Subgroups — #27271742

Order 13568.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276990	C₂	1	1	yes	maximal, characteristic
2	#20272787	C₂	1	1	yes	maximal, characteristic
3	C₅₃×C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic
4	#16811028	—	2	1	no	—
5	#16810449	—	2	1	no	—
6	#16810449	C₄	1	2	yes	—
7	#16810376	—	2	1	no	—
8	C₂×C₃₂×C₅₃	—	2	1	no	—
9	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
10	#13133217	—	4	1	no	—
11	#13133151	—	2	2	no	—
12	#13133151	—	4	1	no	—
13	C₄⋊#88322	—	4	1	no	—
14	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
16	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
17	C₃₂×C₅₃	—	4	1	no	—
18	C₂×Dic₁₀₆	—	8	1	no	—
19	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
20	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
21	C₄²×C₅₃	D₈	1	1	yes	characteristic
22	C₄⋊Dic₅₃	—	4	2	no	—
23	C₄⋊Dic₅₃	—	8	1	no	—
24	C₂×#88322	—	8	1	no	—
25	C₁₆×C₅₃	—	2	1	no	—
26	C₁₆×C₅₃	SD₁₆	1	1	yes	characteristic
27	C₁₆×C₅₃	Q₁₆	1	1	yes	characteristic
28	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
29	C₂×C₄×C₅₃	D₁₆	1	1	yes	characteristic
30	C₂×C₄×C₅₃	SD₃₂	1	1	yes	characteristic
31	C₂×Dic₅₃	—	8	2	no	—
32	Dic₁₀₆	—	16	1	no	—
33	Dic₁₀₆	—	16	1	no	—
34	C₈×C₅₃	—	2	1	no	—
35	C₈×C₅₃	#101	1	1	yes	characteristic
36	C₈×C₅₃	#103	1	1	yes	characteristic
37	#88322	—	16	1	no	—
38	#7450	—	53	1	no	maximal, Sylow
39	C₂²×C₅₃	#357	1	1	yes	characteristic
40	C₄×C₅₃	—	2	1	no	—
41	C₄×C₅₃	#332	1	1	yes	characteristic
42	C₄×C₅₃	#362	1	1	yes	characteristic
43	C₄×C₅₃	D₃₂	1	1	yes	characteristic
44	C₄×C₅₃	SD₆₄	1	1	yes	characteristic
45	Dic₅₃	—	16	2	no	—
46	#2252	—	53	1	no	—
47	C₄⋊C₃₂	D₅₃	1	1	yes	characteristic
48	#1371	—	53	1	no	—
49	C₂×C₅₃	#1364	1	1	yes	characteristic
50	C₂×C₅₃	#1415	1	1	yes	characteristic
51	C₂×C₅₃	#1420	1	1	yes	characteristic
52	#500	—	106	1	no	—
53	C₂×C₃₂	—	2	1	no	—
54	#366	—	53	2	no	—
55	#366	—	106	1	no	—
56	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
57	#335	—	106	1	no	—
58	C₅₃	#7450	1	1	yes	characteristic, complemented, Sylow
59	C₄⋊Q₈	—	212	1	no	—
60	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
61	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
62	#107	—	106	2	no	—
63	#107	—	212	1	no	—
64	C₄⋊C₈	—	212	1	no	—
65	C₄×C₈	D₂₁₂	1	1	yes	characteristic
66	C₃₂	—	4	1	no	—
67	C₂×Q₈	—	424	1	no	—
68	C₂×C₈	—	424	1	no	—
69	C₂×C₈	#11719913	1	1	yes	characteristic
70	C₂×C₈	#11719921	1	1	yes	characteristic
71	C₄⋊C₄	—	424	1	no	—
72	C₄⋊C₄	—	212	2	no	—
73	C₄²	D₄₂₄	1	1	yes	characteristic
74	C₁₆	—	2	1	no	—
75	C₁₆	#11719923	1	1	yes	characteristic
76	C₁₆	#11719925	1	1	yes	characteristic
77	Q₈	—	848	1	no	—
78	Q₈	—	848	1	no	—
79	C₂×C₄	—	424	2	no	—
80	C₂×C₄	#13133139	1	1	yes	characteristic
81	C₂×C₄	D₈₄₈	1	1	yes	characteristic
82	C₂×C₄	#13133162	1	1	yes	characteristic
83	C₈	—	2	1	no	—
84	C₈	—	848	1	no	—
85	C₈	#13133135	1	1	yes	characteristic
86	C₈	#13133140	1	1	yes	characteristic
87	C₂²	#16810425	1	1	yes	characteristic
88	C₄	—	2	1	no	—
89	C₄	—	848	2	no	—
90	C₄	#16810368	1	1	yes	characteristic
91	C₄	#16810440	1	1	yes	characteristic
92	C₄	D₁₆₉₆	1	1	yes	characteristic
93	C₄	#16810462	1	1	yes	characteristic
94	C₂	#20272769	1	1	yes	characteristic
95	C₂	#20272894	1	1	yes	characteristic
96	C₂	#20272909	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16810425
commutator	C₂×C₁₆×C₅₃	C₂×C₄
Frattini	C₄×C₁₆	C₂×D₅₃
Fitting	C₅₃×C₄⋊C₃₂	C₂
socle, prime radical	C₂²×C₅₃	#357
cosocle	C₄×C₁₆×C₅₃	C₂²
hypercenter	C₄⋊C₃₂	D₅₃
hypocenter	C₅₃	#7450
LCS	G ▷ C₂×C₁₆×C₅₃ ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄⋊C₃₂

Complements

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

N	H ≅ G/N	type
58. C₅₃	#7450	semidirect

Maximal subgroups

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

subgroup	index
#20276990	2
#20272787	2
C₅₃×C₄⋊C₃₂	2
#7450	53