Subgroups — #46780953

Order 33664.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30681768	C₂	1	1	yes	maximal, characteristic
2	#30681766	C₂	1	1	yes	maximal, characteristic
3	M₆₄×C₂₆₃	C₂	1	1	yes	maximal, characteristic
4	#22554406	—	2	1	no	—
5	#22554404	—	2	1	no	—
6	C₂×C₁₆×C₂₆₃	C₂²	1	1	yes	characteristic
7	C₃₂×C₂₆₃	C₄	1	2	yes	complemented
8	#16862360	—	4	1	no	—
9	C₄⋊Dic₂₆₃	—	4	1	no	—
10	C₂×C₈×C₂₆₃	D₄	1	1	yes	characteristic
11	C₁₆×C₂₆₃	C₂×C₄	1	1	yes	characteristic
12	C₁₆×C₂₆₃	Q₈	1	1	yes	characteristic
13	C₂×Dic₂₆₃	—	8	1	no	—
14	C₂₆₃⋊C₈	—	8	1	no	—
15	C₂×C₄×C₂₆₃	D₈	1	1	yes	characteristic
16	C₈×C₂₆₃	C₄⋊C₄	1	1	yes	characteristic
17	C₈×C₂₆₃	Q₁₆	1	1	yes	characteristic
18	C₂²×C₂₆₃	SD₃₂	1	1	yes	characteristic
19	C₄×C₂₆₃	#107	1	1	yes	characteristic
20	C₄×C₂₆₃	SD₃₂	1	1	yes	characteristic
21	Dic₂₆₃	—	16	1	no	—
22	C₂×C₂₆₃	—	2	1	no	—
23	C₂×C₂₆₃	#367	1	1	yes	characteristic
24	C₂₆₃	#1426	1	1	yes	characteristic, complemented, Sylow
25	#1426	—	263	1	no	maximal, Sylow
26	M₆₄	D₂₆₃	1	1	yes	characteristic
27	#368	—	263	1	no	—
28	#367	—	263	1	no	—
29	C₂×C₁₆	C₂×D₂₆₃	1	1	yes	characteristic
30	#108	—	526	1	no	—
31	#107	—	526	1	no	—
32	C₃₂	Dic₂₆₃	1	2	yes	complemented
33	M₁₆	—	1052	1	no	—
34	C₂×C₈	D₁₀₅₂	1	1	yes	characteristic
35	C₄⋊C₄	—	1052	1	no	—
36	C₁₆	C₂×Dic₂₆₃	1	1	yes	characteristic
37	C₁₆	Dic₅₂₆	1	1	yes	characteristic
38	C₂×C₄	—	2104	1	no	—
39	C₂×C₄	D₂₁₀₄	1	1	yes	characteristic
40	C₈	—	2104	1	no	—
41	C₈	C₄⋊Dic₂₆₃	1	1	yes	characteristic
42	C₈	Dic₁₀₅₂	1	1	yes	characteristic
43	C₂²	#22554415	1	1	yes	characteristic
44	C₄	—	4208	1	no	—
45	C₄	#22554404	1	1	yes	characteristic
46	C₄	#22554415	1	1	yes	characteristic
47	C₂	—	2	1	no	—
48	C₂	#30681766	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#30681766
commutator	C₁₆×C₂₆₃	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₂₆₃
Fitting	M₆₄×C₂₆₃	C₂
socle	C₂×C₂₆₃	#367
prime radical	C₂²×C₂₆₃	SD₃₂
cosocle	C₂×C₁₆×C₂₆₃	C₂²
hypercenter	M₆₄	D₂₆₃
hypocenter	C₂₆₃	#1426
LCS	G ▷ C₁₆×C₂₆₃ ▷ C₈×C₂₆₃ ▷ C₄×C₂₆₃ ▷ C₂×C₂₆₃ ▷ C₂₆₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
7. C₃₂×C₂₆₃	C₄	semidirect
24. C₂₆₃	#1426	semidirect
32. C₃₂	Dic₂₆₃	semidirect

Maximal subgroups

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

subgroup	index
#30681768	2
#30681766	2
M₆₄×C₂₆₃	2
#1426	263