Subgroups — #20274322

Order 6784.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811000	C₂	1	1	yes	maximal, characteristic
2	#16810971	C₂	1	4	yes	maximal, complemented
3	C₈×#88322	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₄₂₄	—	2	2	no	—
6	#13133214	C₂²	1	2	yes	—
7	#13133137	—	2	8	no	—
8	C₄×#88322	C₂²	1	2	yes	—
9	C₂×Q₁₆×C₅₃	—	2	2	no	—
10	C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
11	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×Q₈×C₅₃	—	2	2	no	—
13	C₂×D₂₁₂	—	2	4	no	—
14	C₂×D₂₁₂	—	2	2	no	—
15	Q₁₆×C₅₃	—	4	4	no	—
16	C₂×C₈×C₅₃	D₄	1	2	yes	—
17	C₄⋊C₄×C₅₃	—	2	4	no	—
18	C₄²×C₅₃	C₂³	1	1	yes	characteristic
19	C₂×#88322	D₄	1	4	yes	—
20	D₄₂₄	—	4	4	no	—
21	C₂²×D₅₃	—	4	2	no	—
22	Q₈×C₅₃	—	4	4	no	—
23	C₂×C₄×C₅₃	—	4	2	no	—
24	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
25	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
26	D₂₁₂	—	4	4	no	—
27	D₂₁₂	—	4	8	no	—
28	C₈×C₅₃	D₈	1	4	yes	—
29	#88322	—	2	4	no	—
30	C₂²×C₅₃	#127	1	1	yes	characteristic
31	C₂×D₅₃	—	8	2	no	—
32	C₂×D₅₃	—	8	4	no	—
33	C₄×C₅₃	—	8	2	no	—
34	C₄×C₅₃	C₂×D₈	1	2	yes	—
35	C₄×C₅₃	#135	1	4	yes	—
36	#1709	—	53	1	no	maximal, Sylow
37	C₂×C₅₃	#493	1	1	yes	characteristic
38	C₂×C₅₃	#495	1	2	yes	—
39	D₅₃	—	16	2	no	—
40	#494	D₅₃	1	1	yes	characteristic, complemented
41	#493	—	53	1	no	—
42	#486	—	53	4	no	—
43	C₈²	—	53	1	no	—
44	C₅₃	#1709	1	1	yes	characteristic, complemented, Sylow
45	C₂×Q₁₆	—	2	2	no	—
46	C₂×D₈	—	106	2	no	—
47	C₄⋊Q₈	C₂×D₅₃	1	2	yes	—
48	#127	—	53	2	no	—
49	#102	—	106	8	no	—
50	C₄×C₈	—	53	2	no	—
51	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
52	C₂×Q₈	—	2	2	no	—
53	C₂×D₄	—	106	4	no	—
54	C₂×D₄	—	106	2	no	—
55	Q₁₆	—	4	4	no	—
56	D₈	—	212	4	no	—
57	C₂×C₈	—	53	4	no	—
58	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
59	C₄⋊C₄	—	2	4	no	—
60	C₄²	C₂²×D₅₃	1	1	yes	characteristic
61	C₂³	—	212	2	no	—
62	Q₈	—	4	4	no	—
63	D₄	—	212	4	no	—
64	D₄	—	212	8	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	D₄×D₅₃	1	2	yes	—
67	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
68	C₈	—	106	4	no	—
69	C₈	#11719922	1	4	yes	—
70	C₂²	—	424	4	no	—
71	C₂²	—	424	2	no	—
72	C₂²	#13133213	1	1	yes	characteristic
73	C₄	—	8	2	no	—
74	C₄	C₂×#11719922	1	2	yes	—
75	C₄	#13133240	1	4	yes	—
76	C₂	—	848	2	no	—
77	C₂	#16810999	1	1	yes	characteristic
78	C₂	#16811008	1	2	yes	—

	subgroup	G/H
center	C₂²	#13133213
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#494	C₂
socle	C₂²×C₅₃	#127
hypercenter	#494	D₅₃
hypocenter	C₅₃	#1709
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

Complements

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

N	H ≅ G/N	type
2. #16810971	C₂	semidirect
3. C₈×#88322	C₂	semidirect
4. C₅₃×#494	C₂	semidirect
40. #494	D₅₃	semidirect
44. C₅₃	#1709	semidirect