Subgroups — #27271956

Order 13568.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20272884	C₂	1	2	yes	maximal
2	C₅₃×#1713	C₂	1	1	yes	maximal, characteristic
3	#16810422	—	2	2	no	—
4	C₅₃×#500	—	2	2	no	—
5	C₅₃×#494	C₄	1	1	yes	characteristic
6	C₅₃×#494	C₂²	1	1	yes	characteristic
7	C₈²×C₅₃	C₄	1	1	yes	characteristic
8	#13133136	—	4	2	no	—
9	C₂×Q₁₆×C₅₃	—	4	1	no	—
10	C₂×Q₁₆×C₅₃	—	4	1	no	—
11	C₄⋊Q₈×C₅₃	—	2	1	no	—
12	C₄⋊Q₈×C₅₃	D₄	1	2	yes	—
13	#107×C₅₃	—	4	4	no	—
14	C₄×C₈×C₅₃	—	2	1	no	—
15	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂×Q₈×C₅₃	—	2	2	no	—
17	C₂×Q₈×C₅₃	—	4	1	no	—
18	Q₁₆×C₅₃	—	8	2	no	—
19	Q₁₆×C₅₃	—	8	2	no	—
20	C₂×C₈×C₅₃	—	2	2	no	—
21	C₂×C₈×C₅₃	—	2	1	no	—
22	C₄⋊C₄×C₅₃	—	4	2	no	—
23	C₄⋊C₄×C₅₃	—	4	2	no	—
24	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
25	#11719917	—	8	2	no	—
26	Q₈×C₅₃	—	8	2	no	—
27	Q₈×C₅₃	—	8	2	no	—
28	C₂×C₄×C₅₃	—	2	1	no	—
29	C₂×C₄×C₅₃	—	4	2	no	—
30	C₂×C₄×C₅₃	—	8	1	no	—
31	C₂×C₄×C₅₃	#99	1	1	yes	characteristic
32	C₈×C₅₃	—	2	4	no	—
33	C₈×C₅₃	—	2	2	no	—
34	#88322	—	16	2	no	—
35	#7532	—	53	1	no	maximal, Sylow
36	C₂²×C₅₃	#353	1	1	yes	characteristic
37	C₄×C₅₃	—	2	2	no	—
38	C₄×C₅₃	—	2	1	no	—
39	C₄×C₅₃	—	8	2	no	—
40	C₄×C₅₃	—	16	1	no	—
41	#1713	D₅₃	1	1	yes	characteristic
42	#1411	—	53	2	no	—
43	C₂×C₅₃	—	2	1	no	—
44	C₂×C₅₃	#1408	1	1	yes	characteristic
45	#500	—	2	2	no	—
46	#494	Dic₅₃	1	1	yes	characteristic
47	#494	C₂×D₅₃	1	1	yes	characteristic
48	#356	—	106	2	no	—
49	C₈²	Dic₅₃	1	1	yes	characteristic
50	C₅₃	#7532	1	1	yes	characteristic, complemented, Sylow
51	C₂×Q₁₆	—	4	1	no	—
52	C₂×Q₁₆	—	4	1	no	—
53	C₄⋊Q₈	—	2	1	no	—
54	C₄⋊Q₈	C₂²⋊D₅₃	1	2	yes	—
55	#107	—	4	4	no	—
56	#101	—	212	2	no	—
57	C₄×C₈	—	2	1	no	—
58	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
59	C₂×Q₈	—	2	2	no	—
60	C₂×Q₈	—	4	1	no	—
61	Q₁₆	—	8	2	no	—
62	Q₁₆	—	8	2	no	—
63	M₁₆	—	424	2	no	—
64	C₂×C₈	—	2	2	no	—
65	C₂×C₈	—	2	1	no	—
66	C₄⋊C₄	—	4	2	no	—
67	C₄⋊C₄	—	4	2	no	—
68	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
69	Q₈	—	8	2	no	—
70	Q₈	—	8	2	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	4	2	no	—
73	C₂×C₄	—	8	1	no	—
74	C₂×C₄	#13133132	1	1	yes	characteristic
75	C₈	—	2	4	no	—
76	C₈	—	2	2	no	—
77	C₈	—	848	2	no	—
78	C₂²	#16810416	1	1	yes	characteristic
79	C₄	—	2	2	no	—
80	C₄	—	2	1	no	—
81	C₄	—	8	2	no	—
82	C₄	—	16	1	no	—
83	C₂	—	2	1	no	—
84	C₂	#20272878	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20272878
commutator	C₄×C₈×C₅₃	C₂×C₄
Frattini	#494	C₂×D₅₃
Fitting	C₅₃×#1713	C₂
socle	C₂×C₅₃	#1408
prime radical	C₂²×C₅₃	#353
cosocle	C₅₃×#494	C₂²
hypercenter	#1713	D₅₃
hypocenter	C₅₃	#7532
LCS	G ▷ C₄×C₈×C₅₃ ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ #494 ◁ #1713

Complements

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

N	H ≅ G/N	type
50. C₅₃	#7532	semidirect

Maximal subgroups

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

subgroup	index
#20272884	2
C₅₃×#1713	2
#7532	53