Subgroups — #27271905

Order 13568.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20272926	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16810342	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2258	C₂	1	1	yes	maximal, characteristic
4	#16810342	—	2	1	no	—
5	#16810342	—	2	1	no	—
6	C₅₃×#504	C₄	1	1	yes	characteristic
7	C₅₃×M₆₄	—	2	1	no	—
8	C₅₃×M₆₄	C₄	1	1	yes	characteristic
9	C₂×C₃₂×C₅₃	—	2	1	no	—
10	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
11	#131×C₅₃	C₈	1	1	yes	characteristic
12	M₃₂×C₅₃	—	2	1	no	—
13	M₃₂×C₅₃	C₈	1	1	yes	characteristic
14	C₂×C₁₆×C₅₃	—	2	1	no	—
15	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₃₂×C₅₃	—	2	1	no	—
17	C₃₂×C₅₃	D₄	1	1	yes	characteristic
18	C₃₂×C₅₃	D₄	1	1	yes	characteristic
19	#41×C₅₃	C₁₆	1	1	yes	characteristic
20	M₁₆×C₅₃	—	2	1	no	—
21	M₁₆×C₅₃	C₁₆	1	1	yes	characteristic
22	C₂×C₈×C₅₃	—	2	1	no	—
23	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
24	C₁₆×C₅₃	—	2	1	no	—
25	C₁₆×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₁₆×C₅₃	M₁₆	1	1	yes	characteristic
27	Q₈×C₅₃	C₃₂	1	1	yes	characteristic
28	D₄×C₅₃	—	2	1	no	—
29	D₄×C₅₃	C₃₂	1	1	yes	characteristic
30	C₂×C₄×C₅₃	—	2	1	no	—
31	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
32	C₈×C₅₃	—	2	1	no	—
33	C₈×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
34	C₈×C₅₃	M₃₂	1	1	yes	characteristic
35	#7514	—	53	1	no	maximal, Sylow
36	C₂²×C₅₃	—	2	1	no	—
37	C₂²×C₅₃	M₆₄	1	1	yes	characteristic
38	C₄×C₅₃	—	2	1	no	—
39	C₄×C₅₃	C₂²⋊C₁₆	1	1	yes	characteristic
40	C₄×C₅₃	C₂×C₃₂	1	1	yes	characteristic
41	#2258	D₅₃	1	1	yes	characteristic
42	M₁₂₈	—	53	1	no	—
43	C₂×C₆₄	—	53	1	no	—
44	C₂×C₅₃	—	2	1	no	—
45	C₂×C₅₃	—	4	1	no	—
46	C₂×C₅₃	C₂²⋊C₃₂	1	1	yes	characteristic
47	#504	Dic₅₃	1	1	yes	characteristic
48	M₆₄	—	2	1	no	—
49	M₆₄	Dic₅₃	1	1	yes	characteristic
50	C₂×C₃₂	—	2	1	no	—
51	C₂×C₃₂	C₂×D₅₃	1	1	yes	characteristic
52	C₆₄	—	106	1	no	—
53	C₆₄	—	106	1	no	—
54	C₅₃	#7514	1	1	yes	characteristic, complemented, Sylow
55	#131	#88322	1	1	yes	characteristic
56	M₃₂	—	2	1	no	—
57	M₃₂	#88322	1	1	yes	characteristic
58	C₂×C₁₆	—	2	1	no	—
59	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
60	C₃₂	—	2	1	no	—
61	C₃₂	C₂²⋊D₅₃	1	1	yes	characteristic
62	C₃₂	C₂²⋊D₅₃	1	1	yes	characteristic
63	#41	#11719908	1	1	yes	characteristic
64	M₁₆	—	2	1	no	—
65	M₁₆	#11719908	1	1	yes	characteristic
66	C₂×C₈	—	2	1	no	—
67	C₂×C₈	C₂×#88322	1	1	yes	characteristic
68	C₁₆	—	2	1	no	—
69	C₁₆	#11719917	1	1	yes	characteristic
70	C₁₆	C₂²⋊Dic₅₃	1	1	yes	characteristic
71	Q₈	#13133123	1	1	yes	characteristic
72	D₄	—	2	1	no	—
73	D₄	#13133123	1	1	yes	characteristic
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	C₂×#11719908	1	1	yes	characteristic
76	C₈	—	2	1	no	—
77	C₈	C₂²⋊#88322	1	1	yes	characteristic
78	C₈	#13133157	1	1	yes	characteristic
79	C₂²	—	2	1	no	—
80	C₂²	#16810457	1	1	yes	characteristic
81	C₄	—	2	1	no	—
82	C₄	C₂²⋊#11719908	1	1	yes	characteristic
83	C₄	C₂×#13133123	1	1	yes	characteristic
84	C₂	—	2	1	no	—
85	C₂	—	4	1	no	—
86	C₂	C₂²⋊#13133123	1	1	yes	characteristic

	subgroup	G/H
center	C₃₂	C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂×C₃₂
Frattini	C₂×C₃₂	C₂×D₅₃
Fitting	C₅₃×#2258	C₂
socle	C₂×C₅₃	C₂²⋊C₃₂
prime radical	#41×C₅₃	C₁₆
cosocle	C₂×C₃₂×C₅₃	C₂²
hypercenter	#2258	D₅₃
hypocenter	C₅₃	#7514
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₃₂ ◁ C₂×C₃₂ ◁ #2258

Complements

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

N	H ≅ G/N	type
1. #20272926	C₂	semidirect
2. C₂×#16810342	C₂	semidirect
54. C₅₃	#7514	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×#16810342	2
C₅₃×#2258	2
#7514	53