Subgroups — #20272885

Order 6784.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810987	C₂	1	1	yes	maximal, characteristic
2	#16810416	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#353	C₂	1	1	yes	maximal, characteristic
4	#13133209	C₄	1	1	yes	characteristic, complemented
5	#13133138	—	2	1	no	—
6	#13133138	—	2	1	no	—
7	#13133132	—	2	1	no	—
8	#13133128	C₄	1	1	yes	characteristic, complemented
9	#127×C₅₃	C₂²	1	1	yes	characteristic
10	#99×C₅₃	—	2	1	no	—
11	C₂×D₄×C₅₃	—	4	1	no	—
12	C₂×D₄×C₅₃	D₄	1	1	yes	characteristic
13	C₂×D₄×C₅₃	D₄	1	1	yes	characteristic
14	C₂²⋊C₄×C₅₃	—	4	1	no	—
15	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂²⋊Dic₅₃	—	4	1	no	—
17	C₄⋊Dic₅₃	—	2	1	no	—
18	#11719919	—	4	1	no	—
19	C₂×#88322	—	2	1	no	—
20	C₂³×C₅₃	—	2	1	no	—
21	C₂³×C₅₃	—	2	1	no	—
22	D₄×C₅₃	—	4	1	no	—
23	D₄×C₅₃	—	4	1	no	—
24	D₄×C₅₃	—	8	2	no	—
25	C₂×C₄×C₅₃	—	2	1	no	—
26	C₂×C₄×C₅₃	—	8	1	no	—
27	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×Dic₅₃	—	4	1	no	—
29	C₂×Dic₅₃	—	8	1	no	—
30	#88322	—	4	1	no	—
31	C₂²×C₅₃	—	4	1	no	—
32	C₂²×C₅₃	—	4	1	no	—
33	C₂²×C₅₃	—	8	1	no	—
34	C₂²×C₅₃	—	8	1	no	—
35	C₂²×C₅₃	#99	1	1	yes	characteristic
36	C₄×C₅₃	—	2	1	no	—
37	C₄×C₅₃	—	4	1	no	—
38	C₄×C₅₃	—	16	1	no	—
39	Dic₅₃	—	8	1	no	—
40	Dic₅₃	—	16	1	no	—
41	#1412	—	53	1	no	maximal, Sylow
42	C₂×C₅₃	—	2	1	no	—
43	C₂×C₅₃	—	8	1	no	—
44	C₂×C₅₃	—	8	1	no	—
45	C₂×C₅₃	#354	1	1	yes	characteristic
46	#490	—	53	1	no	—
47	#353	—	53	1	no	—
48	#353	D₅₃	1	1	yes	characteristic
49	C₅₃	#1412	1	1	yes	characteristic, complemented, Sylow
50	#127	C₂×D₅₃	1	1	yes	characteristic
51	#125	—	53	1	no	—
52	#102	—	106	1	no	—
53	#102	—	106	1	no	—
54	#99	—	2	1	no	—
55	#99	—	106	1	no	—
56	#97	—	53	1	no	—
57	C₂×D₄	—	4	1	no	—
58	C₂×D₄	D₂₁₂	1	1	yes	characteristic
59	C₂×D₄	C₂²⋊D₅₃	1	1	yes	characteristic
60	C₂×C₈	—	106	1	no	—
61	C₄⋊C₄	—	106	1	no	—
62	C₄⋊C₄	—	212	1	no	—
63	C₂²⋊C₄	—	4	1	no	—
64	C₂²⋊C₄	—	212	1	no	—
65	C₄²	C₄×D₅₃	1	1	yes	characteristic
66	C₂³	—	2	1	no	—
67	C₂³	—	2	1	no	—
68	D₄	—	4	1	no	—
69	D₄	—	4	1	no	—
70	D₄	—	8	2	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	8	1	no	—
73	C₂×C₄	—	424	1	no	—
74	C₂×C₄	—	212	1	no	—
75	C₂×C₄	#11719921	1	1	yes	characteristic
76	C₈	—	212	1	no	—
77	C₂²	—	4	1	no	—
78	C₂²	—	4	1	no	—
79	C₂²	—	8	1	no	—
80	C₂²	—	8	1	no	—
81	C₂²	#13133131	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	4	1	no	—
84	C₄	—	16	1	no	—
85	C₄	—	848	1	no	—
86	C₄	—	424	1	no	—
87	C₂	—	2	1	no	—
88	C₂	—	8	1	no	—
89	C₂	—	8	1	no	—
90	C₂	#16810417	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16810417
commutator	C₄²×C₅₃	C₂×C₄
Frattini	#127	C₂×D₅₃
Fitting	C₅₃×#353	C₂
socle	C₂×C₅₃	#354
hypercenter	#353	D₅₃
hypocenter	C₅₃	#1412
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #127 ◁ #353

Complements

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

N	H ≅ G/N	type
4. #13133209	C₄	semidirect
8. #13133128	C₄	semidirect
49. C₅₃	#1412	semidirect