Subgroups — #20274317

Order 6784.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810996	C₂	1	1	yes	maximal, characteristic
2	#16810973	C₂	1	2	yes	maximal, complemented
3	#16810971	C₂	1	2	yes	maximal
4	C₈×#88322	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#498	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11719913	—	2	2	no	—
7	#13133217	C₂²	1	1	yes	characteristic
8	#13133214	C₂²	1	1	yes	characteristic
9	#13133141	—	2	4	no	—
10	#13133137	—	2	4	no	—
11	C₄×#88322	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
13	#106×C₅₃	—	2	2	no	—
14	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₅₃	—	2	2	no	—
16	C₂×D₂₁₂	—	2	2	no	—
17	C₂×D₂₁₂	—	2	1	no	—
18	C₂×Dic₁₀₆	—	2	1	no	—
19	C₂×C₈×C₅₃	D₄	1	2	yes	—
20	C₄⋊C₄×C₅₃	—	2	2	no	—
21	C₄⋊C₄×C₅₃	—	2	2	no	—
22	C₄²×C₅₃	C₂³	1	1	yes	characteristic
23	C₄⋊Dic₅₃	—	2	2	no	—
24	C₂×#88322	D₄	1	4	yes	—
25	#11719913	—	4	4	no	—
26	C₂²×D₅₃	—	4	1	no	—
27	Q₈×C₅₃	—	4	4	no	—
28	C₂×C₄×C₅₃	—	4	2	no	—
29	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
30	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
31	C₂×Dic₅₃	—	4	1	no	—
32	D₂₁₂	—	4	2	no	—
33	D₂₁₂	—	4	4	no	—
34	Dic₁₀₆	—	4	2	no	—
35	C₈×C₅₃	SD₁₆	1	4	yes	—
36	#88322	—	2	4	no	—
37	C₂²×C₅₃	#127	1	1	yes	characteristic
38	C₂×D₅₃	—	8	1	no	—
39	C₂×D₅₃	—	8	2	no	—
40	C₄×C₅₃	—	8	2	no	—
41	C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
42	C₄×C₅₃	#135	1	4	yes	—
43	Dic₅₃	—	8	1	no	—
44	#1708	—	53	1	no	maximal, Sylow
45	C₂×C₅₃	#492	1	1	yes	characteristic
46	C₂×C₅₃	#495	1	2	yes	—
47	D₅₃	—	16	1	no	—
48	#498	D₅₃	1	1	yes	characteristic, complemented
49	#492	—	53	1	no	—
50	#487	—	53	2	no	—
51	#486	—	53	2	no	—
52	C₈²	—	53	1	no	—
53	C₅₃	#1708	1	1	yes	characteristic, complemented, Sylow
54	C₂×SD₁₆	—	106	2	no	—
55	C₄⋊Q₈	—	53	1	no	—
56	C₄⋊Q₈	C₂×D₅₃	1	2	yes	—
57	#127	—	53	1	no	—
58	#106	—	2	2	no	—
59	#103	—	106	4	no	—
60	#102	—	106	4	no	—
61	C₄×C₈	—	53	2	no	—
62	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
63	C₂×Q₈	—	2	2	no	—
64	C₂×Q₈	—	106	1	no	—
65	C₂×D₄	—	106	2	no	—
66	C₂×D₄	—	106	1	no	—
67	SD₁₆	—	212	4	no	—
68	C₂×C₈	—	53	4	no	—
69	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
70	C₄⋊C₄	—	2	2	no	—
71	C₄⋊C₄	—	2	2	no	—
72	C₄⋊C₄	—	106	2	no	—
73	C₄²	C₂²×D₅₃	1	1	yes	characteristic
74	C₂³	—	212	1	no	—
75	Q₈	—	4	4	no	—
76	Q₈	—	212	2	no	—
77	D₄	—	212	2	no	—
78	D₄	—	212	4	no	—
79	C₂×C₄	—	4	2	no	—
80	C₂×C₄	—	212	1	no	—
81	C₂×C₄	D₄×D₅₃	1	2	yes	—
82	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
83	C₈	—	106	4	no	—
84	C₈	#11719924	1	4	yes	—
85	C₂²	—	424	1	no	—
86	C₂²	—	424	2	no	—
87	C₂²	#13133213	1	1	yes	characteristic
88	C₄	—	8	2	no	—
89	C₄	—	424	1	no	—
90	C₄	C₂×#11719924	1	2	yes	—
91	C₄	#13133238	1	4	yes	—
92	C₂	—	848	1	no	—
93	C₂	#16810997	1	1	yes	characteristic
94	C₂	#16811004	1	2	yes	—

	subgroup	G/H
center	C₂²	#13133213
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#498	C₂
socle	C₂²×C₅₃	#127
hypercenter	#498	D₅₃
hypocenter	C₅₃	#1708
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

Complements

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

N	H ≅ G/N	type
2. #16810973	C₂	semidirect
4. C₈×#88322	C₂	semidirect
5. C₅₃×#498	C₂	semidirect
48. #498	D₅₃	semidirect
53. C₅₃	#1708	semidirect