Subgroups — #27271764

Order 13568.

98 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	C₁₆×#88322	C₂	1	1	yes	maximal, characteristic
3	C₅₃×C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133123	—	2	1	no	—
5	C₄×#11719908	C₄	1	1	yes	characteristic
6	C₈×#88322	C₄	1	1	yes	characteristic
7	C₂×C₃₂×C₅₃	—	2	1	no	—
8	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×#11719908	—	2	1	no	—
10	C₄×#88322	C₈	1	1	yes	characteristic
11	C₄×#88322	C₈	1	1	yes	characteristic
12	#13133123	—	4	1	no	—
13	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
14	C₂×C₁₆×C₅₃	Q₈	1	1	yes	characteristic
15	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₃₂×C₅₃	—	4	1	no	—
17	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
18	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
19	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
20	C₂×#88322	—	2	1	no	—
21	C₂×#88322	C₁₆	1	1	yes	characteristic
22	C₂×#88322	C₁₆	1	1	yes	characteristic
23	C₁₆×C₅₃	—	2	1	no	—
24	C₁₆×C₅₃	D₈	1	1	yes	characteristic
25	C₁₆×C₅₃	Q₁₆	1	1	yes	characteristic
26	#11719908	—	2	2	no	—
27	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
28	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
29	C₂×C₄×C₅₃	M₃₂	1	1	yes	characteristic
30	C₈×C₅₃	—	2	1	no	—
31	C₈×C₅₃	#107	1	1	yes	characteristic
32	C₈×C₅₃	#108	1	1	yes	characteristic
33	#88322	—	2	2	no	—
34	#88322	—	2	1	no	—
35	#88322	C₃₂	1	2	yes	complemented
36	#7458	—	53	1	no	maximal, Sylow
37	C₂²×C₅₃	C₄⋊C₁₆	1	1	yes	characteristic
38	C₄×C₅₃	—	2	1	no	—
39	C₄×C₅₃	#335	1	1	yes	characteristic
40	C₄×C₅₃	#364	1	1	yes	characteristic
41	C₄×C₅₃	C₂×C₃₂	1	1	yes	characteristic
42	C₄×C₅₃	M₆₄	1	1	yes	characteristic
43	C₄⋊C₃₂	—	53	1	no	—
44	C₄⋊C₃₂	D₅₃	1	1	yes	characteristic
45	C₈×C₁₆	—	53	1	no	—
46	C₂×C₅₃	#1370	1	1	yes	characteristic
47	C₂×C₅₃	C₄⋊C₃₂	1	1	yes	characteristic
48	C₂×C₅₃	#1422	1	1	yes	characteristic
49	C₂×C₃₂	—	2	1	no	—
50	C₂×C₃₂	—	106	1	no	—
51	C₄×C₁₆	—	53	1	no	—
52	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
53	C₈²	—	53	1	no	—
54	C₅₃	#7458	1	1	yes	characteristic, complemented, Sylow
55	C₂×C₁₆	—	106	1	no	—
56	C₂×C₁₆	Dic₁₀₆	1	1	yes	characteristic
57	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
58	C₄×C₈	—	53	1	no	—
59	C₄×C₈	—	53	1	no	—
60	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
61	C₃₂	—	4	1	no	—
62	C₃₂	—	212	1	no	—
63	C₂×C₈	—	53	1	no	—
64	C₂×C₈	—	53	1	no	—
65	C₂×C₈	—	106	1	no	—
66	C₂×C₈	#11719912	1	1	yes	characteristic
67	C₂×C₈	#11719919	1	1	yes	characteristic
68	C₄²	C₈×D₅₃	1	1	yes	characteristic
69	C₁₆	—	2	1	no	—
70	C₁₆	—	106	2	no	—
71	C₁₆	#11719922	1	1	yes	characteristic
72	C₁₆	#11719925	1	1	yes	characteristic
73	C₂×C₄	#13133146	1	1	yes	characteristic
74	C₂×C₄	C₁₆×D₅₃	1	1	yes	characteristic
75	C₂×C₄	#13133156	1	1	yes	characteristic
76	C₈	—	2	1	no	—
77	C₈	—	53	2	no	—
78	C₈	—	106	2	no	—
79	C₈	—	106	1	no	—
80	C₈	#13133150	1	1	yes	characteristic
81	C₈	#13133152	1	1	yes	characteristic
82	C₂²	#16810441	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	#16810375	1	1	yes	characteristic
85	C₄	#16810443	1	1	yes	characteristic
86	C₄	C₃₂×D₅₃	1	1	yes	characteristic
87	C₄	#16810456	1	1	yes	characteristic
88	C₂	#20272784	1	1	yes	characteristic
89	C₂	#20272910	1	1	yes	characteristic
90	C₂	#20272912	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂×C₃₂
Frattini	C₄×C₁₆	C₂×D₅₃
Fitting	C₅₃×C₄⋊C₃₂	C₂
socle, prime radical	C₂²×C₅₃	C₄⋊C₁₆
cosocle	C₄×C₁₆×C₅₃	C₂²
hypercenter	C₄⋊C₃₂	D₅₃
hypocenter	C₅₃	#7458
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₄×C₁₆ ◁ C₄⋊C₃₂

Complements

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

N	H ≅ G/N	type
35. #88322	C₃₂	semidirect
54. C₅₃	#7458	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
C₁₆×#88322	2
C₅₃×C₄⋊C₃₂	2
#7458	53