Subgroups — #20274331

Order 6784.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811026	C₂	1	4	yes	maximal
2	#16811003	C₂	1	1	yes	maximal, characteristic
3	C₈×#88322	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#494	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₂₁₂	—	2	2	no	—
6	#13133217	C₂²	1	2	yes	—
7	#13133150	—	2	8	no	—
8	C₄×#88322	C₂²	1	2	yes	—
9	C₂×Q₁₆×C₅₃	—	2	2	no	—
10	C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
11	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×Q₈×C₅₃	—	2	2	no	—
13	C₂×Dic₁₀₆	—	2	2	no	—
14	Q₁₆×C₅₃	—	4	4	no	—
15	C₂×C₈×C₅₃	D₄	1	2	yes	—
16	C₄⋊C₄×C₅₃	—	2	4	no	—
17	C₄²×C₅₃	C₂³	1	1	yes	characteristic
18	C₄⋊Dic₅₃	—	2	4	no	—
19	C₂×#88322	D₄	1	4	yes	—
20	Dic₂₁₂	—	4	4	no	—
21	Q₈×C₅₃	—	4	4	no	—
22	C₂×C₄×C₅₃	—	4	2	no	—
23	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
24	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
25	C₂×Dic₅₃	—	4	2	no	—
26	Dic₁₀₆	—	4	4	no	—
27	C₈×C₅₃	D₈	1	4	yes	—
28	#88322	Q₁₆	1	8	yes	complemented
29	C₂²×C₅₃	#127	1	1	yes	characteristic
30	C₄×C₅₃	—	8	2	no	—
31	C₄×C₅₃	C₂×D₈	1	2	yes	—
32	C₄×C₅₃	C₂×Q₁₆	1	4	yes	—
33	Dic₅₃	—	8	2	no	—
34	#1713	—	53	1	no	maximal, Sylow
35	C₂×C₅₃	#493	1	1	yes	characteristic
36	C₂×C₅₃	#494	1	2	yes	—
37	#500	—	53	4	no	—
38	#494	—	53	1	no	—
39	#494	D₅₃	1	1	yes	characteristic
40	C₈²	—	53	1	no	—
41	C₅₃	#1713	1	1	yes	characteristic, complemented, Sylow
42	C₂×Q₁₆	—	2	2	no	—
43	C₂×Q₁₆	—	106	2	no	—
44	C₄⋊Q₈	—	53	2	no	—
45	C₄⋊Q₈	C₂×D₅₃	1	2	yes	—
46	#107	—	106	8	no	—
47	C₄×C₈	—	53	2	no	—
48	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
49	C₂×Q₈	—	2	2	no	—
50	C₂×Q₈	—	106	2	no	—
51	Q₁₆	—	4	4	no	—
52	Q₁₆	—	212	4	no	—
53	C₂×C₈	—	53	4	no	—
54	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
55	C₄⋊C₄	—	2	4	no	—
56	C₄⋊C₄	—	106	4	no	—
57	C₄²	C₂²×D₅₃	1	1	yes	characteristic
58	Q₈	—	4	4	no	—
59	Q₈	—	212	4	no	—
60	C₂×C₄	—	4	2	no	—
61	C₂×C₄	—	212	2	no	—
62	C₂×C₄	D₄×D₅₃	1	2	yes	—
63	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
64	C₈	—	53	8	no	—
65	C₈	#11719922	1	4	yes	—
66	C₂²	#13133213	1	1	yes	characteristic
67	C₄	—	8	2	no	—
68	C₄	—	424	2	no	—
69	C₄	C₂×#11719922	1	2	yes	—
70	C₄	Q₁₆×D₅₃	1	4	yes	—
71	C₂	#16810999	1	1	yes	characteristic
72	C₂	#16811001	1	2	yes	—

	subgroup	G/H
center	C₂²	#13133213
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#494	C₂
socle	C₂²×C₅₃	#127
hypercenter	#494	D₅₃
hypocenter	C₅₃	#1713
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

Complements

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

N	H ≅ G/N	type
28. #88322	Q₁₆	semidirect
41. C₅₃	#1713	semidirect