Subgroups — #28299321

Order 13568.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20289884	C₂	1	2	yes	maximal, complemented
2	C₂×#16810456	C₂	1	1	yes	maximal, characteristic
3	#16811462	—	2	2	no	—
4	C₂×C₁₆×D₅₃	C₄	1	1	yes	characteristic
5	#16810456	—	2	2	no	—
6	C₂×#13133123	C₂²	1	1	yes	characteristic
7	C₂×C₃₂×C₅₃	C₄	1	1	yes	characteristic
8	C₂×C₈×D₅₃	C₈	1	1	yes	characteristic
9	C₂×#11719908	C₈	1	1	yes	characteristic
10	C₁₆×D₅₃	—	2	1	no	—
11	C₁₆×D₅₃	—	2	1	no	—
12	#13133123	D₄	1	2	yes	—
13	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
14	C₃₂×C₅₃	—	2	1	no	—
15	C₂×C₄×D₅₃	C₁₆	1	1	yes	characteristic
16	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
17	C₂×#88322	C₁₆	1	1	yes	characteristic
18	C₈×D₅₃	—	2	1	no	—
19	C₈×D₅₃	—	2	1	no	—
20	C₁₆×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
21	C₁₆×C₅₃	M₁₆	1	1	yes	characteristic
22	#11719908	—	2	1	no	—
23	C₂²×D₅₃	C₃₂	1	1	yes	characteristic
24	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
25	C₂×Dic₅₃	C₃₂	1	1	yes	characteristic
26	C₄×D₅₃	—	2	1	no	—
27	C₄×D₅₃	—	2	1	no	—
28	C₈×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
29	C₈×C₅₃	M₃₂	1	1	yes	characteristic
30	#88322	—	2	1	no	—
31	#7513	—	53	1	no	maximal, Sylow
32	C₂²×C₅₃	C₂×C₃₂	1	1	yes	characteristic
33	C₂×D₅₃	—	2	1	no	—
34	C₂×D₅₃	—	4	1	no	—
35	C₄×C₅₃	C₂²⋊C₁₆	1	1	yes	characteristic
36	C₄×C₅₃	M₆₄	1	1	yes	characteristic
37	Dic₅₃	—	2	1	no	—
38	C₂×M₆₄	—	53	1	no	—
39	M₁₂₈	—	53	2	no	—
40	C₂×C₅₃	—	2	1	no	—
41	C₂×C₅₃	C₂²⋊C₃₂	1	1	yes	characteristic
42	D₅₃	—	4	1	no	—
43	C₂²×C₁₆	—	53	1	no	—
44	M₆₄	—	106	2	no	—
45	C₂×C₃₂	—	53	1	no	—
46	C₂×C₃₂	#6157	1	1	yes	characteristic
47	C₆₄	—	106	2	no	—
48	C₅₃	#7513	1	1	yes	characteristic, complemented, Sylow
49	C₂²×C₈	—	53	1	no	—
50	C₂×C₁₆	—	53	1	no	—
51	C₂×C₁₆	—	106	1	no	—
52	C₂×C₁₆	—	106	1	no	—
53	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
54	C₃₂	—	2	1	no	—
55	C₃₂	—	53	2	no	—
56	C₂²×C₄	—	53	1	no	—
57	C₂×C₈	—	53	1	no	—
58	C₂×C₈	—	106	1	no	—
59	C₂×C₈	—	106	1	no	—
60	C₂×C₈	D₅₃⋊C₈	1	1	yes	characteristic
61	C₁₆	—	106	1	no	—
62	C₁₆	#11719936	1	1	yes	characteristic
63	C₁₆	C₂²⋊#6157	1	1	yes	characteristic
64	C₂³	—	53	1	no	—
65	C₂×C₄	—	53	1	no	—
66	C₂×C₄	—	106	1	no	—
67	C₂×C₄	—	106	1	no	—
68	C₂×C₄	D₅₃⋊C₁₆	1	1	yes	characteristic
69	C₈	—	106	1	no	—
70	C₈	#13133274	1	1	yes	characteristic
71	C₈	#13133292	1	1	yes	characteristic
72	C₂²	—	106	1	no	—
73	C₂²	—	212	1	no	—
74	C₂²	D₅₃⋊C₃₂	1	1	yes	characteristic
75	C₄	—	106	1	no	—
76	C₄	#16811518	1	1	yes	characteristic
77	C₄	#16811556	1	1	yes	characteristic
78	C₂	—	2	1	no	—
79	C₂	—	212	1	no	—
80	C₂	#20289829	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₂²⋊#6157
commutator	C₂²×C₅₃	C₂×C₃₂
Frattini	C₂×C₁₆	C₂×#6157
Fitting	C₂×C₃₂×C₅₃	C₄
socle	C₂×C₅₃	C₂²⋊C₃₂
prime radical	C₂²×D₅₃	C₃₂
cosocle	C₂×#13133123	C₂²
hypercenter	C₂×C₃₂	#6157
hypocenter	C₅₃	#7513
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
1. #20289884	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
#20289884	2
C₂×#16810456	2
#7513	53