Subgroups — #20291112

Order 6784.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₁₆×#6157	C₂	1	2	yes	maximal
2	#16811550	C₂	1	1	yes	maximal, characteristic
3	#16811536	C₂	1	2	yes	maximal
4	#16811510	C₂	1	1	yes	maximal, characteristic
5	Q₃₂×D₅₃	C₂	1	1	yes	maximal, characteristic
6	Q₈×#6157	—	2	2	no	—
7	D₅₃⋊C₁₆	C₂²	1	1	yes	characteristic
8	#13133286	C₂²	1	2	yes	—
9	#13133280	—	2	2	no	—
10	C₈×#6157	C₂²	1	1	yes	characteristic
11	Q₁₆×D₅₃	C₂²	1	2	yes	—
12	Dic₄₂₄	C₄	1	1	yes	characteristic, complemented
13	#13133163	C₄	1	2	yes	complemented
14	C₁₆×D₅₃	C₂²	1	1	yes	characteristic
15	Q₃₂×C₅₃	C₄	1	1	yes	characteristic, complemented
16	Q₈×D₅₃	—	2	2	no	—
17	C₄⋊#6157	—	2	2	no	—
18	C₄⋊#6157	—	4	2	no	—
19	C₄×#6157	—	4	2	no	—
20	C₄×#6157	D₄	1	1	yes	characteristic
21	D₅₃⋊C₈	D₄	1	1	yes	characteristic
22	Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
23	#11719925	—	2	2	no	—
24	Dic₂₁₂	C₂×C₄	1	2	yes	—
25	C₈×D₅₃	C₂³	1	1	yes	characteristic
26	#11719910	—	2	1	no	—
27	C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
28	#11719908	C₂×C₄	1	1	yes	characteristic
29	C₂×#6157	—	4	2	no	—
30	C₂×#6157	D₈	1	2	yes	—
31	Q₈×C₅₃	—	2	2	no	—
32	C₄×D₅₃	—	4	2	no	—
33	C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
34	Dic₁₀₆	—	2	2	no	—
35	Dic₁₀₆	—	4	2	no	—
36	#88324	—	2	1	no	—
37	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	#88322	#41	1	1	yes	characteristic
39	C₂×D₅₃	C₂×D₈	1	1	yes	characteristic
40	#6157	—	2	2	no	—
41	#6157	—	8	2	no	—
42	C₄×C₅₃	—	4	2	no	—
43	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
44	Dic₅₃	—	4	2	no	—
45	Dic₅₃	#135	1	1	yes	characteristic
46	#2176	—	53	1	no	maximal, Sylow
47	C₂×C₅₃	C₄×D₈	1	1	yes	characteristic
48	D₅₃	#510	1	2	yes	—
49	C₂×Q₃₂	—	53	1	no	—
50	C₄×Q₁₆	—	53	2	no	—
51	#367	—	53	1	no	—
52	#358	—	53	2	no	—
53	#346	—	53	1	no	—
54	C₅₃	#2176	1	1	yes	characteristic, complemented, Sylow
55	C₂×Q₁₆	—	53	2	no	—
56	C₄×Q₈	—	106	2	no	—
57	Q₃₂	—	53	2	no	—
58	Q₃₂	—	53	1	no	—
59	Q₃₂	#6157	1	1	yes	characteristic, complemented
60	C₂×C₁₆	—	53	1	no	—
61	C₂×C₁₆	—	53	1	no	—
62	#107	—	53	2	no	—
63	#103	—	106	2	no	—
64	C₄×C₈	—	53	1	no	—
65	C₂×Q₈	—	106	2	no	—
66	Q₁₆	—	53	2	no	—
67	Q₁₆	—	106	2	no	—
68	Q₁₆	C₂×#6157	1	2	yes	—
69	C₂×C₈	—	53	1	no	—
70	C₂×C₈	—	53	1	no	—
71	C₄⋊C₄	—	106	2	no	—
72	C₄⋊C₄	—	212	2	no	—
73	C₄²	—	53	1	no	—
74	C₄²	—	212	2	no	—
75	C₁₆	—	53	1	no	—
76	C₁₆	—	106	1	no	—
77	C₁₆	C₂×#6157	1	1	yes	characteristic
78	Q₈	—	2	2	no	—
79	Q₈	—	106	2	no	—
80	Q₈	—	212	2	no	—
81	C₂×C₄	—	53	1	no	—
82	C₂×C₄	—	53	2	no	—
83	C₂×C₄	—	212	2	no	—
84	C₂×C₄	—	212	2	no	—
85	C₈	—	53	1	no	—
86	C₈	—	106	1	no	—
87	C₈	C₂²×#6157	1	1	yes	characteristic
88	C₂²	—	53	1	no	—
89	C₄	—	4	2	no	—
90	C₄	—	53	1	no	—
91	C₄	—	106	2	no	—
92	C₄	—	424	2	no	—
93	C₄	—	212	2	no	—
94	C₄	D₄×#6157	1	1	yes	characteristic
95	C₂	—	53	2	no	—
96	C₂	D₈×#6157	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×#6157
commutator	C₈×C₅₃	C₂²×C₄
Frattini	C₈	C₂²×#6157
Fitting	Q₃₂×C₅₃	C₄
socle	C₂×C₅₃	C₄×D₈
hypercenter	Q₃₂	#6157
hypocenter	C₅₃	#2176
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

Complements

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

N	H ≅ G/N	type
12. Dic₄₂₄	C₄	semidirect
13. #13133163	C₄	semidirect
15. Q₃₂×C₅₃	C₄	semidirect
54. C₅₃	#2176	semidirect
59. Q₃₂	#6157	semidirect