Subgroups — #20272883

Order 6784.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811002	C₂	1	1	yes	maximal, characteristic
2	#16810422	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#356	C₂	1	1	yes	maximal, characteristic
4	C₂×#11719925	—	2	1	no	—
5	C₂×#11719925	—	2	1	no	—
6	#13133217	C₄	1	1	yes	characteristic
7	#13133136	—	2	1	no	—
8	C₄×#88322	C₄	1	1	yes	characteristic
9	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
10	#101×C₅₃	—	2	1	no	—
11	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic
12	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic
13	C₂×Dic₁₀₆	—	2	1	no	—
14	M₁₆×C₅₃	—	4	1	no	—
15	C₄⋊C₄×C₅₃	—	4	1	no	—
16	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
17	#11719925	—	4	2	no	—
18	#11719925	—	4	2	no	—
19	C₄⋊Dic₅₃	—	4	1	no	—
20	#11719917	—	4	1	no	—
21	C₂×#88322	—	2	1	no	—
22	Q₈×C₅₃	—	4	1	no	—
23	Q₈×C₅₃	—	4	1	no	—
24	C₂×C₄×C₅₃	—	2	1	no	—
25	C₂×C₄×C₅₃	—	2	1	no	—
26	C₂×C₄×C₅₃	—	2	1	no	—
27	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×Dic₅₃	—	4	1	no	—
29	Dic₁₀₆	—	4	1	no	—
30	Dic₁₀₆	—	4	1	no	—
31	C₈×C₅₃	—	8	1	no	—
32	#88322	—	2	2	no	—
33	#88322	—	8	1	no	—
34	C₂²×C₅₃	#99	1	1	yes	characteristic
35	C₄×C₅₃	—	2	2	no	—
36	C₄×C₅₃	—	2	1	no	—
37	C₄×C₅₃	—	4	1	no	—
38	C₄×C₅₃	—	4	1	no	—
39	Dic₅₃	—	8	1	no	—
40	#1411	—	53	1	no	maximal, Sylow
41	C₂×C₅₃	—	2	1	no	—
42	C₂×C₅₃	#353	1	1	yes	characteristic
43	#494	—	53	1	no	—
44	#356	—	53	1	no	—
45	#356	D₅₃	1	1	yes	characteristic
46	C₅₃	#1411	1	1	yes	characteristic, complemented, Sylow
47	C₂×Q₁₆	—	106	1	no	—
48	C₂×Q₁₆	—	106	1	no	—
49	C₄⋊Q₈	—	53	1	no	—
50	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
51	#101	—	2	1	no	—
52	#101	—	106	1	no	—
53	C₄×C₈	—	53	1	no	—
54	C₂×Q₈	—	106	1	no	—
55	C₂×Q₈	D₂₁₂	1	1	yes	characteristic
56	C₂×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
57	Q₁₆	—	212	2	no	—
58	Q₁₆	—	212	2	no	—
59	M₁₆	—	4	1	no	—
60	M₁₆	—	212	1	no	—
61	C₂×C₈	—	106	1	no	—
62	C₄⋊C₄	—	4	1	no	—
63	C₄⋊C₄	—	212	1	no	—
64	C₄²	C₄×D₅₃	1	1	yes	characteristic
65	Q₈	—	4	1	no	—
66	Q₈	—	4	1	no	—
67	Q₈	—	212	1	no	—
68	Q₈	—	212	1	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	212	1	no	—
73	C₂×C₄	#11719921	1	1	yes	characteristic
74	C₈	—	8	1	no	—
75	C₈	—	106	2	no	—
76	C₈	—	424	1	no	—
77	C₂²	#13133131	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	—	2	1	no	—
80	C₄	—	4	1	no	—
81	C₄	—	4	1	no	—
82	C₄	—	424	1	no	—
83	C₂	—	2	1	no	—
84	C₂	#16810415	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16810415
commutator	C₄²×C₅₃	C₂×C₄
Frattini	C₄⋊Q₈	C₂×D₅₃
Fitting	C₅₃×#356	C₂
socle	C₂×C₅₃	#353
hypercenter	#356	D₅₃
hypocenter	C₅₃	#1411
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊Q₈ ◁ #356

Complements

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

N	H ≅ G/N	type
46. C₅₃	#1411	semidirect