Subgroups — #46780821

Order 33664.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30681691	C₂	1	2	yes	maximal
2	C₄×C₁₆×C₂₆₃	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₂₆₃⋊C₈)	—	2	2	no	—
4	C₂×C₁₆×C₂₆₃	C₄	1	1	yes	characteristic
5	C₂×C₁₆×C₂₆₃	C₄	1	1	yes	characteristic
6	C₄×C₈×C₂₆₃	C₂²	1	1	yes	characteristic
7	C₂×C₂₆₃⋊C₈	—	4	2	no	—
8	C₂×C₈×C₂₆₃	C₂×C₄	1	1	yes	characteristic
9	C₂×C₈×C₂₆₃	Q₈	1	1	yes	characteristic
10	C₄²×C₂₆₃	D₄	1	1	yes	characteristic
11	C₁₆×C₂₆₃	—	2	1	no	—
12	C₁₆×C₂₆₃	C₈	1	2	yes	complemented
13	C₂₆₃⋊C₈	—	8	2	no	—
14	C₂×C₄×C₂₆₃	C₄⋊C₄	1	1	yes	characteristic
15	C₂×C₄×C₂₆₃	D₈	1	1	yes	characteristic
16	C₂×C₄×C₂₆₃	Q₁₆	1	1	yes	characteristic
17	C₈×C₂₆₃	—	2	1	no	—
18	C₈×C₂₆₃	C₂×C₈	1	1	yes	characteristic
19	C₈×C₂₆₃	M₁₆	1	1	yes	characteristic
20	C₂²×C₂₆₃	#107	1	1	yes	characteristic
21	C₄×C₂₆₃	—	2	1	no	—
22	C₄×C₂₆₃	C₄⋊C₈	1	1	yes	characteristic
23	C₄×C₂₆₃	#108	1	1	yes	characteristic
24	C₄×C₂₆₃	SD₃₂	1	2	yes	—
25	C₂×C₂₆₃	#335	1	1	yes	characteristic
26	C₂×C₂₆₃	#367	1	1	yes	characteristic
27	C₂×C₂₆₃	#368	1	1	yes	characteristic
28	C₂₆₃	#1372	1	1	yes	characteristic, complemented, Sylow
29	#1372	—	263	1	no	maximal, Sylow
30	C₄×C₁₆	D₂₆₃	1	1	yes	characteristic
31	#335	—	263	2	no	—
32	C₂×C₁₆	Dic₂₆₃	1	1	yes	characteristic
33	C₂×C₁₆	Dic₂₆₃	1	1	yes	characteristic
34	C₄⋊C₈	—	526	2	no	—
35	C₄×C₈	C₂×D₂₆₃	1	1	yes	characteristic
36	C₂×C₈	—	1052	2	no	—
37	C₂×C₈	C₂×Dic₂₆₃	1	1	yes	characteristic
38	C₂×C₈	Dic₅₂₆	1	1	yes	characteristic
39	C₄²	D₁₀₅₂	1	1	yes	characteristic
40	C₁₆	—	2	1	no	—
41	C₁₆	C₂₆₃⋊C₈	1	2	yes	complemented
42	C₂×C₄	C₄⋊Dic₂₆₃	1	1	yes	characteristic
43	C₂×C₄	D₂₁₀₄	1	1	yes	characteristic
44	C₂×C₄	Dic₁₀₅₂	1	1	yes	characteristic
45	C₈	—	2	1	no	—
46	C₈	—	2104	2	no	—
47	C₈	C₂×C₂₆₃⋊C₈	1	1	yes	characteristic
48	C₈	#16862360	1	1	yes	characteristic
49	C₂²	#22554404	1	1	yes	characteristic
50	C₄	—	2	1	no	—
51	C₄	C₄⋊(C₂₆₃⋊C₈)	1	1	yes	characteristic
52	C₄	#22554406	1	1	yes	characteristic
53	C₄	#22554415	1	2	yes	—
54	C₂	#30681691	1	1	yes	characteristic
55	C₂	#30681766	1	1	yes	characteristic
56	C₂	#30681768	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₂₁₀₄
commutator	C₈×C₂₆₃	C₂×C₈
Frattini	C₄×C₈	C₂×D₂₆₃
Fitting	C₄×C₁₆×C₂₆₃	C₂
socle, prime radical	C₂²×C₂₆₃	#107
cosocle	C₄×C₈×C₂₆₃	C₂²
hypercenter	C₄×C₁₆	D₂₆₃
hypocenter	C₂₆₃	#1372
LCS	G ▷ C₈×C₂₆₃ ▷ C₄×C₂₆₃ ▷ C₂×C₂₆₃ ▷ C₂₆₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
12. C₁₆×C₂₆₃	C₈	semidirect
28. C₂₆₃	#1372	semidirect
41. C₁₆	C₂₆₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#30681691	2
C₄×C₁₆×C₂₆₃	2
#1372	263