Subgroups — #20276972

Order 6784.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133160	C₂	1	4	yes	maximal, complemented
2	#16811000	C₂	1	1	yes	maximal, characteristic
3	C₄×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₄₂₄	—	2	1	no	—
6	C₂×D₄₂₄	C₂²	1	2	yes	—
7	#13133214	—	2	1	no	—
8	#13133160	—	2	8	no	—
9	C₂×#11719908	C₂²	1	2	yes	—
10	C₂×Q₁₆×C₅₃	—	2	1	no	—
11	C₂×Q₁₆×C₅₃	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₅₃	—	2	1	no	—
13	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×Q₈×C₅₃	—	2	2	no	—
15	C₂×D₂₁₂	—	2	2	no	—
16	C₂×D₂₁₂	—	4	2	no	—
17	Q₁₆×C₅₃	—	2	4	no	—
18	Q₁₆×C₅₃	—	4	2	no	—
19	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
20	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
21	C₄⋊C₄×C₅₃	—	4	2	no	—
22	C₄²×C₅₃	D₄	1	1	yes	characteristic
23	D₄₂₄	—	2	4	no	—
24	D₄₂₄	—	4	2	no	—
25	#11719908	D₄	1	4	yes	complemented
26	C₂²×D₅₃	—	4	2	no	—
27	Q₈×C₅₃	—	4	2	no	—
28	Q₈×C₅₃	—	4	2	no	—
29	C₂×C₄×C₅₃	—	4	2	no	—
30	C₂×C₄×C₅₃	D₈	1	2	yes	—
31	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
32	D₂₁₂	—	4	2	no	—
33	D₂₁₂	—	4	2	no	—
34	D₂₁₂	—	8	4	no	—
35	C₈×C₅₃	D₈	1	2	yes	—
36	C₈×C₅₃	C₂×D₄	1	2	yes	—
37	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
38	C₂×D₅₃	—	8	2	no	—
39	C₂×D₅₃	—	8	4	no	—
40	C₄×C₅₃	—	8	2	no	—
41	C₄×C₅₃	SD₃₂	1	4	yes	complemented
42	C₄×C₅₃	#127	1	1	yes	characteristic
43	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
44	#2248	—	53	1	no	maximal, Sylow
45	C₂×C₅₃	#493	1	1	yes	characteristic
46	C₂×C₅₃	C₂×SD₃₂	1	2	yes	—
47	D₅₃	—	16	2	no	—
48	C₂×SD₃₂	—	53	4	no	—
49	#494	D₅₃	1	1	yes	characteristic, complemented
50	#493	—	53	1	no	—
51	C₄×C₁₆	—	53	1	no	—
52	C₅₃	#2248	1	1	yes	characteristic, complemented, Sylow
53	C₂×Q₁₆	—	2	1	no	—
54	C₂×Q₁₆	C₂×D₅₃	1	2	yes	—
55	C₂×D₈	—	53	2	no	—
56	C₂×D₈	—	106	1	no	—
57	C₄⋊Q₈	—	2	1	no	—
58	#127	—	106	1	no	—
59	SD₃₂	—	106	8	no	—
60	C₂×C₁₆	—	53	2	no	—
61	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
62	C₂×Q₈	—	2	2	no	—
63	C₂×D₄	—	106	2	no	—
64	C₂×D₄	—	212	2	no	—
65	Q₁₆	—	2	4	no	—
66	Q₁₆	—	4	2	no	—
67	D₈	—	106	4	no	—
68	D₈	—	212	2	no	—
69	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
70	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
71	C₄⋊C₄	—	4	2	no	—
72	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
73	C₁₆	—	53	4	no	—
74	C₂³	—	212	2	no	—
75	Q₈	—	4	2	no	—
76	Q₈	—	4	2	no	—
77	D₄	—	424	4	no	—
78	D₄	—	212	2	no	—
79	D₄	—	212	2	no	—
80	C₂×C₄	—	4	2	no	—
81	C₂×C₄	#11719922	1	2	yes	—
82	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
83	C₈	#11719922	1	2	yes	—
84	C₈	D₄×D₅₃	1	2	yes	—
85	C₂²	—	424	4	no	—
86	C₂²	—	424	2	no	—
87	C₂²	C₂×#11719922	1	1	yes	characteristic
88	C₄	—	8	2	no	—
89	C₄	#13133160	1	4	yes	complemented
90	C₄	#13133213	1	1	yes	characteristic
91	C₄	C₂×#11719922	1	1	yes	characteristic
92	C₂	—	848	2	no	—
93	C₂	#16810999	1	1	yes	characteristic
94	C₂	C₂×#13133160	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#11719922
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#494	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	#494	D₅₃
hypocenter	C₅₃	#2248
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #494

Complements

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

N	H ≅ G/N	type
1. C₂×#13133160	C₂	semidirect
3. C₄×#11719908	C₂	semidirect
4. C₅₃×#494	C₂	semidirect
25. #11719908	D₄	semidirect
41. C₄×C₅₃	SD₃₂	semidirect
49. #494	D₅₃	semidirect
52. C₅₃	#2248	semidirect
89. C₄	#13133160	semidirect