Subgroups — #20273007

Order 6784.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810676	C₂	1	2	yes	maximal
2	#16810345	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#88322	C₄	1	4	yes	—
5	C₂²⋊#88322	C₄	1	4	yes	—
6	C₄×#88322	C₂²	1	4	yes	—
7	C₂²×C₈×C₅₃	C₄	1	2	yes	—
8	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
9	C₄×C₈×C₅₃	C₄	1	2	yes	—
10	C₄×C₈×C₅₃	C₂²	1	2	yes	—
11	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
12	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
13	C₂×C₈×C₅₃	—	2	4	no	—
14	C₂×C₈×C₅₃	C₂×C₄	1	4	yes	—
15	C₄²×C₅₃	C₂×C₄	1	2	yes	—
16	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₄²×C₅₃	C₂³	1	1	yes	characteristic
18	C₂×#88322	C₂×C₄	1	8	yes	—
19	C₂³×C₅₃	C₄²	1	1	yes	characteristic
20	C₂×C₄×C₅₃	—	2	4	no	—
21	C₂×C₄×C₅₃	C₄²	1	2	yes	—
22	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
23	C₂×C₄×C₅₃	M₁₆	1	4	yes	—
24	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
25	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
26	C₈×C₅₃	—	2	4	no	—
27	#88322	—	2	8	no	—
28	C₂²×C₅₃	—	2	2	no	—
29	C₂²×C₅₃	#97	1	2	yes	—
30	C₂²×C₅₃	C₂×C₄²	1	1	yes	characteristic
31	C₄×C₅₃	—	2	2	no	—
32	C₄×C₅₃	#97	1	2	yes	—
33	C₄×C₅₃	C₂×M₁₆	1	2	yes	—
34	C₄×C₅₃	#131	1	4	yes	—
35	#1452	—	53	1	no	maximal, Sylow
36	C₂×C₅₃	—	2	2	no	—
37	C₂×C₅₃	C₂×#97	1	1	yes	characteristic
38	C₂×C₅₃	#405	1	2	yes	—
39	#431	—	53	2	no	—
40	C₂×C₄×C₈	D₅₃	1	1	yes	characteristic
41	#322	—	53	4	no	—
42	C₅₃	#1452	1	1	yes	characteristic, complemented, Sylow
43	C₂²×C₈	Dic₅₃	1	2	yes	—
44	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
45	C₄⋊C₈	—	53	4	no	—
46	C₂²⋊C₈	—	53	4	no	—
47	C₄×C₈	—	53	4	no	—
48	C₄×C₈	Dic₅₃	1	2	yes	—
49	C₄×C₈	C₂×D₅₃	1	2	yes	—
50	C₂²×C₄	C₄×D₅₃	1	2	yes	—
51	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
52	C₂×C₈	—	2	4	no	—
53	C₂×C₈	—	53	8	no	—
54	C₂×C₈	C₂×Dic₅₃	1	4	yes	—
55	C₄²	C₄×D₅₃	1	2	yes	—
56	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
57	C₄²	C₂²×D₅₃	1	1	yes	characteristic
58	C₂³	C₄×Dic₅₃	1	1	yes	characteristic
59	C₂×C₄	—	2	4	no	—
60	C₂×C₄	#11719917	1	4	yes	—
61	C₂×C₄	C₄×Dic₅₃	1	2	yes	—
62	C₂×C₄	C₄×Dic₅₃	1	1	yes	characteristic
63	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
64	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
65	C₈	—	2	4	no	—
66	C₈	—	106	8	no	—
67	C₂²	—	2	2	no	—
68	C₂²	#13133128	1	2	yes	—
69	C₂²	C₂×C₄×Dic₅₃	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	#13133128	1	2	yes	—
72	C₄	C₂×#11719917	1	2	yes	—
73	C₄	#13133223	1	4	yes	—
74	C₂	—	2	2	no	—
75	C₂	C₂×#13133128	1	1	yes	characteristic
76	C₂	#16810578	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂×C₄²
Fitting	C₂×C₄×C₈×C₅₃	C₂
hypercenter	C₂×C₄×C₈	D₅₃
hypocenter	C₅₃	#1452
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #16810345	C₂	semidirect
42. C₅₃	#1452	semidirect