Subgroups — #20272682

Order 6784.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810399	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810374	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#11719908	—	2	1	no	—
5	C₄⋊#88322	—	2	1	no	—
6	C₂×D₈×C₅₃	C₄	1	1	yes	characteristic
7	C₄×D₄×C₅₃	—	2	1	no	—
8	#107×C₅₃	C₄	1	1	yes	characteristic
9	#102×C₅₃	—	2	1	no	—
10	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×D₄×C₅₃	—	2	1	no	—
12	C₂²×C₄×C₅₃	—	4	1	no	—
13	D₈×C₅₃	—	2	1	no	—
14	D₈×C₅₃	C₈	1	2	yes	complemented
15	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
17	C₄⋊C₄×C₅₃	—	2	1	no	—
18	C₂²⋊C₄×C₅₃	—	4	1	no	—
19	C₄²×C₅₃	D₄	1	1	yes	characteristic
20	C₂×#88322	—	4	1	no	—
21	#11719908	—	2	2	no	—
22	C₂³×C₅₃	—	4	1	no	—
23	D₄×C₅₃	—	2	2	no	—
24	D₄×C₅₃	—	4	1	no	—
25	C₂×C₄×C₅₃	—	4	1	no	—
26	C₂×C₄×C₅₃	—	8	2	no	—
27	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
29	C₂×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
30	C₈×C₅₃	—	2	1	no	—
31	C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
32	C₈×C₅₃	M₁₆	1	1	yes	characteristic
33	#88322	—	8	1	no	—
34	C₂²×C₅₃	—	4	2	no	—
35	C₂²×C₅₃	—	8	1	no	—
36	C₂²×C₅₃	—	8	1	no	—
37	C₂²×C₅₃	#102	1	1	yes	characteristic
38	C₄×C₅₃	—	2	1	no	—
39	C₄×C₅₃	—	2	1	no	—
40	C₄×C₅₃	—	8	1	no	—
41	C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
42	C₄×C₅₃	#104	1	1	yes	characteristic
43	#1333	—	53	1	no	maximal, Sylow
44	C₂×C₅₃	—	8	2	no	—
45	C₂×C₅₃	D₄⋊C₈	1	1	yes	characteristic
46	C₂×C₅₃	#360	1	1	yes	characteristic
47	C₂×C₅₃	#361	1	1	yes	characteristic
48	C₄×D₈	D₅₃	1	1	yes	characteristic
49	#346	—	53	1	no	—
50	#334	—	53	1	no	—
51	C₅₃	#1333	1	1	yes	characteristic, complemented, Sylow
52	C₂×D₈	Dic₅₃	1	1	yes	characteristic
53	C₄×D₄	—	2	1	no	—
54	C₂×C₁₆	—	106	1	no	—
55	#107	Dic₅₃	1	1	yes	characteristic
56	C₄⋊C₈	—	106	1	no	—
57	#102	—	2	1	no	—
58	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
59	C₂×D₄	—	2	1	no	—
60	C₂²×C₄	—	4	1	no	—
61	D₈	—	2	1	no	—
62	D₈	#88322	1	2	yes	complemented
63	C₂×C₈	—	212	1	no	—
64	C₂×C₈	C₂×Dic₅₃	1	1	yes	characteristic
65	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
66	C₄⋊C₄	—	2	1	no	—
67	C₂²⋊C₄	—	4	1	no	—
68	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
69	C₁₆	—	106	2	no	—
70	C₂³	—	4	1	no	—
71	D₄	—	2	2	no	—
72	D₄	—	4	1	no	—
73	C₂×C₄	—	4	1	no	—
74	C₂×C₄	—	8	2	no	—
75	C₂×C₄	#11719922	1	1	yes	characteristic
76	C₂×C₄	#11719923	1	1	yes	characteristic
77	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
78	C₈	—	2	1	no	—
79	C₈	—	424	1	no	—
80	C₈	C₂×#88322	1	1	yes	characteristic
81	C₈	#11719917	1	1	yes	characteristic
82	C₂²	—	4	2	no	—
83	C₂²	—	8	1	no	—
84	C₂²	—	8	1	no	—
85	C₂²	#13133138	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	—	2	1	no	—
88	C₄	—	8	1	no	—
89	C₄	C₂²⋊#88322	1	1	yes	characteristic
90	C₄	#13133144	1	1	yes	characteristic
91	C₂	—	8	2	no	—
92	C₂	D₄⋊#88322	1	1	yes	characteristic
93	C₂	#16810433	1	1	yes	characteristic
94	C₂	#16810436	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#13133138
commutator	C₈×C₅₃	C₂×C₈
Frattini	C₄×C₈	C₂×D₅₃
Fitting	C₄×D₈×C₅₃	C₂
socle	C₂²×C₅₃	#102
hypercenter	C₄×D₈	D₅₃
hypocenter	C₅₃	#1333
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×D₈

Complements

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

N	H ≅ G/N	type
1. #16810399	C₂	semidirect
2. #16810374	C₂	semidirect
14. D₈×C₅₃	C₈	semidirect
51. C₅₃	#1333	semidirect
62. D₈	#88322	semidirect