Subgroups — #46780935

Order 33664.

74 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	#30681766	C₂	1	1	yes	maximal, characteristic, complemented
3	M₆₄×C₂₆₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#22554492	—	2	1	no	—
5	Dic₂₁₀₄	C₄	1	1	yes	characteristic, complemented
6	#22554415	—	2	1	no	—
7	D₄₂₀₈	C₄	1	1	yes	characteristic, complemented
8	#22554404	—	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	C₄⋊Dic₂₆₃	—	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	C₂×Dic₂₆₃	—	8	1	no	—
23	C₄×D₂₆₃	—	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	D₁₆	1	1	yes	characteristic
28	C₂×D₂₆₃	—	8	1	no	—
29	C₄×C₂₆₃	#102	1	1	yes	characteristic
30	C₄×C₂₆₃	SD₃₂	1	1	yes	characteristic
31	Dic₂₆₃	—	8	1	no	—
32	Dic₂₆₃	—	16	1	no	—
33	C₂×C₂₆₃	—	2	1	no	—
34	C₂×C₂₆₃	#357	1	1	yes	characteristic
35	D₂₆₃	—	16	1	no	—
36	C₂₆₃	#1418	1	1	yes	characteristic, complemented, Sylow
37	#1418	—	263	1	no	maximal, Sylow
38	#508	—	263	1	no	—
39	M₆₄	D₂₆₃	1	1	yes	characteristic, complemented
40	#367	—	263	1	no	—
41	#135	—	526	1	no	—
42	Q₃₂	—	263	1	no	—
43	SD₃₂	—	526	1	no	—
44	D₁₆	—	263	1	no	—
45	C₂×C₁₆	C₂×D₂₆₃	1	1	yes	characteristic
46	#107	—	526	1	no	—
47	C₃₂	—	2	1	no	—
48	#41	—	1052	1	no	—
49	Q₁₆	—	526	1	no	—
50	SD₁₆	—	1052	1	no	—
51	D₈	—	526	1	no	—
52	C₂×C₈	D₁₀₅₂	1	1	yes	characteristic
53	C₄⋊C₄	—	1052	1	no	—
54	C₁₆	C₄×D₂₆₃	1	1	yes	characteristic
55	C₁₆	C₂²⋊D₂₆₃	1	1	yes	characteristic
56	Q₈	—	1052	1	no	—
57	D₄	—	1052	1	no	—
58	D₄	—	2104	1	no	—
59	C₂×C₄	—	2104	1	no	—
60	C₂×C₄	—	2104	1	no	—
61	C₂×C₄	D₂₁₀₄	1	1	yes	characteristic
62	C₈	(C₂×D₂₆₃)⋊C₄	1	1	yes	characteristic
63	C₈	#16862365	1	1	yes	characteristic
64	C₂²	—	2104	1	no	—
65	C₂²	D₄₂₀₈	1	1	yes	characteristic
66	C₄	—	2104	1	no	—
67	C₄	—	4208	1	no	—
68	C₄	#22554392	1	1	yes	characteristic
69	C₄	#22554415	1	1	yes	characteristic
70	C₂	—	2	1	no	—
71	C₂	—	4208	1	no	—
72	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	#30682370	C₂
cosocle	C₂×C₁₆×C₂₆₃	C₂²
hypercenter	M₆₄	D₂₆₃
hypocenter	C₂₆₃	#1418
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. #30681766	C₂	semidirect
3. M₆₄×C₂₆₃	C₂	semidirect
5. Dic₂₁₀₄	C₄	semidirect
7. D₄₂₀₈	C₄	semidirect
36. C₂₆₃	#1418	semidirect
39. M₆₄	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
#30681766	2
M₆₄×C₂₆₃	2
#1418	263