Subgroups — #20272998

Order 6784.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810676	C₂	1	2	yes	maximal
2	#16810344	C₂	1	2	yes	maximal, complemented
3	C₈×#88322	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₄⋊#88322	C₄	1	4	yes	—
6	C₂²⋊#88322	C₄	1	4	yes	—
7	C₄×#88322	C₂²	1	4	yes	—
8	C₂²×C₈×C₅₃	C₄	1	2	yes	—
9	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
10	C₄×C₈×C₅₃	C₄	1	2	yes	—
11	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
14	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂×C₈×C₅₃	C₈	1	8	yes	complemented
16	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
18	C₄²×C₅₃	C₂×C₄	1	2	yes	—
19	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₄²×C₅₃	C₂³	1	1	yes	characteristic
21	C₂×#88322	C₂×C₄	1	8	yes	—
22	C₂³×C₅₃	C₄²	1	1	yes	characteristic
23	C₂×C₄×C₅₃	—	2	4	no	—
24	C₂×C₄×C₅₃	C₄²	1	2	yes	—
25	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
26	C₂×C₄×C₅₃	C₂×C₈	1	4	yes	—
27	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
28	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
29	C₈×C₅₃	C₂×C₈	1	4	yes	—
30	C₈×C₅₃	C₂×C₈	1	4	yes	—
31	#88322	—	2	8	no	—
32	C₂²×C₅₃	—	2	2	no	—
33	C₂²×C₅₃	C₄×C₈	1	2	yes	—
34	C₂²×C₅₃	C₂×C₄²	1	1	yes	characteristic
35	C₄×C₅₃	—	2	2	no	—
36	C₄×C₅₃	C₄×C₈	1	2	yes	—
37	C₄×C₅₃	C₂²×C₈	1	2	yes	—
38	C₄×C₅₃	#131	1	4	yes	—
39	#1450	—	53	1	no	maximal, Sylow
40	C₂×C₅₃	—	2	2	no	—
41	C₂×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
42	C₂×C₅₃	#405	1	2	yes	—
43	#431	—	53	2	no	—
44	C₂×C₄×C₈	D₅₃	1	1	yes	characteristic
45	#322	—	53	2	no	—
46	C₈²	—	53	2	no	—
47	C₅₃	#1450	1	1	yes	characteristic, complemented, Sylow
48	C₂²×C₈	Dic₅₃	1	2	yes	—
49	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
50	C₄⋊C₈	—	53	4	no	—
51	C₂²⋊C₈	—	53	4	no	—
52	C₄×C₈	—	53	4	no	—
53	C₄×C₈	Dic₅₃	1	2	yes	—
54	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
55	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
56	C₂²×C₄	C₄×D₅₃	1	2	yes	—
57	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
58	C₂×C₈	—	53	8	no	—
59	C₂×C₈	#88322	1	8	yes	complemented
60	C₂×C₈	C₂×Dic₅₃	1	2	yes	—
61	C₂×C₈	C₂×Dic₅₃	1	2	yes	—
62	C₄²	C₄×D₅₃	1	2	yes	—
63	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
64	C₄²	C₂²×D₅₃	1	1	yes	characteristic
65	C₂³	C₄×Dic₅₃	1	1	yes	characteristic
66	C₂×C₄	—	2	4	no	—
67	C₂×C₄	C₂×#88322	1	4	yes	—
68	C₂×C₄	C₄×Dic₅₃	1	2	yes	—
69	C₂×C₄	C₄×Dic₅₃	1	1	yes	characteristic
70	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
71	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
72	C₈	—	106	8	no	—
73	C₈	C₂×#88322	1	4	yes	—
74	C₈	C₂×#88322	1	4	yes	—
75	C₂²	—	2	2	no	—
76	C₂²	C₄×#88322	1	2	yes	—
77	C₂²	C₂×C₄×Dic₅₃	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	C₄×#88322	1	2	yes	—
80	C₄	C₂²×#88322	1	2	yes	—
81	C₄	#13133223	1	4	yes	—
82	C₂	—	2	2	no	—
83	C₂	C₂×C₄×#88322	1	1	yes	characteristic
84	C₂	#16810578	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂×D₅₃
commutator	C₂×C₅₃	C₂×C₄×C₈
Frattini	C₄²	C₂²×D₅₃
Fitting	C₂×C₄×C₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄²
hypercenter	C₂×C₄×C₈	D₅₃
hypocenter	C₅₃	#1450
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #16810344	C₂	semidirect
3. C₈×#88322	C₂	semidirect
15. C₂×C₈×C₅₃	C₈	semidirect
47. C₅₃	#1450	semidirect
59. C₂×C₈	#88322	semidirect