Subgroups — #27271746

Order 13568.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#13133123	C₂	1	1	yes	maximal, characteristic
2	#20272787	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2252	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133123	—	2	1	no	—
5	#16810376	—	2	1	no	—
6	C₅₃×#500	—	2	1	no	—
7	C₅₃×#366	—	2	1	no	—
8	C₅₃×#366	C₄	1	2	yes	—
9	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
10	C₄⋊#88322	—	4	1	no	—
11	#13133123	—	4	1	no	—
12	C₄⋊Q₈×C₅₃	—	4	1	no	—
13	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
15	#107×C₅₃	—	2	2	no	—
16	#107×C₅₃	—	4	1	no	—
17	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
18	C₂×Q₈×C₅₃	—	8	1	no	—
19	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
20	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
21	C₄⋊C₄×C₅₃	—	4	2	no	—
22	C₄⋊C₄×C₅₃	—	8	1	no	—
23	C₄²×C₅₃	D₈	1	1	yes	characteristic
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	Q₈×C₅₃	—	16	2	no	—
29	C₂×C₄×C₅₃	—	8	2	no	—
30	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
31	C₂×C₄×C₅₃	D₁₆	1	1	yes	characteristic
32	C₂×C₄×C₅₃	SD₃₂	1	1	yes	characteristic
33	C₈×C₅₃	—	2	1	no	—
34	C₈×C₅₃	#101	1	1	yes	characteristic
35	C₈×C₅₃	#103	1	1	yes	characteristic
36	#88322	—	16	1	no	—
37	#7451	—	53	1	no	maximal, Sylow
38	C₂²×C₅₃	#357	1	1	yes	characteristic
39	C₄×C₅₃	—	2	1	no	—
40	C₄×C₅₃	—	16	2	no	—
41	C₄×C₅₃	#332	1	1	yes	characteristic
42	C₄×C₅₃	#362	1	1	yes	characteristic
43	C₄×C₅₃	SD₆₄	1	1	yes	characteristic
44	C₄×C₅₃	Q₆₄	1	1	yes	characteristic
45	#2252	D₅₃	1	1	yes	characteristic
46	C₄⋊C₃₂	—	53	1	no	—
47	#1371	—	53	1	no	—
48	C₂×C₅₃	#1364	1	1	yes	characteristic
49	C₂×C₅₃	#1416	1	1	yes	characteristic
50	C₂×C₅₃	#1419	1	1	yes	characteristic
51	#500	—	2	1	no	—
52	C₂×C₃₂	—	106	1	no	—
53	#366	—	2	1	no	—
54	#366	Dic₅₃	1	2	yes	—
55	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
56	#335	—	106	1	no	—
57	C₅₃	#7451	1	1	yes	characteristic, complemented, Sylow
58	C₄⋊Q₈	—	4	1	no	—
59	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
60	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
61	#107	—	2	2	no	—
62	#107	—	4	1	no	—
63	C₄⋊C₈	—	212	1	no	—
64	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
65	C₃₂	—	212	1	no	—
66	C₂×Q₈	—	8	1	no	—
67	C₂×C₈	—	424	1	no	—
68	C₂×C₈	#11719923	1	1	yes	characteristic
69	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
70	C₄⋊C₄	—	4	2	no	—
71	C₄⋊C₄	—	8	1	no	—
72	C₄²	#11719922	1	1	yes	characteristic
73	C₁₆	—	2	1	no	—
74	C₁₆	#11719924	1	1	yes	characteristic
75	C₁₆	#11719925	1	1	yes	characteristic
76	Q₈	—	16	2	no	—
77	C₂×C₄	—	8	2	no	—
78	C₂×C₄	#13133138	1	1	yes	characteristic
79	C₂×C₄	#13133158	1	1	yes	characteristic
80	C₂×C₄	#13133161	1	1	yes	characteristic
81	C₈	—	2	1	no	—
82	C₈	—	848	1	no	—
83	C₈	#13133136	1	1	yes	characteristic
84	C₈	#13133141	1	1	yes	characteristic
85	C₂²	#16810424	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	—	16	2	no	—
88	C₄	#16810369	1	1	yes	characteristic
89	C₄	#16810439	1	1	yes	characteristic
90	C₄	#16810460	1	1	yes	characteristic
91	C₄	#16810463	1	1	yes	characteristic
92	C₂	#20272770	1	1	yes	characteristic
93	C₂	#20272896	1	1	yes	characteristic
94	C₂	#20272905	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16810424
commutator	C₂×C₁₆×C₅₃	C₂×C₄
Frattini	C₄×C₁₆	C₂×D₅₃
Fitting	C₅₃×#2252	C₂
socle, prime radical	C₂²×C₅₃	#357
cosocle	C₄×C₁₆×C₅₃	C₂²
hypercenter	#2252	D₅₃
hypocenter	C₅₃	#7451
LCS	G ▷ C₂×C₁₆×C₅₃ ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2252

Complements

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

N	H ≅ G/N	type
57. C₅₃	#7451	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#13133123	2
#20272787	2
C₅₃×#2252	2
#7451	53