Subgroups — #46780938

Order 33664.

78 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	#30681768	C₂	1	1	yes	maximal, characteristic, complemented
3	M₆₄×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	—
7	#22554406	—	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	#16862360	—	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	C₂₆₃⋊C₈	—	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	SD₃₂	1	1	yes	characteristic
25	C₂×D₂₆₃	—	8	2	no	—
26	C₂×D₂₆₃	—	16	2	no	—
27	C₄×C₂₆₃	#102	1	1	yes	characteristic
28	C₄×C₂₆₃	D₁₆	1	1	yes	characteristic
29	C₂×C₂₆₃	—	2	1	no	—
30	C₂×C₂₆₃	#357	1	1	yes	characteristic
31	D₂₆₃	—	16	2	no	—
32	C₂₆₃	#1419	1	1	yes	characteristic, complemented, Sylow
33	#1419	—	263	1	no	maximal, Sylow
34	C₂×D₁₆	—	263	1	no	—
35	M₆₄	D₂₆₃	1	1	yes	characteristic, complemented
36	#368	—	263	1	no	—
37	C₂×D₈	—	526	1	no	—
38	D₁₆	—	263	2	no	—
39	D₁₆	—	526	1	no	—
40	C₂×C₁₆	C₂×D₂₆₃	1	1	yes	characteristic
41	#108	—	526	1	no	—
42	C₃₂	—	2	1	no	—
43	C₂×D₄	—	1052	1	no	—
44	D₈	—	526	2	no	—
45	D₈	—	1052	1	no	—
46	M₁₆	—	1052	1	no	—
47	C₂×C₈	D₁₀₅₂	1	1	yes	characteristic
48	C₁₆	C₄×D₂₆₃	1	1	yes	characteristic
49	C₁₆	C₂²⋊D₂₆₃	1	1	yes	characteristic
50	C₂³	—	2104	1	no	—
51	D₄	—	1052	2	no	—
52	D₄	—	2104	1	no	—
53	C₂×C₄	D₂₁₀₄	1	1	yes	characteristic
54	C₈	—	2104	1	no	—
55	C₈	(C₂×D₂₆₃)⋊C₄	1	1	yes	characteristic
56	C₈	#16862365	1	1	yes	characteristic
57	C₂²	—	2104	2	no	—
58	C₂²	—	4208	2	no	—
59	C₂²	#22554415	1	1	yes	characteristic
60	C₄	#22554392	1	1	yes	characteristic
61	C₄	D₄₂₀₈	1	1	yes	characteristic
62	C₂	—	2	1	no	—
63	C₂	—	4208	2	no	—
64	C₂	#30681740	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#30681740
commutator	C₁₆×C₂₆₃	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₂₆₃
Fitting	M₆₄×C₂₆₃	C₂
socle	C₂×C₂₆₃	#357
prime radical	C₂×D₄₂₀₈	C₂
cosocle	C₂×C₁₆×C₂₆₃	C₂²
hypercenter	M₆₄	D₂₆₃
hypocenter	C₂₆₃	#1419
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
2. #30681768	C₂	semidirect
3. M₆₄×C₂₆₃	C₂	semidirect
32. C₂₆₃	#1419	semidirect
35. M₆₄	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
#30681768	2
M₆₄×C₂₆₃	2
#1419	263