Subgroups — #20274329

Order 6784.

280 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811000	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810999	C₂	1	4	yes	maximal, complemented
3	C₈×#88322	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₄₂₄	—	2	2	no	—
6	C₂×#11719922	—	2	8	no	—
7	#13133214	C₂²	1	2	yes	—
8	C₄×#88322	C₂²	1	2	yes	—
9	C₂×D₈×C₅₃	—	2	2	no	—
10	#127×C₅₃	C₂²	1	2	yes	—
11	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×D₄×C₅₃	—	2	2	no	—
13	C₂×D₄×C₅₃	—	2	4	no	—
14	C₂×D₂₁₂	—	2	4	no	—
15	C₂×D₂₁₂	—	2	2	no	—
16	D₈×C₅₃	—	4	4	no	—
17	C₂×C₈×C₅₃	D₄	1	2	yes	—
18	C₄²×C₅₃	C₂³	1	1	yes	characteristic
19	#11719922	—	4	16	no	—
20	C₂×#88322	D₄	1	4	yes	—
21	D₄₂₄	—	4	4	no	—
22	C₂³×C₅₃	—	4	2	no	—
23	C₂²×D₅₃	—	4	2	no	—
24	D₄×C₅₃	—	4	8	no	—
25	D₄×C₅₃	—	4	4	no	—
26	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
27	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
28	D₂₁₂	—	4	4	no	—
29	D₂₁₂	—	4	8	no	—
30	C₈×C₅₃	D₈	1	4	yes	complemented
31	#88322	D₈	1	8	yes	complemented
32	C₂²×C₅₃	—	8	4	no	—
33	C₂²×C₅₃	—	8	2	no	—
34	C₂²×C₅₃	#127	1	1	yes	characteristic
35	C₂×D₅₃	—	8	4	no	—
36	C₂×D₅₃	—	8	2	no	—
37	C₄×C₅₃	C₂×D₈	1	4	yes	—
38	C₄×C₅₃	C₂×D₈	1	2	yes	—
39	#1712	—	53	1	no	maximal, Sylow
40	C₂×C₅₃	—	16	2	no	—
41	C₂×C₅₃	#493	1	2	yes	—
42	C₂×C₅₃	#493	1	1	yes	characteristic
43	D₅₃	—	16	2	no	—
44	#493	—	53	4	no	—
45	#493	—	53	1	no	—
46	#493	D₅₃	1	1	yes	characteristic, complemented
47	C₈²	—	53	1	no	—
48	C₅₃	#1712	1	1	yes	characteristic, complemented, Sylow
49	C₂×D₈	—	2	2	no	—
50	C₂×D₈	—	106	8	no	—
51	C₂×D₈	—	106	2	no	—
52	#127	—	53	2	no	—
53	#127	C₂×D₅₃	1	2	yes	—
54	C₄×C₈	—	53	2	no	—
55	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
56	C₂×D₄	—	2	4	no	—
57	C₂×D₄	—	2	2	no	—
58	C₂×D₄	—	106	4	no	—
59	C₂×D₄	—	106	2	no	—
60	D₈	—	4	4	no	—
61	D₈	—	212	16	no	—
62	D₈	—	212	4	no	—
63	C₂×C₈	—	53	4	no	—
64	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
65	C₄²	C₂²×D₅₃	1	1	yes	characteristic
66	C₂³	—	4	2	no	—
67	C₂³	—	212	2	no	—
68	D₄	—	4	8	no	—
69	D₄	—	4	4	no	—
70	D₄	—	212	8	no	—
71	D₄	—	212	4	no	—
72	C₂×C₄	D₄×D₅₃	1	2	yes	—
73	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
74	C₈	—	53	8	no	—
75	C₈	#11719922	1	4	yes	complemented
76	C₂²	—	8	4	no	—
77	C₂²	—	8	2	no	—
78	C₂²	—	424	4	no	—
79	C₂²	—	424	2	no	—
80	C₂²	#13133213	1	1	yes	characteristic
81	C₄	D₈×D₅₃	1	4	yes	—
82	C₄	C₂×#11719922	1	2	yes	—
83	C₂	—	16	2	no	—
84	C₂	—	848	2	no	—
85	C₂	#16810998	1	2	yes	—
86	C₂	#16810999	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#13133213
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#493	C₂
socle	C₂²×C₅₃	#127
hypercenter	#493	D₅₃
hypocenter	C₅₃	#1712
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

Complements

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

N	H ≅ G/N	type
1. #16811000	C₂	semidirect
2. #16810999	C₂	semidirect
3. C₈×#88322	C₂	semidirect
4. C₅₃×#493	C₂	semidirect
30. C₈×C₅₃	D₈	semidirect
31. #88322	D₈	semidirect
46. #493	D₅₃	semidirect
48. C₅₃	#1712	semidirect
75. C₈	#11719922	semidirect