Subgroups — #20272889

Order 6784.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810977	C₂	1	1	yes	maximal, characteristic
2	#16810420	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#354	C₂	1	1	yes	maximal, characteristic
4	#13133209	C₄	1	1	yes	characteristic, complemented
5	#13133141	—	2	1	no	—
6	#13133138	—	2	1	no	—
7	#13133136	—	2	1	no	—
8	C₄×#88322	C₄	1	1	yes	characteristic, complemented
9	#124×C₅₃	C₂²	1	1	yes	characteristic
10	#99×C₅₃	—	2	1	no	—
11	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic
12	C₂×D₄×C₅₃	D₄	1	1	yes	characteristic
13	C₂²⋊C₄×C₅₃	—	4	1	no	—
14	C₂²⋊C₄×C₅₃	—	4	1	no	—
15	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₄⋊Dic₅₃	—	2	1	no	—
17	#11719919	—	4	1	no	—
18	#11719917	—	4	1	no	—
19	C₂×#88322	—	2	1	no	—
20	C₂³×C₅₃	—	2	1	no	—
21	Q₈×C₅₃	—	4	1	no	—
22	D₄×C₅₃	—	4	1	no	—
23	C₂×C₄×C₅₃	—	2	1	no	—
24	C₂×C₄×C₅₃	—	2	1	no	—
25	C₂×C₄×C₅₃	—	8	1	no	—
26	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×Dic₅₃	—	4	1	no	—
28	#88322	—	4	1	no	—
29	#88322	—	8	1	no	—
30	C₂²×C₅₃	—	4	1	no	—
31	C₂²×C₅₃	—	8	1	no	—
32	C₂²×C₅₃	#99	1	1	yes	characteristic
33	C₄×C₅₃	—	2	1	no	—
34	C₄×C₅₃	—	4	1	no	—
35	C₄×C₅₃	—	4	1	no	—
36	C₄×C₅₃	—	16	1	no	—
37	Dic₅₃	—	8	1	no	—
38	#1414	—	53	1	no	maximal, Sylow
39	C₂×C₅₃	—	2	1	no	—
40	C₂×C₅₃	—	8	1	no	—
41	C₂×C₅₃	#354	1	1	yes	characteristic
42	#488	—	53	1	no	—
43	#355	—	53	1	no	—
44	#354	D₅₃	1	1	yes	characteristic
45	C₅₃	#1414	1	1	yes	characteristic, complemented, Sylow
46	#125	—	53	1	no	—
47	#124	C₂×D₅₃	1	1	yes	characteristic
48	#103	—	106	1	no	—
49	#102	—	106	1	no	—
50	#101	—	106	1	no	—
51	#99	—	2	1	no	—
52	C₄×C₈	—	53	1	no	—
53	C₂×Q₈	D₂₁₂	1	1	yes	characteristic
54	C₂×D₄	C₂²⋊D₅₃	1	1	yes	characteristic
55	M₁₆	—	212	1	no	—
56	C₂×C₈	—	106	1	no	—
57	C₄⋊C₄	—	106	1	no	—
58	C₄⋊C₄	—	212	1	no	—
59	C₂²⋊C₄	—	4	1	no	—
60	C₂²⋊C₄	—	4	1	no	—
61	C₄²	C₄×D₅₃	1	1	yes	characteristic
62	C₂³	—	2	1	no	—
63	Q₈	—	4	1	no	—
64	D₄	—	4	1	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	8	1	no	—
68	C₂×C₄	—	212	1	no	—
69	C₂×C₄	#11719921	1	1	yes	characteristic
70	C₈	—	424	1	no	—
71	C₈	—	212	1	no	—
72	C₂²	—	4	1	no	—
73	C₂²	—	8	1	no	—
74	C₂²	#13133131	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	4	1	no	—
77	C₄	—	4	1	no	—
78	C₄	—	16	1	no	—
79	C₄	—	424	1	no	—
80	C₂	—	2	1	no	—
81	C₂	—	8	1	no	—
82	C₂	#16810417	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16810417
commutator	C₄²×C₅₃	C₂×C₄
Frattini	#124	C₂×D₅₃
Fitting	C₅₃×#354	C₂
socle	C₂×C₅₃	#354
hypercenter	#354	D₅₃
hypocenter	C₅₃	#1414
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #124 ◁ #354

Complements

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

N	H ≅ G/N	type
4. #13133209	C₄	semidirect
8. C₄×#88322	C₄	semidirect
45. C₅₃	#1414	semidirect