Subgroups — #46781049

Order 33664.

214 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#16862359	C₂	1	3	yes	maximal
2	#30681659	C₂	1	3	yes	maximal, complemented
3	C₈²×C₂₆₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16862359	C₄	1	6	yes	—
5	C₄²×D₂₆₃	C₂²	1	1	yes	characteristic
6	#22554380	C₄	1	6	yes	—
7	C₄×C₂₆₃⋊C₈	C₂²	1	3	yes	—
8	C₄×C₈×C₂₆₃	C₂²	1	3	yes	—
9	C₂×C₄×D₂₆₃	C₂×C₄	1	3	yes	—
10	#16862359	—	2	12	no	—
11	C₂×C₂₆₃⋊C₈	C₂×C₄	1	6	yes	—
12	C₄×Dic₂₆₃	C₂×C₄	1	3	yes	—
13	C₂×C₈×C₂₆₃	C₂×C₄	1	6	yes	—
14	C₄²×C₂₆₃	C₂³	1	1	yes	characteristic
15	C₂²×D₂₆₃	C₄²	1	1	yes	characteristic
16	C₂×Dic₂₆₃	C₄²	1	3	yes	—
17	C₄×D₂₆₃	—	2	6	no	—
18	C₂₆₃⋊C₈	—	2	6	no	—
19	C₂×C₄×C₂₆₃	C₂²×C₄	1	3	yes	—
20	C₈×C₂₆₃	M₁₆	1	12	yes	complemented
21	C₂²×C₂₆₃	C₂×C₄²	1	1	yes	characteristic
22	C₂×D₂₆₃	—	2	3	no	—
23	C₄×C₂₆₃	C₂×M₁₆	1	6	yes	—
24	Dic₂₆₃	—	2	3	no	—
25	C₂×C₂₆₃	C₄×M₁₆	1	3	yes	—
26	D₂₆₃	—	4	1	no	—
27	C₂₆₃	#1455	1	1	yes	characteristic, complemented, Sylow
28	#1455	—	263	1	no	maximal, Sylow
29	C₄×M₁₆	—	263	3	no	—
30	#322	—	263	3	no	—
31	C₈²	D₂₆₃	1	1	yes	characteristic, complemented
32	C₂×M₁₆	—	263	6	no	—
33	C₂×C₄²	—	263	1	no	—
34	#97	—	263	6	no	—
35	C₄×C₈	—	263	3	no	—
36	C₄×C₈	C₂×D₂₆₃	1	3	yes	—
37	C₂²×C₄	—	263	3	no	—
38	M₁₆	—	526	12	no	—
39	C₂×C₈	—	263	6	no	—
40	C₂×C₈	C₄×D₂₆₃	1	6	yes	—
41	C₄²	—	263	3	no	—
42	C₄²	C₂²×D₂₆₃	1	1	yes	characteristic
43	C₂³	—	263	1	no	—
44	C₂×C₄	—	263	3	no	—
45	C₂×C₄	—	526	6	no	—
46	C₂×C₄	C₂×C₄×D₂₆₃	1	3	yes	—
47	C₈	—	526	6	no	—
48	C₈	#16862359	1	12	yes	complemented
49	C₂²	—	526	3	no	—
50	C₂²	C₄²×D₂₆₃	1	1	yes	characteristic
51	C₄	—	526	3	no	—
52	C₄	C₂×#16862359	1	6	yes	—
53	C₂	—	1052	1	no	—
54	C₂	C₄×#16862359	1	3	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₂₆₃
commutator, socle	C₂²×C₂₆₃	C₂×C₄²
Fitting	C₈²×C₂₆₃	C₂
prime radical	C₂²×D₂₆₃	C₄²
cosocle	C₄²×C₂₆₃	C₂³
hypercenter	C₈²	D₂₆₃
hypocenter	C₂₆₃	#1455
LCS	G ▷ C₂²×C₂₆₃ ▷ C₂₆₃
UCS	C₁ ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
2. #30681659	C₂	semidirect
3. C₈²×C₂₆₃	C₂	semidirect
20. C₈×C₂₆₃	M₁₆	semidirect
27. C₂₆₃	#1455	semidirect
31. C₈²	D₂₆₃	semidirect
48. C₈	#16862359	semidirect

Maximal subgroups

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

subgroup	index
C₄×#16862359	2
#30681659	2
C₈²×C₂₆₃	2
#1455	263