Subgroups — #20276573

Order 6784.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₄₂₄	C₂	1	1	yes	maximal, characteristic
2	#16810717	C₂	1	2	yes	maximal
3	#16810426	C₂	1	2	yes	maximal
4	C₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#366	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₁₂	C₂²	1	2	yes	—
7	#13133182	—	2	2	no	—
8	Dic₄₂₄	C₄	1	4	yes	complemented
9	C₂×#11719908	C₂²	1	1	yes	characteristic
10	#13133140	—	2	2	no	—
11	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	#107×C₅₃	C₂²	1	2	yes	—
14	C₂×Dic₁₀₆	—	2	2	no	—
15	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
16	C₄⋊C₄×C₅₃	—	2	2	no	—
17	#11719919	—	4	2	no	—
18	C₄×Dic₅₃	—	4	2	no	—
19	C₄×Dic₅₃	D₄	1	1	yes	characteristic
20	C₂×#88322	D₄	1	1	yes	characteristic
21	Dic₂₁₂	—	2	2	no	—
22	Dic₂₁₂	C₂×C₄	1	4	yes	—
23	C₁₆×C₅₃	C₂×C₄	1	2	yes	—
24	#11719908	—	2	1	no	—
25	C₂×C₄×C₅₃	—	4	2	no	—
26	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
27	C₂×Dic₅₃	—	4	2	no	—
28	C₂×Dic₅₃	D₈	1	1	yes	characteristic
29	C₂×Dic₅₃	D₈	1	1	yes	characteristic
30	Dic₁₀₆	—	2	4	no	—
31	Dic₁₀₆	—	4	2	no	—
32	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
33	C₈×C₅₃	#41	1	1	yes	characteristic
34	#88322	—	2	1	no	—
35	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
36	C₄×C₅₃	—	8	2	no	—
37	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
38	C₄×C₅₃	#135	1	1	yes	characteristic
39	Dic₅₃	—	2	1	no	—
40	Dic₅₃	—	4	4	no	—
41	Dic₅₃	Q₃₂	1	2	yes	complemented
42	C₄×Q₃₂	—	53	1	no	maximal, Sylow
43	C₂×C₅₃	C₄×D₈	1	1	yes	characteristic
44	C₂×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
45	C₂×C₅₃	#508	1	1	yes	characteristic
46	C₂×Q₃₂	—	53	1	no	—
47	C₄×Q₁₆	—	53	2	no	—
48	#366	D₅₃	1	1	yes	characteristic
49	#358	—	53	2	no	—
50	C₄×C₁₆	—	53	1	no	—
51	C₅₃	C₄×Q₃₂	1	1	yes	characteristic, complemented, Sylow
52	C₂×Q₁₆	—	53	2	no	—
53	C₄×Q₈	—	106	2	no	—
54	Q₃₂	—	53	4	no	—
55	C₂×C₁₆	—	53	1	no	—
56	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
57	#107	C₂×D₅₃	1	2	yes	—
58	#103	—	106	2	no	—
59	C₄×C₈	—	53	1	no	—
60	C₂×Q₈	—	106	2	no	—
61	Q₁₆	—	53	4	no	—
62	Q₁₆	—	106	2	no	—
63	C₂×C₈	—	53	1	no	—
64	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
65	C₄⋊C₄	—	2	2	no	—
66	C₄⋊C₄	—	212	2	no	—
67	C₄²	—	53	1	no	—
68	C₄²	—	212	2	no	—
69	C₁₆	—	106	1	no	—
70	C₁₆	C₄×D₅₃	1	2	yes	—
71	Q₈	—	106	4	no	—
72	Q₈	—	212	2	no	—
73	C₂×C₄	—	4	2	no	—
74	C₂×C₄	—	53	1	no	—
75	C₂×C₄	—	53	1	no	—
76	C₂×C₄	—	212	2	no	—
77	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
78	C₈	—	106	1	no	—
79	C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
80	C₈	#11719949	1	1	yes	characteristic
81	C₂²	D₈×D₅₃	1	1	yes	characteristic
82	C₄	—	8	2	no	—
83	C₄	—	53	2	no	—
84	C₄	—	106	1	no	—
85	C₄	—	212	4	no	—
86	C₄	#13133179	1	1	yes	characteristic
87	C₄	#13133240	1	1	yes	characteristic
88	C₂	#16810705	1	1	yes	characteristic
89	C₂	Q₃₂×D₅₃	1	1	yes	characteristic
90	C₂	#16811052	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×D₅₃
commutator	C₈×C₅₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#366	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	#366	D₅₃
hypocenter	C₅₃	C₄×Q₃₂
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

Complements

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

N	H ≅ G/N	type
8. Dic₄₂₄	C₄	semidirect
41. Dic₅₃	Q₃₂	semidirect
51. C₅₃	C₄×Q₃₂	semidirect