Subgroups — #20276971

Order 6784.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133162	C₂	1	4	yes	maximal, complemented
2	#16811003	C₂	1	1	yes	maximal, characteristic, complemented
3	#16811000	C₂	1	1	yes	maximal, characteristic
4	C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Dic₂₁₂	—	2	1	no	—
6	C₂×Dic₂₁₂	C₂²	1	2	yes	—
7	C₂×D₄₂₄	—	2	1	no	—
8	C₂×D₄₂₄	C₂²	1	2	yes	—
9	#13133217	—	2	1	no	—
10	#13133214	—	2	1	no	—
11	#13133162	—	2	8	no	—
12	C₂×C₁₆×C₅₃	C₂²	1	2	yes	—
13	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×D₂₁₂	—	2	2	no	—
15	C₂×D₂₁₂	—	4	2	no	—
16	C₂×Dic₁₀₆	—	2	2	no	—
17	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
18	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
19	C₄²×C₅₃	D₄	1	1	yes	characteristic
20	C₄⋊Dic₅₃	—	4	2	no	—
21	Dic₂₁₂	—	2	4	no	—
22	Dic₂₁₂	—	4	2	no	—
23	D₄₂₄	—	2	4	no	—
24	D₄₂₄	—	4	2	no	—
25	C₁₆×C₅₃	D₄	1	4	yes	complemented
26	C₂²×D₅₃	—	4	2	no	—
27	C₂×C₄×C₅₃	D₈	1	2	yes	—
28	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
29	C₂×Dic₅₃	—	4	2	no	—
30	D₂₁₂	—	4	2	no	—
31	D₂₁₂	—	4	2	no	—
32	D₂₁₂	—	8	4	no	—
33	Dic₁₀₆	—	4	2	no	—
34	Dic₁₀₆	—	4	2	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	SD₃₂	1	4	yes	complemented
41	C₄×C₅₃	#127	1	1	yes	characteristic
42	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
43	Dic₅₃	—	8	2	no	—
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	—	53	1	no	—
50	#493	—	53	1	no	—
51	C₄×C₁₆	D₅₃	1	1	yes	characteristic, complemented
52	C₅₃	#2248	1	1	yes	characteristic, complemented, Sylow
53	C₂×Q₁₆	—	53	2	no	—
54	C₂×Q₁₆	—	106	1	no	—
55	C₂×D₈	—	53	2	no	—
56	C₂×D₈	—	106	1	no	—
57	C₄⋊Q₈	—	106	1	no	—
58	#127	—	106	1	no	—
59	SD₃₂	—	106	8	no	—
60	C₂×C₁₆	C₂×D₅₃	1	2	yes	—
61	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
62	C₂×Q₈	—	106	2	no	—
63	C₂×D₄	—	106	2	no	—
64	C₂×D₄	—	212	2	no	—
65	Q₁₆	—	106	4	no	—
66	Q₁₆	—	212	2	no	—
67	D₈	—	106	4	no	—
68	D₈	—	212	2	no	—
69	C₂×C₈	D₂₁₂	1	1	yes	characteristic
70	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
71	C₄⋊C₄	—	212	2	no	—
72	C₄²	D₂₁₂	1	1	yes	characteristic
73	C₁₆	D₂₁₂	1	4	yes	complemented
74	C₂³	—	212	2	no	—
75	Q₈	—	212	2	no	—
76	Q₈	—	212	2	no	—
77	D₄	—	424	4	no	—
78	D₄	—	212	2	no	—
79	D₄	—	212	2	no	—
80	C₂×C₄	—	212	2	no	—
81	C₂×C₄	D₄₂₄	1	2	yes	—
82	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
83	C₈	D₄₂₄	1	2	yes	—
84	C₈	C₂×D₂₁₂	1	2	yes	—
85	C₂²	—	424	4	no	—
86	C₂²	—	424	2	no	—
87	C₂²	C₂×D₄₂₄	1	1	yes	characteristic
88	C₄	—	424	2	no	—
89	C₄	#13133162	1	4	yes	complemented
90	C₄	#13133214	1	1	yes	characteristic
91	C₄	C₂×D₄₂₄	1	1	yes	characteristic
92	C₂	—	848	2	no	—
93	C₂	#16811000	1	1	yes	characteristic
94	C₂	C₂×#13133162	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₄₂₄
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₄×C₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	C₄×C₁₆	D₅₃
hypocenter	C₅₃	#2248
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#13133162	C₂	semidirect
2. #16811003	C₂	semidirect
4. C₄×C₁₆×C₅₃	C₂	semidirect
25. C₁₆×C₅₃	D₄	semidirect
40. C₄×C₅₃	SD₃₂	semidirect
51. C₄×C₁₆	D₅₃	semidirect
52. C₅₃	#2248	semidirect
73. C₁₆	D₂₁₂	semidirect
89. C₄	#13133162	semidirect