Subgroups — #20273310

Order 6784.

226 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#11719917	C₂	1	2	yes	maximal
2	#16810365	C₂	1	4	yes	maximal, complemented
3	C₅₃×#533	C₂	1	1	yes	maximal, characteristic
4	C₂×#11719917	—	2	2	no	—
5	#13133128	C₂²	1	4	yes	—
6	C₄×#88322	—	2	2	no	—
7	C₄⋊Q₈×C₅₃	C₄	1	2	yes	—
8	#125×C₅₃	C₂²	1	2	yes	—
9	C₂²⋊Q₈×C₅₃	—	2	2	no	—
10	C₄×Q₈×C₅₃	—	2	2	no	—
11	#117×C₅₃	C₄	1	2	yes	—
12	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×Q₈×C₅₃	—	2	2	no	—
14	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₅₃	D₄	1	2	yes	—
16	C₄⋊C₄×C₅₃	—	2	2	no	—
17	C₄⋊C₄×C₅₃	—	2	4	no	—
18	C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
19	C₂²⋊C₄×C₅₃	—	2	2	no	—
20	C₄²×C₅₃	—	2	2	no	—
21	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₅₃	D₄	1	2	yes	—
23	C₄²×C₅₃	C₂³	1	1	yes	characteristic
24	#11719917	—	2	8	no	—
25	C₂×#88322	—	2	4	no	—
26	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
27	Q₈×C₅₃	—	4	4	no	—
28	C₂×C₄×C₅₃	—	2	2	no	—
29	C₂×C₄×C₅₃	—	2	4	no	—
30	C₂×C₄×C₅₃	—	2	4	no	—
31	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
32	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
35	#88322	—	2	8	no	—
36	C₂²×C₅₃	—	2	1	no	—
37	C₂²×C₅₃	—	2	2	no	—
38	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
39	C₄×C₅₃	—	2	1	no	—
40	C₄×C₅₃	—	2	2	no	—
41	C₄×C₅₃	—	4	4	no	—
42	C₄×C₅₃	#101	1	2	yes	—
43	C₄×C₅₃	#104	1	4	yes	—
44	#1526	—	53	1	no	maximal, Sylow
45	C₂×C₅₃	—	4	1	no	—
46	C₂×C₅₃	C₂×#101	1	1	yes	characteristic
47	C₂×C₅₃	C₂×#104	1	2	yes	—
48	#533	D₅₃	1	1	yes	characteristic
49	C₄×M₁₆	—	53	2	no	—
50	#330	—	53	4	no	—
51	C₅₃	#1526	1	1	yes	characteristic, complemented, Sylow
52	C₂×M₁₆	—	106	2	no	—
53	C₄⋊Q₈	Dic₅₃	1	2	yes	—
54	#125	C₂×D₅₃	1	2	yes	—
55	C₂²⋊Q₈	—	2	2	no	—
56	C₄×Q₈	—	2	2	no	—
57	#117	Dic₅₃	1	2	yes	—
58	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
59	#97	—	53	4	no	—
60	C₄×C₈	—	106	2	no	—
61	C₂×Q₈	—	2	2	no	—
62	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
63	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
64	M₁₆	—	106	8	no	—
65	C₂×C₈	—	106	4	no	—
66	C₄⋊C₄	—	2	4	no	—
67	C₄⋊C₄	—	2	2	no	—
68	C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
69	C₂²⋊C₄	—	2	2	no	—
70	C₄²	—	2	2	no	—
71	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
72	C₄²	C₂²⋊D₅₃	1	2	yes	—
73	C₄²	C₂²×D₅₃	1	1	yes	characteristic
74	C₂³	C₂²⋊Dic₅₃	1	1	yes	characteristic
75	Q₈	—	4	4	no	—
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	2	4	no	—
78	C₂×C₄	—	2	4	no	—
79	C₂×C₄	C₂²⋊Dic₅₃	1	2	yes	—
80	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
81	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
82	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
83	C₈	—	106	8	no	—
84	C₂²	—	2	1	no	—
85	C₂²	—	2	2	no	—
86	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
87	C₄	—	2	1	no	—
88	C₄	—	2	2	no	—
89	C₄	—	4	4	no	—
90	C₄	#13133136	1	2	yes	—
91	C₄	#13133144	1	4	yes	—
92	C₂	—	4	1	no	—
93	C₂	C₂×#13133136	1	1	yes	characteristic
94	C₂	C₂×#13133144	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#533	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	#533	D₅₃
hypocenter	C₅₃	#1526
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

Complements

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

N	H ≅ G/N	type
2. #16810365	C₂	semidirect
51. C₅₃	#1526	semidirect