Subgroups — #46780817

Order 33664.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₂₆₃⋊C₁₆)	C₂	1	2	yes	maximal
2	C₈²×C₂₆₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₆₃⋊C₁₆	—	2	2	no	—
4	C₄×C₈×C₂₆₃	C₄	1	1	yes	characteristic
5	C₄×C₈×C₂₆₃	C₄	1	1	yes	characteristic
6	C₄×C₈×C₂₆₃	C₂²	1	1	yes	characteristic
7	C₂₆₃⋊C₁₆	—	4	2	no	—
8	C₂×C₈×C₂₆₃	—	2	1	no	—
9	C₂×C₈×C₂₆₃	C₈	1	1	yes	characteristic
10	C₂×C₈×C₂₆₃	C₈	1	1	yes	characteristic
11	C₂×C₈×C₂₆₃	D₄	1	1	yes	characteristic
12	C₂×C₈×C₂₆₃	Q₈	1	1	yes	characteristic
13	C₄²×C₂₆₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₂₆₃	C₄⋊C₄	1	1	yes	characteristic
15	C₂×C₄×C₂₆₃	C₂×C₈	1	1	yes	characteristic
16	C₂×C₄×C₂₆₃	M₁₆	1	1	yes	characteristic
17	C₈×C₂₆₃	—	2	2	no	—
18	C₈×C₂₆₃	—	2	1	no	—
19	C₈×C₂₆₃	—	2	1	no	—
20	C₈×C₂₆₃	C₁₆	1	2	yes	complemented
21	C₈×C₂₆₃	D₈	1	1	yes	characteristic
22	C₈×C₂₆₃	Q₁₆	1	1	yes	characteristic
23	C₂²×C₂₆₃	C₄⋊C₈	1	1	yes	characteristic
24	C₄×C₂₆₃	—	2	1	no	—
25	C₄×C₂₆₃	#107	1	1	yes	characteristic
26	C₄×C₂₆₃	#108	1	1	yes	characteristic
27	C₄×C₂₆₃	C₂×C₁₆	1	1	yes	characteristic
28	C₄×C₂₆₃	M₃₂	1	1	yes	characteristic
29	C₂×C₂₆₃	#335	1	1	yes	characteristic
30	C₂×C₂₆₃	C₄⋊C₁₆	1	1	yes	characteristic
31	C₂×C₂₆₃	#364	1	1	yes	characteristic
32	C₂₆₃	#1370	1	1	yes	characteristic, complemented, Sylow
33	#1370	—	263	1	no	maximal, Sylow
34	C₄⋊C₁₆	—	263	2	no	—
35	C₈²	D₂₆₃	1	1	yes	characteristic
36	C₂×C₁₆	—	526	2	no	—
37	C₄×C₈	Dic₂₆₃	1	1	yes	characteristic
38	C₄×C₈	Dic₂₆₃	1	1	yes	characteristic
39	C₄×C₈	C₂×D₂₆₃	1	1	yes	characteristic
40	C₂×C₈	—	2	1	no	—
41	C₂×C₈	C₂₆₃⋊C₈	1	1	yes	characteristic
42	C₂×C₈	C₂₆₃⋊C₈	1	1	yes	characteristic
43	C₂×C₈	D₁₀₅₂	1	1	yes	characteristic
44	C₂×C₈	Dic₅₂₆	1	1	yes	characteristic
45	C₄²	C₂×Dic₂₆₃	1	1	yes	characteristic
46	C₁₆	—	1052	2	no	—
47	C₂×C₄	C₄⋊Dic₂₆₃	1	1	yes	characteristic
48	C₂×C₄	C₂×C₂₆₃⋊C₈	1	1	yes	characteristic
49	C₂×C₄	#16862360	1	1	yes	characteristic
50	C₈	—	2	2	no	—
51	C₈	—	2	1	no	—
52	C₈	—	2	1	no	—
53	C₈	C₂₆₃⋊C₁₆	1	2	yes	complemented
54	C₈	D₂₁₀₄	1	1	yes	characteristic
55	C₈	Dic₁₀₅₂	1	1	yes	characteristic
56	C₂²	C₄⋊(C₂₆₃⋊C₈)	1	1	yes	characteristic
57	C₄	—	2	1	no	—
58	C₄	#22554404	1	1	yes	characteristic
59	C₄	#22554406	1	1	yes	characteristic
60	C₄	C₂×C₂₆₃⋊C₁₆	1	1	yes	characteristic
61	C₄	#22554410	1	1	yes	characteristic
62	C₂	#30681691	1	1	yes	characteristic
63	C₂	C₄⋊(C₂₆₃⋊C₁₆)	1	1	yes	characteristic
64	C₂	#30681759	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₁₀₅₂
commutator	C₄×C₂₆₃	C₂×C₁₆
Frattini	C₄×C₈	C₂×D₂₆₃
Fitting	C₈²×C₂₆₃	C₂
socle, prime radical	C₂²×C₂₆₃	C₄⋊C₈
cosocle	C₄×C₈×C₂₆₃	C₂²
hypercenter	C₈²	D₂₆₃
hypocenter	C₂₆₃	#1370
LCS	G ▷ C₄×C₂₆₃ ▷ C₂×C₂₆₃ ▷ C₂₆₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈²

Complements

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

N	H ≅ G/N	type
20. C₈×C₂₆₃	C₁₆	semidirect
32. C₂₆₃	#1370	semidirect
53. C₈	C₂₆₃⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊(C₂₆₃⋊C₁₆)	2
C₈²×C₂₆₃	2
#1370	263