Subgroups — #20272628

Order 6784.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810647	C₂	1	2	yes	maximal
2	C₅₃×#423	C₂	1	1	yes	maximal, characteristic
3	C₂×#11719917	C₄	1	4	yes	—
4	C₄⋊#88322	—	2	2	no	—
5	C₄⋊#88322	—	2	2	no	—
6	C₂×M₁₆×C₅₃	C₄	1	2	yes	—
7	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
8	C₄⋊C₈×C₅₃	—	2	2	no	—
9	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
10	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
11	M₁₆×C₅₃	—	4	2	no	—
12	C₂×C₈×C₅₃	—	2	2	no	—
13	C₄²×C₅₃	D₄	1	1	yes	characteristic
14	C₄²×C₅₃	D₄	1	2	yes	—
15	C₄²×C₅₃	Q₈	1	1	yes	characteristic
16	#11719917	—	4	4	no	—
17	C₂×#88322	—	2	4	no	—
18	C₂³×C₅₃	C₄²	1	1	yes	characteristic
19	C₂×C₄×C₅₃	—	2	2	no	—
20	C₂×C₄×C₅₃	—	2	1	no	—
21	C₂×C₄×C₅₃	—	2	2	no	—
22	C₂×C₄×C₅₃	—	2	1	no	—
23	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
24	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
26	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
27	C₈×C₅₃	—	4	2	no	—
28	#88322	—	4	4	no	—
29	C₂²×C₅₃	—	2	2	no	—
30	C₂²×C₅₃	—	2	1	no	—
31	C₂²×C₅₃	#95	1	1	yes	characteristic
32	C₄×C₅₃	—	2	2	no	—
33	C₄×C₅₃	—	2	1	no	—
34	C₄×C₅₃	#100	1	2	yes	—
35	C₄×C₅₃	#100	1	1	yes	characteristic
36	C₄×C₅₃	#101	1	2	yes	—
37	C₄×C₅₃	#101	1	1	yes	characteristic
38	#1309	—	53	1	no	maximal, Sylow
39	C₂×C₅₃	—	4	1	no	—
40	C₂×C₅₃	#343	1	2	yes	—
41	C₂×C₅₃	#343	1	1	yes	characteristic
42	#423	—	53	2	no	—
43	#423	D₅₃	1	1	yes	characteristic
44	C₅₃	#1309	1	1	yes	characteristic, complemented, Sylow
45	C₂×M₁₆	—	53	4	no	—
46	C₂×M₁₆	Dic₅₃	1	2	yes	—
47	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
48	C₄⋊C₈	—	2	2	no	—
49	C₄⋊C₈	—	106	2	no	—
50	C₄⋊C₈	—	106	2	no	—
51	C₂²×C₄	C₄×D₅₃	1	2	yes	—
52	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
53	M₁₆	—	4	2	no	—
54	M₁₆	—	212	4	no	—
55	C₂×C₈	—	2	2	no	—
56	C₂×C₈	—	106	4	no	—
57	C₄²	Dic₁₀₆	1	1	yes	characteristic
58	C₄²	D₂₁₂	1	1	yes	characteristic
59	C₄²	C₂²⋊D₅₃	1	2	yes	—
60	C₂³	C₄×Dic₅₃	1	1	yes	characteristic
61	C₂×C₄	—	2	2	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	2	no	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	#11719919	1	2	yes	—
66	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
67	C₂×C₄	#11719921	1	2	yes	—
68	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
69	C₈	—	4	2	no	—
70	C₈	—	212	4	no	—
71	C₂²	—	2	2	no	—
72	C₂²	—	2	1	no	—
73	C₂²	#13133124	1	1	yes	characteristic
74	C₄	—	2	2	no	—
75	C₄	—	2	1	no	—
76	C₄	#13133133	1	2	yes	—
77	C₄	#13133134	1	1	yes	characteristic
78	C₄	#13133135	1	2	yes	—
79	C₄	#13133136	1	1	yes	characteristic
80	C₂	—	4	1	no	—
81	C₂	#16810391	1	2	yes	—
82	C₂	#16810392	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#13133124
commutator	C₂³×C₅₃	C₄²
Frattini	C₂×C₄²	C₂×D₅₃
Fitting	C₅₃×#423	C₂
socle	C₂²×C₅₃	#95
hypercenter	#423	D₅₃
hypocenter	C₅₃	#1309
LCS	G ▷ C₂³×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

Complements

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

N	H ≅ G/N	type
44. C₅₃	#1309	semidirect