Subgroups — #20274384

Order 6784.

198 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811026	C₂	1	2	yes	maximal, complemented
2	#16811003	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810969	C₂	1	2	yes	maximal
4	#16810344	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#492	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₁₂	—	2	2	no	—
7	#13133217	C₂²	1	2	yes	—
8	#13133150	—	2	4	no	—
9	#13133138	—	2	4	no	—
10	C₄×#88322	C₂²	1	2	yes	—
11	C₂×SD₁₆×C₅₃	—	2	2	no	—
12	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
13	#127×C₅₃	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₅₃	—	2	1	no	—
16	C₂×D₄×C₅₃	—	2	1	no	—
17	C₂×D₄×C₅₃	—	2	2	no	—
18	C₂×Dic₁₀₆	—	2	2	no	—
19	SD₁₆×C₅₃	—	4	4	no	—
20	C₂×C₈×C₅₃	D₄	1	2	yes	—
21	C₄⋊C₄×C₅₃	—	2	2	no	—
22	C₄²×C₅₃	C₂³	1	1	yes	characteristic
23	C₄⋊Dic₅₃	—	2	2	no	—
24	C₄⋊Dic₅₃	—	2	2	no	—
25	C₂×#88322	D₄	1	4	yes	—
26	Dic₂₁₂	—	4	4	no	—
27	C₂³×C₅₃	—	4	1	no	—
28	Q₈×C₅₃	—	4	2	no	—
29	D₄×C₅₃	—	4	2	no	—
30	D₄×C₅₃	—	4	4	no	—
31	C₂×C₄×C₅₃	—	4	1	no	—
32	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
33	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
34	C₂×Dic₅₃	—	4	2	no	—
35	Dic₁₀₆	—	4	4	no	—
36	C₈×C₅₃	D₈	1	4	yes	—
37	#88322	—	2	4	no	—
38	C₂²×C₅₃	—	8	1	no	—
39	C₂²×C₅₃	—	8	2	no	—
40	C₂²×C₅₃	#127	1	1	yes	characteristic
41	C₄×C₅₃	—	8	1	no	—
42	C₄×C₅₃	C₂×D₈	1	2	yes	—
43	C₄×C₅₃	#137	1	4	yes	—
44	Dic₅₃	—	8	2	no	—
45	#1722	—	53	1	no	maximal, Sylow
46	C₂×C₅₃	—	16	1	no	—
47	C₂×C₅₃	#493	1	1	yes	characteristic
48	C₂×C₅₃	#497	1	2	yes	—
49	#500	—	53	2	no	—
50	#494	—	53	1	no	—
51	#492	D₅₃	1	1	yes	characteristic
52	#486	—	53	2	no	—
53	#322	—	53	1	no	—
54	C₅₃	#1722	1	1	yes	characteristic, complemented, Sylow
55	C₂×Q₁₆	—	106	2	no	—
56	C₂×SD₁₆	—	2	2	no	—
57	C₄⋊Q₈	—	53	2	no	—
58	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
59	#127	C₂×D₅₃	1	1	yes	characteristic
60	#107	—	106	4	no	—
61	#102	—	106	4	no	—
62	C₄×C₈	—	53	2	no	—
63	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
64	C₂×Q₈	—	2	1	no	—
65	C₂×Q₈	—	106	2	no	—
66	C₂×D₄	—	2	2	no	—
67	C₂×D₄	—	2	1	no	—
68	Q₁₆	—	212	4	no	—
69	SD₁₆	—	4	4	no	—
70	C₂×C₈	—	53	4	no	—
71	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
72	C₄⋊C₄	—	2	2	no	—
73	C₄⋊C₄	—	106	2	no	—
74	C₄⋊C₄	—	106	2	no	—
75	C₄²	C₂²×D₅₃	1	1	yes	characteristic
76	C₂³	—	4	1	no	—
77	Q₈	—	4	2	no	—
78	Q₈	—	212	4	no	—
79	D₄	—	4	2	no	—
80	D₄	—	4	4	no	—
81	C₂×C₄	—	4	1	no	—
82	C₂×C₄	—	212	2	no	—
83	C₂×C₄	D₄×D₅₃	1	2	yes	—
84	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
85	C₈	—	106	4	no	—
86	C₈	#11719922	1	4	yes	—
87	C₂²	—	8	2	no	—
88	C₂²	—	8	1	no	—
89	C₂²	#13133213	1	1	yes	characteristic
90	C₄	—	8	1	no	—
91	C₄	—	424	2	no	—
92	C₄	C₂×#11719922	1	2	yes	—
93	C₄	#13133246	1	4	yes	—
94	C₂	—	16	1	no	—
95	C₂	#16810999	1	1	yes	characteristic
96	C₂	#16811014	1	2	yes	—

	subgroup	G/H
center	C₂²	#13133213
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#492	C₂
socle	C₂²×C₅₃	#127
hypercenter	#492	D₅₃
hypocenter	C₅₃	#1722
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

Complements

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

N	H ≅ G/N	type
1. #16811026	C₂	semidirect
2. #16811003	C₂	semidirect
4. #16810344	C₂	semidirect
54. C₅₃	#1722	semidirect