16810456	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	212	1	no	—
88	C₂	#20272852	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	#11719921
commutator	C₂²×C₅₃	C₂×C₃₂
Frattini	C₂×C₃₂	C₂×D₅₃
Fitting	C₅₃×M₁₂₈	C₂
socle	C₂×C₅₃	C₂²⋊C₃₂
prime radical	C₂²×D₅₃	C₃₂
cosocle	C₂×C₃₂×C₅₃	C₂²
hypercenter	M₁₂₈	D₅₃
hypocenter	C₅₃	#7513
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₁₆ ◁ C₂×C₃₂ ◁ M₁₂₈

Complements

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

N	H ≅ G/N	type
2. #20272926	C₂	semidirect
3. C₅₃×M₁₂₈	C₂	semidirect
44. M₁₂₈	D₅₃	semidirect
55. C₅₃	#7513	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810456	2
#20272926	2
C₅₃×M₁₂₈	2
#7513	53