Subgroups — #20289735

Order 6784.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811546	C₂	1	1	yes	maximal, characteristic
2	#16811546	C₂	1	1	yes	maximal, characteristic
3	#16811022	C₂	1	1	yes	maximal, characteristic
4	#13133291	—	2	1	no	—
5	#13133291	—	2	1	no	—
6	#13133209	C₄	1	1	yes	characteristic
7	#13133150	—	2	1	no	—
8	#13133148	—	2	1	no	—
9	C₄×#88322	C₂²	1	1	yes	characteristic
10	#125×C₅₃	C₄	1	1	yes	characteristic
11	C₄⋊C₄×C₅₃	—	2	1	no	—
12	C₄⋊C₄×C₅₃	—	4	1	no	—
13	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
14	C₄⋊Dic₅₃	—	2	1	no	—
15	#11719919	—	4	1	no	—
16	C₂×#88322	D₄	1	1	yes	characteristic
17	C₂×#88322	D₄	1	1	yes	characteristic
18	#11719910	—	4	1	no	—
19	#11719910	—	4	1	no	—
20	C₂×C₄×C₅₃	—	2	1	no	—
21	C₂×C₄×C₅₃	—	4	1	no	—
22	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×Dic₅₃	—	4	1	no	—
24	#88322	SD₁₆	1	1	yes	characteristic
25	#88322	SD₁₆	1	1	yes	characteristic
26	#88322	Q₁₆	1	1	yes	characteristic
27	#88322	Q₁₆	1	1	yes	characteristic
28	C₂²×C₅₃	#100	1	1	yes	characteristic
29	C₄×C₅₃	—	4	1	no	—
30	C₄×C₅₃	—	8	1	no	—
31	C₄×C₅₃	#103	1	1	yes	characteristic
32	C₄×C₅₃	#103	1	1	yes	characteristic
33	Dic₅₃	—	8	1	no	—
34	#1366	—	53	1	no	maximal, Sylow
35	C₂×C₅₃	—	2	1	no	—
36	C₂×C₅₃	#333	1	1	yes	characteristic
37	#499	—	53	1	no	—
38	#364	—	53	1	no	—
39	#364	—	53	1	no	—
40	C₅₃	#1366	1	1	yes	characteristic, complemented, Sylow
41	#125	—	53	1	no	—
42	#125	#6157	1	1	yes	characteristic
43	M₃₂	—	106	1	no	—
44	M₃₂	—	106	1	no	—
45	#107	—	106	1	no	—
46	#106	—	106	1	no	—
47	C₄×C₈	—	53	1	no	—
48	C₂×C₈	—	53	1	no	—
49	C₂×C₈	—	53	1	no	—
50	C₄⋊C₄	—	2	1	no	—
51	C₄⋊C₄	—	4	1	no	—
52	C₄⋊C₄	—	106	1	no	—
53	C₄⋊C₄	—	212	1	no	—
54	C₄²	C₂×#6157	1	1	yes	characteristic
55	C₁₆	—	212	1	no	—
56	C₁₆	—	212	1	no	—
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	4	1	no	—
59	C₂×C₄	—	212	1	no	—
60	C₂×C₄	C₂²⋊#6157	1	1	yes	characteristic
61	C₈	—	53	1	no	—
62	C₈	—	53	1	no	—
63	C₈	—	53	1	no	—
64	C₈	—	53	1	no	—
65	C₂²	#13133277	1	1	yes	characteristic
66	C₄	—	4	1	no	—
67	C₄	—	8	1	no	—
68	C₄	—	424	1	no	—
69	C₄	#13133280	1	1	yes	characteristic
70	C₄	#13133280	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	#16811483	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16811483
commutator	C₄²×C₅₃	C₂×C₄
Frattini	C₄²	C₂×#6157
Fitting	#125×C₅₃	C₄
socle	C₂×C₅₃	#333
hypercenter	#125	#6157
hypocenter	C₅₃	#1366
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
40. C₅₃	#1366	semidirect