Subgroups — #20276592

Order 6784.

172 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₈₄₈	C₂	1	1	yes	maximal, characteristic
2	C₄×D₄₂₄	C₂	1	2	yes	maximal, complemented
3	#16810451	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810425	C₂	1	2	yes	maximal, complemented
5	C₅₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄₂₄	C₂²	1	2	yes	—
7	C₄×D₂₁₂	—	2	2	no	—
8	D₈₄₈	C₄	1	4	yes	complemented
9	#13133151	C₂²	1	2	yes	—
10	#13133139	—	2	2	no	—
11	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×D₂₁₂	—	2	2	no	—
15	C₂×C₄×D₅₃	—	4	2	no	—
16	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
17	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
18	C₄²×C₅₃	D₄	1	1	yes	characteristic
19	#11719921	—	4	2	no	—
20	C₄⋊Dic₅₃	—	2	2	no	—
21	D₄₂₄	—	2	2	no	—
22	D₄₂₄	C₂×C₄	1	4	yes	complemented
23	C₁₆×C₅₃	—	2	1	no	—
24	C₁₆×C₅₃	C₂×C₄	1	2	yes	complemented
25	C₂²×D₅₃	—	4	2	no	—
26	C₂×C₄×C₅₃	D₈	1	2	yes	—
27	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
28	C₂×Dic₅₃	—	4	2	no	—
29	D₂₁₂	—	2	4	no	—
30	D₂₁₂	—	4	2	no	—
31	C₄×D₅₃	—	8	4	no	—
32	C₈×C₅₃	—	2	1	no	—
33	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
34	C₈×C₅₃	#41	1	1	yes	characteristic
35	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
36	C₂×D₅₃	—	4	4	no	—
37	C₂×D₅₃	—	8	2	no	—
38	C₂×D₅₃	—	8	2	no	—
39	C₄×C₅₃	—	2	2	no	—
40	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
41	C₄×C₅₃	#135	1	1	yes	characteristic
42	Dic₅₃	—	8	2	no	—
43	#2177	—	53	1	no	maximal, Sylow
44	C₂×C₅₃	C₄×D₈	1	1	yes	characteristic
45	C₂×C₅₃	#509	1	2	yes	—
46	D₅₃	—	8	4	no	—
47	C₂×D₁₆	—	53	1	no	—
48	C₄×D₈	—	53	2	no	—
49	#367	—	53	1	no	—
50	#357	—	53	2	no	—
51	#346	D₅₃	1	1	yes	characteristic, complemented
52	C₅₃	#2177	1	1	yes	characteristic, complemented, Sylow
53	C₂×D₈	—	53	2	no	—
54	C₄×D₄	—	106	2	no	—
55	D₁₆	—	53	4	no	—
56	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
57	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
58	#107	—	53	2	no	—
59	#102	—	106	2	no	—
60	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
61	C₂×D₄	—	106	2	no	—
62	C₂²×C₄	—	212	2	no	—
63	D₈	—	53	4	no	—
64	D₈	—	106	2	no	—
65	C₂×C₈	D₂₁₂	1	1	yes	characteristic
66	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
67	C₄⋊C₄	—	106	2	no	—
68	C₂²⋊C₄	—	212	2	no	—
69	C₄²	D₂₁₂	1	1	yes	characteristic
70	C₁₆	—	2	1	no	—
71	C₁₆	C₄×D₅₃	1	2	yes	complemented
72	C₂³	—	212	2	no	—
73	D₄	—	106	4	no	—
74	D₄	—	212	2	no	—
75	C₂×C₄	—	424	4	no	—
76	C₂×C₄	—	212	2	no	—
77	C₂×C₄	D₄₂₄	1	2	yes	—
78	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
79	C₈	—	2	1	no	—
80	C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
81	C₈	#11719945	1	1	yes	characteristic
82	C₂²	—	424	2	no	—
83	C₂²	—	424	2	no	—
84	C₂²	—	212	4	no	—
85	C₂²	C₂×D₄₂₄	1	1	yes	characteristic
86	C₄	—	2	2	no	—
87	C₄	—	424	2	no	—
88	C₄	C₄×D₂₁₂	1	1	yes	characteristic
89	C₄	#13133239	1	1	yes	characteristic
90	C₂	—	424	4	no	—
91	C₂	C₄×D₄₂₄	1	1	yes	characteristic
92	C₂	#16811061	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₄₂₄
commutator	C₈×C₅₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#346	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	#346	D₅₃
hypocenter	C₅₃	#2177
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
2. C₄×D₄₂₄	C₂	semidirect
3. #16810451	C₂	semidirect
4. #16810425	C₂	semidirect
5. C₅₃×#346	C₂	semidirect
8. D₈₄₈	C₄	semidirect
22. D₄₂₄	C₂×C₄	semidirect
24. C₁₆×C₅₃	C₂×C₄	semidirect
51. #346	D₅₃	semidirect
52. C₅₃	#2177	semidirect
71. C₁₆	C₄×D₅₃	semidirect