Subgroups — C₂×#16810457 #20277007

Order 6784.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810457	C₂	1	4	yes	maximal, direct product factor
2	C₂×#13133123	C₂	1	2	yes	maximal, complemented
3	C₂²×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
4	#13133123	C₂²	1	4	yes	complemented
5	C₂²×C₈×C₅₃	C₄	1	1	yes	characteristic
6	C₂×C₁₆×C₅₃	C₄	1	1	yes	characteristic
7	C₂×C₁₆×C₅₃	C₄	1	2	yes	—
8	C₂×C₁₆×C₅₃	C₂²	1	2	yes	—
9	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₅₃	C₈	1	1	yes	characteristic
11	C₂×C₈×C₅₃	C₈	1	2	yes	—
12	C₂×C₈×C₅₃	C₈	1	1	yes	characteristic
13	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
14	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₁₆×C₅₃	C₂×C₄	1	2	yes	—
16	C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₁₆×C₅₃	C₂³	1	1	yes	characteristic
18	C₂³×C₅₃	C₁₆	1	1	yes	characteristic
19	C₂×C₄×C₅₃	C₁₆	1	2	yes	—
20	C₂×C₄×C₅₃	C₁₆	1	1	yes	characteristic
21	C₂×C₄×C₅₃	C₂×C₈	1	2	yes	—
22	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
23	C₈×C₅₃	C₂×C₈	1	2	yes	—
24	C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
25	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
26	C₂²×C₅₃	—	2	2	no	—
27	C₂²×C₅₃	C₂×C₁₆	1	2	yes	—
28	C₂²×C₅₃	C₂×C₁₆	1	1	yes	characteristic
29	C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
30	C₄×C₅₃	C₂×C₁₆	1	2	yes	—
31	C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
32	C₂×M₆₄	—	53	1	no	maximal, Sylow
33	C₂×C₅₃	—	2	2	no	—
34	C₂×C₅₃	M₆₄	1	2	yes	complemented
35	C₂×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
36	C₂²×C₁₆	D₅₃	1	1	yes	characteristic
37	M₆₄	—	53	4	no	—
38	C₂×C₃₂	—	53	2	no	—
39	C₅₃	C₂×M₆₄	1	1	yes	characteristic, complemented, Sylow
40	C₂²×C₈	Dic₅₃	1	1	yes	characteristic
41	C₂×C₁₆	Dic₅₃	1	1	yes	characteristic
42	C₂×C₁₆	Dic₅₃	1	2	yes	—
43	C₂×C₁₆	C₂×D₅₃	1	2	yes	—
44	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
45	C₃₂	—	53	4	no	—
46	C₂²×C₄	#88322	1	1	yes	characteristic
47	C₂×C₈	#88322	1	1	yes	characteristic
48	C₂×C₈	#88322	1	2	yes	—
49	C₂×C₈	C₂×Dic₅₃	1	2	yes	—
50	C₂×C₈	C₂×Dic₅₃	1	1	yes	characteristic
51	C₁₆	C₂×Dic₅₃	1	2	yes	—
52	C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
53	C₁₆	C₂²×D₅₃	1	1	yes	characteristic
54	C₂³	#11719908	1	1	yes	characteristic
55	C₂×C₄	#11719908	1	2	yes	—
56	C₂×C₄	#11719908	1	1	yes	characteristic
57	C₂×C₄	C₂×#88322	1	2	yes	—
58	C₂×C₄	C₂×#88322	1	1	yes	characteristic
59	C₈	C₂×#88322	1	2	yes	—
60	C₈	C₂×#88322	1	1	yes	characteristic
61	C₈	C₂²×Dic₅₃	1	1	yes	characteristic
62	C₂²	—	2	2	no	—
63	C₂²	C₂×#11719908	1	2	yes	—
64	C₂²	C₂×#11719908	1	1	yes	characteristic
65	C₄	C₂×#11719908	1	1	yes	characteristic
66	C₄	C₂×#11719908	1	2	yes	—
67	C₄	C₂²×#88322	1	1	yes	characteristic
68	C₂	—	2	2	no	—
69	C₂	#16810457	1	2	yes	direct product factor
70	C₂	C₂²×#11719908	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂×D₅₃
commutator	C₂×C₅₃	C₂²×C₁₆
Frattini	C₁₆	C₂²×D₅₃
Fitting	C₂²×C₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₁₆
hypercenter	C₂²×C₁₆	D₅₃
hypocenter	C₅₃	C₂×M₆₄
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
1. #16810457	C₂	direct (N × H)
2. C₂×#13133123	C₂	semidirect
4. #13133123	C₂²	semidirect
34. C₂×C₅₃	M₆₄	semidirect
39. C₅₃	C₂×M₆₄	semidirect
69. C₂	#16810457	direct (N × H)

Subgroups — C2×#16810457 #20277007

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16810457 #20277007