Subgroups — #46780932

Order 33664.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30682370	C₂	1	1	yes	maximal, characteristic
2	#30681768	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₂×C₂₆₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#22554492	—	2	1	no	—
5	Dic₂₁₀₄	C₄	1	1	yes	characteristic
6	#22554415	—	2	1	no	—
7	D₄₂₀₈	C₄	1	1	yes	characteristic
8	#22554406	—	2	1	no	—
9	C₂×C₁₆×C₂₆₃	C₂²	1	1	yes	characteristic
10	C₃₂×C₂₆₃	—	2	1	no	—
11	#16862376	—	4	1	no	—
12	Dic₁₀₅₂	—	2	1	no	—
13	#16862365	—	4	1	no	—
14	D₂₁₀₄	—	2	1	no	—
15	#16862360	—	4	1	no	—
16	C₂×C₈×C₂₆₃	D₄	1	1	yes	characteristic
17	C₁₆×C₂₆₃	C₂×C₄	1	1	yes	characteristic
18	C₁₆×C₂₆₃	D₄	1	1	yes	characteristic
19	Dic₅₂₆	—	4	1	no	—
20	D₁₀₅₂	—	4	1	no	—
21	C₂²⋊D₂₆₃	—	8	1	no	—
22	C₄×D₂₆₃	—	8	1	no	—
23	C₂₆₃⋊C₈	—	8	1	no	—
24	C₂×C₄×C₂₆₃	D₈	1	1	yes	characteristic
25	C₈×C₂₆₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₂₆₃	SD₁₆	1	1	yes	characteristic
27	C₂²×C₂₆₃	SD₃₂	1	1	yes	characteristic
28	C₂×D₂₆₃	—	8	1	no	—
29	C₄×C₂₆₃	#102	1	1	yes	characteristic
30	C₄×C₂₆₃	D₁₆	1	1	yes	characteristic
31	Dic₂₆₃	—	8	1	no	—
32	C₂×C₂₆₃	—	2	1	no	—
33	C₂×C₂₆₃	#357	1	1	yes	characteristic
34	D₂₆₃	—	16	1	no	—
35	C₂₆₃	#1417	1	1	yes	characteristic, complemented, Sylow
36	#1417	—	263	1	no	maximal, Sylow
37	#508	—	263	1	no	—
38	C₂×C₃₂	D₂₆₃	1	1	yes	characteristic, complemented
39	#368	—	263	1	no	—
40	#135	—	526	1	no	—
41	Q₃₂	—	263	1	no	—
42	SD₃₂	—	526	1	no	—
43	D₁₆	—	263	1	no	—
44	C₂×C₁₆	C₂×D₂₆₃	1	1	yes	characteristic
45	#108	—	526	1	no	—
46	C₃₂	—	2	1	no	—
47	#41	—	1052	1	no	—
48	Q₁₆	—	526	1	no	—
49	SD₁₆	—	1052	1	no	—
50	D₈	—	526	1	no	—
51	M₁₆	—	1052	1	no	—
52	C₂×C₈	D₁₀₅₂	1	1	yes	characteristic
53	C₁₆	C₄×D₂₆₃	1	1	yes	characteristic
54	C₁₆	C₂²⋊D₂₆₃	1	1	yes	characteristic
55	Q₈	—	1052	1	no	—
56	D₄	—	1052	1	no	—
57	D₄	—	2104	1	no	—
58	C₂×C₄	—	2104	1	no	—
59	C₂×C₄	D₂₁₀₄	1	1	yes	characteristic
60	C₈	—	2104	1	no	—
61	C₈	(C₂×D₂₆₃)⋊C₄	1	1	yes	characteristic
62	C₈	#16862365	1	1	yes	characteristic
63	C₂²	—	2104	1	no	—
64	C₂²	#22554415	1	1	yes	characteristic
65	C₄	—	2104	1	no	—
66	C₄	#22554392	1	1	yes	characteristic
67	C₄	D₄₂₀₈	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	4208	1	no	—
70	C₂	#30681740	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄₂₀₈
commutator	C₁₆×C₂₆₃	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₂₆₃
Fitting	C₂×C₃₂×C₂₆₃	C₂
socle	C₂×C₂₆₃	#357
prime radical	#30682370	C₂
cosocle	C₂×C₁₆×C₂₆₃	C₂²
hypercenter	C₂×C₃₂	D₂₆₃
hypocenter	C₂₆₃	#1417
LCS	G ▷ C₁₆×C₂₆₃ ▷ C₈×C₂₆₃ ▷ C₄×C₂₆₃ ▷ C₂×C₂₆₃ ▷ C₂₆₃
UCS	C₁ ◁ C₄ ◁ 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. #30681768	C₂	semidirect
3. C₂×C₃₂×C₂₆₃	C₂	semidirect
35. C₂₆₃	#1417	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
#30682370	2
#30681768	2
C₂×C₃₂×C₂₆₃	2
#1417	263