Subgroups — #20283796

Order 6784.

396 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#88322	C₂	1	8	yes	maximal, complemented
2	#16810676	C₂	1	6	yes	maximal
3	C₂×C₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#88322	C₂²	1	12	yes	—
5	C₂²⋊#88322	C₂²	1	4	yes	—
6	C₄×#88322	C₂²	1	12	yes	—
7	C₂²×Q₈×C₅₃	C₄	1	1	yes	characteristic
8	C₄×Q₈×C₅₃	C₄	1	4	yes	—
9	C₄×Q₈×C₅₃	C₂²	1	4	yes	—
10	C₂×C₄⋊C₄×C₅₃	C₄	1	3	yes	—
11	C₂×C₄²×C₅₃	C₂²	1	3	yes	—
12	C₂×Q₈×C₅₃	C₈	1	8	yes	complemented
13	C₂×Q₈×C₅₃	C₂×C₄	1	4	yes	—
14	C₂²×C₄×C₅₃	C₂×C₄	1	3	yes	—
15	C₂²×C₄×C₅₃	C₂×C₄	1	3	yes	—
16	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
17	C₄⋊C₄×C₅₃	C₂×C₄	1	12	yes	—
18	C₄²×C₅₃	C₂×C₄	1	6	yes	—
19	C₄²×C₅₃	C₂³	1	6	yes	—
20	C₂×#88322	C₂³	1	8	yes	—
21	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
22	Q₈×C₅₃	C₂×C₈	1	16	yes	—
23	C₂×C₄×C₅₃	—	2	2	no	—
24	C₂×C₄×C₅₃	—	4	3	no	—
25	C₂×C₄×C₅₃	C₂×C₈	1	12	yes	—
26	C₂×C₄×C₅₃	C₂²×C₄	1	6	yes	—
27	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₄×C₅₃	C₂²×C₄	1	6	yes	—
29	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
30	#88322	—	2	8	no	—
31	C₂²×C₅₃	—	2	1	no	—
32	C₂²×C₅₃	—	2	1	no	—
33	C₂²×C₅₃	C₂²×C₈	1	2	yes	—
34	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
35	C₄×C₅₃	—	2	1	no	—
36	C₄×C₅₃	—	2	6	no	—
37	C₄×C₅₃	C₂²×C₈	1	12	yes	—
38	C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
39	#2982	—	53	1	no	maximal, Sylow
40	C₂×C₅₃	—	2	2	no	—
41	C₂×C₅₃	#519	1	1	yes	characteristic
42	C₂×C₅₃	C₂³×C₈	1	1	yes	characteristic
43	C₂×C₅₃	#568	1	1	yes	characteristic
44	C₂×C₄×Q₈	D₅₃	1	1	yes	characteristic
45	C₈×Q₈	—	53	8	no	—
46	#431	—	53	6	no	—
47	C₅₃	#2982	1	1	yes	characteristic, complemented, Sylow
48	C₂²×Q₈	Dic₅₃	1	1	yes	characteristic
49	C₄×Q₈	Dic₅₃	1	4	yes	—
50	C₄×Q₈	C₂×D₅₃	1	4	yes	—
51	C₂×C₄⋊C₄	Dic₅₃	1	3	yes	—
52	C₂×C₄²	C₂×D₅₃	1	3	yes	—
53	C₄⋊C₈	—	53	12	no	—
54	C₂²⋊C₈	—	53	4	no	—
55	C₄×C₈	—	53	12	no	—
56	C₂×Q₈	#88322	1	8	yes	complemented
57	C₂×Q₈	C₂×Dic₅₃	1	4	yes	—
58	C₂²×C₄	C₂×Dic₅₃	1	3	yes	—
59	C₂²×C₄	C₂×Dic₅₃	1	3	yes	—
60	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
61	C₂×C₈	—	53	8	no	—
62	C₄⋊C₄	C₂×Dic₅₃	1	12	yes	—
63	C₄²	C₂×Dic₅₃	1	6	yes	—
64	C₄²	C₂²×D₅₃	1	6	yes	—
65	C₂³	C₂²×Dic₅₃	1	1	yes	characteristic
66	Q₈	C₂×#88322	1	16	yes	—
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	—	4	3	no	—
69	C₂×C₄	C₂×#88322	1	12	yes	—
70	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
71	C₂×C₄	C₂²×Dic₅₃	1	6	yes	—
72	C₂×C₄	C₂²×Dic₅₃	1	6	yes	—
73	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
74	C₈	—	106	8	no	—
75	C₂²	—	2	1	no	—
76	C₂²	—	2	1	no	—
77	C₂²	C₂²×#88322	1	2	yes	—
78	C₂²	C₂³×Dic₅₃	1	1	yes	characteristic
79	C₄	—	2	1	no	—
80	C₄	—	2	6	no	—
81	C₄	C₂²×#88322	1	12	yes	—
82	C₄	#13133265	1	2	yes	—
83	C₂	—	2	2	no	—
84	C₂	#16811101	1	1	yes	characteristic
85	C₂	C₂³×#88322	1	1	yes	characteristic
86	C₂	#16811390	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator	C₂×C₅₃	C₂³×C₈
Fitting	C₂×C₄×Q₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂³×C₄
hypercenter	C₂×C₄×Q₈	D₅₃
hypocenter	C₅₃	#2982
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. Q₈×#88322	C₂	semidirect
12. C₂×Q₈×C₅₃	C₈	semidirect
47. C₅₃	#2982	semidirect
56. C₂×Q₈	#88322	semidirect