Subgroups — #20276320

Order 6784.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133157	C₂	1	2	yes	maximal
2	#16810401	C₂	1	4	yes	maximal, complemented
3	C₅₃×#405	C₂	1	1	yes	maximal, characteristic
4	#13133157	C₄	1	4	yes	complemented
5	#13133157	C₂²	1	4	yes	—
6	C₂×#11719908	C₄	1	4	yes	complemented
7	C₂×M₁₆×C₅₃	C₄	1	1	yes	characteristic
8	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	#117×C₅₃	C₄	1	1	yes	characteristic
10	#97×C₅₃	C₂²	1	2	yes	—
11	C₄×C₈×C₅₃	C₄	1	2	yes	—
12	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
13	M₁₆×C₅₃	—	2	2	no	—
14	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
15	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
18	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
19	C₄⋊C₄×C₅₃	—	2	1	no	—
20	C₂²⋊C₄×C₅₃	—	2	1	no	—
21	C₄²×C₅₃	C₂×C₄	1	2	yes	—
22	#11719908	C₂×C₄	1	8	yes	—
23	C₂³×C₅₃	M₁₆	1	1	yes	characteristic
24	C₂×C₄×C₅₃	—	2	2	no	—
25	C₂×C₄×C₅₃	C₄²	1	2	yes	—
26	C₂×C₄×C₅₃	M₁₆	1	1	yes	characteristic
27	C₂×C₄×C₅₃	M₁₆	1	2	yes	—
28	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
29	C₈×C₅₃	—	2	2	no	—
30	C₈×C₅₃	C₄²	1	1	yes	characteristic
31	C₈×C₅₃	C₄²	1	1	yes	characteristic
32	C₈×C₅₃	C₂²×C₄	1	2	yes	—
33	C₂²×C₅₃	—	4	1	no	—
34	C₂²×C₅₃	#97	1	2	yes	—
35	C₂²×C₅₃	C₂×M₁₆	1	1	yes	characteristic
36	C₄×C₅₃	—	4	2	no	—
37	C₄×C₅₃	#97	1	2	yes	—
38	C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
39	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
40	#2110	—	53	1	no	maximal, Sylow
41	C₂×C₅₃	—	2	1	no	—
42	C₂×C₅₃	—	2	2	no	—
43	C₂×C₅₃	C₂×#97	1	1	yes	characteristic
44	C₂×M₃₂	—	53	2	no	—
45	#405	D₅₃	1	1	yes	characteristic
46	#347	—	53	4	no	—
47	C₅₃	#2110	1	1	yes	characteristic, complemented, Sylow
48	C₂×M₁₆	Dic₅₃	1	1	yes	characteristic
49	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
50	#117	Dic₅₃	1	1	yes	characteristic
51	M₃₂	—	53	4	no	—
52	M₃₂	—	53	4	no	—
53	C₂×C₁₆	—	53	4	no	—
54	#97	C₂×D₅₃	1	2	yes	—
55	C₄×C₈	Dic₅₃	1	2	yes	—
56	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
57	M₁₆	—	2	2	no	—
58	C₂×C₈	C₄×D₅₃	1	2	yes	—
59	C₂×C₈	C₄×D₅₃	1	2	yes	—
60	C₂×C₈	C₂×Dic₅₃	1	2	yes	—
61	C₂×C₈	C₂×Dic₅₃	1	1	yes	characteristic
62	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
63	C₄⋊C₄	—	2	1	no	—
64	C₂²⋊C₄	—	2	1	no	—
65	C₄²	C₂×Dic₅₃	1	2	yes	—
66	C₁₆	—	53	8	no	—
67	C₂³	#11719917	1	1	yes	characteristic
68	C₂×C₄	—	2	2	no	—
69	C₂×C₄	#11719917	1	1	yes	characteristic
70	C₂×C₄	#11719917	1	2	yes	—
71	C₂×C₄	C₄×Dic₅₃	1	2	yes	—
72	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
73	C₈	—	2	2	no	—
74	C₈	C₄×Dic₅₃	1	1	yes	characteristic
75	C₈	C₄×Dic₅₃	1	1	yes	characteristic
76	C₈	C₂×C₄×D₅₃	1	2	yes	—
77	C₂²	—	4	1	no	—
78	C₂²	#13133128	1	2	yes	—
79	C₂²	C₂×#11719917	1	1	yes	characteristic
80	C₄	—	4	2	no	—
81	C₄	#13133128	1	2	yes	—
82	C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
83	C₄	C₂×#11719917	1	1	yes	characteristic
84	C₂	—	2	1	no	—
85	C₂	—	2	2	no	—
86	C₂	C₂×#13133128	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₄×Dic₅₃
commutator	C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#405	C₂
socle	C₂×C₅₃	C₂×#97
hypercenter	#405	D₅₃
hypocenter	C₅₃	#2110
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
2. #16810401	C₂	semidirect
4. #13133157	C₄	semidirect
6. C₂×#11719908	C₄	semidirect
47. C₅₃	#2110	semidirect