Subgroups — #46780926

Order 33664.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄₂₀₈	C₂	1	1	yes	maximal, characteristic
2	#30681764	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₂×C₂₆₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₂₁₀₄	—	2	1	no	—
5	D₄₂₀₈	—	2	1	no	—
6	D₄₂₀₈	C₄	1	2	yes	complemented
7	#22554404	—	2	1	no	—
8	C₂×C₁₆×C₂₆₃	C₂²	1	1	yes	characteristic
9	C₃₂×C₂₆₃	—	2	1	no	—
10	C₂×D₁₀₅₂	—	4	1	no	—
11	D₂₁₀₄	—	2	2	no	—
12	D₂₁₀₄	—	4	1	no	—
13	C₄⋊Dic₂₆₃	—	4	1	no	—
14	C₂×C₈×C₂₆₃	D₄	1	1	yes	characteristic
15	C₁₆×C₂₆₃	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₂₆₃	D₄	1	1	yes	characteristic
17	C₂²×D₂₆₃	—	8	1	no	—
18	D₁₀₅₂	—	4	2	no	—
19	D₁₀₅₂	—	8	1	no	—
20	C₂×Dic₂₆₃	—	8	1	no	—
21	C₂×C₄×C₂₆₃	D₈	1	1	yes	characteristic
22	C₈×C₂₆₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₈×C₂₆₃	SD₁₆	1	1	yes	characteristic
24	C₂²×C₂₆₃	D₁₆	1	1	yes	characteristic
25	C₂×D₂₆₃	—	8	2	no	—
26	C₂×D₂₆₃	—	16	1	no	—
27	C₂×D₂₆₃	—	16	1	no	—
28	C₄×C₂₆₃	#102	1	1	yes	characteristic
29	C₄×C₂₆₃	SD₃₂	1	1	yes	characteristic
30	Dic₂₆₃	—	16	1	no	—
31	C₂×C₂₆₃	#357	1	1	yes	characteristic
32	C₂×C₂₆₃	D₃₂	1	1	yes	characteristic
33	C₂×C₂₆₃	SD₆₄	1	1	yes	characteristic
34	D₂₆₃	—	16	2	no	—
35	C₂₆₃	#1415	1	1	yes	characteristic, complemented, Sylow
36	#1415	—	263	1	no	maximal, Sylow
37	C₂×D₁₆	—	263	1	no	—
38	C₂×C₃₂	D₂₆₃	1	1	yes	characteristic, complemented
39	#366	—	263	1	no	—
40	C₂×D₈	—	526	1	no	—
41	D₁₆	—	263	2	no	—
42	D₁₆	—	526	1	no	—
43	C₂×C₁₆	C₂×D₂₆₃	1	1	yes	characteristic
44	#107	—	526	1	no	—
45	C₃₂	—	2	1	no	—
46	C₂×D₄	—	1052	1	no	—
47	D₈	—	526	2	no	—
48	D₈	—	1052	1	no	—
49	C₂×C₈	D₁₀₅₂	1	1	yes	characteristic
50	C₄⋊C₄	—	1052	1	no	—
51	C₁₆	C₄×D₂₆₃	1	1	yes	characteristic
52	C₁₆	C₂²⋊D₂₆₃	1	1	yes	characteristic
53	C₂³	—	2104	1	no	—
54	D₄	—	1052	2	no	—
55	D₄	—	2104	1	no	—
56	C₂×C₄	—	2104	1	no	—
57	C₂×C₄	D₂₁₀₄	1	1	yes	characteristic
58	C₈	(C₂×D₂₆₃)⋊C₄	1	1	yes	characteristic
59	C₈	#16862365	1	1	yes	characteristic
60	C₂²	—	2104	2	no	—
61	C₂²	—	4208	1	no	—
62	C₂²	—	4208	1	no	—
63	C₂²	D₄₂₀₈	1	1	yes	characteristic
64	C₄	—	4208	1	no	—
65	C₄	#22554392	1	1	yes	characteristic
66	C₄	#22554415	1	1	yes	characteristic
67	C₂	—	4208	2	no	—
68	C₂	#30681740	1	1	yes	characteristic
69	C₂	D₈₄₁₆	1	1	yes	characteristic
70	C₂	#30681777	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄₂₀₈
commutator	C₁₆×C₂₆₃	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₂₆₃
Fitting	C₂×C₃₂×C₂₆₃	C₂
socle	C₂²×C₂₆₃	D₁₆
prime radical	C₂×D₄₂₀₈	C₂
cosocle	C₂×C₁₆×C₂₆₃	C₂²
hypercenter	C₂×C₃₂	D₂₆₃
hypocenter	C₂₆₃	#1415
LCS	G ▷ C₁₆×C₂₆₃ ▷ C₈×C₂₆₃ ▷ C₄×C₂₆₃ ▷ C₂×C₂₆₃ ▷ C₂₆₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
2. #30681764	C₂	semidirect
3. C₂×C₃₂×C₂₆₃	C₂	semidirect
6. D₄₂₀₈	C₄	semidirect
35. C₂₆₃	#1415	semidirect
38. C₂×C₃₂	D₂₆₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×D₄₂₀₈	2
#30681764	2
C₂×C₃₂×C₂₆₃	2
#1415	263