Subgroups — #20272544

Order 6784.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810676	C₂	1	2	yes	maximal
2	C₅₃×#376	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#88322	—	2	2	no	—
4	C₂²⋊#88322	—	2	2	no	—
5	C₄×#88322	C₄	1	4	yes	complemented
6	C₂×C₄²×C₅₃	C₄	1	2	yes	—
7	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
8	#95×C₅₃	—	2	2	no	—
9	C₂²×C₄×C₅₃	—	2	2	no	—
10	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
11	C₂²×C₄×C₅₃	D₄	1	2	yes	—
12	C₄²×C₅₃	—	2	2	no	—
13	C₄²×C₅₃	C₈	1	4	yes	complemented
14	C₄²×C₅₃	C₂×C₄	1	2	yes	—
15	C₄²×C₅₃	D₄	1	1	yes	characteristic
16	C₄²×C₅₃	Q₈	1	1	yes	characteristic
17	C₂×#88322	—	2	4	no	—
18	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
19	C₂×C₄×C₅₃	—	2	4	no	—
20	C₂×C₄×C₅₃	—	4	2	no	—
21	C₂×C₄×C₅₃	—	4	4	no	—
22	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
23	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
24	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
25	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
26	C₂×C₄×C₅₃	C₂×C₈	1	2	yes	—
27	C₂×C₄×C₅₃	M₁₆	1	2	yes	—
28	#88322	—	4	4	no	—
29	C₂²×C₅₃	—	2	2	no	—
30	C₂²×C₅₃	#95	1	1	yes	characteristic
31	C₂²×C₅₃	C₂²⋊C₈	1	2	yes	—
32	C₄×C₅₃	—	2	2	no	—
33	C₄×C₅₃	—	2	2	no	—
34	C₄×C₅₃	—	4	4	no	—
35	C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
36	C₄×C₅₃	#97	1	1	yes	characteristic
37	C₄×C₅₃	C₄⋊C₈	1	2	yes	—
38	#1274	—	53	1	no	maximal, Sylow
39	C₂×C₅₃	—	2	2	no	—
40	C₂×C₅₃	#336	1	1	yes	characteristic
41	C₂×C₅₃	H(3,ℤ/4)	1	2	yes	—
42	#431	—	53	2	no	—
43	#376	D₅₃	1	1	yes	characteristic
44	C₅₃	#1274	1	1	yes	characteristic, complemented, Sylow
45	C₂×C₄²	Dic₅₃	1	2	yes	—
46	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
47	C₄⋊C₈	—	106	2	no	—
48	C₂²⋊C₈	—	106	2	no	—
49	C₄×C₈	—	53	4	no	—
50	#95	—	2	2	no	—
51	C₂²×C₄	—	2	2	no	—
52	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
53	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
54	C₂×C₈	—	106	4	no	—
55	C₄²	—	2	2	no	—
56	C₄²	#88322	1	4	yes	complemented
57	C₄²	Dic₁₀₆	1	1	yes	characteristic
58	C₄²	C₄×D₅₃	1	2	yes	—
59	C₄²	D₂₁₂	1	1	yes	characteristic
60	C₂³	C₂²⋊Dic₅₃	1	1	yes	characteristic
61	C₂×C₄	—	2	4	no	—
62	C₂×C₄	—	4	2	no	—
63	C₂×C₄	—	4	4	no	—
64	C₂×C₄	C₂×#88322	1	2	yes	—
65	C₂×C₄	#11719917	1	2	yes	—
66	C₂×C₄	C₄×Dic₅₃	1	1	yes	characteristic
67	C₂×C₄	#11719919	1	2	yes	—
68	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
69	C₂×C₄	#11719921	1	2	yes	—
70	C₈	—	212	4	no	—
71	C₂²	—	2	2	no	—
72	C₂²	#13133124	1	1	yes	characteristic
73	C₂²	C₂²⋊#88322	1	2	yes	—
74	C₄	—	2	2	no	—
75	C₄	—	2	2	no	—
76	C₄	—	4	4	no	—
77	C₄	C₄×#88322	1	1	yes	characteristic
78	C₄	#13133128	1	1	yes	characteristic
79	C₄	C₄⋊#88322	1	2	yes	—
80	C₂	—	2	2	no	—
81	C₂	#16810378	1	1	yes	characteristic
82	C₂	#16810379	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₄×Dic₅₃
commutator	C₄×C₅₃	C₄×C₈
Frattini	C₂×C₄²	C₂×D₅₃
Fitting	C₅₃×#376	C₂
socle	C₂²×C₅₃	#95
hypercenter	#376	D₅₃
hypocenter	C₅₃	#1274
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #376

Complements

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

N	H ≅ G/N	type
5. C₄×#88322	C₄	semidirect
13. C₄²×C₅₃	C₈	semidirect
44. C₅₃	#1274	semidirect
56. C₄²	#88322	semidirect