Subgroups — #20272881

Order 6784.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810997	C₂	1	1	yes	maximal, characteristic
2	#16810422	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#356	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11719924	—	2	1	no	—
5	C₂×#11719924	—	2	1	no	—
6	#13133214	C₄	1	1	yes	characteristic
7	#13133136	—	2	1	no	—
8	C₄×#88322	C₄	1	1	yes	characteristic
9	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
10	#101×C₅₃	—	2	1	no	—
11	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic
12	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic
13	C₂×D₂₁₂	—	2	1	no	—
14	C₂×D₂₁₂	—	4	1	no	—
15	M₁₆×C₅₃	—	4	1	no	—
16	C₄⋊C₄×C₅₃	—	4	1	no	—
17	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
18	#11719924	—	4	2	no	—
19	#11719924	—	4	2	no	—
20	#11719917	—	4	1	no	—
21	C₂×#88322	—	2	1	no	—
22	C₂²×D₅₃	—	4	1	no	—
23	Q₈×C₅₃	—	4	1	no	—
24	Q₈×C₅₃	—	4	1	no	—
25	C₂×C₄×C₅₃	—	2	1	no	—
26	C₂×C₄×C₅₃	—	2	1	no	—
27	C₂×C₄×C₅₃	—	2	1	no	—
28	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
29	D₂₁₂	—	4	1	no	—
30	D₂₁₂	—	4	1	no	—
31	D₂₁₂	—	8	2	no	—
32	C₈×C₅₃	—	8	1	no	—
33	#88322	—	2	2	no	—
34	#88322	—	8	1	no	—
35	C₂²×C₅₃	#99	1	1	yes	characteristic
36	C₂×D₅₃	—	8	2	no	—
37	C₂×D₅₃	—	8	1	no	—
38	C₄×C₅₃	—	2	2	no	—
39	C₄×C₅₃	—	2	1	no	—
40	C₄×C₅₃	—	4	1	no	—
41	C₄×C₅₃	—	4	1	no	—
42	#1410	—	53	1	no	maximal, Sylow
43	C₂×C₅₃	—	2	1	no	—
44	C₂×C₅₃	#353	1	1	yes	characteristic
45	D₅₃	—	16	1	no	—
46	#492	—	53	1	no	—
47	#356	—	53	1	no	—
48	#356	D₅₃	1	1	yes	characteristic, complemented
49	C₅₃	#1410	1	1	yes	characteristic, complemented, Sylow
50	C₂×SD₁₆	—	106	1	no	—
51	C₂×SD₁₆	—	106	1	no	—
52	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
53	#127	—	53	1	no	—
54	#101	—	2	1	no	—
55	#101	—	106	1	no	—
56	C₄×C₈	—	53	1	no	—
57	C₂×Q₈	D₂₁₂	1	1	yes	characteristic
58	C₂×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
59	C₂×D₄	—	106	1	no	—
60	C₂×D₄	—	212	1	no	—
61	SD₁₆	—	212	2	no	—
62	SD₁₆	—	212	2	no	—
63	M₁₆	—	4	1	no	—
64	M₁₆	—	212	1	no	—
65	C₂×C₈	—	106	1	no	—
66	C₄⋊C₄	—	4	1	no	—
67	C₄²	C₄×D₅₃	1	1	yes	characteristic
68	C₂³	—	212	1	no	—
69	Q₈	—	4	1	no	—
70	Q₈	—	4	1	no	—
71	D₄	—	424	2	no	—
72	D₄	—	212	1	no	—
73	D₄	—	212	1	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	#11719921	1	1	yes	characteristic
78	C₈	—	8	1	no	—
79	C₈	—	106	2	no	—
80	C₈	—	424	1	no	—
81	C₂²	—	424	2	no	—
82	C₂²	—	424	1	no	—
83	C₂²	#13133131	1	1	yes	characteristic
84	C₄	—	2	2	no	—
85	C₄	—	2	1	no	—
86	C₄	—	4	1	no	—
87	C₄	—	4	1	no	—
88	C₂	—	2	1	no	—
89	C₂	—	848	1	no	—
90	C₂	#16810415	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16810415
commutator	C₄²×C₅₃	C₂×C₄
Frattini	C₄⋊Q₈	C₂×D₅₃
Fitting	C₅₃×#356	C₂
socle	C₂×C₅₃	#353
hypercenter	#356	D₅₃
hypocenter	C₅₃	#1410
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊Q₈ ◁ #356

Complements

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

N	H ≅ G/N	type
2. #16810422	C₂	semidirect
3. C₅₃×#356	C₂	semidirect
48. #356	D₅₃	semidirect
49. C₅₃	#1410	semidirect