Subgroups — #20276590

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	#16810705	C₂	1	2	yes	maximal, complemented
3	#16810423	C₂	1	2	yes	maximal, complemented
4	#16810398	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#367	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄₂₄	C₂²	1	2	yes	—
7	#13133179	—	2	2	no	—
8	D₈₄₈	C₄	1	4	yes	complemented
9	C₂×#11719908	C₂²	1	1	yes	characteristic
10	#13133137	—	2	2	no	—
11	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	#107×C₅₃	C₂²	1	2	yes	—
14	C₂×D₂₁₂	—	2	2	no	—
15	C₂×C₄×D₅₃	—	4	2	no	—
16	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
17	C₄⋊C₄×C₅₃	—	2	2	no	—
18	#11719921	—	4	2	no	—
19	C₄×Dic₅₃	D₄	1	1	yes	characteristic
20	C₂×#88322	D₄	1	1	yes	characteristic
21	D₄₂₄	—	2	2	no	—
22	D₄₂₄	C₂×C₄	1	4	yes	complemented
23	C₁₆×C₅₃	C₂×C₄	1	2	yes	complemented
24	#11719908	—	2	1	no	—
25	C₂²×D₅₃	—	4	2	no	—
26	C₂×C₄×C₅₃	—	4	2	no	—
27	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
28	C₂×Dic₅₃	D₈	1	2	yes	—
29	D₂₁₂	—	2	4	no	—
30	D₂₁₂	—	4	2	no	—
31	C₄×D₅₃	—	8	4	no	—
32	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
33	C₈×C₅₃	#41	1	1	yes	characteristic
34	#88322	—	2	1	no	—
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	—	8	2	no	—
40	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
41	C₄×C₅₃	#135	1	1	yes	characteristic
42	Dic₅₃	—	2	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	D₅₃	1	1	yes	characteristic, complemented
50	#357	—	53	2	no	—
51	#346	—	53	1	no	—
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	—	53	1	no	—
57	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
58	#107	C₂×D₅₃	1	2	yes	—
59	#102	—	106	2	no	—
60	C₄×C₈	—	53	1	no	—
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	—	53	1	no	—
66	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
67	C₄⋊C₄	—	2	2	no	—
68	C₂²⋊C₄	—	212	2	no	—
69	C₄²	—	53	1	no	—
70	C₁₆	—	106	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	—	4	2	no	—
76	C₂×C₄	—	53	2	no	—
77	C₂×C₄	—	424	4	no	—
78	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
79	C₈	—	106	1	no	—
80	C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
81	C₈	#11719949	1	1	yes	characteristic
82	C₂²	—	424	2	no	—
83	C₂²	—	424	2	no	—
84	C₂²	—	212	4	no	—
85	C₂²	D₈×D₅₃	1	1	yes	characteristic
86	C₄	—	8	2	no	—
87	C₄	—	106	2	no	—
88	C₄	#13133179	1	1	yes	characteristic
89	C₄	#13133240	1	1	yes	characteristic
90	C₂	—	424	4	no	—
91	C₂	#16810705	1	1	yes	characteristic
92	C₂	#16811060	1	2	yes	—

	subgroup	G/H
center	C₂²	D₈×D₅₃
commutator	C₈×C₅₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#367	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	#367	D₅₃
hypocenter	C₅₃	#2177
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

Complements

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

N	H ≅ G/N	type
2. #16810705	C₂	semidirect
3. #16810423	C₂	semidirect
4. #16810398	C₂	semidirect
5. C₅₃×#367	C₂	semidirect
8. D₈₄₈	C₄	semidirect
22. D₄₂₄	C₂×C₄	semidirect
23. C₁₆×C₅₃	C₂×C₄	semidirect
49. #367	D₅₃	semidirect
52. C₅₃	#2177	semidirect
71. C₁₆	C₄×D₅₃	semidirect