Subgroups — #20289783

Order 6784.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811633	C₂	1	2	yes	maximal
2	C₂×#13133156	C₂	1	1	yes	maximal, characteristic
3	#13133296	—	2	2	no	—
4	#13133286	—	2	2	no	—
5	#13133285	C₄	1	4	yes	complemented
6	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
7	#13133156	—	2	2	no	—
8	C₂×#11719908	C₄	1	1	yes	characteristic, complemented
9	C₂×C₁₆×C₅₃	C₄	1	1	yes	characteristic, complemented
10	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
11	C₂²⋊#6157	—	4	2	no	—
12	C₄⋊#6157	—	2	4	no	—
13	C₄×#6157	—	4	2	no	—
14	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂×#88322	Q₈	1	1	yes	characteristic
16	C₈×D₅₃	C₂×C₄	1	2	yes	—
17	C₈×D₅₃	D₄	1	2	yes	—
18	C₁₆×C₅₃	—	2	1	no	—
19	#11719908	—	2	1	no	—
20	C₂²×D₅₃	D₈	1	1	yes	characteristic
21	C₂×#6157	—	4	4	no	—
22	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
23	C₂×Dic₅₃	Q₁₆	1	1	yes	characteristic
24	C₄×D₅₃	C₂²⋊C₄	1	2	yes	—
25	C₄×D₅₃	SD₁₆	1	2	yes	—
26	C₈×C₅₃	C₄²	1	1	yes	characteristic
27	C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
28	#88322	C₄⋊C₄	1	2	yes	—
29	C₂²×C₅₃	#107	1	1	yes	characteristic
30	C₂×D₅₃	—	4	1	no	—
31	C₂×D₅₃	#102	1	2	yes	—
32	#6157	—	8	4	no	—
33	C₄×C₅₃	#95	1	1	yes	characteristic
34	C₄×C₅₃	#106	1	1	yes	characteristic
35	Dic₅₃	#103	1	2	yes	—
36	#1384	—	53	1	no	maximal, Sylow
37	C₂×C₅₃	—	2	1	no	—
38	C₂×C₅₃	#340	1	1	yes	characteristic
39	D₅₃	—	2	2	no	—
40	C₂×M₃₂	—	53	1	no	—
41	#427	—	53	2	no	—
42	C₅₃	#1384	1	1	yes	characteristic, complemented, Sylow
43	C₂²×C₈	—	53	1	no	—
44	#117	—	106	2	no	—
45	M₃₂	—	106	2	no	—
46	C₂×C₁₆	—	53	1	no	—
47	C₂×C₁₆	#6157	1	1	yes	characteristic, complemented
48	#107	—	106	2	no	—
49	#106	—	53	4	no	—
50	C₂²×C₄	—	53	1	no	—
51	C₂×C₈	—	53	1	no	—
52	C₂×C₈	—	53	2	no	—
53	C₂×C₈	—	53	2	no	—
54	C₂×C₈	C₂×#6157	1	1	yes	characteristic
55	C₄⋊C₄	—	106	4	no	—
56	C₂²⋊C₄	—	212	2	no	—
57	C₄²	—	212	2	no	—
58	C₁₆	—	2	1	no	—
59	C₁₆	—	106	1	no	—
60	C₂³	—	53	1	no	—
61	C₂×C₄	—	53	1	no	—
62	C₂×C₄	—	53	2	no	—
63	C₂×C₄	—	53	2	no	—
64	C₂×C₄	—	212	4	no	—
65	C₂×C₄	C₄⋊#6157	1	1	yes	characteristic
66	C₈	—	53	2	no	—
67	C₈	C₄×#6157	1	1	yes	characteristic
68	C₈	C₂²⋊#6157	1	1	yes	characteristic
69	C₂²	—	53	2	no	—
70	C₂²	—	212	1	no	—
71	C₂²	#13133286	1	1	yes	characteristic
72	C₄	—	53	2	no	—
73	C₄	—	424	4	no	—
74	C₄	(C₂×#6157)⋊C₄	1	1	yes	characteristic
75	C₄	#13133285	1	1	yes	characteristic
76	C₂	—	2	1	no	—
77	C₂	—	106	2	no	—
78	C₂	#16811497	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#13133285
commutator	C₈×C₅₃	C₄²
Frattini	C₂×C₈	C₂×#6157
Fitting	C₂×C₁₆×C₅₃	C₄
socle	C₂×C₅₃	#340
hypercenter	C₂×C₁₆	#6157
hypocenter	C₅₃	#1384
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
5. #13133285	C₄	semidirect
8. C₂×#11719908	C₄	semidirect
9. C₂×C₁₆×C₅₃	C₄	semidirect
42. C₅₃	#1384	semidirect
47. C₂×C₁₆	#6157	semidirect