Subgroups — C₂×#16811556 #20291137

Order 6784.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811556	C₂	1	4	yes	maximal, direct product factor
2	C₂×#13133267	C₂	1	2	yes	maximal, complemented
3	C₂×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#13133267	C₂²	1	4	yes	complemented
5	C₂×C₈×D₅₃	C₄	1	1	yes	characteristic
6	C₂×#11719908	C₂²	1	1	yes	characteristic
7	C₁₆×D₅₃	C₄	1	2	yes	—
8	C₁₆×D₅₃	C₂²	1	2	yes	—
9	C₂×C₁₆×C₅₃	C₄	1	1	yes	characteristic
10	C₂×C₄×D₅₃	C₈	1	1	yes	characteristic
11	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
12	C₂×#88322	C₈	1	1	yes	characteristic
13	C₈×D₅₃	C₈	1	2	yes	—
14	C₈×D₅₃	C₂×C₄	1	2	yes	—
15	C₁₆×C₅₃	C₂×C₄	1	2	yes	—
16	#11719908	C₂×C₄	1	1	yes	characteristic
17	#11719908	C₂³	1	1	yes	characteristic
18	C₂²×D₅₃	C₁₆	1	1	yes	characteristic
19	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
20	C₂×Dic₅₃	C₁₆	1	1	yes	characteristic
21	C₄×D₅₃	C₁₆	1	2	yes	—
22	C₄×D₅₃	C₂×C₈	1	2	yes	—
23	C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
24	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
25	#88322	C₂×C₈	1	2	yes	—
26	C₂²×C₅₃	C₂×C₁₆	1	1	yes	characteristic
27	C₂×D₅₃	—	2	2	no	—
28	C₂×D₅₃	C₂×C₁₆	1	2	yes	—
29	C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
30	C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
31	Dic₅₃	C₂×C₁₆	1	2	yes	—
32	C₂×M₆₄	—	53	1	no	maximal, Sylow
33	C₂×C₅₃	M₆₄	1	2	yes	complemented
34	C₂×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
35	D₅₃	—	2	2	no	—
36	C₂²×C₁₆	—	53	1	no	—
37	M₆₄	—	53	4	no	—
38	C₂×C₃₂	—	53	2	no	—
39	C₅₃	C₂×M₆₄	1	1	yes	characteristic, complemented, Sylow
40	C₂²×C₈	—	53	1	no	—
41	C₂×C₁₆	—	53	1	no	—
42	C₂×C₁₆	—	53	2	no	—
43	C₂×C₁₆	—	53	2	no	—
44	C₂×C₁₆	#6157	1	1	yes	characteristic
45	C₃₂	—	53	4	no	—
46	C₂²×C₄	—	53	1	no	—
47	C₂×C₈	—	53	1	no	—
48	C₂×C₈	—	53	2	no	—
49	C₂×C₈	—	53	2	no	—
50	C₂×C₈	C₂×#6157	1	1	yes	characteristic
51	C₁₆	—	53	1	no	—
52	C₁₆	—	53	1	no	—
53	C₁₆	C₂×#6157	1	2	yes	—
54	C₂³	—	53	1	no	—
55	C₂×C₄	—	53	1	no	—
56	C₂×C₄	—	53	2	no	—
57	C₂×C₄	—	53	2	no	—
58	C₂×C₄	D₅₃⋊C₈	1	1	yes	characteristic
59	C₈	—	53	2	no	—
60	C₈	D₅₃⋊C₈	1	1	yes	characteristic
61	C₈	C₂²×#6157	1	1	yes	characteristic
62	C₂²	—	53	2	no	—
63	C₂²	—	106	2	no	—
64	C₂²	D₅₃⋊C₁₆	1	1	yes	characteristic
65	C₄	—	53	2	no	—
66	C₄	D₅₃⋊C₁₆	1	1	yes	characteristic
67	C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
68	C₂	—	106	2	no	—
69	C₂	#16811556	1	2	yes	direct product factor
70	C₂	C₂×D₅₃⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×#6157
commutator	C₂×C₅₃	C₂²×C₁₆
Frattini	C₈	C₂²×#6157
Fitting	C₂×C₁₆×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₁₆
hypercenter	C₂×C₁₆	#6157
hypocenter	C₅₃	C₂×M₆₄
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
1. #16811556	C₂	direct (N × H)
2. C₂×#13133267	C₂	semidirect
4. #13133267	C₂²	semidirect
33. C₂×C₅₃	M₆₄	semidirect
39. C₅₃	C₂×M₆₄	semidirect
69. C₂	#16811556	direct (N × H)

Subgroups — C2×#16811556 #20291137

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16811556 #20291137