Subgroups — #20276114

Order 6784.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810671	C₂	1	3	yes	maximal, complemented
2	C₂×#13133133	C₂	1	3	yes	maximal
3	C₅₃×#338	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×D₂₁₂	C₂²	1	1	yes	characteristic
5	C₂×#11719917	C₂²	1	3	yes	—
6	#13133152	—	2	6	no	—
7	#13133133	—	2	6	no	—
8	C₂×M₁₆×C₅₃	C₂²	1	3	yes	—
9	C₂³×D₅₃	Q₈	1	2	yes	—
10	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
11	C₂×D₂₁₂	—	2	3	no	—
12	C₂×D₂₁₂	—	2	3	no	—
13	M₁₆×C₅₃	—	2	3	no	—
14	M₁₆×C₅₃	—	2	3	no	—
15	C₂×C₈×C₅₃	—	2	3	no	—
16	#11719917	—	2	3	no	—
17	#11719917	D₄	1	6	yes	—
18	C₂×#88322	—	2	3	no	—
19	C₂³×C₅₃	C₂×Q₈	1	1	yes	characteristic
20	C₂²×D₅₃	—	2	6	no	—
21	C₂²×D₅₃	—	8	2	no	—
22	C₂×C₄×C₅₃	C₂×D₄	1	3	yes	—
23	C₂×C₄×C₅₃	#41	1	3	yes	—
24	D₂₁₂	—	4	3	no	—
25	D₂₁₂	—	4	1	no	—
26	C₈×C₅₃	—	4	3	no	—
27	#88322	—	2	6	no	—
28	C₂²×C₅₃	—	4	1	no	—
29	C₂²×C₅₃	C₂²⋊Q₈	1	3	yes	—
30	C₂×D₅₃	—	4	2	no	—
31	C₂×D₅₃	—	8	6	no	—
32	C₄×C₅₃	#121	1	3	yes	—
33	C₄×C₅₃	#126	1	1	yes	characteristic
34	#2069	—	53	1	no	maximal, Sylow
35	C₂×C₅₃	—	2	3	no	—
36	C₂×C₅₃	#396	1	1	yes	characteristic
37	D₅₃	—	8	2	no	—
38	#430	—	53	3	no	—
39	C₂×#100	—	53	3	no	—
40	#338	D₅₃	1	1	yes	characteristic, complemented
41	C₅₃	#2069	1	1	yes	characteristic, complemented, Sylow
42	C₂²×D₄	—	53	1	no	—
43	C₂×M₁₆	—	53	3	no	—
44	C₂×M₁₆	C₂×D₅₃	1	3	yes	—
45	#108	—	106	6	no	—
46	#100	—	106	6	no	—
47	C₂⁴	—	53	2	no	—
48	C₂×D₄	—	106	3	no	—
49	C₂×D₄	—	106	3	no	—
50	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
51	M₁₆	—	2	3	no	—
52	M₁₆	—	2	3	no	—
53	M₁₆	—	53	6	no	—
54	M₁₆	—	106	3	no	—
55	C₂×C₈	—	2	3	no	—
56	C₂×C₈	—	106	3	no	—
57	C₂³	—	106	6	no	—
58	C₂³	—	424	2	no	—
59	C₂³	Q₈×D₅₃	1	1	yes	characteristic
60	D₄	—	212	3	no	—
61	D₄	—	212	1	no	—
62	C₂×C₄	#11719945	1	3	yes	—
63	C₂×C₄	D₄×D₅₃	1	3	yes	—
64	C₈	—	4	3	no	—
65	C₈	—	106	6	no	—
66	C₂²	—	4	1	no	—
67	C₂²	—	424	6	no	—
68	C₂²	—	212	2	no	—
69	C₂²	#13133195	1	3	yes	—
70	C₄	#13133190	1	3	yes	—
71	C₄	#13133211	1	1	yes	characteristic
72	C₂	—	2	3	no	—
73	C₂	—	424	2	no	—
74	C₂	#16810547	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16810547
commutator	C₂²×C₄×C₅₃	C₂³
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#338	C₂
socle	C₂×C₅₃	#396
hypercenter	#338	D₅₃
hypocenter	C₅₃	#2069
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #338

Complements

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

N	H ≅ G/N	type
1. #16810671	C₂	semidirect
3. C₅₃×#338	C₂	semidirect
40. #338	D₅₃	semidirect
41. C₅₃	#2069	semidirect