Subgroups — #20272626

Order 6784.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810680	C₂	1	2	yes	maximal
2	C₅₃×#423	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#88322	—	2	2	no	—
4	C₂²⋊#88322	C₄	1	4	yes	—
5	#13133128	—	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	1	yes	characteristic
10	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
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	C₂×#88322	—	2	4	no	—
17	C₂³×C₅₃	C₄²	1	1	yes	characteristic
18	C₂×C₄×C₅₃	—	2	1	no	—
19	C₂×C₄×C₅₃	—	2	1	no	—
20	C₂×C₄×C₅₃	—	4	2	no	—
21	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
22	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
24	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
25	C₈×C₅₃	—	4	2	no	—
26	#88322	—	4	4	no	—
27	C₂²×C₅₃	—	2	1	no	—
28	C₂²×C₅₃	—	2	2	no	—
29	C₂²×C₅₃	#95	1	1	yes	characteristic
30	C₄×C₅₃	—	2	1	no	—
31	C₄×C₅₃	—	2	2	no	—
32	C₄×C₅₃	—	2	2	no	—
33	C₄×C₅₃	#100	1	1	yes	characteristic
34	C₄×C₅₃	#101	1	1	yes	characteristic
35	#1307	—	53	1	no	maximal, Sylow
36	C₂×C₅₃	—	4	1	no	—
37	C₂×C₅₃	#343	1	1	yes	characteristic
38	C₂×C₅₃	#344	1	2	yes	—
39	#432	—	53	2	no	—
40	#423	D₅₃	1	1	yes	characteristic
41	C₅₃	#1307	1	1	yes	characteristic, complemented, Sylow
42	C₂×M₁₆	Dic₅₃	1	2	yes	—
43	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
44	C₄⋊C₈	—	2	2	no	—
45	C₄⋊C₈	—	106	2	no	—
46	C₂²⋊C₈	—	53	4	no	—
47	#97	—	106	2	no	—
48	C₂²×C₄	C₄×D₅₃	1	2	yes	—
49	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
50	M₁₆	—	4	2	no	—
51	C₂×C₈	—	2	2	no	—
52	C₂×C₈	—	106	4	no	—
53	C₄²	Dic₁₀₆	1	1	yes	characteristic
54	C₄²	D₂₁₂	1	1	yes	characteristic
55	C₄²	C₂²⋊D₅₃	1	2	yes	—
56	C₂³	C₄×Dic₅₃	1	1	yes	characteristic
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	4	2	no	—
60	C₂×C₄	#11719919	1	2	yes	—
61	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
62	C₂×C₄	#11719921	1	2	yes	—
63	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
64	C₈	—	4	2	no	—
65	C₈	—	212	4	no	—
66	C₂²	—	2	1	no	—
67	C₂²	—	2	2	no	—
68	C₂²	#13133124	1	1	yes	characteristic
69	C₄	—	2	1	no	—
70	C₄	—	2	2	no	—
71	C₄	—	2	2	no	—
72	C₄	#13133134	1	1	yes	characteristic
73	C₄	#13133136	1	1	yes	characteristic
74	C₂	—	4	1	no	—
75	C₂	#16810392	1	1	yes	characteristic
76	C₂	#16810395	1	2	yes	—

	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	#1307
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
41. C₅₃	#1307	semidirect